Postdoctoral fello, University of Wurzburg, Germany
Theoretical and numerical aspects of inverse problems related to medical imaging and optimal control
My primary research area is on theoretical and numerical methods inversion of the circular, elliptic, broken-ray, spherical and conical Radon transform with centers on a circle or sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical imaging like ultrasound, single-scattering optical tomography, photoacoustic tomography and intravascular imaging. My aim is to derive analytical inversion formulae for reconstruction of a function from its Radon transform with partial radial data. Moreover, I also devise efficient computational algorithms to simulate the reconstruction procedure such that they are fast and robust even in presence of noisy data.
Another field of my research is the study of optimal control problems related to stochastic models in the frameworks of partial differential equations (Fokker-Planck, Liouville, piecewise deterministic processes). Such models arise in controlling crowd motion, transportation problems and differential games. My research is to build effective models, providing theoretical results of existence and uniqueness of optimal controls and devising higher order numerical algorithms to solve the optimality systems using non-smooth methods. In addition, I provide a comprehensive numerical analysis (higher order error estimates in L^2 and L^1 norms) for the underlying state equations which are either parabolic or hyperbolic scalar and systems.
Abstract: We consider optical flow estimation of flows with vorticity governed by 2D incompressible Euler and Navier–Stokes equations . A vorticity-streamfunction formulation and optimization techniques are used. We use Helmholtz decomposition of the velocity field and prove existence of an unique velocity and vorticity field for the linearized vorticity equations. Discontinuous galerkin finite elements are used to solve the vorticity equation for Euler’s flow to efficiently track discontinuous vortices. Finally we test our method with two vortex flows governed by Euler and Navier–Stokes equations at high Reynolds number which support our theoretical results.
Pub.: 23 Sep '15, Pinned: 02 Jun '17
Abstract: A variational approach is used to recover fluid motion governed by Stokes and Navier–Stokes equations. Unlike previous approaches where optical flow method is used to track rigid body motion, this new framework aims at investigating incompressible flows using optical flow techniques. We formulate a minimization problem and determine conditions under which unique solution exists. Numerical results using finite element method not only support theoretical results but also show that Stokes flow forced by a potential are recovered almost exactly. A variational approach is used to recover fluid motion governed by Stokes and Navier–Stokes equations. Unlike previous approaches where optical flow method is used to track rigid body motion, this new framework aims at investigating incompressible flows using optical flow techniques. We formulate a minimization problem and determine conditions under which unique solution exists. Numerical results using finite element method not only support theoretical results but also show that Stokes flow forced by a potential are recovered almost exactly.
Pub.: 07 Dec '16, Pinned: 02 Jun '17
Abstract: We present an efficient and novel numerical algorithm for inversion of transforms arising in imaging modalities such as ultrasound imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing, and radar imaging with circular acquisition geometry. Our algorithm is based on recently discovered explicit inversion formulas for circular and elliptical Radon transforms with radially partial data derived by Ambartsoumian, Gouia-Zarrad, Lewis and by Ambartsoumian and Krishnan. These inversion formulas hold when the support of the function lies on the inside (relevant in ultrasound imaging, thermoacoustic and photoacoustic tomography, non-destructive testing), outside (relevant in intravascular imaging), both inside and outside (relevant in radar imaging) of the acquisition circle. Given the importance of such inversion formulas in several new and emerging imaging modalities, an efficient numerical inversion algorithm is of tremendous topical interest. The novelty of our non-iterative numerical inversion approach is that the entire scheme can be pre-processed and used repeatedly in image reconstruction, leading to a very fast algorithm. Several numerical simulations are presented showing the robustness of our algorithm.
Pub.: 17 Dec '14, Pinned: 02 Jun '17
Abstract: The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed and robustness of our method in the case of both noise-free and noisy data.
Pub.: 07 Mar '16, Pinned: 02 Jun '17
Abstract: We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm based on our inversion formula and demonstrate its accuracy and efficiency on several numerical examples.
Pub.: 14 Feb '17, Pinned: 02 Jun '17