טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentNossek Raz Zvi
SubjectNonlinear Analysis: Eigenvalue Problems and Spectral
Applications
DepartmentDepartment of Electrical Engineering
Supervisor Dr. Guy Gilboa
Full Thesis textFull thesis text - English Version


Abstract

Nonlinear variational methods have become very powerful tools for many image processing tasks. Recently a new line of research has emerged, dealing with nonlinear

eigenfunctions induced by convex functionals. This has provided new insights and

better theoretical understanding of convex regularization and introduced new processing methods. However, the theory of nonlinear eigenvalue problems is still at its infancy. We present a new way that can generate nonlinear eigenfunctions of the form T(u) = λu, where T(u) is a nonlinear operator and λ in R is the eigenvalue. We also demonstrate that the framework of nonlinear spectral decompositions, based on variational regularization, is very well suited for several image manipulation and fusion tasks.

In the case of nonlinear eigenfunction generation we develop the theory

where T(u) is a subgradient element of a regularizing one-homogeneous functional, such as total-variation (TV) or total-generalized-variation (TGV). We introduce two flows: a forward flow and an inverse flow. The forward flow monotonically smooths the solution (with respect to the regularizer) and simultaneously increases the L2 norm. The inverse flow has the opposite characteristics. For both flows, the steady state depends on the initial condition, thus different initial conditions yield different eigenfunctions. This enables a deeper investigation into the space of nonlinear eigenfunctions, allowing to produce numerically diverse examples, which may be unknown yet. In addition we suggest an indicator to measure the affinity of a function to an eigenfunction and relate it to pseudo-eigenfunctions in the linear case.

For performing the image manipulation task we regard the total variation functional

as the regularizer and show it is very well suited for this task. The well-localized and edge-preserving spectral TV decomposition allows to select different scale elements of a certain image and to transfer particular features (such as wrinkles in a face) from one image to the other. We demonstrate the effectiveness of the proposed approach in several numerical experiments, including a comparison to the competing techniques of Poisson image editing and image osmosis.