Image Decomposition with G-norm Weighted by Total Symmetric Variation

2025-03-28

AI Summary

This work addresses the problem of texture leakage into contour edges in image cartoon–texture decomposition. To resolve this, we propose a novel variational model that integrates total symmetric variation (TSV) regularization with a weighted Meyer G-norm. Our key contribution is the first incorporation of TSV into cartoon–texture decomposition, leveraging its ability to precisely characterize region boundaries for strict separation between contours and interior texture. We establish theoretical existence of solutions under the joint constraint of bounded variation (BV) and bounded TSV. To handle the nonconvex optimization, we design an efficient operator-splitting algorithm. Experimental results demonstrate that our method significantly improves cartoon–texture separation accuracy and contour preservation across diverse image classes, outperforming state-of-the-art approaches.

Technology Category

• Computer Vision: Segmentation
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In this paper, we propose a novel variational model for decomposing images into their respective cartoon and texture parts. Our model characterizes certain non-local features of any Bounded Variation (BV) image by its Total Symmetric Variation (TSV). We demonstrate that TSV is effective in identifying regional boundaries. Based on this property, we introduce a weighted Meyer's $G$-norm to identify texture interiors without including contour edges. For BV images with bounded TSV, we show that the proposed model admits a solution. Additionally, we design a fast algorithm based on operator-splitting to tackle the associated non-convex optimization problem. The performance of our method is validated by a series of numerical experiments.

Problem

• Decompose images into cartoon and texture parts
• Identify texture interiors excluding contour edges
• Solve non-convex optimization for image decomposition

Innovation

• Total Symmetric Variation identifies regional boundaries
• Weighted Meyer's G-norm excludes contour edges
• Operator-splitting algorithm solves non-convex optimization