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On the importance of the $\varepsilon$-regularization of the distribution-dependent Mumford-Shah model for hyperspectral image segmentation

Jan-Christopher Cohrs, Benjamin Berkels

TL;DR

The paper analyzes a distribution-dependent Mumford–Shah model for hyperspectral image segmentation with an $\varepsilon$-regularized covariance term. It proves the existence of minimizers for the $\varepsilon$-regularized functional $J_\varepsilon$ and establishes $\Gamma$-convergence to a limit $J_0$ as $\varepsilon \to 0$, showing that minimizers may fail to exist in the limit. A concrete example demonstrates loss of closedness and minimizer existence in $J_0$, highlighting the crucial role of eigenvalue regularization. Additionally, a formula for the minimal eigenvalues of covariance estimates is derived, indicating when the data itself enforces positive eigenvalues and when regularization is indispensable. Overall, the work underscores the necessity of $\varepsilon$-regularization for feasibility and well-posedness in this class of segmentation models and informs potential directions to modify the functional to retain minimizer existence without regularization.

Abstract

Recently, the distribution-dependent Mumford-Shah model for hyperspectral image segmentation was introduced. It approximates an image based on first and second order statistics using a data term, that is built of a Mahalanobis distance plus a covariance regularization, and the total variation as spatial regularization. Moreover, to achieve feasibility, the appearing matrices are restricted to symmetric positive definite ones with eigenvalues exceeding a certain threshold. This threshold is chosen in advance as a data-independent parameter. In this article, we study theoretical properties of the model. In particular, we prove the existence of minimizers of the functional and show its $Γ$-convergence when the threshold regularizing the eigenvalues of the matrices tends to zero. It turns out that in the $Γ$-limit we lose the guaranteed existence of minimizers; and we give an example of an image where the $Γ$-limit indeed has no minimizer. Finally, we derive a formula for the minimum eigenvalues of the covariance matrices appearing in the functional that hints under which conditions the functional is able to handle the data without regularizing the eigenvalues. The results of this article demonstrate the significance and importance of the eigenvalue regularization to the model and that it cannot be dropped without substantial modifications.

On the importance of the $\varepsilon$-regularization of the distribution-dependent Mumford-Shah model for hyperspectral image segmentation

TL;DR

The paper analyzes a distribution-dependent Mumford–Shah model for hyperspectral image segmentation with an -regularized covariance term. It proves the existence of minimizers for the -regularized functional and establishes -convergence to a limit as , showing that minimizers may fail to exist in the limit. A concrete example demonstrates loss of closedness and minimizer existence in , highlighting the crucial role of eigenvalue regularization. Additionally, a formula for the minimal eigenvalues of covariance estimates is derived, indicating when the data itself enforces positive eigenvalues and when regularization is indispensable. Overall, the work underscores the necessity of -regularization for feasibility and well-posedness in this class of segmentation models and informs potential directions to modify the functional to retain minimizer existence without regularization.

Abstract

Recently, the distribution-dependent Mumford-Shah model for hyperspectral image segmentation was introduced. It approximates an image based on first and second order statistics using a data term, that is built of a Mahalanobis distance plus a covariance regularization, and the total variation as spatial regularization. Moreover, to achieve feasibility, the appearing matrices are restricted to symmetric positive definite ones with eigenvalues exceeding a certain threshold. This threshold is chosen in advance as a data-independent parameter. In this article, we study theoretical properties of the model. In particular, we prove the existence of minimizers of the functional and show its -convergence when the threshold regularizing the eigenvalues of the matrices tends to zero. It turns out that in the -limit we lose the guaranteed existence of minimizers; and we give an example of an image where the -limit indeed has no minimizer. Finally, we derive a formula for the minimum eigenvalues of the covariance matrices appearing in the functional that hints under which conditions the functional is able to handle the data without regularizing the eigenvalues. The results of this article demonstrate the significance and importance of the eigenvalue regularization to the model and that it cannot be dropped without substantial modifications.

Paper Structure

This paper contains 8 sections, 5 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The distribution-dependent Mumford--Shah model
  3. Model description of epsilonAMS
  4. Existence of minimizers of epsilonAMS
  5. Gamma convergence of J epsilon when epsilon to zero
  6. Loss of closedness of Pepsilon when epsilon to zero
  7. Bound for eigenvalues of covariance estimates
  8. Conclusion

Key Result

lemma 1

Let $(A_n)_{n\in\mathbb{N}}$ be a sequence of measurable, bounded subsets of $\mathbb{R}^d$ and $A \subseteq \mathbb{R}^d$ measurable and bounded. If $(A_n)_{n\in\mathbb{N}}$ converges to $A$ in measure in $\Omega$, then also the measures of $(A_n)_{n\in\mathbb{N}}$ restricted to $\Omega$ converge t

Figures (1)

  • Figure 1: The two figures show samples from two exemplary spectral distributions of $g^m\vert_{\mathcal{O}_l}$ for a segment $\mathcal{O}_l\subseteq \Omega$ and images $g^m\in L^\infty(\Omega, \mathbb{R}^L)$, $m\in \{1,2\}$. The arrows $v_{l,i}^m \in \mathbb{R}^L$ and $v_{l,j}^m \in \mathbb{R}^L$ are the $i$-th and $j$-th eigenvector of the symmetric and positive-semidefinite covariance matrix $\Sigma_l^m \in \mathbb{R}^{L\times L}$. The projections of the sampled spectra onto those vectors are plotted. In the top figure, there is variation of the spectra in both directions $v_{l,i}^1$ and $v_{l,j}^1$, which results in estimates for the $i$-th and $j$-th eigenvalue of $\Sigma_l^1$ being bounded from below with a positive bound (cf. \ref{['eq:minimum-eigenvalue-covariance-J-0']}) and hence $\Sigma_l^1 \in \mathcal{P}$ (provided that we also have variation in the directions of all other eigenvectors $v_{l,t}^1$ for $t\in \{1,\dots, L\}\setminus \{i,j\}$). In the bottom figure, there is only variation of the spectra in direction $v_{l,j}^2$, not in direction $v_{l,i}^2$. This results in an estimate for the $i$-th eigenvalue of $\Sigma_l^2$ equal to $0$, which is not admissible anymore since then $\Sigma_l^2 \notin \mathcal{P}$. Technically, we have $a = 0$ in \ref{['eq:minimum-eigenvalues-function']} in this situation, making the function unbounded from below and \ref{['eq:minimum-eigenvalue-covariance-J-0']} infeasible.

Theorems & Definitions (11)

  • definition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • theorem 1
  • proof
  • definition 2
  • theorem 2
  • ...and 1 more