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Contour Flow Constraint: Preserving Global Shape Similarity for Deep Learning based Image Segmentation

Shengzhe Chen, Zhaoxuan Dong, Jun Liu

TL;DR

This paper addresses the challenge of preserving global shape similarity in deep image segmentation by introducing a contour flow constraint that links segmentation contours to ground-truth contours. It establishes a mathematical framework where global shape similarity is equivalent to an orthogonality condition between the segmentation gradient and a ground-truth contour flow, and provides two practical implementations: a shape loss that can be added to any segmentation network and an unrolled variational model that yields the CFSSnet architecture. The approach yields consistent improvements in boundary-related metrics (BD, BDSD) and robustness to noise across diverse medical imaging datasets, while also enabling a 3D extension via a 3D contour-flow term. The CFSSnet demonstrates how unrolled optimization can embed principled shape priors into network architectures, offering a principled, architecture-agnostic path to better shape-preserving segmentation with potential for multi-class extensions and topology-aware enhancements.

Abstract

For effective image segmentation, it is crucial to employ constraints informed by prior knowledge about the characteristics of the areas to be segmented to yield favorable segmentation outcomes. However, the existing methods have primarily focused on priors of specific properties or shapes, lacking consideration of the general global shape similarity from a Contour Flow (CF) perspective. Furthermore, naturally integrating this contour flow prior image segmentation model into the activation functions of deep convolutional networks through mathematical methods is currently unexplored. In this paper, we establish a concept of global shape similarity based on the premise that two shapes exhibit comparable contours. Furthermore, we mathematically derive a contour flow constraint that ensures the preservation of global shape similarity. We propose two implementations to integrate the constraint with deep neural networks. Firstly, the constraint is converted to a shape loss, which can be seamlessly incorporated into the training phase for any learning-based segmentation framework. Secondly, we add the constraint into a variational segmentation model and derive its iterative schemes for solution. The scheme is then unrolled to get the architecture of the proposed CFSSnet. Validation experiments on diverse datasets are conducted on classic benchmark deep network segmentation models. The results indicate a great improvement in segmentation accuracy and shape similarity for the proposed shape loss, showcasing the general adaptability of the proposed loss term regardless of specific network architectures. CFSSnet shows robustness in segmenting noise-contaminated images, and inherent capability to preserve global shape similarity.

Contour Flow Constraint: Preserving Global Shape Similarity for Deep Learning based Image Segmentation

TL;DR

This paper addresses the challenge of preserving global shape similarity in deep image segmentation by introducing a contour flow constraint that links segmentation contours to ground-truth contours. It establishes a mathematical framework where global shape similarity is equivalent to an orthogonality condition between the segmentation gradient and a ground-truth contour flow, and provides two practical implementations: a shape loss that can be added to any segmentation network and an unrolled variational model that yields the CFSSnet architecture. The approach yields consistent improvements in boundary-related metrics (BD, BDSD) and robustness to noise across diverse medical imaging datasets, while also enabling a 3D extension via a 3D contour-flow term. The CFSSnet demonstrates how unrolled optimization can embed principled shape priors into network architectures, offering a principled, architecture-agnostic path to better shape-preserving segmentation with potential for multi-class extensions and topology-aware enhancements.

Abstract

For effective image segmentation, it is crucial to employ constraints informed by prior knowledge about the characteristics of the areas to be segmented to yield favorable segmentation outcomes. However, the existing methods have primarily focused on priors of specific properties or shapes, lacking consideration of the general global shape similarity from a Contour Flow (CF) perspective. Furthermore, naturally integrating this contour flow prior image segmentation model into the activation functions of deep convolutional networks through mathematical methods is currently unexplored. In this paper, we establish a concept of global shape similarity based on the premise that two shapes exhibit comparable contours. Furthermore, we mathematically derive a contour flow constraint that ensures the preservation of global shape similarity. We propose two implementations to integrate the constraint with deep neural networks. Firstly, the constraint is converted to a shape loss, which can be seamlessly incorporated into the training phase for any learning-based segmentation framework. Secondly, we add the constraint into a variational segmentation model and derive its iterative schemes for solution. The scheme is then unrolled to get the architecture of the proposed CFSSnet. Validation experiments on diverse datasets are conducted on classic benchmark deep network segmentation models. The results indicate a great improvement in segmentation accuracy and shape similarity for the proposed shape loss, showcasing the general adaptability of the proposed loss term regardless of specific network architectures. CFSSnet shows robustness in segmenting noise-contaminated images, and inherent capability to preserve global shape similarity.

Paper Structure

This paper contains 42 sections, 4 theorems, 23 equations, 9 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Learning Based and Model Based Segmentation
  4. Shape Prior in Image Segmentation
  5. Unrolling
  6. The Proposed Global Shape Similarity Preserving Constraint
  7. General Variational Segmentation Framework
  8. The Signed Distance Function for Ground Truth
  9. Global Shape Similarity by Contours
  10. Contour Flow Constraint
  11. Implementation I: Shape Loss for any Deep-learning based Segmentation Framework
  12. Implementation II: Deep Neural Network by Unrolling Variational Model
  13. Variational Segmentation Model with Global Shape Similarity
  14. CFSSnet -- Unrolled DCNN Preserving Global Shape Similarity
  15. Backbone
  16. ...and 27 more sections

Key Result

Corollary 1

(Boundary similarity) Contour similarity $\delta_{u}\sim \delta_{g}$ leads to boundary similarity between $\Omega_u,\Omega_{g}$. Specifically, for each branch $\delta_{u}^{\gamma,m}$ of $\partial\Omega_u$, $\forall i,j\in \delta_{u}^{\gamma,m},$$D(i,\partial\Omega_{g}) = D(j,\partial\Omega_{g}).$$\p

Figures (9)

  • Figure 1: An intuition of the concept of global shape similarity of the "C" letter.
  • Figure 2: Intuitive illustration of some important concepts in this paper.
  • Figure 3: A toy example to illustrate the effectiveness of the proposed global shape similarity condition.
  • Figure 4: Architecture of CFSSnet and its components. "ConvBR" is short for "Convolution-BatchNorm-ReLU".
  • Figure 5: Segmentation results of benchmark networks trained with cross entropy loss alone and together with the proposed shape loss.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Corollary 1
  • Lemma 1
  • proof
  • Definition 4
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 2 more