Use of penalty terms in gradient-based iterative reconstruction schemes for optical tomography

Abstract

It is well known that the reconstruction problem in optical tomography is ill-posed. In other words, many different spatial distributions of optical properties inside the medium can lead to the same detector readings on the surface of the medium under consideration. Therefore, the choice of an appropriate method to overcome this problem is of crucial importance for any successful optical tomographic image reconstruction algorithm. In this work we approach the problem within a gradient-based iterative image reconstruction scheme. The image reconstruction is considered to be a minimization of an appropriately defined objective function. The objective function can be separated into a least-square-error term, which compares predicted and actual detector readings, and additional penalty terms that may contain a priori information about the system. For the efficient minimization of this objective function the gradient with respect to the spatial distribution of optical properties is calculated. Besides presenting the underlying concepts in our approach to overcome ill-posedness in optical tomography, we will show numerical results that demonstrate how prior knowledge, represented as penalty terms, can improve the reconstruction results.

Three-Dimensional Diffuse Optical Tomography with a priori Anatomical Information

“…[8][9][10][11][12] to incorporate a priori information. Introducing penalty functions 11 and uniform [2][3][4][5] or spatially varying regularization 12 terms in the inverse problem formulation is another way of incorporating a priori information. However, in these studies, rather than specific information about the unknown image, generic probability density functions or regularization terms have been used.…”

Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: Near-infrared fluorescence tomography

“…For example, Tikhonov regularization (11) involves a regularization parameter that is added to the diagonal of the matrix to be inverted to achieve diagonal dominance; this parameter is spatially invariant and must be carefully selected to ensure convergence, or can be updated dynamically by using heuristics, as in the Levenberg-Marquardt method (12). Improved results have been demonstrated by using spatially variant regularization based on a priori assumptions of system noise (13), or by using a penalty function based on a priori assumptions on the final parameter distribution (10). Others have used estimates of measurement error based on shot-noise statistics to weight measurements (14).…”

“…10). For example, Tikhonov regularization (11) involves a regularization parameter that is added to the diagonal of the matrix to be inverted to achieve diagonal dominance; this parameter is spatially invariant and must be carefully selected to ensure convergence, or can be updated dynamically by using heuristics, as in the Levenberg-Marquardt method (12).…”

Rapid convergence to the inverse solution regularized with Lorentzian distributed function for near-infrared continuous wave diffuse optical tomography

“…Some studies [20][21][22] used gradient-based iterative image reconstruction schemes consisting of the minimization of an appropriately defined objective function separated into both least squares of errors and additional penalty terms containing a priori information. This gradient-based iterative image reconstruction method uses the gradient of the objective function in a line-minimization scheme to provide subsequent guesses of the spatial distribution of the optical properties for the forward model, and the reconstruction of these properties is completed once a minimum of this objective function is found.…”