# Facial recognition and the open mouth problem

Imported: 10 Mar '17 | Published: 27 Nov '08

Michael Bronstein, Alexander Bronstein, Ron Kimmel

USPTO - Utility Patents

## Abstract

A method of cropping a representation of a face for electronic processing, said method comprising: selecting a first geodesic contour about an invariant reference point on said face, setting a region within said first geodesic contour as a first mask, selecting a second geodesic contour about a boundary of said identified first region, setting a region within said second geodesic contour as a second mask, and forming a final mask from a union of said first mask and said second mask.

## Description

### RELATIONSHIP TO EXISTING APPLICATIONS

This application is a Divisional of U.S. patent application Ser. No. 11/058,171, filed on Feb. 16, 2005 which is a Continuation-In-Part of U.S. patent application Ser. No. 10/284,281 filed on Oct. 31, 2002, now U.S. Pat. No. 6,947,579, issued on Sep. 20, 2005, which claims priority from U.S. Provisional Patent Application No. 60/416,243 filed on Oct. 7, 2002, the contents of all of which are hereby incorporated by reference.

### FIELD AND BACKGROUND OF THE INVENTION

The present invention relates to facial recognition and the open mouth problem and, more particularly, but not exclusively to a method and apparatus for three dimensional facial recognition that can distinguish between different expressions of the same face on the one hand and different faces on the other hand.

An important question in understanding the structure of facial appearance, is what are the invariants of a human face under various expressions. The change of a face by expressions makes modelling of the face and extraction of features that are not influenced by those changes a challenging task.

It was previously suggested to treat faces as deformable surfaces in the context of Riemannian geometry, and modelling facial expressions as near-isometric transformations of the facial surface. The method of bending-invariant canonical forms was used to construct a representation of the faces, invariant to such isometric transformations. The isometric model, however, has difficulties in handling facial expressions that change the topology of the facial surface. One such example is comparing a face with an open mouth to one with a closed mouth.

That is, how can someone's face be given a unique description, that does not change by his or her expression. Important examples include the problem of face recognition in computer vision, texture mapping for facial animation in computer graphics, emotion interpretation in psychology, and measurement of geometric parameters of the face in cosmetic surgery. The variability of the face appearance due to the non-rigid structure of the human face makes this a non-trivial task and challenges for a convenient model to analyze the nature of facial expressions. In previous work we proposed an isometric model for the face geometry, according to which expressions can be approximated by metric preserving transformations of the facial surface. The use of such a model allows us to use the bending-invariant canonical forms in order to construct an expression-invariant representation of the face. Our isometric model was shown to be applicable to strong facial expressions, but it implicitly assumes that facial expressions are topology-preserving. Thus we had a problem comparing faces with an open mouth to those with a closed one, which is a case in which the topology is not preserved.

FIG. 1A demonstrates this phenomenon by showing the geodesics 3, 4 and 5 between two points on the upper and the lower lips of a face 2. As long as the mouth is closed, the geodesics cross the lips without any significant change, even for extreme facial expressions, compare geodesics 3 and 4. However, opening the mouth changes completely the length of the minimal geodesics. In this case, the geodesic between the two lips passes along the lip contour, as indicated by reference numeral 5. In other words, our previous model always required that in all expressions the mouth has to be either always closed or always open.

There is thus a widely recognized need for, and it would be highly advantageous to have, a facial recognition system devoid of the above limitations, and able to recognize the face irrespective of whether the mouth is open or closed, overcoming the fact that the two cases are topologically different.

### SUMMARY OF THE INVENTION

According to one aspect of the present invention there is provided a method of cropping a representation of a face for electronic processing, said method comprising:

selecting a first geodesic contour about an invariant reference point on said face,

setting a region within said first geodesic contour as a first mask,

selecting a second geodesic contour about a boundary of said identified first region,

setting a region within said second geodesic contour as a second mask, and

In an embodiment, said identifying comprises texture processing.

An embodiment may comprise cropping said face by:

According to a second embodiment of the present invention there is provided apparatus for cropping a representation of a face for electronic processing, said face having differential mouth opening states including an open mouth state, a closed mouth state and at least one intermediate state therebetween, the apparatus comprising:

a first masking unit for selecting a first geodesic contour about an invariant reference point on said face, and setting a region within said first geodesic contour as a first mask,

a second masking unit for selecting a second geodesic contour about a boundary of said identified mouth region, and setting a region within said second geodesic contour as a second mask, and

a third masking unit for forming a final mask from a union of said first mask and said second mask, thereby to provide cropping of said face such as to obtain a full facial region and negligible background within said cropping for any of said mouth opening states.

Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. The materials, methods, and examples provided herein are illustrative only and not intended to be limiting.

Implementation of the method and system of the present invention involves performing or completing certain selected tasks or steps manually, automatically, or a combination thereof. Moreover, according to actual instrumentation and equipment of preferred embodiments of the method and system of the present invention, several selected steps could be implemented by hardware or by software on any operating system of any firmware or a combination thereof. For example, as hardware, selected steps of the invention could be implemented as a chip or a circuit. As software, selected steps of the invention could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In any case, selected steps of the method and system of the invention could be described as being performed by a data processor, such as a computing platform for executing a plurality of instructions.

### DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present embodiments comprise a solution to the open mouth problem that operates by enforcing a constraint to the topology on the facial surface, in particular about the mouth region. In one embodiment the constraint is that the mouth is always closed, in effect gluing the lips. In an alternative embodiment the constraint is that the mouth is always open, and this is achieved by disconnecting the lips. The disconnection can be achieved by introducing a cut in the surface when the mouth is closed. In either case the constraint provides for a geodesic that passes vertically through the lip region to follow the same path regardless of whether the mouth is open or closed.

The examples herein focus on the second, disconnecting, solution, but the skilled person will appreciate that the principles described herein are readily applicable to the first, glued, solution as well. The following description presents the concept of topologically-constrained canonical forms, that allow us to construct a representation of the face that is insensitive to facial expressions, and remains so irrespective of changes of state between closed and open mouths.

The principles and operation of a facial recognition system according to the present invention may be better understood with reference to the drawings and accompanying description.

Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of the components set forth in the following description or illustrated in the drawings. The invention is capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting.

Reference is now made to FIG. 1B, which shows the same series of faces as in FIG. 1A, which vary in the extent to which the mouths are open. According to the present embodiment the region of the mouth is excluded from consideration when drawing geodesics, so that the representation always treats the mouth as open, and the geodesic goes around the excluded region. Therefore it makes no difference whether the mouth is open or shut. The geodesic 6 always goes around the mouth, and the mouth open state now ceases to have topological relevance for the model.

Reference is now made to FIG. 1C, which is a simplified diagram illustrating how the above operation may be performed. A face 8 is converted into a cloud of sample points. The sample points avoid the mouth area, since the mouth area is excluded. Furthermore all geodesics avoid the mouth area so that even when the sample points move, the relationships between the sample points, as measured by the linking geodesics does not change very much. Thus the geometric relationship between the sample points does not change significantly whatever the expressions of the face. FIGS. 1D and 1E show absolute and relative changes respectively in distances for changes in expression of the same face. The distance measurement used is discussed in more detail hereinbelow.

The issue is now considered in greater detail. Reference is now made to FIG. 2A, which is a simplified diagram showing apparatus for obtaining 3-Dimensional data of a geometric body for classification, including matching, according to a first preferred embodiment of the present invention. The preferred embodiments relate specifically to matching of faces but the skilled person will be aware that the principles of the present invention are applicable to any geometric body having a three-dimensional structure.

Apparatus 10 comprises a three dimensional scanner 12 for obtaining three-dimensional topographical data of the body. Several types of scanner are described hereinbelow together with brief discussions of how to process data therefrom in some of the cases.

Data from the three-dimensional scanner 12 is passed to a triangulator 14. The triangulator may perform triangulation on the data received from the scanner in order to generate a three-dimensional triangulated manifold to represent the topological features of the body. The exact operation of the triangulator to derive the manifold may vary depending on the way in which the 3D information is gathered. In certain cases the manifold may be formed directly from the gathered data without the need for any intervening processing stage. The manifold preferably represents all of the three-dimensional topology of the body and therefore is in theory sufficient for allowing matching. However, in practice, direct comparisons using the triangulated manifold have a number of disadvantages, as will be demonstrated in the results given hereinbelow. They requires a large amount of calculation. Matching does not distinguish reliably between different faces. Moreover matching generally fails when the same face has a different expression and matching is unreliable even when the same face is posed at a different angle.

Embodiments of the present invention therefore preferably include four additional processing stages, the first of which is a preprocessor 16. Preprocessor 16 takes a reference point in order to determine an orientation around the manifold. A reference point which is relatively easy to find automatically from a manifold of a face is the tip of the nose. Other possible reference points include centers of eyeballs and the center of the mouth. Once the preprocessor has found the tip of the nose (FIG. 8) it is able to orientate itself with respect to the rest of the face and then parts of the face whose geometry is particularly susceptible to expressions, hereinafter referred to as soft regions, can be ignored. Parts of the face that are invariant with change of expression and the like, hereinafter hard regions, can be retained or even emphasized. Further relative reference points can be added, as shown in FIG. 9 below. As will be explained in greater detail below, the definition of soft regions is not fixed. For some methods and in some circumstances soft regions to be excluded may include all of the lower region of the face around the mouth. It is to be noted that the exclusion referred to here is the exclusion of regions from adding reference points, and is not the same as lip cropping discussed hereinbelow. In other cases less drastic exclusions may be considered. In one embodiment, soft regions are removed using a geodesic mask. The mask may be applied separately to a texture map of the face and a depth map of the face.

Following the preprocessor is a subsampler 18. The subsampler 18 takes the preprocessed manifold and removes points so as to produce a less well defined manifold, but one which still defines the essential geometry of the face it is desired to match. In preferred embodiments, the user is able to select the number of points to trade off between accurate matchinga large number of pointsand faster processinga smaller number of points. As will be discussed in greater detail below, a preferred embodiment of the sub-sampler uses the Voronoi subsampling technique which begins at an initial point or vertex on the manifold and then adds the point or vertex having the greatest distance therefrom. The procedure is repeated iteratively until the selected number of points are included. Preferably the technique uses geodesic distances, which may be obtained using the fast marching method for the triangulated domain (FMM-TD), as described below.

Following the subsampler is a geodesic converter 20. The geodesic converter 20 receives the list of points of the subsampled manifold and calculates a vector for each pair of points. The vectors are expressed as geodesic distances, and the fast marching algorithm for the triangulated domain is again used to obtain the geodesic distances in as efficient a manner as possible.

Following the geodesic converter is a multi-dimensional scaler 22, which takes the matrix of the geodesic distances calculated by the geodesic converter 20, referred to below as the distance matrix, and forms a low dimensional Euclidean representation of the series of geodesic distances, using multi-dimensional scaling. Multi-dimensional scaling is discussed in detail below. The low dimensional Euclidean representation provides a bending invariant representation of the geometric body, as will be explained in the discussion on multi-dimensional scaling below. The use of such a bending invariant representation ensures that the matching process is not fooled by, for example, scans of the head at different angles.

The output 24 of the multi-dimensional scalar is a representation of the 3D face in terms of Euclidean distances between surface points, referred to hereinbelow as the canonical form representation.

Reference is now made to FIG. 2B, which is a simplified diagram showing a matching apparatus for matching two faces using the canonical form output as described above. The matcher 30 may be a continuation of the apparatus 10 or may be supplied as a separate unit. The matcher 30 comprises a distance calculator 32, which takes as input two canonical form representations 34 and 36, and calculates a distance therebetween. The distance calculation may use any suitable method for comparison of the canonical forms for each of the faces to be matched. A straightforward approach is to measure a distance between two sets of points, using, for example, the Hausdorff metric. However, the Hausdorf metric based method is computationally extensive.

An alternative approach, used in the present embodiments, takes the first m eigenvalues obtained from the MDS procedure to provide coordinates in a low-dimensional feature space. Although the dominant eigenvalues do not describe the canonical form entirely, it is reasonable that similar faces have similar eigenvalues (and thus form clusters in the feature space). A distance is calculated between the geometric bodies, or, as will be described below, plotted on a graph of the feature space and a thresholder 38, which is connected subsequently to the distance calculator, thresholds the calculated distance to determine the presence or absence of a cluster in the feature space, the cluster indicating a match. In the embodiments described in detail herein, the first three Eigenvalues are taken and are plotted in a three dimensional feature space.

A quantitative validation of our topologically-constrained isometric model is possible by tracking a set of feature points on the facial surface and measuring how the distances between them change due to facial expressions, while ensuring that the topology of the surface is preserved. Unfortunately, there are very few points that can be located accurately on a human face. In order to overcome this difficulty, we placed 133 white round markers (approximately 2 mm in diameter) as invariant fiducial points, onto our subject's face. These markers are explained with respect to FIG. 1C, which shows a facial expressions experiment. On the leftmost side of the figure is a first facial image 8 with the markers. Then, moving towards the right are examples of four facial expressions with the same marked reference points, but the reference points have moved due to the changing expressions of the subject. The markers are selected so as to be easily detected under a variety of facial expressions as exemplified by the four faces towards the left in the figure. In the experiment, sixteen faces were in fact used, between them having weak, medium and strong facial expressions, including open and shut mouth. Lips were cut out, in the manner discussed with respect to FIG. 1B above, to enforce a fixed topology. The reference points were manually labelled; then the Fast Marching algorithm was used to compute the geodesic distances between them. For details of the Fast Marching algorithm see R. Kimmel and J. A. Sethian. Computing geodesic on manifolds. In Proc. US National Academy of Science, volume 95, pages 8431-8435, 1998, the contents of which are hereby incorporated herein by reference. As explained, due to the constraint that all geodesics must run around the mouth irrespective of whether the mouth is actually open or not, the distances did not change that much for different mouth open states.

In order to quantify the changes of the distances due to expressions, we use two measures: the absolute error w.r.t the reference distances (the reference distances were averaged on the neutral expressions): iabs=didiref, and the relative error irel=(didiref)/diref (here di denotes the i-th distance and diref is the corresponding reference distance). The distributions of abs and rel are shown in FIGS. 1D and 1E respectively. The standard deviation of abs is 5:89 mm for geodesic distances and 12:03 mm for Euclidean ones (a difference of 104:3%). The standard deviation of rel was 15:85% for geodesic distances and 39:6% for Euclidean ones (a difference of 150:2%).

The conclusion of this experiment is two-fold. First, the changes of the geodesic distances due to facial expressions are insignificant for exaggerated expressions, which justifies the present model. Secondly, Euclidean distances are much more sensitive to changes due to facial expressions compared to the geodesic ones. This observation will be reinforced by the results presented hereinbelow, where we compare our method to a method that treats facial surfaces as rigid objects.

Preprocessing

The geometry of the face is acquired by a range camera and has to be processed before the canonical form is computed. The raw range data is given as a cloud of points, that can be triangulated or represented as a parametric manifold as shown in FIG. 1C. Preprocessing according to the present embodiments is illustrated in the flow chart of FIG. 3. As illustrated the process includes three steps:

(i) geometric processing of the facial surface S3.1;

(ii) lip cropping S3.2; and

(iii) cropping the whole face S3.3.

Geometric processing S3.1 is shown in greater detail in the flow chart of FIG. 4 to include three stages of spike removal S4.1, hole filling S4.2 and selective smoothing, S4.3. For details on the three stages of geometric processing see M. Bronstein, A. Bronstein, and R. Kimmel. Three-dimensional face recogntion. Technical Report CIS-2004-04, Dept. of Computer Science, Technion, Israel, 2003, the contents of which are hereby incorporated by reference.

Lip cropping is shown in greater detail in the simplified flow chart of FIG. 5. Lip cropping is preferably performed by first segmenting the lips based on the texture information S5.1, and then removing the corresponding points in the 3D data S5.2.

The whole face cropping procedure is performed using a modification of the geodesic mask proposed in our U.S. patent application Ser. No. 10/284,281. The unmodified version is shown in FIG. 6. The key idea of the geodesic mask computation is locating an invariant source point on the face (e.g. the tip of the nose), S6.1, and measuring an equidistant (in sense of the geodesic distances) contour around it, S6.2. The geodesic mask is defined in S6.3 as the interior of this contour and all points outside the contour are removed. This allows us to crop the facial surface in a consistent manner, insensitively to facial expressions. In case of topological changes, however, the geodesic mask looses its invariance, as the geodesic distances are influences by the hole created at the mouth location. That is to say, if the user opens his mouth, the chin rotates downwardly away from the front of the face, and the boundary geodesic tends therefore to lose the bottom of the chin.

Reference is now made to FIG. 7, which is a simplified flow chart of a method of carrying out whole face cropping which overcomes the above problem. As shown in FIG. 7 the embodiment uses two masks. A first mask is computed in stages S7.1 to S7.4. in which an equidistant region around the nose tip is computed on the facial surface as before and ignoring the fact that the lips are cut.

A second mask is then defined in stages S7.5 to S7.7, in which an equidistant region around the lips is used. In S7.5 we use all the points along the lip perimeter as source points. In S7.6 an equidistant geodesic contour is set around the lips and in S7.7 the region within the contour is set as a second mask. In stage 7.8 a union of the first and second masks is calculated and the final geodesic mask is set in stage 7.9 as that union. Typically the cropped surface contains N3000 points.

FIG. 8 is a simplified illustration of a manifold and shows the selection of an invariant point, the nose tip 60, which is an easy point for a machine to find and use as a reference. As will be shown below, the system can be given improved reliability by using two such points, for example the nose tip and the center of the bridge of the nose.

FIG. 9 shows a facial manifold with a cloud of points developed from an initial reference point such as that shown in FIG. 8.

FIG. 10 is a worked example, on a manifold, of the geodesic mask computation of FIGS. 3-7 and its insensitivity to facial expressions. As shown in face 10a, computation of the geodesic mask requires setting up of equidistant contours from first fixed sources, in this case equidistant contours are measured from two sources, one located at the tip of the nose and the second being the center of the bridge of the nose (left). In face 10b, equidistant contours are measured from sources located on the perimeter of the lips. In face 10c the final mask is obtained as a union of the two regions. In the second row: faces 10d to 10g show the final masks computed in the same way but with different initial expressions or degrees of open mouth. The examples show that the final mask is an effective capture of the outline of the face whatever the extent of the mouth being open. That is to say the geodesic mask is shown to have good insensitivity to facial expressions.

4 Topologically-Constrained Canonical Forms

Isometric transformations preserve intrinsic or geodesic distances among all points. Our model of the face thus assumes that the geodesic distance dij between any given two points xi; xj on the facial surface remains unchanged. The first problem is that a discrete set of points {xi}i=1N j=1 can be ordered arbitrarily, and thus the matrix D=(dij) is invariant up to some permutation of the rows and columns. Moreover, when sampling a surface, there is no guarantee that the surface will be sampled at similar points, nor even that the number of points in two surfaces is necessarily the same. This makes the computation of such an invariant impractical.

An alternative is to avoid dealing explicitly with the matrix of geodesic distances, and replace the curved surface with its Riemannian metric by a different dual surface. Intrinsic geodesic distances are replaced by Euclidean ones in the embedding space. The new surface, approximates the original one in some optimal way. Such a procedure is calledflat embedding, and the resulting set of points in the Euclidean space is called the bending invariant canonicalform of the face. Unlike the matrix D, the canonical form (though inevitably being an approximate representation of the original surface) is invariant, up to rotation, translation and reflection, which is much easier to deal with when trying to match two non-rigid surfaces.

We would like to find the most isometric embedding, the one that deforms the manifold distances the least. In practice, we have a finite discrete set of N manifold samples {xi}i=1N (represented as a 3 X N matrix X=(xn; . . . ; xN)) and a set of N2 mutual geodesic distances between these samples. We consider a mapping of the form : (M,d)(m,d), which maps the manifold samples xi into points xi in an m-dimensional Euclidean space, such that the geodesic distances dij are replaced by Euclidean ones dij=xixj2.

The embedding error can be measured as a discrepancy between the geodesic and the resulting Euclidean distances using some norm,

s(X; D)=DD(X),(1)

where X=(x1; . . . ; xN) is an m X N matrix representing the points in the embedding space,

and D is the matrix of mutual Euclidean distances depending on the points configuration X. The function (1) is referred to hereinbelow as stress.

Finding the best approximate flat embedding is possible by minimization of s(X; D) with respect to X. A family of algorithms used to carry out such an approximate flat embedding is usually referred to as multidimensional scaling (MDS). These algorithms differ in the choice of the embedding error criterion and the numerical method used for its minimization.

One straightforward possibility is to choose the norm in (1) to be Euclidean, and thus have the metric distortion defined as a sum of squared differences

$s ( X ; D ) = i j ( d ij - d ij ) 2 , ( 2 )$

and the MDS is posed as a least-squares (LS) problem and is known as LS-MDS. We used the iterative SMACOF algorithm to compute the canonical forms. The SMACOF algorithm is disclosed in I. Borg and P. Groenen. Modern multidimensional scalingtheory and applications, Springer-Verlag, Berlin Heidelberg New York, 1997; and also in J. De-Leeuw. Recent developments in statistics, chapter Applications of convex analysis to multidimensional scaling, pages 133-145. North-Holland, Amsterdam, 1977.

When the embedding is performed into a space with m3 dimensions, the canonical form can be plotted as a surface. FIG. 11 depicts canonical forms of one subject with different facial expressions. More particularly FIG. 11, faces 11a-11m shows examples of topologically-constrained canonical forms of faces with strong facial expressions, including expressions with closed, open, and partially open mouth. For comparison, faces 11k and 11m are a canonical form of a different subject.

It demonstrates that although the facial surface changes are substantial, the changes between the corresponding canonical forms are insignificant.

Additional objects, advantages, and novel features of the present invention will become apparent to one ordinarily skilled in the art upon examination of the following examples, which are not intended to be limiting. Additionally, each of the various embodiments and aspects of the present invention as delineated hereinabove and as claimed in the claims section below finds experimental support in the following examples.

The above-described approach was tested on a data set containing 102 instances of 6 subjects and one mannequin (Eve). The set of faces is shown in FIG. 12, in which all the faces have neutral expressions. As illustrated in FIG. 13 with just a single face, the human subjects appear in the data base with different facial expressions, classified into 7 groups (neutral expression +6 expressions with open or shut mouth). Furthermore as shown in FIG. 14 the expressions each appear in three strengths, weak, medium, and strong. A neutral expression is the natural posture of the face with a closed mouth; strong expressions are extreme postures that obviously rarely occur in real life. Head rotations of up to about 10 degrees were allowed.

Using the database, an experiment was carried out to test the sensitivity of the topologically constrained canonical forms to extreme facial expressions including open and closed mouth.

As a reference, the facial surfaces were also compared as rigid objects. Surface matching (both the facial surfaces and the canonical forms) was carried out by comparing their moments.

The surface is described as a vector of its 56 high-order moments, and the Euclidean norm is used to measure the distance between these vectors.

FIGS. 15A and 15B are graphical representations of the results of the experiment (15A) and the control (15B). More particularly FIGS. 15A and 15B depict a low-dimensional representation (obtained by MDS) of the dissimilarities (distances) between faces. Each symbol on the plot represents a face; colors denote different subjects; the symbol's shape represents the facial expression and its size represents the expression strength. Ideally, clusters corresponding to each subject should be as close as possible (meaning that the representation is insensitive to facial expressions) and as distant as possible from other subjects, which means that the representation allows us to discriminate between different subjects, and not between different expression. That is, the within class distances should be as small as possible, while the between classes distances should be as large as possible. It can be seen that using straightforward rigid surface matching as in the control in FIG. 15B, the clusters overlap, implying that variability due to facial expressions is larger than due to the subject's identity. On the other hand, using topologically-constrained canonical forms as in FIG. 15A, we obtain tight and distinguishable clusters.

Table 1 below is a description of the facial expressions in the data set used in the experiment of FIG. 15 and the inter-cluster to intra-cluster dissimilarity ratio using original and canonical surface matching. The triple asterisk in the table denotes an artificial subject.

Subject Color Neut Weak Med Str Shut Open Michael Red 2 2 10 6 6 14 2.55 17.10 Eyal green 1 2 8 5 2 14 1.35 8.61 Guy ma- 3 4 6 4 5 12 1.44 10.64 genta Mitya Yel- 2 9 7 5 6 17 2.40 14.77 low Boaz d. Cyan 3 3 3 1 6 4 1.30 3.01 David d. Ma- 1 2 4 3 1 9 0.97 8.65 genta Eve*** Black 6 6 0.11 0.70

Table 1, final two columns, shows for the canonical forms case and the control respectively, the values of the ratio of the maximum inter-cluster to minimum intracluster dissimilarity.

$k = max i , j C k ij min i C k , j C k ij , ( 3 )$

in which Ck denotes indexes of the k-th subject's faces, C=kCk and ij denotes dissimilarities between faces i and j) for facial and canonical surface matching. This criterion is convenient being scale-invariant; it measures the tightness of each cluster of faces that belong to the same subject and its distance from other clusters. Ideally, k should tend to zero. The approach of the present embodiments, 9th column, outperforms rigid facial surface matching, the 10th column, by up to 790:3% in the sense of k.

### CONCLUSIONS

In our previous patent application referred to hereinabove we used canonical forms to solve the problem of 3D face recognition.

In the present application we carry out a generalization of that approach to handle facial expressions with an open mouth. The method of topologically-constrained canonical forms is applicable to facial expressions with both closed and open mouth, and can thus handle extreme facial expressions. The resulting representation is especially suitable for 3D face recognition in a natural environment, for example, when a subject is speaking.

It is expected that during the life of this patent many relevant 3D data gathering and topology measurement devices and systems as well as devices, systems and methods for measuring contours and carrying out comparisons based on representations of three-dimensional data will be developed and the scope of the corresponding terms herein are intended to include all such new technologies a priori.

It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination.

Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims. All publications, patents and patent applications mentioned in this specification are herein incorporated in their entirety by reference into the specification, to the same extent as if each individual publication, patent or patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention.

## Claims

1. A method of cropping a representation of a face for electronic processing, said method comprising:
selecting a first geodesic contour about an invariant reference point on said face,
setting a region within said first geodesic contour as a first mask,
selecting a second geodesic contour about a boundary of said identified first region,
setting a region within said second geodesic contour as a second mask, and
selecting a first geodesic contour about an invariant reference point on said face,
setting a region within said first geodesic contour as a first mask,
selecting a second geodesic contour about a boundary of said identified first region,
setting a region within said second geodesic contour as a second mask, and