Mumford-Shah type variational models and fast algorithms for image segmentation
Ying Gu · 2013
dual algorithm is at least as efficient as the Chambolle's algorithm and is relatively accurate.We demonstrate that the new method also provides a viable alternative for image restoration.(ii) We propose a direct global minimization method for multiclass labeling and multiphase image segmentation (see Chapter 3).Different from the existing methods, we work directly with the binary setting without using convex relaxation, which is thereby termed as a direct approach.We provide the sufficient and necessary conditions to guarantee a global optimum, and present efficient algorithms based on a reduction of the intermediate unknowns from the augmented Lagrangian formulation.As a result, the underlying algorithms involve significantly fewer parameters and unknowns than the naive use of augmented Lagrangian-based methods, so they are fast and easy to implement.Furthermore, they can produce a global optimum under mild conditions.(iii) We present a robust edge detection method by using the modified Mumford-Shah model (see Chapter 4).This model immerses an edge into a narrow band surrounding it, and measures the edge length by TV of the associated binary level-set function.The situation that the edge set could be of measure zero in the limiting process presents significant challenges for minimization.We propose several ideas to surmount the obstacles, which include the convex relaxation technique, splitting-penalty method, the proximity algorithm and split Bregman method.The approach can also be extended to color image processing, where limited results are available.(iv) We propose a new decomposition model for multiphase piecewise smooth image segmentation (see Chapter 5).By decomposing an image into an interpolation of piecewise constant part, smooth component and noise part, we incorporate these components into the Mumford-Shah model.This only requires to solve the smooth component on the whole domain.With the Abstract 15 entropy penalization, we construct efficient algorithms which include the Fourier method, smoothed Chambolle-dual algorithm and proximity algorithm.Numerical results are provided to show the advantages over the existing method.ψ or {ψ i } level-set function χ χ = (χ 1 , χ 2 , • • • , χ m ), the characteristic functions m ∑ i=1 ∫ Ω |c i -I| 2 χ i dx +