Imported: 17 Feb '17 | Published: 01 Aug '06
USPTO - Utility Patents
Enhancement of contours in images that are noisy or otherwise corrupted is important in medical imaging, scanning for weapons detection, and many other fields. Here, the Curve Indicator Random Field (CIRF) is used as a model of uncorrupted images of contours for constructing linear, quadratic and cubic Volterra filters involving a number of adjustable parameters.
This application claims the benefit of U.S. Provisional Application No. 60/329,264, filed Oct. 15, 2001.
The present application is related to the present inventor's application entitled “Biased Curve Indicator Random Field Filters for Enhancement of Contours In Images”, which was filed on the same day as the present application, and is incorporated herein by reference.
The present invention relates to the enhancement of the appearance of contours in images, and to methods and systems for enhancing contours of noisy or otherwise corrupted images.
Noisy images are encountered in many fields. For example, medical images by ultrasound, x-ray, and MRI are often so noisy that it is difficult to accurately perceive the critical contours of organs, bones, and medical instruments that are obscured by the noise. Noisy images are also an important problem in many other areas, such as radar, weapon detection during airport inspection of luggage, and the like. These needs have motivated efforts to deal with very difficult machine vision problems using computerized mathematical methods for contour enhancement in noisy images.
As will be understood, image enhancement involves manipulation of image data values in order to improve the appearance of the data as perceived by humans, as well as to render the data more suitable for computer analysis of the image. Various image enhancement processes have been proposed. For example, U.S. Pat. No. 4,803,736, issued Feb. 7, 1989 to Grossberg et al approaches contour perception by use of mask filters. Other prior art references will be discussed hereinafter, along with a detailed explanation of the present invention and concepts related thereto. That discussion will assist a deeper understanding of the present invention, which involves use of Curve Indicator Random Field (CIRF) filtering for contour enhancement in noisy images.
I. Biased CIRF:
The present invention broadly provides a new biased CIRF filtering method of enhancing contours in a noisy image. The novel method comprises the steps of:
According to a preferred embodiment, U, μ, ν, C, f, and b are functions of (r_{1},r_{2}), where r_{1 }comprises the two-dimensional space (x,y) or the three-dimensional space (x,y,z) and where r_{2 }comprises at least one of local direction, curvature, torsion, blur scale, color, texture, contrast and other properties used to form a local description of a contour passing a point of said two-dimensional space (x,y) or said three-dimensional space (x,y,z), respectively.
According to another embodiment.
c=ε[m−A(U)],
where
According to another embodiment, U, μ, ν, C, f, and b are functions of (x, y, θ), and:
According to another embodiment, U, μ, ν, C, f, and b are functions of (x, y, θ, κ), and:
Preferably, the method further comprises the step of:
Step (c) may preferably be carried out by transforming U(r) into an image I(x,y,z) by integrating U or selecting maxima of U over at least one spatial dimension other than (x,y,z). For example, one may choose U(r)=U(x,y,z,θ,φ), where θ and φ define a direction in three dimensions, and I(x,y,z) comprises either an integral of U over θ and φ or a selection of maxima of U over θ and φ. The transformation of U(r) into a planar image I(x,y) may be done by integrating U or selecting maxima of U over at least one spatial dimension other than (x,y). In particular, one may use U(r)=U(x,y,θ), and I(x,y) may then comprise either an integral of U over θ or a selection of maxima of U over θ.
According to a preferred embodiment, the method may include the further step of displaying the image I(s) on a suitable display, such as a workstation monitor.
The invention also provides a computer readable medium (such as a magnetic disc, tape, or optical disc) which includes computer instructions for carrying out any of the methods disclosed herein of enhancing contours in a noisy image.
The invention further provides a system, of the biased CIRF type, for enhancing contours in a noisy image. The aforesaid system comprises:
According to preferred embodiments, the computer may be enabled to carry out the various transformations on the various functions disclosed hereinabove with respect to embodiments of the biased CIRF method.
In particular, according to a preferred embodiment, the inventive system comprises a computer for transforming U(r) into an image I(s), wherein r represents more dimensions than s. While this computer may be a different computer than the computer discussed above for transforming m into U, it will be understood that the same computer, suitably programmed, can serve both purposes.
According to another preferred embodiment, the system may further comprise a display, such as computer monitor, for displaying the aforesaid image I(s).
II. Volterra:
The present invention broadly provides a new Volterra filtering method of enhancing contours in a noisy image; the aforesaid Volterra method comprises the steps of:
It will be seen that when a_{2}=a_{3}=0, a linear filter is defined. Similarly, when a_{3}=0, a quadratic filter is defined.
According to a preferred embodiment, U is a function of (r_{1},r_{2}), and wherein r_{1 }comprises the two-dimensional space (x,y) or the three-dimensional space (x,y,z,), and r_{2 }comprises at least one of a local direction, curvature, torsion, blur scale, color, texture, contrast and other properties used to form a local description of a contour passing a point of said two-dimensional space (x,y) or said three-dimensional space (x,y,z), respectively.
According to another preferred embodiment, the novel Volterra method comprises the step of:
According to a preferred embodiment, the method may include the further step of displaying the image I(s) on a suitable display, such as a workstation monitor.
The invention also provides a computer readable medium (such as a magnetic disc or tape) which includes computer instructions for carrying out any of the Volterra filtering methods disclosed hereinabove of enhancing contours in a noisy image.
The invention provides a system, of the Volterra type, for enhancing contours in a noisy image. The aforesaid system comprises:
Preferably, U is a function of (r_{1},r_{2}), wherein r, comprises the two-dimensional space (x,y) or the three-dimensional space (x,y,z,), and wherein r_{2 }comprises at least one of local direction, curvature, torsion, blur scale, color, texture, contrast and/or other properties used to form a local description of a contour passing a point of said two-dimensional space (x,y) or said three-dimensional space (x,y,z), respectively.
To define a linear filter, one may select a_{2}=a_{3}=0. For a quadratic filter, one should set a_{3}=0.
In particular, according to a preferred embodiment, the inventive system comprises a computer for transforming U(r) into an image I(s), wherein r represents more dimensions than s. While this computer may be a different computer than the computer discussed above for transforming m into U, it will be understood that the same computer, suitably programmed, can serve both purposes.
This transformation of U(r) into an image I(x,y,z) by integrating U or selecting maxima of U over at least one spatial dimension other than (x,y,z). For example, one may choose U(r)=U(x,y,z,θ,φ), where θ and φ define a direction in three dimensions, and I(x,y,z) comprises either an integral of U over θ and φ or a selection of maxima of U over θ and φ. The transformation of U(r) into a planar image I(x,y) may be done by integrating U or selecting maxima of U over at least one spatial dimension other than (x,y). In particular, one may use U(r)=U(x,y,θ), and I(x,y) may then comprise either an integral of U over θ or a selection of maxima of U over θ.
According to another preferred embodiment, the system may further comprise a display, such as computer monitor, for displaying the aforesaid image I(s).
The following discussion will explore related concepts and will outline results obtained in exemplary applications of the biased CIRF and Volterra filtering methods. A table of notations used is included for the convenience of the reader. A final example is given in order to succinctly outline the steps used in these two filtering methods.
Section 1
1.1 Discussion
There is a semantic gap between images and the visual structures suggested by them. On the one hand, an image is a distributed array of numbers representing, for example, a field of spatially distributed light intensity. On the other, the “objects” depicted by these images are discrete (e.g., a person, a car, or a ship's wake) and form a kind of alphabet for our descriptions and interactions with the world (e.g., “the person on the left is approaching the car on the right,” or “the ship's wake suggests the presence of a ship”). Bridging the gap between these qualitatively distinct sorts of descriptions—performing the signal-to-symbol transition—is at the heart of early vision. The present application focusses on one such transition: obtaining contour structure from images.
Most research on inferring image contours was inspired by the Gestalt psychologists who introduced a number of informal principles of perceptual organization to describe the perceived grouping of visual elements in images. For contours, most relevant is the principle of good continuation, which asserts that among the available ways of continuing a contour, the smoothest one should be chosen. Good continuation is used in grouping as a way of weighting associations of contour fragments; since these fragments must somehow already be known, grouping assumes that part of the signal-to-symbol transition has already been solved.
The cause of this difficulty ties in the predominantly discrete nature of thinking about perceptual organization. For example, in one standard grouping demonstration, a subject is presented with a stimulus consisting of perhaps a dozen dots, and the theoretical task is to explain why the subject chose one particular partitioning of the dots among the finite set of possibilities. In illusory contour perception as well, the stimulus is a number of inducing endpoints, where one seeks to understand the (perceptual) selection of one of a finite set of arrangements. The “units” of perceptual organization here (dots, endpoints) are discrete, and typically few in number. Artificial intelligence has reinforced this view, where the units of perceptual organization become the “atoms” of
Unfortunately, natural images are ambiguous, and not only in terms of the groupings of discrete units: even the existence or absence of these units is uncertain. For the inference of contour structure, this ambiguity implies that the true space of units is the uncountably infinite set of all possible curve groups; explicitly enumerating them is unthinkable. To cross this both semantic and now practical divide between images and contours, we introduce an intermediate structure—the curve indicator random field—which will allow us to bring perceptual organization closer to signal processing by formalizing what one can mean by an image of curves. In particular, we shall study and exploit the statistical structure of this field and then derive filters for extracting contour information from noisy images.
To motivate our model, consider the situation of an artist sketching a scene. Using a pen, the artist draws contours on the paper; this motion can be formalized mathematically as a curve, or mapping of a parameter, the current time in the duration of the stroke, to the point in the plane where the tip of the pen is located. Such pen motion is actually recorded in current hand-held electronic pen devices such as the Palm Pilot. A separate time-sequence of pen locations can be recorded for each stroke; together this set of curves provides a description of the artist's activity. Another way of describing this situation is the drawing itself; the ink records a trace of the pen's motion, but discards the curves' parameterizations. The drawing might be called an “ink field,” where the value of the field at a point specifies how much ink was deposited there; this field has zero value except along the sketched contours. Combined with the randomness inherent in the act of drawing, this ink field will inspire our definition of the curve indicator random field, or
Given the apparently slight distinction between the set of contour points and the
There are two major sources of inspiration for the present invention. The first is the set of relaxation labeling (S. W. Zucker, R. Hummel, and A. Rosenfeld. An application of relaxation labelling to line and curve enhancement. IEEE Trans. Computers, C-26:393–403, 922–929, 1977) approaches to contour computation, which explicitly emphasized the need to include contextual interaction in inference, in contrast to classical edge detection techniques that regard each edge independently (followed by a linking procedure perhaps) (J. Canny. A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8:679–698, 1986). These approaches formalize contour inference as a labeling problem using a notion of label compatibility derived from good continuation formalized using orientation or co-circularity. The second source of inspiration is the work on curves of least energy or elastica and its probabilistic expression as stochastic completion fields (L. Williams and D. Jacobs. Local parallel computation of stochastic completion fields. Neural Computation, 9(4):859–881, 1997). We have introduced the
This discussion is organized as follows. Because the signal-to-symbol transition is fraught with uncertainty, we shall use probabilistic techniques throughout. We begin in section 2 with a review of Markov process models of the random contours that will drive the
section 4 explores how we can exploit the statistics of the
1.2 Contributions
This invention presents a number of results and novel ideas for understanding and inferring curve structure in images. Specifically, we
What class of contour models is broad enough to capture curve properties relevant for vision and yet both mathematically and computationally tractable? We start this section by arguing that Markov processes generally satisfy these needs, while acknowledging situations where they do not. We then review the theory of Markov processes and a process by Mumford (supra) that represents the basic local geometry of a curve up to its tangent. Because curvature is also significant in vision, we then introduce a Markov process in curvature. Finally, we suggest other Markov processes that are useful for vision.
We now establish some notation and terminology. Recall that a planar curve is a function taking a parameter tεto a point γ(t):=(x(t),y(t)) in the plane ^{2}. (Also recall that the notation A:=B means that the symbol A is defined as the expression B, whereas A=:B means the opposite.) The local tangent vector ({dot over (x)},{dot over (y)}) has direction θ (Directions are angles over the circle [0, 2π), while orientations are angles over the half-circle [0, π).), defined as θ:=arg({dot over (x)}+{dot over (y)}), where is the imaginary unit √{square root over (−1)}, the dot denotes differentiation with respect to the arc-length parameter t, and arg z denotes the counterclockwise angle from the real axis to z in the complex plane. Without loss of generality, we assume that the tangent ({dot over (x)},{dot over (y)}) (where it exists) has unit length, i.e., {dot over (x)}^{2}+{dot over (y)}^{2}=1. Curvature κ is equal to {dot over (θ)}, the rate of change of direction.
Into this deterministic picture we now inject the uncertainty of vision: since images are ambiguous, so too will be the structures within it. In other words, any given structure in a particular image can be viewed as a realization of a random variable. (A random variable may be a scalar, vector or function (i.e., a process or field). A random process has only one parameter, while a random field generally has several.) In this way, any given curve γ is thus a realization of a random curve Γ. (Upper and lower case letters will be used to denote random variables and their realizations (samples), respectively. Capitals also denote operators later in this disclosure.) Since Γ=Γ(t) is a one-parameter family of random planar vectors, Γ is a random process by definition.
2.2 Why Markov Processes?
By making randomness explicit, we clear a path for overcoming it. While the true curve is uncertain, the curve random process has a lawful distribution that can be exploited to support contour inferences. Put loosely, we can bias our inferences towards the probable curves and away from the rare. The rich variety of possible image curves may appear to suggest, however, that the structure of this distribution is either too simple to be useful (e.g., “uniform”) or too complex to be practical (i.e., an arbitrary distribution over the infinite-dimensional space of distributions of functions). As suggested by the Gestalt law of good continuation, the truth lies is somewhere in between: although many contours can occur, most have some regular structure; e.g., object boundaries usually have a degree of smoothness. The class of Markov processes is ideal for describing such local statistical regularity of visual curves, while remaining tractable.
A random process R=R_{t}=R(t) is Markovian (or a Markov process) if and only if the distribution of the future {R_{t}: t>t_{0}} of the process, conditioned on the past {R_{t}: t≦t_{0}}, is equal to the distribution of the future, conditioned only on the present (or state) R_{t}_{0}. We can construct various Markov process models R for approximating our unknown curve Γ. For example, letting the state of the process be just the planar position of the curve, we could set R(t)=(X, Y)(t)=Γ(t) with realization r(t)=(x, y)(t)=γ(t)ε^{2}. But more information can be represented about the contour via the process of lifting, where extra components are appended to the state to take the two-dimensional curve to a curve in a higher-dimensional space. Thus richer examples of Markov processes can be created by lifting, such as Mumford's process with direction with the three-dimensional state R(t)=(X, Y, Θ)(t)=(Γ, Θ)(t) and realization r(t)=(x, y, θ)(t)=(γ,θ)(t)ε^{2}×. The space ^{2}× is also called (x, y, θ)-space, the unit tangent bundle, and orientation space. Another example of lifting, but involving a mapping from two to four dimensions, is provided by the Markov process with curvature that we introduce here with R(t)=(X, Y, Θ, K)(t)=(Γ, Θ, K)(t) with realization r(t)=(x, y, θ,κ)(t)=(γ, θ, κ)(t)ε^{2}××. These examples show that not only can the present state R(t) of the Markov process represent the present value Γ(t) of the unknown curve, it also can capture (uncertain) local differential information such as direction and curvature. (In a discrete-time setting there is the distinction between first-order and higher-order processes; by a suitable change of state space this distinction is irrelevant and so we consider first-order Markov processes only.) In addition to the curve's local geometry, the state can represent other local information such as blur scale, intensity (on one or both sides of the curve), contrast, color, texture, material information, edge classification (edge vs. line, etc.), and indeed any property that is locally definable.
Merely setting up the state vector is insufficient to characterize a Markov process: the transition probabilities are also necessary. Such probabilities describe how the present state influences the future; for visual contours, the transition probabilities therefore model how local properties extend along the curve. For example, in a world of infinitely long straight lines, the local position and direction determine those elsewhere along the curve; additionally including curvature is not necessary. For a world of very smooth contours, higher-derivative information than curvature may reliably predict nearby positions along the curves. But for fractal boundaries, even local direction would be a poor predictor of the contour nearby a point along the curve. Transition probabilities can model all of these situations, and can therefore be used to gauge the usefulness of particular local properties: those that “persist” for some distance along the curve will support the inference of these curves that we study in section 4. In a sense, Markov processes both formalize the Gestalt concept of good continuation and generalize it to non-geometrical visual cues such as contrast; the transition probabilities quantitatively characterize the relative power of these cues. Markov processes are not even limited to the modeling of such generalized good continuations; they can also specify how good continuations break down, e.g., at discontinuities (see section 2.6).
2.2.1 Limitations of Markov Processes
While Markov processes do capture a variety of crucial properties of contours, it is important to realize that non-local cues are not explicitly included, such as closure, symmetry, simplicity (no self-intersections), and junctions (intersections between possibly distinct contours). The fundamental difficulty with all of these properties is their global nature: for a random process with any of these properties, the entire process history directly influences the future. For a Markov process, however, only the present state directly affects the future; the past has only indirect effects. Fortunately, there are several tools using which we can exploit global contour properties without abandoing Markov processes: spectral techniques, alternative shape representations, and the curve indicator random field (section 3.2.1). Moreover, in section 7.1 we suggest another way to represent both symmetry and closure using a Markov process model of the skeleton representation of the boundary.
Two other global contour properties are simplicity and junctions. For example, given a parameterization of the contour, one has to traverse the entire domain of parameterization to determine whether the contour crossed itself (U. Montanari. On the optimum dection of curves in noisy pictures. Comm. ACM, 14:335–345, 1971). As we shall see in section 3.2.1, this global computation in terms of the parameterization becomes local in terms of the
Thus, it can be seen that while Markov processes in themselves do not convey global contour properties, they are not incompatible with global contour information when captured appropriately. We now consider one other global contour property: length.
2.2.2 Length Distributions
Not only do contours vary in their local properties, they also vary in length. While one may consider a large variety of length distributions, as a first approximation we assert a memoryless property: whether the contour continues beyond a certain point does not depend on how long it already is. We also assume that the length is independent of other aspects of the contour. This implies an exponential density p_{T}(t)=α exp(−αt) for the unknown contour length T, where λ:=α^{−1 }is the average contour length. This length distribution is equivalent to a constant killing (or decay rate) α at each point in the state-space (see 2.7).
One may view the exponential distribution over contour length as conservative in two respects. First, the memoryless property is an independence assumption: the probability that the curve will continue for a certain length is independent of its current length. Second, the exponential distribution can be used to represent highly ambiguous lengths. Consider what occurs as λ is increased to infinity: the slope of the density becomes increasingly flat, and so the length distribution increasingly appears uniform over any given length interval.
To model more general length distributions we can consider either (1) non-stationary Markov processes on, the same state-space or (2) stationary Markov processes on an augmented state-space. As we shall require stationarity in section 3, we could still take the second approach where we would add a length component to the state vector and modify the transition probabilities to monotonically increase the length component with process time. Richer length distributions would be captured by a non-constant decay rate, i.e., killing depends on the length component of the state. For example, to bias toward curves with length less than t* one would have a small killing rate for states with length component less than t*, and a large killing rate for states with a large length component.
With this capacity for more varied length distributions, we continue our presentation of stationary Markov processes with exponentially distributed lengths. We shall concentrate on contour geometry in continuous spaces; later we discretize in preparation for the formal development of the contour filters we derive in section 4.
2.3 Planar Brownian Motion
Perhaps the simplest contour model is Brownian motion in the plane. Although we shall not be using it in our filters, Brownian motion is historically important and forms a benchmark for more complex random processes. Intuitively, it can represent the random motion of microscopic particles. Mathematically, we can use the notation of stochastic differential equations (
dX=σdW^{(1)}, dY=σdW^{(2)}, (2.1)
where W^{(1) }and W^{(2) }are independent, standard Brownian motions and σ=σ_{BM}>0 is the “standard deviation in the derivatives”. (For reference, a standard Brownian motion W is a stationary (continuous) Markov process on where the increments are independent and increment W_{t}−W_{t′} is a zero-mean Gaussian random variable of variance |t−t|.) Planar Brownian motion is an example of a diffusion process and is characterized by the transition probability p_{R(t)|R(0)}(r,t|r_{0}), which is the conditional probability density that the particle is located at r at time t given that it started at r_{0 }at time 0. We write the transition probability as p=p(x, y, t)=p_{R(t)|R(0)}(r|r_{0}), where r_{0}:=(x_{0}, y_{0}) is the initial position of the process, and they satisfy the following Fokker-Planck diffusion equation:
where the initial condition is p_{R(0)}(r)=δ(r−r_{0})=p(x, y, 0)=δ(x−x_{0})δ(y−y_{0}), δ(x) is the Dirac δ-distribution and the spatial boundary conditions must be appropriately specified. The partial differential operator is known as the (infinitesimal) generator of the Markov process and fully characterizes the infinitesimal motion of the particle's probability density. The other Markov processes we shall see are distinguished by their generators.
While Brownian motion has the advantage of capturing the continuity of contours (since realizations are continuous), it has two difficulties for vision. First, sample paths of Brownian motion frequently self-intersect. Not only does this not correspond to actual contours in the world, it also makes the job of filtering more difficult, as we shall see in section 4.6. Second, planar Brownian motion does not represent the local geometry of contours. In particular, since Brownian paths are differentiable almost-nowhere, the tangent does not exist (except on a set of measure zero). Thus planar Brownian motion is a poor model of good continuation, elongated receptive fields in the visual cortex, and the observed edge statistics in natural images, where tangent information is important. The need to capture the local geometry of contours leads us to more elaborate Markov processes.
2.4 Brownian Motion in Direction
To motivate their model for local contour smoothness, Mumford (supra) imagined a particle at R(t)=(X_{t}, Y_{t}, Θ_{t})ε^{2}× whose direction Θ_{t }is slightly perturbed at each time instant t before taking its next step forward. Mathematically, this particle can be described by a Markov process with the stochastic differential equation
where σ=σ_{κ} bounds the direction perturbations and W is standard Brownian motion (on the circle ). In this way a fundamental descriptor of the local geometry of curves—the tangent (cos θ, sin θ)—is captured by lifting a planar curve to ^{2}× and making curvature a white noise. This drives the contour randomly while ensuring that the tangent exists. This direction process has the transition density p=p(x, y, θ, t) (at time t) that satisfies the following Fokker-Planck diffusion equation:
where the initial condition is p(x, y, θ, 0)=δ(x−x_{0})δ(y−y_{0})δ(θ−θ_{0}), p is periodic in θ, and appropriate boundary conditions in x and y must be specified (see section 5.2.1). Weighting by the exponential length distribution, we can illustrate these diffusions by taking an integral over time. For a discussion of numerical techniques for the direction process, see section 5.2.1. Mumford (supra) suggested the direction process as a model for occluded and illusory contours. The covariances of the curve indicator random field (Corollary 4) based on the direction process also resemble several formalizations of good continuation and horizontal interactions in the visual cortex.
As Mumford (supra) has shown, the mode of the distribution of this direction process is described by an elastica, or that planar curve which minimizes the following elastica functional:
where β=(2σ^{2})^{−1 }and α=λ^{−1 }is the constant killing. The elastica functional measures the bending-energy of thin splines. Mumford (supra) derived formulas for elastica in terms of theta functions. In computer vision, elasticae have been used to model illusory contours and the completion of boundary gaps (S. Ullman. Filling-in gaps: The shape of subjective contours and a model for their generation. Biological Cybernetics, 25:1–6, 1976).
2.5 Brownian Motion in Curvature
Inspired by Gestalt psychology, most previous studies of contour smoothness have focused on good continuity in orientation, that is to say, curves with varying orientation—high curvature—are rejected, and, conversely, straighter curves are enhanced. This is naturally phrased in terms of the elastica functional (2.5) on curves that minimizes curvature. Now we introduce a Markov process model that instead aims to enforce good continuation in curvature, and thus minimizes changes in curvature.
2.5.1 An Analogy
To understand why we believe that good continuation in curvature is important, imagine the situation of a bug trying to “track” a curved contour. Suppose the bug is special in that it can only “search” for its next piece of contour in a cone in front of it centered around its current predicted position and direction (i.e., orientation with polarity). This strategy is appropriate so long as the contour is relatively straight. However, when the bug is on a portion of the contour veering to the right, it will constantly waste time searching to the left, perhaps even mistracking completely if the curvature is too large. In estimation terms, the errors of our searching bug are correlated, a tell-tale clue that the assumption that the contour is straight is biased. A good model would only lead to an unavoidable “uncorrelated” error. We discuss a Markov process that models not only the contour's direction, but also its local curvature.
It may appear that one may avoid these problems altogether by allowing a higher bound on curvature. However, this forces the bug to spend more time searching in a larger cone. In stochastic terms, this larger cone is amounts to asserting that the current (position, direction) state has a weaker influence on the next state; in other words, the prior on contour shape is weaker (less peaked or broader). But a weaker prior will be less able to counteract a weak likelihood (high noise): it will not be robust to noise. Thus we must accept that good continuation models based only on contour direction are forced to choose between allowing high curvature or high noise; they cannot have it both ways.
Although studying curvature is hardly new in vision, modeling it probabilistically began only recently. The closest work in spirit is relaxation labeling (P. Parent and S. W. Zucker. Trace inference, curvature consistency, and curve detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(8):823–839, August 1989). Co-circularity (Parent and Zucker, supra) is related to curvature, and was used in Markov random field models for contour enhancement (L. Herault and R. Horaud. Figure-ground discrimination: A combinatorial optimization approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(9):899–914, 1993).
We now formally introduce our Markov process in curvature and its diffusion equation; we then present example impulse responses, which act like the “bent” search cones. We also relate the mode of the distribution for the curvature process to an energy functional on smooth curves.
2.5.2 The Mathematics of the Curvature Process
Consider the Markov process that results from making curvature a Brownian motion. This process has state R(t)=(X, Y, Θ, K)(t), realization r=(x, y, θ, κ)ε^{2}××, and can be described by the following stochastic differential equation:
where σ=σ_{{dot over (κ)}} is the “standard deviation in curvature change” (see section 2.5.3) and W denotes standard Brownian motion. The corresponding Fokker-Planck diffusion equation for the bug's probability density, is
where
where p=p(x, y, θ, κ, t)=p_{R(t)|R(0)}(r|r_{0}). Observe that this partial differential equation (
2.5.3 What is the Mode of the Distribution of the Curvature Process?
To get more insight into our random process in curvature, we now consider one of the simplest aspects of its probability distribution: its mode. Given the succinct expression of the mode of the direction process in terms of the elastica functional (2.5), we ask: Is there a corresponding functional for the curvature process? If so, what is its form?
To answer these questions, first we discretize our random curve into N subsections. Then we write out the distribution and observe the discretization of a certain integral that will form our desired energy functional.
Recall that length T is exponentially distributed with mean λ=α^{−1}. Each step of the N-link approximation to the curve therefore has length Δt:=T/N. Using the definition of the t-derivatives, for example,
we can make the approximation X_{i+1}≈X_{i}+Δt{dot over (X)}, where i is the link index. Recalling the stochastic differential equation (2.7), we therefore let the curvature process be approximated in discrete time by
X_{i+1}=X_{i}+Δt cos Θ_{i}, Y_{i+1}=Y_{i}+Δt sin Θ_{i}, Θ_{i+1}=Θ_{i}+ΔtK_{i},
where i=1, . . . , N. Because Brownian motion has independent increments whose standard deviation grows with the square root √{square root over (Δt)} of the time increment Δt, the change in curvature for the discrete process becomes
K_{i+1}=K_{i}+√{square root over (Δt)}ε_{i},
where {ε_{i}} is an independent and identically distributed set of 0-mean, Gaussian random variables of standard deviation σ=σ_{{dot over (κ)}}. Now let the discrete contour be denoted by
γ_{N}={(X_{i}, Y_{i}, Θ_{i}, K_{i}):i=0, . . . , N}.
Given an initial point p_{0}=(x_{0}, y_{0}, θ_{0}, κ_{0}), the probability density for the other points is
which, by substitution, is proportional to
We immediately recognize
as an approximation to
and so we conclude that
p(γ_{N}|p_{0})→p(γ|p_{0})∝e^{−E(γ) }as N→∞,
where the energy E(γ) of (continuous) curve γ is
and where β=(2σ^{2})^{−1 }and α=λ^{−1}.
Maximizers of the distribution p(γ) for the curvature random process are planar curves that minimize the functional E(γ), which we call the Euler spiral functional (B. B. Kimia, I. Frankel, and A.-M. Popescu. Euler spiral for shape completion. In K. Boyer and S. Sarkar, editors, Perceptual Organization for Artificial Vision Systems, pages 289–309. Kluwer Academic, Boston, 2000). When at α=0—when there is no penalty on length—, such curves are known as Euler spirals. A key aspect of the Euler spiral functional (2.8) is that it penalizes changes in curvature, preferring curves with slowly varying curvature. In contrast, the elastica functional (2.5) penalizes curvature itself, and therefore allows only relatively straight curves, to the dismay of the imaginary bug of section 2.5.1. This Euler spiral energy is also loosely related to the notion of co-circularity: the coefficient β would play a role of weighting the “compatibility distance” between a pair of sites (x, y, θ, κ), curvature while the coefficient α would affect the “transport distance” between them. In other words, the integral over {dot over (κ)}^{2 }in E(γ) is a measure of how non-co-circular two tangents are, while the second integral measures how far apart they are, defined as the length of γ, the smooth curve joining them.
2.6 Other Markov Process Models
At the start of this section we suggested the use of Markov processes for a host of contour cues other than the local geometry captured by the direction and curvature processes. The simplest way of extending a Markov process to include this extra information, say contrast C, is to add a component to the state vector and assume that it is a Brownian motion independent of the geometry and other factors. Smoothness in C can be enforced by letting the derivative Ċ be a Brownian motion (or even the Ornstein-Uhlenbeck (D. Mumford. Algebraic Geometry and Its Applications, chapter Elastica and Computer Vision, pages 491–506. Springer-Verlag, 1994) process suggested in 3-dimensions next), and then appending two components to the state vector (one for the contrast itself, the other for its derivative). Even dependencies among different sources of information, say intensity and geometry, can be modeled using Markov processes, but this is more complex.
While a preferred application of the present invention is curves in the plane, the framework of the invention applies unmodified to higher dimensional images. In medical imaging of vascular structures in
dX^{(i)}=σdW^{(i)}, i=1, 2, 3,
where {W^{(i)}: i=1, 2, 3} are independent, standard Brownian motions. For a random 3-dimensional space curve with smoothness up to the local tangent (similar to the (planar) direction process of section 2.4), Mumford (supra) suggested a Markov process obtained by integrating a Brownian motion on ^{2 }(the sphere of unit-length tangent vectors). A simpler model would not enforce the unit-length constraint for the tangent vector; instead, the tangent could be 3-dimensional Brownian motion. A slightly more general version of this idea includes a force (of spring constant ε) to ensure that the tangent vector (the particle's velocity V=(V^{(1)}, V^{(2)}, V^{(3)})) does not stray too high:
where we can eliminate the force by setting ε to zero. The velocity component processes V^{(i) }are known as Ornstein-Uhlenbeck processes. In two-dimensions, these processes were suggested for vision by Mumford (supra). The extension of process (2.9) to include curvature and torsion information is straightforward.
In addition to these Gaussian noise-driven Markov processes, one can construct Markov processes driven by other, heavy-tailed forms of noise. It should be understood that Levy processes (stationary, independent-increment Markov processes) can be used as a generalized replacement for Brownian motion in all of the above Markov processes. In this way discontinuities, or “jumps”, in direction (i.e., corners), curvature, and even contrast can be included. We focus on diffusion operators here both because they clearly relate to the differential geometry of curves and because they correspond to the qualitative structure of the edge correlation patterns that can be observed in natural images. When approximated with a discrete state-space, all stationary Markov processes, including those with jumps, fit within our filtering framework.
2.7 A Working Formalism: Continuous-Time, Discrete-State Markov Processes
Having set out the kind of processes we shall ultimately use as uncertain contour models, we need to transition from continuous state-spaces to discrete ones that will support computations relevant for digital images. In this section we therefore briefly establish the notation and basic theory of discrete-state Markov processes.
With a slight abuse of notation, we let R_{t}, where 0≦t<T, denote any stationary continuous-time Markov process taking values in a finite set I of cardinality |I|. Sites or states within I will be denoted i and j. We view I as a discrete approximation to our continuous state-spaces ^{2}×, ^{2}××, etc., where our contour random processes takes values. For example, (discrete) state i would represent some continuous state (x, y, θ) for the direction process, and R_{t }denotes the (discrete) curve Markov process. As in section 2.2.2, the random variable T is exponentially-distributed with mean value λ, and represents the approximate length of the contour. To ensure finiteness of the expressions in section 3, we further assume λ<∞.
The (discrete-space) generator of the Markov process R_{t }is the |I|×|I| matrix L=(l_{ij}), and is the instantaneous rate of change of the probability transition matrix P(t)=(p_{ij})(t) for R_{t}. Since we assume that the Markov process is stationary, the generator L does not vary with time t. One can obtain the discrete-space generator L by discretizing the continuous-space generator appropriately; details are provided in section 5.2.
To enforce the exponential distribution over lifetime (length) T of each particle, we construct a killed Markov process with generator Q=L−αI, where α:=λ^{−1 }as in section 2.2.2. (Formally, we do this by adding a single “death” state to the discrete state space I. When t≧T, the process enters and it cannot leave.) Slightly changing our notation again, we shall now use R_{t }to mean the killed Markov process with generator Q.
The Green's operator G=(g_{ij}) of the (killed) Markov process is the matrix ∫_{0}^{∞}e^{Qt}dt=∫_{0}^{∞}P(t)e^{−αt}dt, where P(t)=e^{Lt }(e^{A }denotes the matrix exponential of matrix A). (Formally, G is the Green's operator (or Green's matrix) when α=0, and is known as the resolvent, resolvent matrix, or resolvent operator for α≧0.) The (i, j)-entry g_{ij }in the Green's matrix represents the expected amount of time that the Markov process R_{t }spent in j (before death) given that the process started in i. The following is a well-known connection between the generator Q (an approximate differential operator) and the Green's operator G (an approximate integral operator):
Lemma 1. G=−Q^{−1}.
Having studied (parameterized) contour models, we remove the parameterization in the next section to construct our prior for contour inference from images.
Section 3
3.1 The Curve Indicator Random Field
Given a Markov process for modeling individual contours, we now define a curve indicator random field (
Several independence assumptions play a role in our model. First, we use a memoryless property on contour length T (section 2.2.2), which led to an exponential density for T. Second, we observe that independence of contours is a reasonable approximation for modeling elongated curves. Tree branch contours, for example, statistically interact primarily where they meet; due to the local process of growth, the individual branches wander largely independently. Occlusion is apparently a counter-example, although if we were interested in an “x-ray” view of contour structure, independence may still be a useful approximation. (We shall weaken the independence assumption in section 3.2.1, where contour interactions are described.) Finally, the unknown number of contours is assumed to be Poisson-distributed: in the related context of random point fields, this distribution is natural where disjoint regions contain independent numbers of points.
3.2 Defining the CIRF
Following section 2.7, we focus on continuous-time, discrete-state stationary killed Markov processes R_{t}, where 0≦t<T. Let {condition} denote the (indicator) function that takes on value 1 if condition is true, and the value 0 otherwise. With this notation, we can define the curve indicator random field V for a single curve as
Observe that V_{i }is the (random) amount of time that the Markov process spent in state i. In particular, V_{i }is zero unless the Markov process passed through site i. In the context of Brownian motion or other symmetric processes, V is variously known as the occupation measure or the local time of R_{t}.
Generalizing to multiple curves, we pick a random number N and then choose N independent copies R_{t}_{1}^{(1)}, . . . , R_{t}_{N}^{(N) }of the Markov process R_{t}, with independent lengths T_{1}, . . . , T_{N}, each distributed as T. To define the multiple curve
Definition 1. The curve indicator random field U is defined as
Thus U_{i }is the total amount of time that all of the Markov processes spent in site i. Again, observe that this definition satisfies the desiderata for an ideal edge/line map suggested in section 1: (1) non-zero valued where the contours are, and (2) zero-valued elsewhere. We emphasize that the curve indicator random field is a different object than the curves used to produce it. First, the
3.2.1 Global to Local: Contour Intersections and the CIRF
In addition to representing the loss of contour parameterization, the
To understand why V_{i}^{2 }measures self-intersection, consider the expression in parentheses on the right: if the process R was in site i at time t, then this expression counts the total amount of time spent in site i=R_{t}; otherwise it is zero. There are two components to this time: (1) the (unavoidable) holding time of R_{t }in site i (i.e., how much time spent in i for the visit including time t) and (2) the remainder, which evidently must be the time spent in site i before or after the visit that included time t, i.e., the self-intersection time of R_{t}=i. Taking the integral over t, we therefore see that V_{i}^{2 }is a measure of self-intersection for site i over the entire contour.
A similar calculation can be carried out for the multiple-curve case U_{i}^{2}, where we will get terms with integrals over the same curve as above (to measure self-intersection), or over a pair of distinct curves where instead we get an analogous intersection or crossing measure. The key is to recognize that a global contour property that requires checking the entire parameterization—intersection—becomes a local property (squaring) of the
3.3 Moments of the Single-Curve CIRF
Probabilistic models in vision and pattern recognition have been specified in a number of ways. For example, Markov random field models are specified via clique potentials and Gaussian models are specified via means and covariances. Here, instead of providing the distribution of the curve indicator random field itself, we derive its moment generating functional, from which all moments are then computed.
Although we are interested in the statistics of the general curve indicator random field U, we first consider the simpler, single-curve case. The first step (Prop. 1) is to derive all the moments of the single-curve
To prepare for studying these moments, we first recognize the rather pathological nature of the
It is also convenient to use this distribution-theoretic formalism for studying the
where c(i)=c_{i}. The integral ∫c(R_{t})dt is known as an additive functional of Markov process R_{t}. In the following, we let α:=λ^{−1 }to simplify expressions. We also introduce a final weighting ν(R_{T−}) on the state of the curve just before death; ν can be used to encourage the curve to end in certain states over others. (We use the notation T− to represent the left limit approaching T from below, i.e., ν(R_{T−})=lim_{t/T}ν(R_{t}).) Let _{i}Z denote the expected value of the random variable Z given that R_{0}=i, and let _{μ}Z denote the same expectation except given that {R_{0}=i}=μ_{i}, where μ is the distribution of the initial point of the curve. To reduce the clutter of many brackets we adopt the convention that the expectation operator applies to all multiplied (functions of) random variables to its right: e.g., ƒ(X)g(Y):=[ƒ(X)g(Y)]. We now provide a formula for the moments of <c, V>, a “probed” single-curve
Proposition 1. The k-th moment of <c, V> with initial distribution μ and final weighting ν=ν(i)=ν_{i}, iεI is:
_{μ}<c, V>^{k}ν(R_{T−})=αk!<μ, (GC)^{k}Gν>, (3.2)
where C=diag c.
3.4 Moment Generating Functional of the Single-Curve CIRF: A Feynman-Kac Formula
To obtain a formula for the moment generating functional (
Proposition 2. For all cε^{|I|} such that |c| is sufficiently small,
_{μ} exp<c, V>ν(R_{T−})=α<μ, G(c)ν>.
Observe that to evaluate the Feynman-Kac formula, one must solve the linear system (Q+C)h+ν=0 for h. This equation will become a key component in the filters of section 4; numerical methods are described in section 5.2 and section 5.3.
3.4.1 Sufficient Condition for MGF Convergence: Khas'minskii's Condition
The convergence of the Feynman-Kac formula can be characterized by the Khas'-minskii condition, as follows.
Proposition 3. The moment generating functional of the (single-curve) curve indicator random field V converges if ∥G|c|∥_{∞}<1.
The Khas'minskii condition is easy to check: one just takes the componentwise absolute value of c (which later will be an input), and then “blur” it with the Green's operator G. If any component of the result is greater than 1, the moment generating functional may not converge. As we shall see in section 5.3, this condition will be exploited in our numerical methods for filtering.
3.4.2 Initial and Final Weightings
As the interpretation of the “final weighting” ν above may seem mysterious, we now restrict μ and ν to be finite measures satisfying the normalization constraint 1=<μ, Gν>. (If this equality is not satisfied, one need only divide by a suitable normalizing constant.)
Corollary 1. Suppose that the joint distribution over initial and final positions is {R_{0}=i, R_{T−}=j}=μ_{i}g_{i,j}ν_{j}. Then the moment generating functional of V, with this joint distribution over initial and final states, is:
exp<c, V>=<μ, G(c)ν>. (3.3)
Remark 1. Although not studied here, it is interesting to consider the problem of finding those measures μ, ν that induce a {R_{0}, R_{T−}} with prescribed marginals {R_{0}} and {R_{T−}} over initial and final states respectively.
Since we are most interested in the case where μ and ν are spatially constant, we can forego answering this question now.
The next corollary shows all of the (joint) moments of V. Let perm_{k }denote the set of permutations of the integers 1, . . . , k.
Corollary 2. If k≧1 the k-th (joint) moment of V at sites i_{1}, . . . , i_{k }is:
3.5 Multiple-Curve Moment Generating Functional
In order to model more than one curve in an image, we need a joint distribution over both the number of curves and the curves (and the corresponding
3.5.1 The Poisson Measure Construction
We begin with a finite measure P: F→_{+} over the measure space (Ω, F), where F is a σ-algebra, and _{+} denotes the nonnegative real numbers. Intuitively, the finite measure P is the (un-normalized) distribution over “points” ωεΩ, where in this disclosure ω is a curve realization (i.e., a Markov process realization) and Ω is the set of all possible curves. We shall now define a probability distribution over random configurations ω=(ω_{1}, . . . , ω_{N})εCon(Ω):={Ω^{0}=Ø, Ω^{1}=Ω, Ω^{2}=Ω×Ω, Ω^{3}, . . . }, where each ω_{n }is a curve in Ω and N is the random number of curves. In our context, Ω^{0 }is the 0-curve configuration (no curves), Ω^{1 }are the one-curve configurations (the set of single curves), Ω^{2 }are the two-curve configurations (the set of pairs of curves), and so on. We now compute the Poisson point measure via its expectation F on any (measurable) function F: Con(Ω)→(clearly this defines a probability distribution for we could take F as an indicator over any (measurable) subset of Con(Ω) to get its probability).
Proposition 4. Suppose N is a Poisson deviate with mean P(Ω). Further suppose that the points ω_{1}, . . . , ω_{n }are (conditionally) independent and identically distributed with P(·)/F(Ω), given N=n. Then:
3.5.2 Application to the MGF of the Curve Indicator Random Field for Multiple Curves
We now finally consider the joint distribution over many curves. Suppose there are N contours on average, and that μ and ν are finite measures on I (vectors in ^{|I|}), characterizing the initial and final positions, respectively, of the Markov processes {R_{t}^{(n)}: n=1, . . . , N}. As before, these measures satisfy the normalization constraint <μ, Gν>=1.
For general-purpose contour enhancement, we typically have no a-priori preference for the start and end locations of each contour, and so we would set these measures proportional to the constant vector 1=(1, . . . , 1). One can show that by letting μ_{i}=|I|^{−1}, ν_{i}=λ^{−1}, ∀iεI, the above constraint is satisfied. We now state a key theoretical result.
Proposition 5. Suppose the inital distribution μ and final weighting ν satisfy the normalization constraint <μ, Gν>=1. Then the moment generating functional of the curve indicator random field U is
exp<c, U>=exp<μ, N(G(c)−G)ν>.
3.5.3 Cumulants of the CIRF
While Prop. 5 may seem abstract, it is actually very useful. First observe that how similar in form it is to the single-curve case. More importantly, with Prop. 5 we can compute the higher-order cumulants of U (recall that the moments define the cumulants and vice versa):
Corollary 3. If k≧1, the k-th (joint) cumulant of the curve indicator random field U at sites i_{1}, . . . , i_{k }is
The cumulant formula has a simple interpretation. First recall that the Green's operator entry g_{ij }is the expected amount of time spent by R_{t }in state j given that it started in i. For any ordering of the k points we take the product of the g_{ij}'s for the successive points in order (the first and last factors deal with the initial and final points). Since the contour could have passed through the points in any order, all permutations must be considered.
Letting G* denote the transpose of G, we can rephrase corollary 3 to show the mean and covariance of the
Corollary 4. Suppose that μ_{i}=|I|^{−1}, ν_{i}=λ^{−1}, ∀iεI, and let η=Nλ|I|^{−1}. The mean of the curve indicator random field U is U_{i}=η, ∀iεI. The covariance matrix of U is cov U=η(G+G*).
While the covariance pattern for the direction process expresses good continuation as local straightness, the covariance pattern for the curvature process captures the local circularity of contours.
Note that by Corollary 3, the cumulants of U of order greater than two are generally not zero, proving the following:
Corollary 5. The curve indicator random field U is non-Gaussian.
Although non-Gaussianity is often difficult to handle, in our case it is easier because we have an explicit, tractable expression of the
Section 4
4.1 Filtering Corrupted CIRFs
In this section we focus on a central objective of this invention:
Suppose an unknown number of contours are located within a set of sites I. Consider the question Q_{i}: “Did a curve pass through location i?”
Task: Given a corrupted image of the curves in I, compute the optimal answer to question Q_{i}, for all i in I.
Unfortunately, this task definition leaves a number of crucial questions unanswered, such as:
In this section we first consider the conceptual issues raised by questions (1) through (4) above, leading to a clearer formal specification of what will become our filtering Task. We then handle question (5) by providing derivations of several linear and nonlinear filters that approximately satisfy our exact specification of filtering noisy contour images.
Before addressing these issues, let us consider whether it is worth the effort to perform this Task at all. Certainly boundary representations of our natural world have been with us for a long time: even early cave drawings exhibited clear contour structure, and artists have obsessed over lines ever since. In computer vision and image processing, researchers have been interested in curves to support a host of computations, including object recognition, tracking, and image compression. However, the natural task would seem to be that of estimating the curves themselves, not whether a curve had passed through a given point. Because we shall return to this issue in section 4.10, at this point we only mention that successfully performing our Task supports the computation of the curves themselves via an edge-linking procedure.
On the surface, satisfying our objective appears startlingly simple: it seems one should simply take one pass over the image, where for each site i one makes a decision, perhaps based on a pixel value or local edge operator output. Unfortunately, in a wide variety of situations the results of this procedure are unsatisfactory, and a number of attempts at more sophisticated computations have proved insufficient to sway the field into depending upon low-level contour information. This invention tackles this contour finding problem at the root by defining the problem exactly in a well-understood probabilistic inference framework, and then systematically deriving procedures for (approximately) solving it.
We begin by relating the questions {Q_{i}} to the curve indicator random field. Because the correct answer to Q_{i }is “yes” or “no,” and because we are uncertain about the correct answer, we can associate a binary random variable with site i, whose value will be 1 if a curve passed through site i and 0 if not. Recalling that the curve indicator random field U_{i }is the total amount of time spent at i by any curve, the correct answer to Q_{i }can be exactly represented by the binary curve indicator random field U_{i}^{>0}:={U_{i}>0}, where i is in I. Notice that U^{>0 }captures a complete notion of what we mean by an ideal edge or line map, or what we shall call a (perfect) image of curves, or contour image. Let the measurement vector m denote the given corrupted contour image. Because we are uncertain about the exact pattern of corruption, it is natural to view m as a realization of a measurement random field M. Therefore our Task is one of determining U^{>0}, given the measurements m.
4.2 Bayes Decision Theory
Realize that our Task mandates that we make a particular decision about U^{>}^{0}, i.e., that we form some estimate of U^{>0}. Because U^{>0 }and M are random, our Task therefore falls under the purview of Bayes decision theory, which we review now.
Suppose we seek to make a decision D about an unknown state of nature X=xεX distributed with prior p(x), given only observation Y=y (conditioned upon the unknown x) distributed with likelihood p(y|x). The loss function loss(x, D) specifies the penalty incurred by making decision D given that the state of nature was in fact x. Because this decision is based upon the observations, we know D is a mapping associating the decision D(y) to the observation y. We focus on that class of decision problems where the decision D is an estimate of the unknown x. The goal in Bayes decision theory is to select that (Bayes) decision rule or estimator {tilde over (x)}={tilde over (x)}(y) which minimizes the expected loss given y, or
{tilde over (x)}(y):=arg min [loss(X, x)|y]. (4.1)
The Bayes risk [loss(X, {tilde over (x)}(Y))] is the overall expected loss (averaged over all observations Y) for the Bayes estimator.
4.2.1 Previous Estimation Problems
Bayesian and related techniques for estimation have a long history in signal process, image processing, and computer vision. These methods emphasize the tension between fidelity to observation (as defined by the likelihood) and some assumed structure or regularity in the unknown (as defined by the prior). Early work was with 1-dimensional signals for communications applications. The Wiener filter exploited knowledge of signal and noise correlations to minimize mean squared error. For the Kalman-Bucy filter, the unknown signal (a function of time) is modeled by a (possibly time-varying) linear stochastic system (with finite dimensional state-space) and the observations have additive Gaussian noise; the filter provides an optimal estimate of the state of the system given the past observations. Hidden Markov models (
Extending these statistical techniques to general image processing applications requires making the transition from random processes to random fields. For additive Gaussian observations with a Gaussian prior, the extension from the 1-dimensional case to 2-dimensional image restoration applications is straightforward. An interesting application of Bayesian methods relates to the restoration of
To relate the Bayes decision framework to our Task, we note that the unknown state of nature X above is our contour image U^{>0 }and the observation Y above is the measurement random field M. This answers two of the questions we began with in this section. First, corruption of an image of contours is simply the random mapping taking U^{>0 }to M, and is completely characterized by the likelihood p(m|u^{>0}). Second, Bayes decision theory does not provide a unique notion of optimality, because the loss function must be specified. We shall return to this latter issue in section 4.2.2.
The requirement of making a decision sets Bayes decision theory apart from related work such as “pure Bayesianism,” which merely reports the posterior distribution p(x|y) without making a decision. Decision theory is similar to game theory both in that an action will be taken and that the loss function is qualitatively comparable to the payoffs; the difference lies in the notions of randomness employed. In game theory the task is often to determine the equilibria, or marginal distributions over the unknowns X_{i}, i=1, . . . , n; the observations Y can be used to bias toward particular strategies via strategy-specific payoffs. In decision theory, the observations are included via the likelihood.
The likelihood is the centerpiece of inferences made by the classical statistical school, where the existence of the (Bayesian) prior is denied. In the context of edge detection in computer vision, this amounts to an assertion that all binary functions of i are equally preferable—a uniform prior on edge maps—; only the likelihood and loss function can be used to make the decision. If the measurements are local and conditionally independent (i.e., p(m|u^{>0})=Π_{i}p(m_{i}|u_{i}^{>0})), which is a common assumption, and if the loss is additive (see section 4.2.2), then one is forced to do sitewise thresholding. The primary difficulty with this uniform prior is that it completely ignores the rich structure of images of curves. Not only are contour images often sparse, but their non-zero regions have a stereotyped behaviour: they are curves! Even the relatively simple Markov process-based
4.2.2 Loss Functions: which Best is Best?
Given this background, we now proceed to apply Bayes estimation for optimally determining U^{>0}. However, we still need to select a loss function to make our answer unique. It is well-known that the loss function can significantly influence an estimate when the posterior is not peaked, so we shall take some care in our choice. But because we are seeking a binary random field, our loss function is limited to the set of mappings from {0, 1}^{|I|}×{0, 1}^{|I| }to . Even that set of possibilities is huge, so we focus on two opposite extremes: the 0/1 and the (symmetric, uniform) additive losses.
The 0/1 loss assigns a penalty of 1 if the estimate is different from the true unknown, and 0 only if the estimate is correct. It is a standard exercise to show that the Bayes estimate under a 0/1 loss is exactly the maximum a posteriori (
Unfortunately, the basic property that the 0/1 loss function underlying
Symmetric, uniform, additive loss functions take a sum of site-wise errors (for binary images) and therefore their Bayes risk is the minimum average number of site-wise errors. Unlike the 0/1 loss, this site-wise error count does not abruptly change with any error at all, rather it gradually increases as the estimate becomes poorer. Perhaps it is the existence of two notions of “error rate” that has helped maintain the popularity of
4.2.3 Minimum Mean Squared Error Filtering
For binary image problems, this additive loss function is equivalent to the squared-error loss, or the sum of the squares of the differences between the true unknown and the estimate. Thus minimizing the additive loss amounts to choosing that binary field that minimizes the mean squared error (
where the last line follows from the second because the terms in the sum of positive numbers can be optimized independently. Thus the additive loss provides an added benefit in complexity: we have a separation of the joint estimation problem over 2^{|I|} possibilities into |I| single-site problems of two possibilities each. Also, the |I| separate expectations are taken with respect to the marginal distributions p(u_{i}^{>0}|m), not the complicated joint distribution. In real-valued problems, one can expect the marginals to be better behaved (smoother) than the joint, since each marginalization is an integration process. (In this binary problem, the marginals are so “smooth” they are a function of a single number!) Thus MMSE estimation many not only be more relevant to image analysis than
4.2.4 Bounding the MMSE Estimate of the Binary CIRF with a Thresholded Posterior Mean of the CIRF
Now that we have refined our original Task to that of computing the
Because the squared-error loss is additive, we saw in (4.3) that we can focus on the problem of finding the MMSE estimate (m) at site i, which can be expressed as
Thus the
Before pursuing that relationship, we first study an intermediary, the hitting random field H, where H_{i}ε{0, 1, 2, . . .} is the total number times that site i was hit by any curve Markov process. Observe that the binary
U_{i}^{>0}={H_{i}>0}, (4.5)
and so any function of U^{>0 }is also a function of H. In particular, while it might appear most natural to assume that our measurements are a corrupted version of the binary
Let τ_{i }denote the average amount of time that the curve Markov process spends in site i each time it hits i (without conditioning on m). It can be shown that τ_{i}=−1/q_{ii}, where Q=(q_{ij}) is the generator of the killed Markov process (if the Markov processes is spatially homogeneous (see section 4.7.3), then τ_{i }does not vary with i). The following lemma relates the posterior means of the curve indicator and hitting random fields by the scalar factor τ_{i}.
Lemma 2. Suppose that m is a realization of the measurement random field M=M(H, N), where noise N is independent of the curve Markov processes (and thus the hitting random field H and the
_{m}U_{i}=τ_{i}_{m}H_{i}, ∀iεI,
where τ_{i }is the average time spent in site i per visit.
Given this connection between U and H, we are now in a position to show that the thresholded posterior mean of the
Proposition 6. Suppose that m is a realization of the measurement random field M=M(H, N), where noise N is independent of the curve Markov processes. Then the
_{i}(m)={_{m}U_{i}^{>0}≧½}≦{_{m}U_{i}≧τ_{i}/2}, ∀iεI, (4.6)
where τ_{i }is the average time spent in site i per visit.
Proposition 6 enables us to connect the (non-binary)
It is well-known that the posterior mean is the
4.3 Observation Models and Likelihoods
Although the
4.3.1 The Gaussian Likelihood
The likelihood represents how our ideal contour image relates to the measurements, and therefore it characterizes the corruptions in local edge and line operator responses, or how observed contours are non-ideal. For example, edges are blurry, in the sense that edge operators have an associated scale. Equally important is the random noise in edge measurements: even for intensity images with high signal-to-noise ratio (SNR), edges can still be noisy for two reasons. Most fundamentally, edge detection requires differentiation, which amplifies noise. In addition, when the local contrast (i.e., signal) is low, the local SNR drops for fixed sensor noise.
One way of specifying such sources of corruption is to write an observation equation that relates the unknowns to the observations. Although one can consider much more general models, both for concreteness and simplicity we assume that the measurements are a linear function of the
We stress that the linear operator B is quite flexible: not only can it be a standard blur operator, but it can also represent more fundamental aspects of observation. For example, in the case where I is the discretized three-dimensional (x, y, θ)-space and M is a two-dimensional image, B would be the (discrete) linear projection that integrates along θ. It can also capture the image derivatives used in local edge and line operators as well, and thus the operator B need not even perform blurring. Noise structure can be partially captured by non-diagonal noise covarainace operator Σ_{N}, i.e., the correlated noise can be handled.
One may object with this likelihood model on the grounds that the measurement is a now function of the
4.3.2 Log-Likelihoods with Quadratic Dependence on the CIRF
The Gaussian is an example of a more general class of quadratic log-likelihoods with
ln p(m|u)=ε<a+Au, u>+constant, (4.8)
where A is a symmetric, nonnegative definite matrix, and vector a and the constant are functions of the measurement vector m. To relate this class to the Gaussian likelihood (4.7), we set
The coefficient ε is introduced to capture the degree of noise; when the noise is white (Σ_{N}=σ_{N}^{2}I), this coefficient could be used to represent the inverse noise variance, or ε=σ_{N}^{−2}. Although for the Gaussian, the vector a is linear in the measurements, the quadratic log-likelihood allows a to be a general functions of m. Indeed, all of the nonlinear filters we shall be presenting allow a to be any (measurable) function of m. Some non-Gaussian observation models can be handled using this freedom in a and A by a change of variables to a Gaussian model. For example, multiplicative noise can be modeled using a logarithmic change of variables to additive Gaussian white noise. We present another non-Gaussian example of the quadratic log-likelihood next.
4.3.3 Exploiting Empirical Distributions of Local Operators
Suppose we have training data in the form of a number of curve images and their corrupted counterparts. How could we use information gathered from this data to empirically derive our likelihood function?
One way of doing this is to construct an observation model that only considers whether a contour hit a site or not by determining the “on” and “off” edge responses. To formalize this, we set up a likelihood in terms of the binary
Letting τ again be the average amount of time that the Markov process spends in a site per hit of the site, observe that U_{i}|τ and U_{i}^{>0 }are equal on average, as long as each site gets hit at most once. Ignoring this latter condition for a moment, one can heuristically try to create a likelihood conditioned on u by replacing u^{>0 }with u/τ above: ln p(m|u)≈εΣ_{i}a_{i}u_{i}+constant(m)=ε<a, u>+constant(m), where
To ensure normalizability, we include a term ε<Au, u> in the likelihood, where
and so obtain a valid quadratic log-likelihood in the form of (4.8). Observe that this log-likelihood has quadratic dependence on u, but possibly nonlinear dependence on m. The norm-squared ∥u∥^{2}=−2<Au, u> measures the self-interaction of the
4.4 The Posterior Distribution
In this section we set up some notation for gluing together the likelihood with the prior to form the posterior. We write the
where p(m):=∫p(m|u)P(du). If U had a density p_{U}(u), then P(du)=p_{U}(u)du. Instead we use the general notation P(du) for the distribution of U since a density exists for
P_{(ε)}(du)=Z_{(ε)}^{−1 }exp(ε<a+Au, u>)P(du),
where Z_{(ε)}:=∫exp(ε<a+Au, u>)P(du) and the subscript with ε indicates the quadratic log-likelihood. Note that although we do not have an explicit expression for the prior P(du) (and therefore no explicit expression for this posterior), we can indirectly use the prior for U through its moment generating functional (
4.5 The Linear Case
Although one may consider all statistical properties of the curve indicator random field, even the pair-wise relations express information critical to enhancing curves. As Gaussian random fields are fully defined by their means and covariances, in this section we simplify our contour filtering problem by replacing the
Combined with the quadratic log-likelihood (4.8) we get the following quadratic log-posterior (implying the posterior is Gaussian):
Since the mean is equal to the mode for the Gaussian, we can obtain the posterior mean for the Gaussian approximation to the
which is equal to (Σ_{U}^{−1}+B*Σ_{N}^{−1}B)^{−1}(B*Σ_{N}^{−1}m+Σ_{U}^{−1}μ_{U}) for the Gaussian log-likelihood corresponding to the observation model M BU+N, as in section 4.3. The resulting posterior mean is a linear filtering of the measurements m by the operator (Σ_{U}^{−1}+B*Σ_{N}^{−1}B)^{−1}B*Σ_{N}^{−1 }(plus a fixed vector).
For the Gaussian likelihood case, this linear filter is known in image restoration as a Wiener filter. The novelty in this application of the Wiener filter is that it operates in the higher-dimensional spaces where curves live, not two-dimensional intensity images. For the direction Markov process, this amounts to linear filtering in position (x, y) and orientation θ, and therefore our linear filter is an oriented Wiener filter. The domain of the random fields we are filtering is not the (discretized) image plane ^{2}, but instead the unit-length tangent bundle ^{2}×. When applied to the curvature Markov process, we obtain linear filtering in ^{2}××. Other Markov process curve models will lead to filtering in other spaces.
Note again the presence of the measurement operator B. Using it, we can explicitly capture the imprecision in edges due to the finite aperture of edge operators. Clearly, we seek an estimate of the underlying curve indicator random field in which the edge strength is localized in position and direction, i.e., we wish to de-blur edge measurement fields on ^{2}×. Our filter is able to perform such de-blurring by combining the blur operator B, the noise model, and curve indicator random field covariance.
4.5.1 Why Simple Cells and Local Edge Operators are Elongated
When the existence of cells in primate visual cortex that respond to edge and bar stimuli was found, a connection between physiology and geometry was established. Unfortunately, no theory deriving the qualitative features of the receptive field for even their “simple cell” has been provided. Recall that there are two key “directions” of the receptive field: normal and tangential to the major axis of the receptive field. Since the receptive fields of simple cells resemble the kernels of (oriented) local edge and line operators, understanding the rationale for one explains the other. The normal profile is generally understood functionally as a kind of blurred differentiation (Canny, supra) using the theory of regularization. The tangential profile, with its elongated structure, is more poorly understood. Nonetheless many have designed elongated filters where the tangential profile is explained using a variety of theories, such as an uncertainty principle, Hilbert space representations, or increased noise immunity. The
But does the curve indicator random field provide more than a contour-based explanation for elongated linear filtering? To answer this question, we shall have to go beyond the connection between linear filtering and second-order statistics.
4.6 Volterra Filters and Higher-Order Statistics
Volterra filters constitute one class of nonlinear filters that subsumes linear filters. Even though most work on Volterra filtering is done on time-domain sequences, we shall apply these ideas to multidimensional signals (functions on ^{2}×, for example) or on vectors m=(m_{i}), where i is a site in I. By definition, the output of a Volterra filter is a polynomial function of the input signal. Although Volterra filters are (generally) nonlinear, they can always be represented as a linear function of products of the input m. A k-th order Volterra filter is a linear function of 0- through k-fold products of the input; for example, a quadratic (second-order) Volterra filter can be written as a linear combination of a constant, the input m_{i }at site i, and the products m_{i}m_{j }of input at sites i and j. A linear filter (plus a constant) is simply a first-order Volterra filter.
Analogous to the connection between
The generality of Volterra filtering can be cumbersome, for one must specify a large number of coefficients in designing one. In filtering a vector of length n (mathematically, a mapping from ^{n }to ^{n}), a linear filter requires O(n^{2}) coefficients, whereas a general k-order Volterra filter requires O(n^{k+1}) coefficients. For applications in vision, the appropriate length n can be truly large: for images n is on the order of a million, and for the discretized ^{2}×, n can be even a hundred times that. Not only is directly solving the linear system for the
Symmetries and structural simplifications of any kind are essentially the best ways for taming the generality of the Volterra filter. One obvious symmetry is homogeneity: the kernel of the filter is the same irrespective of which portion of the image it is applied. This brings the number of coefficients need down to O(n^{k}) from O(n^{k+1}) for the k-th order Volterra filter. Such homogeneity of the filter is a consequence of the statistical homogeneity of the unknown signal and noise, and amounts to stationarity for one-dimensional signals. Other simplifications arise when the unknown signal U is in some parametric family (e.g., ARMA models in one-dimension) or when the Volterra filter is assumed to have a particular structure, such as factorizability or a tensor product basis representation. In vision, quadratic filters have been applied for local contour detection independent of phase (e.g., the contour's edge vs. line characteristics), and general polynomial filters have been linked to the characterization and discrimination of textures. In the case of contour filtering, we shall now see that the special structure of the
4.7 Deriving CIRF Volterra Filters via Perturbation Expansion
Here we derive linear, quadratic and even cubic Volterra filters for the
Suppose we have a random vector X=(X^{i}) (superscripts here index the component of the vector and do not represent taking a power) with prior distribution P(dx) (with known cumulants) and posterior distribution P_{(ε)}(dx)=Z_{(ε)}^{−1 }exp(εf(x))P(dx), where ε is a small real number, f(x)=(a+Ax, x) is a real-valued function of x, a=(a_{i}) is a constant vector, A=(a_{ij}) is a symmetric, negative definite linear operator, and the normalizing constant Z_{(ε) }is equal to ∫exp(εf (x))P(dx). Observe that this situation applies exactly to the
and
Note that we do know all the cumulants of the
Recalling that the
Now, observe that the (posterior) mean can be computed by differentiating the logarithm of the normalizing constant, or
Therefore to get a Taylor series for the posterior mean, we need only get a Taylor series for ln Z_{(ε) }and then differentiate with respect to εa_{i}.
Where do we get a Taylor series for ln Z_{(ε)}? Observe that the normalizing constant Z_{(ε) }is just the moment generating functional of the random vector Y=f(X) in disguise, or, equivalently, Y has the cumulant generating functional
But the coefficients in a Taylor series expansion of a cumulant generating functional are by definition the cumulants, and therefore we need to obtain the cumulants of Y=f(X), a (quadratic) polynomial in X. Unfortunately, the cumulants of Y are not directly known; we must somehow calculate them in terms of the (known) cumulants of X.
4.7.1 Tensors and Generalized Cumulants
Without appropriate tools, calculating the cumulants of a polynomial can be extraordinarily difficult. To aid in this task, a tensor formalism can be used for representing moments, cumulants, and so-called generalized cumulants. For the purposes of this discussion, a tensor is an indexed array of numbers, for example a_{ij }or b_{k}^{ij}, where indices such as i, j, k can appear as superscripts or subscripts and vary over a set of index values, e.g., iεI.
Tensors are multiplied together by multiplying the corresponding array entries (as numbers), as in a_{i}X^{i }or b_{ij}d_{i}, although we shall generally adopt the summation convention. In this convention, an index that appears in both a subscript and a superscript is assumed to be summed over; e.g.,
An index that appears twice or more as a superscript, but not as a subscript (or vice versa) is not assumed to be summed over; so, for example, we do not assume that g^{ij}g^{jk }is equal to
but we do assume that
The summation convention often makes for pithy expressions, but we shall not hesitate from writing out the sum explicitly if it helps to reduce ambiguity.
Moments and cumulants can conveniently be represented using tensors, as follows. The first moment X^{i }will be denoted by κ_{X}^{i }and the second moment X^{i}X^{j }by κ_{X}^{ij}. More generally, the k-th moment X^{i}^{1 }. . . X^{i}^{k }of the i_{1}-th through i_{k}-th components of X will be indicated by κ_{X}^{i}^{1}^{i}^{k}=κ^{i}^{1}^{i}^{k}, where the subscript X will be dropped if understood. Cumulants will be written in a similar fashion, except that commas will separate the indices; we define κ^{i,j,k}:=cum{X^{i}, X^{j}, X^{k}}, for example (where X is understood). In general, the k-th cumulant of X^{i}^{1 }through X^{i}^{k }is denoted by κ^{i}^{1, . . . ,}^{i}^{k}:=cum{X^{i}^{1}, . . . , X^{i}^{k}}. Observe that κ^{i }denotes both the first moment and first cumulant of X^{i}, which is appropriate since they are equal.
A generalized cumulant of random vector X is simply a cumulant of products of the components of X, such as cum{X^{i}, X^{j }X^{k}}. The following is a convenient notation
κ^{ij,k,lmn}:=cum{X^{i}X^{j}, X^{k}, X^{l}X^{m}X^{n}} (4.12)
for generalized cumulants, where the commas in the superscript separate blocks of indices, with each block corresponding to a product of components of X and where these products are separated by commas in the (ordinary) cumulant (as on the right hand side of (4.12)). Due to the symmetry of both (ordinary) cumulants and products of components in their indices, the generalized cumulant is symmetric in its arguments in two senses: (1) the order of the blocks is irrelevant and (2) the order of the indices within each block is irrelevant. Indices cannot be moved from one block to another, however, and therefore κ^{i,jk }is equal to κ^{kj,i }but not equal to κ^{ij,k}, for example. Observe that an ordinary cumulant of k-th order is a generalized cumulant with k blocks of indices, with just one index per block, or κ^{i}^{1, . . . }^{i}^{k}. In contrast, a k-th order moment is a simply a generalized cumulant with one block of k indices, or κ^{i}^{1}^{i}^{k}. Thus the generalized cumulant indeed is a generalization of the statistical constructions of moments and cumulants.
A generalized cumulant can be expressed in a formula as a sum of products of ordinary cumulants. Because we use this formula later, we shall set it up here. Observe that a generalized cumulant with α blocks involving a total of β random variables can be written as κ(), where :={v_{1}, . . . , v_{α}} is a partition of β indices into α non-empty blocks v_{1}, . . . , v_{α} of indices. Now, any such partition can be written as a graph with β vertices, where one vertex is for each random variable or random vector component. All vertices corresponding to each block v_{j }of the partition will be connected by edges, but no two vertices in distinct blocks will be linked by an edge. In this way, a partition corresponds to a graph that is a union of α complete graphs, but with no two such complete subgraphs being connected. Thus will also denote this corresponding graph. Given any other partition ′ of the same set of indices (and its corresponding graph), the edge sum ⊕′ is a graph in the same vertices where the set of edges is the union of the edges for and ′. Now we can state a formula for the generalized cumulant in terms of products of ordinary cumulants:
where the sum extends over all n-block partitions ′:{v′_{1}, . . . , v′_{n}} such that ⊕′ is a connected graph. The factors κ(v′_{j}) in the sum are ordinary cumulants in the indices in block v′_{j }of the partition ′.
To illustrate this result, consider the generalized cumulant corresponding to the partition ={{1, 2}, {3, 4}, {5}, {6}} with α=4 blocks and β=6 indices. While each block of partition ′={{1, 3}, {2, 4}, {5, 6}} joins a block of , the edge sum ⊕′ is not connected (because {5, 6} is disconnected from the other nodes) and so there is no term in the sum in (4.13) corresponding to ′. However, the partition ″={{1, 2, 3}, {4, 5, 6}} does contribute the term κ({1, 2, 3})κ({4, 5, 6})=κ^{1,2,3}κ^{4,5,6 }to the sum because ⊕″ is connected. Using the formula, one obtains, for example,
κ^{i,jk}=κ^{i,j,k}+κ^{i,j}κ^{k}+κ^{i,k}κ^{j}=κ^{i,j,k}+κ^{i,j}κ^{k}[2],
where the bracketed number on the right of a term is the number of partitions of similar form that should be included in the sum. In this case, κ^{i,j}κ^{k}[2] equals κ^{i,j}κ^{k}+κ^{i,k}κ^{k}. Although this bracket notation can be ambiguous (e.g., note that term κ^{j,k}κ^{i}, although of similar form, is not implied by the above use of κ^{i,j}κ^{k}[2]), it is useful in avoiding truly long enumerated lists. Another example of this formula is
κ^{i,j,kl}=κ^{i,j,k,l}+κ^{k}κ^{i,j,l}[2]+κ^{i,k}κ^{j,l}[2].
Applied to the third moment κ^{ijk}, the formula requires a sum over all partitions of {i, j, k} (because ={{i, j, k}} is already connected); and applied to ordinary cumulant κ^{i,j,k,l }(with ={{1}, {2}, {3}, {4}}), the formula is satisfied by exactly one partition ′={{1, 2, 3, 4}} connecting all nodes. This formula succinctly expresses generalized cumulants in terms of a sum of products of ordinary cumulants.
4.7.2 Cumulants of Polynomials
With the generalized cumulant formula, we now need only express the cumulants of our polynomial Y=f(X) in terms of the generalized cumulants of X. By the linearity of expectation and of cumulants, cumulants of homogeneous polynomials such as a_{i}X^{i }and a_{ij}X^{i}X^{j }can be computed straightforwardly. For example, [a_{i}X^{i}]=cum{a_{i}X^{i}}=a_{i}cum{X^{i}}=a_{i}κ^{i }and
using the generalized cumulant formula. Although one can compute the cumulants of general polynomial transformations of a random variable, here we focus on our case of interest, Y=<a+AX, X>=a_{i}X^{i}+a_{ij}X^{i}X^{j}=:(A_{1}+A_{2})X=: PX, where we use the notation P=A_{1}+A_{2 }for our polynomial operator with linear part A_{1 }and quadratic part A_{2}. One can express the cumulant generating functional in terms of the cumulant tensors κ of X using the formal expansion
CGF_{Y}(ε)=exp(εP)κ=(1+εP+ε^{2}PP/2!+ε^{3}PPP/3!+ . . . )κ.
The terms in this expansion are determined by expanding the polynomial operator into a sum of homogeneous polynomials, as follows:
Observe there is only one block of indices in the generalized cumulants in Pκ, but there are two blocks for the generalized cumulants in PPκ. Higher order terms behave similarly, as we now observe in our sought-for expression of the cumulant generating function of the polynomial Y in terms of the generalized cumulants of X:
This expression can be further simplified by observing that all of the bracketed numbers representing permutations become scalar factors by relabeling indices appropriately. To obtain the posterior mean _{(ε)}X, by (4.10) and (4.11) we need only differentiate
Expression (4.14) can be used directly to compute the posterior mean of a signal (with known cumulants) that is corrupted by linear blur and (the limit of) large, possibly correlated, Gaussian noise (for more general observation models, see section 4.3.2). One can derive linear (O(ε) terms), quadratic (O(ε^{2}) terms), cubic (O(ε^{3}) terms) and other higher-order Volterra filters by systematically computing all of the generalized cumulants of X in terms of its ordinary cumulants.
4.7.3 The CIRF: A Tractable Special Case
In the general case, the expansion of formula (4.14) is a lengthy and tedious task: each generalized cumulant will become a possibly large number of products of ordinary cumulants. In addition, the resulting computation naively suggested by the summation convention would have high-order polynomial complexity, where the order depends on the number of terms in the perturbation expansion, e.g., O(n^{7}) operations for the cubic filter, which has all the terms explicitly listed in (4.14). For n=10^{6}, applying the cubic filter would naively require on the order of (10^{6})^{7}=10^{42 }operations. It is little wonder, therefore, that this formula has not appeared in high-noise signal processing or vision applications before.
Fortunately, the contour images modeled by the
Spatial Homogeneity
For general purpose contour enhancement, we assume that any knowledge about contour endpoint locations will be conveyed by the contour image itself. Equivalently, one is indifferent to the starting and ending locations of each contour. As introduced in section 3.5.2, this amounts to setting the initial distribution μ and final weighting ν proportional to a constant, e.g., μ_{i}=|I|^{−1},ν_{i}=λ^{−1}, for all sites iεI. Using the notion of contour density η:=Nλ/|I| (the average amount of contour per site) and observing that G1=λ1, a simple calculation shows that the k-th cumulant of the spatially homogeneous
(Superscripts are used here in keeping with the tensor notation and summation convention.) In addition, the Markov process R_{t }itself will be assumed to be spatially homogeneous in the sense that shifts in the initial conditions give rise to corresponding shifts in the solution to the corresponding Fokker-Planck diffusion. As a consequence, the Green's operator G=(g^{ij}) is constant along the diagonal, i.e., g^{ii}=:d, for all iεI.
Low Contour Density
Another fact that we shall exploit is that, in a large number of images, contours constitute a small fraction of the total image. For the
The key consequence of low contour density is that the generalized cumulants in (4.14) can be approximated with one corresponding joint cumulant, as follows. Recall formula (4.13) for expanding generalized cumulant κ(), where graph ={v_{1}, . . . , v_{α}} has vertices i_{1}, . . . , i_{k}, into a sum of products of ordinary cumulants corresponding to partitions ′ such that ⊕′ is connected. Observe that the complete graph ″={{i_{1}, . . . , i_{k}}} (with only one block in the corresponding partition) always leaves ⊕″ connected, since ″ is connected. Thus there will always be the term κ(″)=κ^{i}^{1}, . . . ,^{i}^{k }(a single ordinary cumulant in all the vertices of ) in formula (4.13), and by (4.15) this term will be O(η). All other terms in the generalized cumulant expansion will be products of ordinary cumulants, and therefore will be O(η^{2}) As a result, with an error of O(η^{2}) we shall replace all of the generalized cumulants in (4.14) with the corresponding joint ordinary cumulant. For example, κ^{r,ij}=κ^{r,i,j}+O(η^{2}) and κ^{r,i,jk,lm}=κ^{r,i,j,k,l,m}+O(η^{2}) for the low-density
Self-Avoidingness of Individual Contours
Suppose our Markov processes were perfectly straight. Then these “curves” would never return to their starting points even if they were infinitely long. Smooth contours approximately obey this relationship for short lengths, and our Markov process contour models have similar behaviour. The direction process, for example, rarely returns to its starting position; the curvature process only returns after a full, smooth, loop, unlike planar Brownian motion, which often self-intersects immediately after passing through a site. This “approximately self-avoiding” nature of our Markov process models will be exploited in deriving the Volterra filters, allowing many terms to be neglected, as we shall see now.
4.7.4 Diagram Method for Evaluating the Posterior Mean
Let us now evaluate the terms in parentheses in the posterior mean expansion (4.14). For the case of additive Gaussian white noise, recall that
After replacing the generalized cumulants with ordinary cumulants via the low contour density assumption, our task is to compute terms such as a_{i}a_{j}κ^{r,i,j }and a_{i}δ_{jk}δ_{lm}κ^{r,i,j,k,l,m}. Also, to avoid unnecessary distraction, we suppress all multiplicative powers of
from the a_{ij}s and the scalar multiplication by η in the
To support the error-prone task of explicitly summing over each permutation and to aid the intuition, we shall construct a diagram corresponding to each permutation. Let us start with the simple case of a_{i}κ^{r,i}=a_{i}g^{ir}+a_{i}g^{ir}, using (4.15). This sum has two terms, the first corresponding to the permutation (r, i). For each term we construct our diagram as a graph with one node per index, and with directed edges corresponding to the order of the permutation (part (a) of FIG. 1). Observe that no node occurs twice in the permutation, and so the diagram just represents an ordered, simple path along the nodes. We then embellish the diagram (part (b) of FIG. 1) to illustrate how it corresponds to the expression a_{i}g^{ri}: node r (the “readout”) is identified with a hollow dot (“observation”), and the nodes corresponding to the coefficient a_{i }are marked with an “X” to denote an “amplification,” “boost,” or “interaction” with input a. The term a_{i}g^{r,i }corresponding to this diagram (part (a) of FIG. 1) represents the action of operator G on vector a and evaluated at r, or (Ga)^{r}; observe that the edge is directed away from r. Conversely, the term a_{i}g^{ir=(G*)}^{ri}a_{i }corresponding to permutation (i, r) represents the action of operator G* on vector a and evaluated at r, with the edge directed towards r. One can check that this pattern persists for more complex diagrams and so we can write the operator G along each link directed away from r and G* along each link directed towards r. Dropping the r to represent the vectorial result, we work inwards toward the readout r on the diagram to simply write Ga for permutation (r, i), and, from part (c) of FIG. 1, G*a for (i, r).
For term a_{i}a_{j}κ^{r,i,j}, we need to consider three kinds of permutations of the indices r, i, and j. Permutation (r, i, j) corresponds to a_{i}a_{j}g^{ri}g^{ij}, which can be evaluated as g^{ri}a_{i}(Ga)^{i}=g^{ri}(diag a Ga)^{i}=G diag a Ga because the componentwise product a_{i}b_{i }of any vectors a and b can always be written as (diag a b)_{i}, the i-th component of the matrix diag a times vector b. This result can be read directly from part (d) of FIG. 1 by working from node j towards the readout r: the input a comes in first at j, is operated on by G, the result Ga is componentwise multiplied by a (equivalently, operated upon by diag a), which is finally operated upon by G, to produce G diag a Ga. Since we get the same diagram back when nodes i and j are swapped, to permutation (r, j, i) we get the same result G diag a Ga. Permutation (j, i, r) has the diagram shown in part (e) of FIG. 1, which is the same as the reversal of the diagram for permutation (r, i, j), and so corresponds to G* diag a G*a. Again, nodes i and j can be swapped to produce a G* diag a G*a for permutation (i, j, r). The readout r need not be at the end of the ordering, as in permutation (j, r, i) corresponding to term a_{i}a_{j}g^{jr}g^{ri}=(G*a)^{r}(Gr)^{r}. Letting a·b denote the vector which is the componentwise product of vectors a and b, i.e., (a·b)_{i}=a_{i}b_{i}, the vector result from permutation (j, r, i) can therefore be written as G*a·Ga:=(G*a)·(Ga). This can also be derived from the diagram by splitting the diagram (part (f) of FIG. 1) at r, separately treating the two resulting sub-paths (one directed into r, the other directed away from r) as before, and then taking the componentwise product as the result. Note that i and j can be swapped as before for another G*a·Ga. Gathering together the results from parts (d), (e), and (f) of FIG. 1, we obtain
a_{i}a_{j}κ^{r,i,j}=(2G diag aGa+2G* diag a G*a+2G*a·Ga)^{r}.
In hindsight, this result can be derived most easily from the appropriate diagrams.
In addition to terms with only coefficients of form a_{i}, we also have to deal with terms that have a Kronecker delta, such a_{i}δ_{jk}κ^{r,i,j,k}. Because the Kronecker delta δ_{jk }evaluates to 1 only when the two indices have equal value, and zero otherwise, the coefficient δ_{jk }represents the identity of indices j and k. In the diagram this amounts to the collapsing of nodes j and k into a new combined node {j, k}, giving rise to a loop. For permutation (r, i, j, k), we therefore get the diagram (a) in FIG. 2 for the expression a_{i}δ_{jk}g^{ri}g^{ij}g^{jk}=g^{ri}a_{i}Σ_{j}g^{ij}g^{jj}. To evaluate this diagram, we start at node {j, k}. We first realize that, due to spatial homogeneity, the G is constant along the diagonal, or g^{jj}=d=(d1)_{j }for all j. On the diagram this can be represented by replacing the loop with an “input” of d1 at node {j, k} (all one-edge loops can be replaced in this way). Working backwards towards r, we find that permutation (r, i, j, k) evaluates to G diag a Gd1=dλGa, since G1=λ1. Having evaluated this diagram we know how to evaluate similar diagrams. First, the reversal of nodes j and k produces dλGa for permutation (r, i, k, j). Second, reversing all of the arrows (for both orderings of j and k) causes the substitution of operator G* for G.
For permutation (j, k, r, i) we must evaluate a_{i}δ_{jk}g^{jk}g^{kr}g^{ri}. Explicitly summing over indices we obtain (Ga)^{r}Σ_{j}g^{rj}g^{jj}=dλ(Ga)^{r}. But from the diagram (part (b) of FIG. 2), we can just take the componentwise product of the results due to the path towards and the path away from r, with input a at node i and input d1 (for the loop) at node {j, k}. We then get the vectorial result Ga·dG1=dλGa. Again, nodes j and k can be swapped and the entire path can be reversed (G becomes G*).
For permutation (r, j, k, i), we need to evaluate a_{i}δ_{jk}g^{rj}g^{jk}g^{ki}. Again, the loop becomes an input d1 at {j, k}, and so we can use the diagram (part (c) of FIG. 2) to obtain the result G diag d1Ga=dG1Ga=dGGa.
Now, observe that G is a positive operator for the Markov processes in which we are interested. Given that there is only one positive eigenvector for a positive operator and that it corresponds to the maximum eigenvalue, the equation G1=λ1 implies that λ is the maximum eigenvalue because 1 is a positive eigenvector of G. This implies that λ^{−1}G is a contraction mapping that only passes constant vectors unchanged; non-constant vectors will shrink. Equivalently, ∥Gy∥<∥λy∥ unless y is a constant vector.
Because of this, we view the result dGGa=dGy for permutation (r, j, k, i) as smaller than the result dλGa=dλy for permutation (j, k, r, i), where y=Ga is generally non-constant for typical inputs a. Observe that such contraction mappings will correspond to repeated operators (GG, G*G*, etc., without a “boost” or “amplification” from a between them) whenever there is a loop located between a regular input a (located at i for permutation (r, j, k, i)) and the output r. As a first approximation, we shall therefore neglect dGGa and all other permutations that have a Kronecker delta index between an input node and the output r; we shall restrict our attention to those permutations where all of the indices with subscripted inputs a form an uninterrupted sequence (in the permutation) to the output r. Let us call this criterion for neglecting terms the input-output-coupling rule.
In addition to this rule for neglecting loops, there is another that is due to the approximately self-avoiding nature of our Markov processes. We illustrate this rule with permutation (r, j, i, k) (part (d) of FIG. 2), or term a_{i}δ_{jk}g^{rj}g^{ji}g^{ik}=Σ_{j}g^{rj}(G^{ji}(G*)^{ji})a_{i}=(G(G·G*)a)^{r}, where A·B denotes the componentwise product of the two operators A and B. Here, the effective input at combined node {j, k} due to the 2-edge loop {{j, k}, i} is (G·G*)a. We have “broken” the loop at i and taken the componentwise product of the operators that correspond to the two distinct paths from i to {j, k}; this is the action of operator G·G* on vector a. For approximately self-avoiding processes, observe that G·G* is approximately d^{2}I: each row is the componentwise product of the corresponding rows of G and G*, which, by spatial homogeneity, is just the componentwise product of the Green's operator kernels for the forward operator G and the backward operator G*. For the directed Markov process in ^{2}×, the forward operator is large only on one side of the impulse, and the backward operator is large only on the other, and thus the componentwise product is only significant at the origin, which corresponds to the common diagonal g^{ii}=d. For the curvature process in ^{2}××, the forward and backward kernels may overlap for impulses of high curvature, suggesting the need for a bound on curvature. Thus permutation (r, j, i, k) evaluates to the simple expression d^{2}Ga, which is clearly smaller than the dλGa expression for permutation (r, i, j, k), since d=τ (the average amount of time the contour spends in a site as it passes through) is certainly much less than the total average contour length λ. This example suggests the 1-link-loop rule: neglect those permutations that correspond to diagrams where the largest loop has more than one link or edge.
4.7.5 Computational Significance of Approximately Self-Avoiding Processes
It is interesting to note that for Brownian motion (a symmetric process, with G*=G) there is perfect overlap between the forward and backward kernels, so the above argument would not apply and therefore we should not invoke the 1-link-loop rule to neglect such terms with inputs inside a loop. This suggests a novel justification for orientation in vision: using approximately self-avoiding Markov processes (e.g. the directed process in ^{2}×) gives rise to a reduction in computational complexity compared to non-self-avoiding processes such as planar Brownian motion. Evolution might exploit this fact since those organisms that expend fewer resources on contour computations are more likely to survive. Observe that this is rather counter-intuitive because it is the more elaborate representation (^{2}×), not the simpler (^{2}), that has the smaller computational burden. Also note that this complexity argument is distinct from the statistical argument as to whether contours in the world do tend to smoothly continue in orientation.
Returning to the calculation, both the 1-link-loop rule and the input-output-coupling rule force us to drop permutation (r, j, i, k), but only the 1-link-loop rule applies will cause us to neglect permutation (j, r, i, k) (part (e) of FIG. 2). The latter permutation, which has the output r within the loop can be evaluated by splitting the loop at i, and then taking the componentwise product of the operators for the two paths to the output r. Since there is no other loop at {j, k}, we view it as having the input 1. (This evaluation of a loop at the output is only necessary if the output is part of the loop.) The result is (G·G*G*)a, which we assume to be small.
Bigger loops (with more inputs a) can be handled by breaking the loop at some node and then forming the appropriate componentwise product operator, giving rise to effective inputs {(G diag a G)·G*}a or {G·(G* diag a G* diag a G*)}a at the combined node. Where there are loops within loops, we recursively evaluate from the innermost loops by replacing each loop with the effective input at the corresponding combined node and working toward the outermost loops. For the purpose of evaluating the high-noise linear, quadratic, and cubic filters, the largest loop will have only two internal inputs, corresponding to, for example, effective inputs {(G diag a G)·G*}a and its reversal {G·(G* diag a G*)}a. The use of approximately self-avoiding processes will allow us to apply the 1-link-loop rule to neglect these terms because, for example, each column of the “amplified” forward operator G diag a G probably has little overlap with the corresponding column of operator G*. For filters of much higher-order than cubic, the 1-link-loop rule may not be appropriate even for approximately self-avoiding processes because the kernel for a long sequence of alternating forward operators and amplifications (e.g., G diag a G diag a . . . G) may overlap with G*; for example, a may represent an edge map of a sufficiently smooth, closed contour. (For the direction process, if there is no diffusion in direction (σ_{κ}=0) and space is unbounded, which corresponds to straight line trajectories, even higher-order kernels will never overlap.) Again, we shall apply the 1-link-loop rule for our
More complicated diagrams than those above can arise; for example, there may be no strict nesting of the loops. However, using the 1-link-loop rule we shall neglect the terms corresponding to such diagrams. These more complex diagrams can represent highly improbable self-intersection patterns—tortuous paths—so neglecting them seems reasonable.
In summary, we use a diagram-based technique for evaluating all the permutations in the terms of the posterior mean expansion (4.14) for
this diagram-based calculation was used to derive the following.
Result 1 (High-noise MMSE CIRF Volterra Filters). Suppose that the curve indicator random field U (for approximately self-avoiding Markov processes) is corrupted with additive Gaussian white noise of variance σ_{N}^{2}=: ε^{−1 }to produce measurement vector m. Let ζ:=dλ, where d=g^{ii }and λ is the average curve length. Then the minimum mean-squared error estimate of U given m has the following approximate asymptotic expansion as σ_{N}→∞(ε→0):
Linear Filter:
ũ^{(1)}=η{1−2εζ+ε(Gm+G*m)}+O(ηε^{2}+η^{2}ε) (4.16)
Quadratic Filter:
Cubic Filter:
We shall see in section 5 that by implementing the operator G in the Fourier domain, its computational complexity is O(n log n) for the direction process, where n=|I|. Since multiplying a diagonal matrix with a vector has O(n) complexity, the net complexity for all of these filters is therefore O(n log n) for the direction process. This is far better than for the general k-th order Volterra filters, which have O(n^{k+1}) complexity.
One could attempt to improve upon these results by pursuing higher-order expansions. However, instead of following that tedious, difficult path, we now turn to a very different approach for deriving filters that is based on the
4.8 The Biased CIRF Approximation of the Posterior
In addition to deriving high-noise Volterra filters by using the low-order cumulants of the prior, we can also compute filters by approximating the posterior with a more manageable distribution. We shall now develop a related approximation for the
Because by Bayes' rule any posterior is a weighted version of the prior, it is reasonable to base our approximation on the prior. Although the prior itself will not suffice as an approximation, since it is not even a function of the measurements, the prior can be modified or “biased” by a function of the measurements to make a useful approximation of the posterior. To see how to do this, recall from section 3.5.2 that we have a tractable expression for the
4.8.1 Cumulants of the Biased CIRF
To compute the cumulants of the biased
For example, to compute the mean of a biased
where ∇_{c }denotes the gradient with respect to c. Substituting the
Since G_{c}:=−(Q+diag c)^{−1}, to continue the calculation we must understand how to take derivatives of inverses of operators, which we study in the following lemma.
Lemma 3. Suppose matrices A=(a_{ij}) and B=(b_{ij}) are functions of some parameter α, i.e., A=A(α), B=B(α). Letting primes (′) denote (componentwise) derivatives d/dα with respect to α, the following formulas hold:
(AB)′=A′B+AB′, (4.19)
(A^{−1})′=−A^{−1}A′A^{−1}. (4.20)
Returning to the calculation of the mean of the biased
G_{c}D_{i}G_{c}, where D_{i }is a matrix with a one at the (i, i)-entry and zeroes elsewhere. The matrix D_{i }can be expressed as the outer product δ_{i}δ*_{i }of discrete impulse vectors δ_{i }at site i, where (δ_{i})_{j}=δ_{ij}. Therefore
proving the following:
Proposition 7. The mean of the
κ_{c}^{i}=_{c}U_{i}=N<μ, G_{c}D_{i}G_{c}ν>. (4.21)
Equivalently, the mean (vector) of this biased
_{c}U=N(G*_{c}μ)·(G_{c}ν). (4.22)
The higher-order cumulants can also be tractably characterized as follows:
Proposition 8. The k-th cumulant at sites i_{1}, . . . , i_{k }of the
where (g_{c}^{ij}):=G_{c }is the Markov process Green 's operator with bias c.
4.8.2 Solving for the Biased CIRF
Given this manageable characterization of the cumulants of the biased
where the second term on the last line is simply the cumulant generating functional of (a−c′+AU, U) (with respect to measure P_{c}(du)) evaluated at ε, similar to (4.11) except the measure on U is now biased by c. Now, as for the Volterra filters, we shall develop the
ln _{c}e^{ε<a−c′+AU,U>}=ln(1+ε<a−c′+AU,U>+O(ε^{2}=_{c}ε<a−c′+AU,U>+O(ε^{2}).
Because the latter expectation is just −<c, _{c}U>+ε<a, _{c}U>+εΣ_{j}_{c}[(AU)_{j}U_{j}], we can rewrite (4.24) as
(4.25)
Since in Z_{(ε)}, which characterizes the true posterior, does not depend on the bias c of our approximate posterior P_{c}, differentiating ln Z_{(ε) }with respect to any function of c will be zero. In particular, we can set up a system of equations in c by taking the derivative of (4.25) with respect to _{c}U_{i}=κ_{c}^{i}, a function of c. We consider two special cases now.
No Blur Case
Consider the situation where there is no blur in the observations and the noise is white. Thus we set
and (4.25) becomes
(4.26)
Differentiating with respect to κ_{c}^{i }and applying the chain rule twice, the first two terms on the right hand side become
because
Using formula (4.23) twice, note that κ_{c}^{jj}=2g_{c}^{jj}NΣ_{i′,j′}μ_{i′}ν_{j′}g_{c}^{i′j}g_{c}^{jj′}=2g_{c}^{jj}κ_{c}^{j}, because the two permutations are identical. So by differentiating (4.25) with respect to κ_{c}^{i}, we get: 0=−c_{i}+ε(a_{i}−g_{c}^{ii}+_{c}U_{i})+O(ε^{2}), proving the following result:
Result 2. Suppose the posterior distribution of the observed
c=ε[a−_{c}U−d(c)]+O(ε^{2}), (4.28)
where d(c) is the diagonal of the Green's operator G_{c }with bias c, or d(c)_{i}:=g_{c}^{ii}.
As discussed in section 4.3.1, we can apply this result to the case where the
Observe that (4.28) is a coupled nonlinear system in the vector c of |I| unknowns; c is expressed as a function not only of the exogenous a, but also of the (generally nonlinear) functions of itself: _{c}U (see Proposition 7) and d(c). This equation describes c as proportional to a balance between positive influence due to input a and negative influence due to the sum of the mean of the biased
Large Blur Case
Another system of equations can be derived in an opposite extreme, where the symmetric negative definite matrix A is proportional to an operator that performs a great deal of blur. In that case, the blurred field AU is essentially a constant vector, and therefore cov_{c}((AU)_{j}, U_{j})≈0. Using ∇_{c}_{U}<_{c}AU, _{c}U>=2A_{c}U and the above calculations for the no-blur case, we get the following system of equations.
Result3. Suppose the posterior distribution of the observed
c≈ε[a+2A_{c}U]+O(ε^{2}). (4.29)
This large blur result provides a nonlinear system of equations in the bias c. It differs from (4.28) by the presence of the blur operator A acting on the mean and the absence of the closure measure d(c). By including blur, we can better describe a number of situations, as outlined in section 4.3
Also note that for both the non-blur and large blur cases, the biased
Combined with the formula for the mean of the biased
Observation (Biased CIRF Approximation of the Posterior Mean of the CIRF). Suppose a
_{m}U≈Nf·b, (4.30)
where f and b solve the following coupled system:
Qf+ε diag[a+2NA(f·b)]f+ν=0 (4.31)
Q*b+ε diag[a+2NA(f·b)]b+μ=0. (4.32)
For reference, we include the above filtering equations for the continuum-limit of the above biased
Observation (Reaction-Diffusion-Convection Equation for the Direction Process). Suppose a
Also note that for A∝I, the nonlinear coupled PDE is in fact a reaction-diffusion-convection equation. Since c≈ε[α+2NA(f·b)], we see that pattern formation arises from c^{+}=max(c, 0) (the positive part of bias c), pattern repression comes from −c^{−}=min(c, 0) (the negative part of c), diffusion comes from the second derivatives, and convection is due to the first derivatives.
4.9 Comparison with Completion Fields
Perceptual organization has raised the possibility of discovering a rigorous foundation for classical demonstrations such as illusory contours. This gave rise to work on contour saliency both as a model of illusory contour formation and for contour organization. We shall focus on drawing relationships to stochastic completion fields (Williams and Jacobs, supra) and eigensources because of their focus on Markov processes such as the direction process.
To make the connection, we begin with the biased
Now, we observe even though we have dropped the interaction operator A, the resulting filter is still a nonlinear input-to-output mapping, and in two distinct ways. First, we are taking the product of the forward and backward solutions. Second, the input arises as a modification to the diagonal of linear operator Q. In particular, we have f=(1−GC)^{−1}Gν=Σ_{k=0}^{∞}(GC)^{k}Gν, which says that f is an infinite polynomial in c, i.e., f is nonlinear in c. If we ignore the constant term (k=0) and all nonlinear terms (k>1), what remains is the linear term GCGν, which is proportional to Gc if ν is a constant vector and G is spatially homogeneous (as it is for the direction process). We recognize Gc as the source field (with sources c∝CGν), and, by the same argument, G*c is the sink field (with sinks c∝CG*μ, where μ is also a constant vector); their product Gc·G*c is then a stochastic completion field. To have different sinks and sources one can adjust μ and ν.
To relate to eigensources, let C:=diag a and thus write the forward equation as: f−εGCf−Gν=0. Setting ν=0 and assuming a_{i}>0, for all i, we can define W^{2}:=C to get GW^{2}f=σ_{N}^{2}f, where we recall ε=σ_{N}^{−2 }(see section 4.3.2). Defining h=Wf, we write GWh=σ_{N}^{2}W^{−1}h, or
WGWh=σ_{N}^{2}h.
This is an eigenvalue equation, where σ_{N}^{2 }is now interpreted as an eigenvalue of WGW. We see, therefore, that the
4.10 What Can One Do with a Filtered CIRF?
It may seem odd that we have sought to (merely) filter a field of local contour measurements, producing yet another field. If one objective of vision is to explicitly represent structures with increasing abstraction, this task may seem to postpone the most difficult work. To return to a question brought up at the start of this section, why not extract the contours directly from the image, instead of just enhancing a contour image?
There are a number of reasons why one might focus on processing a field. First, this approach captures a notion of global information beyond that of a single curve, by explicitly expressing the idea of an image of contours, where both the interactions among the contours and the blur due to imaging is not ignored. In contrast, previous approaches for computing curves using Markov process contour models (Montanari, supra), implicitly assume that there is only a single contour. By emphasizing the field we can combine the richer contour models of the single-contour schemes with the contour interactions implicit in the line processes used in
More pragmatically, simple post-processing of the filtered
Since information is lost in thresholding, however, for more sophisticated contour processing, might require the (un-thresholded)
Section 5
5.1 Numerical Methods
Having worked out formulas for
5.2 Applying the Green's Operator by Solving the Forward Equation
To compute the action f of the Green's operator G upon a vector h, we can—equivalently—solve for f in the forward equation
(αI−L)f=h, (5.1)
since Q=L−αI and G=−Q^{−1}. Now, all of the theory in sections 3 and 4 was based on continuous-time, discrete-space Markov processes, to avoid many measure-theoretic distractions. However, as we saw in section 2, the Markov processes we are most interested in live in continuous spaces such as ^{2}×. In general, by designing a Markov process via an appropriate stochastic differential equation, we can immediately obtain the process's generator from the corresponding Fokker-Planck equation. When the Markov process is a diffusion process, is a partial differential operator acting on functions of continuous space, not vectors. To make the transition from the continuous to discrete domains, we let the discrete-space generator L be some (reasonable) discretization of the continuous-space generator . Strictly speaking, therefore, the
5.2.1 Direction Process
Here we solve the (discrete) forward equation for Mumford's direction process by first specifying the related direction process forward
Recalling the continuous-space generator of the direction process from (2.4), the corresponding forward
where f=f(x, y, θ), h=h(x, y, θ), and σ=σ_{κ}>0. Observe that this is a degenerate elliptic
The Dirichlet and Neumann boundary conditions may be more natural for contour enhancement, as periodic boundary conditions can lead to “wrap around” artifacts.
At least two major approaches can be taken in numerically solving
A second approach to solving the direction process forward
Before developing our method, we observe that trigonometric bases correspond to the three different boundary conditions. Specifically, the sine, cosine, and complex exponential bases correspond to the Dirichlet, Neumann, and periodic boundary conditions respectively. Despite the periodicity of the complex exponential basis, we apply it here both because of the above advantages and to simplify the presentation. To reduce the negative artifacts of periodicity in (x, y), we can often solve the equation on a larger domain than necessary (after suitably padding h), and then extract the region of interest from the solution f.
We now detail our direct method. Let denote the imaginary unit √{square root over (−1)}. The complex exponential having periods x_{max}, y_{max}, and 2π and at frequencies j, k, and l in the x-, y-, and θ-dimensions, respectively, is
Let the Fourier coefficient {circumflex over (f)}_{jkl }be defined as the inner product <f, φ_{jkl}> of a function f with a complex exponential φ_{jkl}; ĥ_{jkl }is defined similarly. Now we can evaluate, term-by-term, the inner product of the complex exponential with the direction process forward
Using integration by parts and periodicity in x, we know
this result (“taking the adjoint of the x-derivative”) holds analogously in y and in θ. By both taking the ajoint of the θ-derivative twice and applying the chain rule twice, we get
Using the formula cos θ=(e^{θ}+e^{−}^{θ})/2 and taking the adjoint of the x-derivative, we can write
as
Similarly, we get
using the formula sin θ=(e^{θ}−e^{−}^{θ})/(2). This proves the following.
Proposition 9. Suppose h(x, y, θ) is periodic in x, y, and θ with periods x_{max}, y_{max}, and 2π, respectively. Then the solution f=f(x, y, θ) of (5.2) is also periodic and, for each j, kε, the Fourier coefficients {{circumflex over (f)}_{jkl}:Iε} and {ĥ_{jkl}:lε} satisfy the following tridiagonal linear system indexed by lε:
To go from the above infinite linear system to a manageable finite one, we assume that h is bandlimited. Then, without loss, we can represent h by its uniform samples h_{j′k′l′}:=h(j′Δx, k′Δy, l′Δθ), with indices j′ε{0, . . . , J−1}, k′ε{0, . . . , K−1}, and l′ε{0, . . . , L−1}, where positive integers J, K, and L are the total numbers of frequencies (including both nonnegative and negative frequencies) in x, y, and θ, respectively, such that the periods satisfy x_{max}=JΔx, y_{max}=KΔy, and 2π=LΔθ. Prop. 9 then implies f is bandlimited in x and y. Due to the bandlimitedness of h in θ and the fact that the direction process forward
where
and
This system can be solved for {circumflex over (f)}_{j,k,}. in O(L) operations. Using the
When the input—before discretization—is not bandlimited, aliasing and oscillation can occur both in the (discrete) input and solution. For example, if the discretization of h is a discrete impulse (a “1” at one position, with “0”s elsewhere), the corresponding continuous-space h is not a Dirac δ-distribution, since the δ-distribution is not band-limited. Instead, the continuous-space h corresponding to a discrete impulse is a sinc function centered on the impulse and with zero crossings at all other sample locations. The (standard) sinc function has a period of infinity. For finite periods, qualitatively similar functions arise. Between the sample locations the sinc oscillates above and below zero. If the discrete impulse is shifted by a non-integral number of samples, which can only fairly mean that the underlying continuous signal—the sinc function—is so shifted and then resampled, the discretization will no longer be a discrete impulse, but will have (discrete) oscillations. This phenomenon also appears in the solution of the direction process forward
The corresponding backward equation (αI−L*)f=h requires taking the adjoint of the direction process generator, which in this case amounts to flipping the signs on the first derivatives. Obeying these sign changes, the backward equations can be solved similarly.
Instead of numerically solving the direction process forward
Results
To demonstrate the performance of the direct method we compared it with the first-order finite differencing method. To solve the direction process forward
In the first experiment, we illustrate the isotropy that results from the accuracy of the direct method, using a slightly blurred (twice convolved with a [1 1] box filter in the x- and y-directions) impulse as h. This experiment shows that at both 0° and 45° the direct method produces similar, but rotated results, whereas the first-order method result is much more diffuse for the 45° direction than for 0°. The difference is especially apparent close to the impulse: at 45° the first-order method result is not nearly as focused as the direct method result. See FIG. 3. In contour filtering applications, accurate results are especially important nearby the impulse, where most of the filtering action occurs. In computer vision, the damaging effect of the discrete grid has often been observed, where vertical and horizontal edges are much more easily enhanced than oblique ones. The direct method overcomes this anisotropy in the context of the direction process.
For the second experiment, we show the oscillations that occur in the direct solution when the (sampling of) h is a discrete impulse (FIG. 4). Here, the discrete impulse is the sampling of a (periodic) sinc function, giving rise to the oscillations observed in these solutions to the direction process forward
The next experiment was designed to illustrate the accuracy of the direct solution compared to finite differencing. A fixed stimulus h was chosen, e.g. a Gaussian, and the discretization was varied on a regular schedule to see the improvement in solution accuracy with successive doublings of spatial resolution. FIG. 5 shows a comparison of solution improvement with increased spatial resolution. The direction process forward
To put the direct and first-order methods into clearer contrast, we re-plotted columns (b) and (c) of FIG. 5 to show, for each method, the rate at which the solution is changing (which we assume to mean improving) with increasing resolution. To save storage space, computation, and, most importantly, achieve a solution accuracy as close to machine precision as desired, it is important that the solution improves quickly as a function of increasing resolution. For each method, we measured the relative change in the solution over the common 8×8 subgrid from scale k to scale k+1 as a function of scale. Let h_{8×8}(2^{k}) be the solution obtained on the 8×8 subgrid at resolution 2^{k}×2_{k}. The relative change in solution is
See FIG. 7, which shows the rate of improvement in solution with increase in resolution. For the first-order method, we observe that the relative change in solution is greater than 10% for the 8×8 to 16×16 resolution change. This relative change in solution decreases very slightly, so that after four doublings of resolution, the relative change is still several percent. For the direct method, the results are dramatic: at first the relative change in solution is about the same as for the first-order method. But by the 64×64 to 128×128 resolution change, the relative change is on the order of machine precision (about 10^{−15}). The small improvement for the next increase in resolution furthur illustrates that the machine precision limit has been achieved. The direct and first-order methods were implemented in Python with the numerical Python library and C extensions under GNU/Linux.
5.2.2 Curvature Process
To compute the action f of the Green's operator G for the curvature process on a vector h, we recall the curvature process generator (2.7) to set up the curvature process forward
where σ=σ_{{dot over (κ)}}>0. The action of the Green's operator G is then defined as our numerical solution to (5.5). To balance performance and simplicity, we use a combination of finite differencing and basis projection. To keep the desired rotational isotropy and solution focus, we project f with respect to only the x-, y-, and θ-dimensions on the same complex exponential basis that we used for the direction process to obtain {circumflex over (f)}_{jkl}(κ):=<f(., ., ., κ); φ_{jkl}>_{2}_{×}_{ }for frequencies j, k, and l. The κ-dimension is discretized into M uniform samples using finite differencing, as follows. We first write {circumflex over (f)}_{jkl;m′}={circumflex over (f)}jkl(m′Δκ), where κ=m′Δκ for integer m′ and stepsize Δκ>0. The input h is discretized similarly. Then we approximate
with (g_{m′−1}−2g_{m′}+g_{m′+1})/(Δκ)^{2}, for g_{m′}:=f(x, y, θ, m′Δκ).
Taking the (x, y, θ)-projection of (5.5) onto the complex exponential basis, we can reuse the direction process results, noting that <∂f/∂θ, φ_{jkl}>_{2}_{×}=l{circumflex over (f)}_{jkl;m′}. We assume the same bandlimitedness (with respect to (x, y, θ)) of h as in the direction process, and so are led to solve for the lowest J, K, L frequencies in (x, y, θ) of f in terms of those of h. Combined with the finite difference approximation in κ, we now need to solve
The same periodic boundary condition in l is used as in the direction process case. We use zero boundary conditions in m′ to put an upper bound on curvature.
Observe that for each j and k, equation (5.6) is an (L·M)×(L·M) sparse linear system in l and m′. Although the computationally advantageous tridiagonality of the directional process case has been lost, we note that we are often interested in small L and M (the numbers of discrete directions and curvatures, respectively). The overall complexity of this system is O((JKLM)log(JKLM))+O(JK(LM)^{3}), using LU decomposition to solve the linear system.
5.3 Applying the Biased Green's Operator
In this section we present a method for computing the action f of the biased Green's operator G_{c }on a vector h. Equivalently, we must solve the linear system
−(Q+C)f=h (5.7)
where Q=αI−L and C is a (possibly non-constant) diagonal operator with c=(c_{i}) along the diagonal. See section 3.4 for how this equation arises in the Feynman-Kac formula. Because of Khas'sminskii's condition (Proposition 3), we know that there is a unique solution to this equation as long as c is sufficiently small.
We first assume that we are given a method for solving −Qf=h, i.e.:
f=Gh.
Such methods were just provided in section 5.2.1 and section 5.2.2 for the direction and curvature processes, respectively. Observe that G is a discretization of an integral operator, as it is the inverse of a discretization αI−L of differential operator α−. Loosely, recall that, for the direction process, G represents a kind of blurring of a directed edge map (function of (x, y, θ)) in the tangent direction θ.
Now, instead of solving the (5.7) as (a discretization of) a
A_{1}f:=G(−Q−C)f=(I−GC)f=Gh=: b_{1}.
This procedure is known as preconditioning a linear system. Since we multiplied −Q−C by G on the left, we shall say that A_{1}u=b_{1 }is preconditioned on the left. One can instead make the substitution f=Gv and solve the linear system:
A_{2}v:=(I−CG)v=(−Q−C)Gv=(−Q−C)f=h=:b_{2}.
Since G appears to the right of −Q−C, we shall say that A_{1}u=b_{1 }is preconditioned on the right. The idea behind preconditioning is that we turn a system that may be badly behaved (such as a differential equation) into one which is better behaved, such as an integral equation.
To solve Au=b, where A=A_{i}, b=b_{i}, i=1, 2, we will invoke the conjugate gradient method. Unfortunately, this method requires a symmetric and positive definite linear system, but Q and therefore A is not symmetric. One technique for circumventing the asymmetry of A is to multiply (5.7) on the left by its transpose A*, forming the normal equations
A*Au=A*b, (5.8)
where A*A is symmetric because (A*A)*=A*(A*)*=A*A and positive definite because x*A*Ax=(Ax)*(Ax)=∥Ax∥^{2 }>0 for nonsingular A unless x=0. Now, we can apply the (iterative) conjugate gradient method directly to (5.8) in order to solve for u in the sense that the relative residual ∥A*Au−A*b∥/∥A*b∥ is reduced to less than some tolerance level. We can then apply the above definitions to get f itself.
5.4 Solving for the Biased CIRF Approximation of the Posterior
In section 4.8 we derived coupled nonlinear systems for filtering. Because of the presence of blur in a number of applications, we elected to solve the nonlinear system in equations (4.31) and (4.32). Left multiplying these by Q^{−1 }and Q*^{−1 }respectively, we get the following system:
f−εGCf+ν′=0 (5.9)
b−εG*Cb+μ′=0 (5.10)
where C:=diag(a+2NA(f·b)), ν′=Gν, and μ′=G*μ. To solve this nonlinear system, we adopted a continuation strategy in ε (K. Georg. Matrix-free numerical continuation and bifurcation. Numerical Functional Analysis and Optimization, 22:303–320, 2001). Specifically, we began with ε small, solved the system, and then used that solution to get an initial condition for the system with ε slightly larger. We proceeded in this fashion until we reached the value of ε required. To solve this system for any such fixed ε, we used Newton's method, where often only a few iterations are required because of the Newton method's powerful local convergence properties. To bootstrap this continuation process with a starting initial condition (corresponding to the smallest value of ε), we set C=diag(a) (i.e., neglecting the nonlinearity) and then solved the resulting decoupled equations, which evidently are the biased
Section 6,
6.1 Results
In this section we illustrate the performance of
6.2 Contour Filtering in ^{2}× with Direction
The first test of the
Because of the computational efficiency of the Volterra filters, we were able to test them on a number of examples. To assess the robustness of the Volterra filter in the presence of noise, we evaluated responses to images of a horizontal straight line corrupted by additive Gaussian noise of variance σ_{N}^{2}. We characterized the degree of noisiness using the peak signal-to-noise ratio, which, for an image I taking values in the range I_{min }to I_{max}, is
expressed in decibels. To map from the planar image to a field on discrete ^{2}×, we simply copied the image to each direction, so that our measurements m=m(x, y, θ) were constant in θ. We were able to select this mapping because the images were purely contours embedded in noise. More generally an oriented local edge or line operator would be applied, as we shall see when we consider natural images.
To avoid the aliasing effect illustrated in FIG. 4, for all experiments in this section we convolved the input with a Gaussian having a standard deviation of 1 pixel in all dimensions before applying a
where <ũ_{1 }, ũ_{2 }> is the Euclidean inner product between ũ_{1 }and ũ_{2}, and ∥u∥=√{square root over (<u, u>)}. FIG. 10 shows noise performance of direction process-based
To study the effect that the parameters of the direction process have on our filtering results, we created an image containing curves of different curvatures and lines with gaps, and then applied the direction process-based
We then tested the filter's response on both open and closed curves. FIG. 12 shows the filtering of crossing contours with direction process-based Volterra filters (ε=10.5, ζ=0, σ_{κ}=⅕, λ=100, and 32 directions). Image before adding white noise is at the top right; the noisy image (peak
To test the behaviour of the filter at crossings more strenuously, we created an image with several nearby crossings. FIG. 13 shows how crossing lines are teased apart with the cubic Volterra filter. From left to right in row (a): original image, image corrupted with additive white Gaussian noise (peak
We next applied the Volterra filters to two natural images. Instead of simply copying the image over directions to form the input, we obtained the filter input m from the output of the logical/linear operator for negative-contrast lines (with parameter degree d=0.1 in Iverson's implementation (code and settings available at http://www.ai.sri.com/{tilde over ( )}leei/loglin.html.), and other values at their default). The first example is of a guide wire imaged fluoroscopically during surgery. FIG. 14 shows a fluoroscopic image of guide wire (left), logical/linear negative-contrast line responses (center) and their thresholding (right). Finding the wire in noise is particularly important, because surgery can take hours, all the while exposing the patient to radiation. Reducing such radiation exposures unfortunately also increases the noise level and makes local filtering (e.g., edge and line operators) perform more poorly. With the
Another example is the ship's wake shown in FIG. 16 (top left). For this demonstration, we work with the magnified section shown in FIG. 16 (top right), and then obtain logical/linear negative-contrast line responses (with degree 0.1) and their thresholding (second row). Responses to high-noise Volterra filters (Result 1) for the direction process-based
To test the biased
6.3 Contour Filtering in ^{2}××with Curvature
In addition to
As argued in section 2.5, the bug with a direction-only search cone would mis-track on a contour as curvature builds up. To make this point computationally, we considered an image of an Euler spiral extending from a straight line segment. (We used formula (16.7) of Kimia et al (Kimia et al, supra), and created the plot in Mathematica with all parameters 0, except γ=0.1 (Kimia et al's notation). The resulting plot was grabbed, combined with a line segment, blurred with a Gaussian, and then subsampled.) FIG. 19 shows Euler spiral images without noise (top left) and with noise (top right) are processed with high-noise
Section 7
7.1 Conclusion
In this disclosure we have introduced a field structure intermediate between contours and the images that suggest them. This curve indicator random field was studied for Markov process contour models, leading to a practical description of its joint statistical structure via a moment generating functional. By relating contours to images, the model provides a way to verify whether it is in agreement with natural image structure; in this way a role for curvature was found, motivating our introduction of a Markov process in contour curvature. We believe that the
Enhancing contour structure in noisy images was characterized as
Besides enabling explicit calculations, the
Surfaces would be the most important immediate extension of this work; one could define an analogous surface indicator random field (
As discussed herein, the purpose of the invention is to enhance the appearance of contours in images. By enhancement, we loosely refer to the reduction of noise or corruption along with an increase in brightness and smoothening of the apparent contours in the image. The basic strategy is to formulate this as a problem of restoration: we seek to estimate the clean sketch which is most closely related to the given noisy image. Our formal model of the clean sketch is what we call the curve indicator random field (
While this formulation of the enhancement of contours as the restoration of a noisy sketch can be applied to many different forms of images, we apply it here to a typical case where the (input) image comes from a camera (e.g. comprising a CCD sensor). The form of this input is a two-dimensional array of numbers {h(x,y), x=1, . . . , n_{x}, y=1, . . . , n_{y}}, where each number h(x,y) is an intensity value (e.g. pixel). If there were little noise in the image, then brighter pixels, or larger values of h(x,y), would indicate contour points, and smaller values would correspond to the background, as indicated by the image at 1 in FIG. 20. In many realistic situations, unfortunately, noise is significant, so that large values of h(x,y) may occur at non-contour locations and small values may occur at the true contour locations, meaning that h(x,y) is not a reliable indicator of the contour. Therefore an example of the typical input image h(x,y) that our invention is intended to enhance is shown at 2 in FIG. 20. To overcome this noise, our method exploits the basic property of sketches that points along a curve are drawn smoothly: the pen does not leave the paper as a curve is drawn, and the pen generally follows a smooth path. Thus the local direction of the contour at each point is important, as nearby points along the contour will roughly align in this direction. For this reason we shall calculate a three-dimensional array {U(x,y,θ), x=1, . . . , n_{x}, y=1, . . . , n_{y}, θ=1, . . . , n_{θ}} where the third dimension θ is used to distinguish various local directions. This array U(x,y,θ) is the estimate of the curve indicator random field (CIRF). Once U(x,y,θ) is computed, one can immediately calculate an enhanced two-dimensional image (although some applications in computer vision may not require this conversion). We will elaborate on this procedure (2-d to 3-d conversion, the calculation of the 3-d array U(x,y,θ), and the final 3-d to 2-d conversion) in a deliberately simplified fashion. In general, however, our invention gives the user a great deal of flexibility in specifying various assumptions that influence this procedure, such as whether blur is present in the imaging process, the precise notion of contour smoothness, the presence of corners in the curves, the modeling of contours as contrast edges versus bright contours on a dark background, the faithfulness of the assumptions to the particulars of the application, and the like.
From the CCD image h(x,y) we can define the three-dimensional array {m(x,y,θ), x=1, . . . , n_{x}, y=1, . . . , n_{y}, θ=1, . . . , n_{θ}}, by copying the two-dimensional array over the third (θ) axis: m(x,yθ)=h(x,y), for all x, y, and θ to form the images shown at 3 of FIG. 20. The unknown U is then computed, by transformation with a computer, from m using either the biased CIRF filtering method or the Volterra filtering method. For this example, one can set the operator Q to a discretization of the partial differential operator
by means of the finite-differencing technique described in Williams and Jacobs (supra), where standard techniques for the numerical solution of partial differential equations are applied. The parameter σ_{θ} defines a bound on the curvature of contours in the noisy image (curvature is the rate of change of contour direction θ per unit length of contour). The parameter α is the reciprocal of the expected length of contour associated with the noisy image.
Biased CIRF Method
On a suitably programmed computer, we compute U by solving the coupled pair of nonlinear equations
Qf+ε(c·f)+1=0,
Q*b+ε(c·b)+1=0,
for the unknown f and b (each is a real-valued three-dimensional array of the same size as m), where c=m−f·b and ε is a parameter which comprises a measure of the reciprocal noise variance. Once these equations are solved for f and b, we let
U(x,y,θ)=(f·b)(x,y,θ)=f(x,y,θ)b(x,y,θ),
the product of the numbers f(x,y,θ) and b(x,y,θ). In comparing this use of the biased CIRF filter to the equations at the end of section 4.8 in the observation entitled “reaction-diffusion-convection equation for the direction process”, note that here we have set a=m, A=−½ I and Ñ=μ=v=1. In FIG. 20, a set of images, indicated at 4, depicts the solution U(x,y, θ) to this pair of coupled equations where each image represents that function of planar position (x,y) corresponding to a particular choice of direction θ. A final enhanced 2-dimensional image, indicated at 5 in FIG. 20, is immediately formed by taking the maximum value along θ for each position (x,y):g(x,y)=max_{θ} U(x,y,θ) and can be displayed for viewing on a suitable monitor or other display device. If the noise variance is unknown, ε can be used as a user-adjustable control parameter. For this purpose, one computes g resulting from various values of ε and selects the value of ε that appears to produce (via display of the image g) the best enhancement of the contours and reduction of noise. Smaller values of ε will produce less contrast and the greatest noise reduction, while large values of ε will produce clearer contours but with less noise reduction.
If ε is sufficiently small and one has a good inital guess of f and b, then one can solve the coupled pair of equations using an inexact Newton method, a standard technique for solving large, sparse nonlinear systems. The technique has two nested loops. In the outer loop one performs Newton's method by linearizing the coupled system around the current guess of f and b, resulting in a linear system of the form Hz=s, where H is an appropriate square 2n_{x }n_{y }n_{θ}-by-2n_{x }n_{y }n_{θ□} matrix, s and z are vectors of length 2n_{x }n_{y }n_{θ□}, and z is unknown (z is a concatenation of the unknown f and b). The inner loop is an iterative method for solving the said linear system. Since H is not generally symmetric positive definite, one can instead solve the normal equations H*Hz=H*s for z, where H* is the transpose of H. One then can use the conjugate gradient method to iteratively solve this latter linear system inexactly (but to any desired precision) for z and thus f and b. Then one can perform another iteration of the outer loop with a new current guess of f and b (from the linear system approximate solution). This outer iteration can be performed until the change in f and b is sufficiently small. One skilled in the art of numerical methods can readily estimate whether f and b are converging quadratically (i.e., whether the change in solution in the current Newton iteration is proportional to the square of the change in solution in the previous Newton iteration).
Since ε is not generally known to be small nor are good initial guesses for f and b necessarily available, one can solve the coupled pair of nonlinear equations using a continuation method, where the parameter ε is increased from a small value incrementally until the correct value is reached. For very small ε the nonlinearity is negligible and one can simply solve the nonlinear system as a linear system by ignoring the nonlinearity, i.e., by setting c=εm. In this latter case, the initial guess for f and b is not important for the conjugate gradient method. Then is increased and f and b are used to provide a new initial guess for the nonlinear system: the above inexact Newton's method can be applied to improve this guess until convergence of the Newton iterations. Then ε can be iteratively increased and the nonlinear coupled system solved using the previous estimate of f and b to define an initial estimate for the current iteration's application of the inexact Newton's method. These are standard techniques for someone skilled in the art of numerical methods.
Volterra Method
Similarly, to apply the Volterra method, the array m is substituted into the Volterra formula:
U=a_{0}+a_{1}[Gm+G*m]+a_{2}[G(diag m)Gm+Gm·G*m+G*(diag m)G*m]+a_{3}[G(diag m)G(diag m)Gm+G(diag m)Gm·G*m+Gm·G*(diag m)G*m+G*(diag m)G*(diag m)G*m],
to compute U by transformation with a computer. This formula is a generalization of that disclosed in equation (4.18) in that the coefficients in front of each bracketed expression above are allowed to take arbitrary values. As in the biased CIRF method, the final enhanced image g(x,y) is immediately formed for viewing on a suitable display by taking the maximum value along θ for each position (x,y):g(x,y)=max_{θ} U(x,y,θ). One method for applying the formula is to first compute and store each of the bracketed expressions in the formula, and then adjust the parameters a_{0 }to a_{3 }via a graphical user interface in which there are controls for these parameters and a window depicting the corresponding enhanced image g. The only difficulty in computing the bracketed expressions in the formula is the computation of the action a=G v of the matrix G on some vector v for a number of different choices of v. Since G=−Q^{−1}, we can instead solve the linear system Qa+v=0 for the vector a. The said solution a can be obtained via an application of the conjugate gradient method on the normal equations Q*Qa+Qv=0. Another, more rapid, technique is suggested in section 5.1.1 hereinabove.
It can be seen that there are many variations on specific implementations of the biased CIRF and Volterra approaches of the present invention, as defined by the appended claims, which are intended to encompass all variations that are inventive contributions suggested to persons skilled in the art of machine vision and numerical analysis. By way of example, such persons will readily understand that the various functions and equations may be implemented in terms of continuous and discrete variables.