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Image segmentation, which aims to extract interesting objects from a given image, is one fundamental task in image processing and computer vision. Among several popular variational models for this purpose, we focus on the seminal Mumford and Shah (MS) model in this thesis. Build upon the MS model, we present various modifications that adapt to different segmentation tasks, and develop efficient minimization algorithms for the proposed modified models. Below, we guide you through the main contents and contributions of this dissertation. (i) We develop efficient dual-algorithm for two-phase image segmentation using TV-Allen-Cahn model (see Chapter 2). The main purpose is to overcome the difficulties resulted from the non-differentiability of the total variation (TV) term, and effectively handle the constraints of the binary level-set function. The use of a splitting-penalty method results in TV-Allen-Cahn type models associated with a very attractive “double-well” potential, which allows for the implementation of the Chambolle’s dual algorithm. Moreover, we present a new dual algorithm based on an edge-featured penalty of the dual variable, which only requires to solve a vectorial Allen-Cahn type equation with linear ∇(div)-diffusion rather than fully nonlinear diffusion in the Chambolle’s approach. Consequently, efficient numerical algorithms such as time-splitting method and Fast Fourier Transform (FFT) can be implemented. Various numerical tests show that two dual algorithms are much faster and stabler than the primal gradient descent approach, and the new

Mumford-Shah type variational models and fast algorithms for image segmentation