An Efficient Algorithm for the Piecewise Affine-Linear Mumford-Shah Model Based on a Taylor Jet Splitting

Area of Science:

• Computer Vision
• Image Processing
• Computational Mathematics

Background:

• The Mumford-Shah model is a fundamental tool in image segmentation.
• Existing algorithms for the piecewise affine Mumford-Shah model face challenges in efficiency and initialization.
• Approximation methods are crucial for practical applications.

Purpose of the Study:

• To propose an efficient algorithm for approximate solutions of the piecewise affine Mumford-Shah model.
• To introduce a novel reformulation of the optimization problem using Taylor jets.
• To develop an exact and efficient solver for linewise segmented jet estimation problems.

Main Methods:

• Reformulation of the optimization problem using Taylor jets.
• A splitting approach to decompose the problem into linewise segmented jet estimation.
• Development of an exact and efficient solver for the subproblems.

Main Results:

• The proposed algorithm efficiently computes approximate solutions.
• The method directly yields a partition without initialization.
• Experimental results demonstrate lower computation times compared to state-of-the-art methods.
• Solutions obtained often exhibit lower functional values.

Conclusions:

• The novel algorithm offers a significant improvement in efficiency and solution quality for the Mumford-Shah model.
• The use of Taylor jets and a splitting approach provides a robust and parallelizable method.
• This work advances the state-of-the-art in image segmentation algorithms.