Radial-basis-function level-set-based regularized Gauss–Newton-filter reconstruction scheme for dynamic shape tomography

Abstract

The dynamic reconstruction problem in tomographic imaging is encountered in several applications, such as species determination, the study of blood flow through arteries/veins, motion compensation in medical imaging, and process tomography. The reconstruction method of choice is the Kalman filter and its variants, which, however, are faced by issues of filter tuning. In addition, since the time-propagation models of physical parameters are typically very complex, most of the time, a random walk model is considered. For geometric deformations, affine models are typically used. In our work, with the objectives of minimizing tuning issues and reconstructing time-varying geometrically deforming features of interest with affine in addition to pointwise-normal scaling motions, a novel level-set-based reconstruction scheme for ray tomography is proposed for shape and electromagnetic parameters using a regularized Gauss-Newton-filter-based scheme. We use an implicit Hermite-interpolation-based radial basis function representation of the zero level set corresponding to the boundary curve. Another important contribution of the paper is an evaluation of the shape-related Frechet derivatives that does not need to evaluate the pointwise Jacobian (the ray-path matrix in our ray-tomography problem). Numerical results validating the formulation are presented for a straight ray-based tomographic reconstruction. To the best of our knowledge, this paper presents the first tomographic reconstruction results in these settings.

Parameterized level-set based pharmacokinetic fluorescence optical tomography using the regularized Gauss–Newton filter

“…This state (concentration) and parameter (pharmacokinetic as well as shape parameters) estimation problem can be solved using either stochastic estimation schemes, such as EKF and its variants, 9,10,22,57 or deterministic schemes, such as the GN filter. 39,58 In our work, we propose an iteratively regularized deterministic GN-filter in a trust-region setting for our reconstructions. We define the state vector at time j as follows:…”

“…(23) and measurement Eq. (26) by solving the following regularized nonlinear least squares problem using the GN method: 39,58 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 7 ; 6 3 ; 4 3 9Θ…”

“…The nonlinear least squares problem can be solved by an iterative regularization scheme using either line search or trust region methodologies. 39,59,60 A regularized GN update p Θ solves at the current iterate Θ:…”

“…The following relative measure 39 of the residual is used as a reflection of "how much" of the residual is left in the range of the augmented Jacobian and serves as a useful stopping criterion: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 8 ; 3 2 6 ; 7 1 9…”

“…A dynamic optical tomography problem in straight path ray-tomography was also solved 39 using RBF level-set objectboundary representations and GN filter based estimation of the dynamic shape and optical parameters. Nonparameterized pointwise-specified level-sets are used to implicitly represent the boundary of time varying fluorescence yield 40 in fluorescence molecular tomography.…”