The piecewise smooth Mumford-Shah functional on an arbitrary graph

Area of Science:

• Computer Vision
• Image Analysis
• Computational Mathematics

Background:

• The Mumford-Shah functional is a cornerstone in image analysis for segmentation and filtering.
• Current optimization relies on active contours and gradient descent, prone to initialization issues and local minima.

Purpose of the Study:

• To reformulate the Mumford-Shah functional on arbitrary graphs.
• To apply combinatorial optimization techniques for improved efficiency and solution quality.
• To explore new applications beyond traditional image segmentation.

Main Methods:

• Reformulation of the Mumford-Shah functional on a graph structure.
• Application of combinatorial optimization algorithms.
• Comparison with gradient descent-based level set methods.
• Inclusion of regularization for image reconstruction outside boundaries.

Main Results:

• The graph formulation and combinatorial optimization achieve lower energy solutions.
• The new method significantly reduces computation time compared to gradient descent.
• Demonstrated effectiveness in new applications like point clustering and nonuniform image filtering.

Conclusions:

• Graph-based combinatorial optimization offers a superior alternative for Mumford-Shah functional minimization.
• This approach overcomes limitations of traditional gradient descent methods.
• The formulation enables broader applicability of the Mumford-Shah functional.