Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model

Area of Science:

• Computer Vision
• Image Processing
• Mathematical Modeling

Background:

• The Mumford-Shah model is a cornerstone in image segmentation, crucial for joint image restoration and contour detection.
• Existing approximations often struggle with computational complexity and specific functional requirements.
• There is a need for efficient and robust algorithms that handle nonsmooth penalizations.

Purpose of the Study:

• To propose a general discrete formulation of the Mumford-Shah functional suitable for nonsmooth penalizations.
• To develop a novel Semi-Linearized Proximal Alternated Minimization (SL-PAM) algorithm with convergence guarantees.
• To evaluate the performance of SL-PAM against state-of-the-art methods for various image processing tasks.

Main Methods:

• Formulation of a discrete Mumford-Shah functional adapted for nonsmooth penalizations.
• Development and application of the Proximal Alternating Linearized Minimization (PALM) algorithm.
• Derivation and implementation of a novel Semi-Linearized Proximal Alternated Minimization (SL-PAM) algorithm.
• Comparative analysis with convex relaxations and Ambrosio-Tortorelli functional variants.

Main Results:

• The proposed discrete Mumford-Shah formulation meets the requirements for PALM and SL-PAM algorithms.
• The SL-PAM algorithm demonstrates faster convergence compared to the original PALM algorithm.
• SL-PAM achieves competitive results in Gaussian/Poisson denoising, image restoration, and RGB-color denoising.

Conclusions:

• The developed SL-PAM algorithm provides an efficient and effective approach for image segmentation, restoration, and denoising.
• This work advances discrete Mumford-Shah functional approximations, offering practical improvements over existing methods.
• The novel SL-PAM algorithm shows significant potential for complex image processing applications.