טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBiton Shai
SubjectAdaptive Anisotropic Total Variation - A Nonlinear Spectral
Analysis
DepartmentDepartment of Electrical Engineering
Supervisor Professor Guy Gilboa
Full Thesis textFull thesis text - English Version


Abstract

A fundamental concept in solving inverse problems is the use of regularizers, which yield more physical and less-oscillatory solutions. Total variation (TV) has been widely used as an edge-preserving regularizer. However, objects are often over-regularized by TV, becoming blob-like convex structures of low curvature. This phenomenon was explained mathematically in the analysis of Andreu et al. They have shown that a TV regularizer can spatially preserve perfectly sets which are nonlinear eigenfunctions of the form λu ∂JTV(u), where ∂JTV(u) is the TV subdifferential. For TV, these shapes are convex sets of low curvature. 

A compelling approach to better preserve structures is to use anisotropic functionals, which adapt the regularization in an image-driven manner, with strong regularization along edges and low across them.

This follows earlier ideas of Weickert on anisotropic diffusion, which do not stem directly from functional minimization. Adaptive anisotropic TV (A2TV) was successfully used in several studies in the past decade. However, until now there is no theory formulating the type of structures which can be perfectly preserved. In this thesis we address this question.


We rely on a recently developed theory of Burger et al on nonlinear spectral analysis of one-homogeneous functionals. We have that eigenfunction sets, admitting λu ∂JA2TV(u), are perfectly preserved under A2TV-flow or minimization with L2 square fidelity. We thus investigate these eigenfunctions theoretically and numerically. We prove non-convex sets can be eigenfunctions in certain conditions and provide numerical results which characterize well the relations between the degree of local anisotropy of the functional and the admitted maximal curvature. A nonlinear spectral representation is formulated, where shapes are well preserved and can be manipulated effectively. Finally, examples of possible applications related to shape manipulation and guided regularization of medical and depth data are shown.


A real life application of the A2TV in the field of optoacoustics is presented in detail. A2TV based reconstruction of medical data for arteries-like shapes can greatly improve image quality and robustness to noise and model inaccuracies.