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An Image Segmentation Model with Transformed Total Variation

Elisha Dayag, Kevin Bui, Fredrick Park, Jack Xin

TL;DR

The paper tackles robust image segmentation under noise by combining a fuzzy multiphase segmentation framework with transformed total variation (TTV) regularization on the gradient. It formulates a TTV-regularized Mumford–Shah-type energy and solves it with an ADMM scheme that leverages a closed-form TL1 proximal operator, enabling nonconvex regularization. Empirical results on retinal vessels and brain MRI-like data show that TTV often matches or surpasses TV, TV^p, and AITV in accuracy (as measured by DICE and Jaccard) and remains computationally competitive, particularly for narrow structures. The work provides a scalable, edge-preserving segmentation method with potential extensions to color images and other imaging modalities.

Abstract

Based on transformed $\ell_1$ regularization, transformed total variation (TTV) has robust image recovery that is competitive with other nonconvex total variation (TV) regularizers, such as TV$^p$, $0<p<1$. Inspired by its performance, we propose a TTV-regularized Mumford--Shah model with fuzzy membership function for image segmentation. To solve it, we design an alternating direction method of multipliers (ADMM) algorithm that utilizes the transformed $\ell_1$ proximal operator. Numerical experiments demonstrate that using TTV is more effective than classical TV and other nonconvex TV variants in image segmentation.

An Image Segmentation Model with Transformed Total Variation

TL;DR

The paper tackles robust image segmentation under noise by combining a fuzzy multiphase segmentation framework with transformed total variation (TTV) regularization on the gradient. It formulates a TTV-regularized Mumford–Shah-type energy and solves it with an ADMM scheme that leverages a closed-form TL1 proximal operator, enabling nonconvex regularization. Empirical results on retinal vessels and brain MRI-like data show that TTV often matches or surpasses TV, TV^p, and AITV in accuracy (as measured by DICE and Jaccard) and remains computationally competitive, particularly for narrow structures. The work provides a scalable, edge-preserving segmentation method with potential extensions to color images and other imaging modalities.

Abstract

Based on transformed regularization, transformed total variation (TTV) has robust image recovery that is competitive with other nonconvex total variation (TV) regularizers, such as TV, . Inspired by its performance, we propose a TTV-regularized Mumford--Shah model with fuzzy membership function for image segmentation. To solve it, we design an alternating direction method of multipliers (ADMM) algorithm that utilizes the transformed proximal operator. Numerical experiments demonstrate that using TTV is more effective than classical TV and other nonconvex TV variants in image segmentation.
Paper Structure (4 sections, 1 theorem, 24 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 4 sections, 1 theorem, 24 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proposed Approach
  3. Experimental Results
  4. Conclusion

Key Result

Lemma 1

Given $x \in \mathbb{R}^n$, the optimal solution to eq:tl1_prox is

Figures (3)

  • Figure 1: Original images for testing. (a)-(c) Retinal vessel images from the DRIVE dataset staal2004ridge. Image size is $584 \times 565$ with pixel intensities 191 (vessel) and 104 (background). (d)-(g) Brain images from the BrainWeb dataset aubert2006new. Image size is $104 \times 87$ with pixel intensities 10 (background), 48 (cerebrospinal fluid), 106 (grey matter), and 154 (white matter).
  • Figure 2: Segmentation results of Figures \ref{['fig:vessel1']}-\ref{['fig:vessel2']} (after normalization) corrupted by Gaussian noise of mean 0 and variance 0.01.
  • Figure 3: Segmentation results of Figure \ref{['fig:brain4']} (after normalization) corrupted by Gaussian noise of mean 0 and variance 0.04. First row is CSF, second row is GM, and third row is WM.

Theorems & Definitions (1)

  • Lemma 1: zhang2017minimization