Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matrix-Valued LogSumExp Approximation for Colour Morphology

Marvin Kahra, Michael Breuß, Andreas Kleefeld, Martin Welk

TL;DR

This paper presents an analysis of a novel approach that replaces the supremum within a morphological operation with the LogExp approximation of the maximum for matrix-valued colours and has the advantage of extending the associativity of dilation from the one-dimensional to the higher-dimensional case.

Abstract

Mathematical morphology is a part of image processing that uses a window that moves across the image to change certain pixels according to certain operations. The concepts of supremum and infimum play a crucial role here, but it proves challenging to define them generally for higher-dimensional data, such as colour representations. Numerous approaches have therefore been taken to solve this problem with certain compromises. In this paper we will analyse the construction of a new approach, which we have already presented experimentally in paper [Kahra, M., Breuß, M., Kleefeld, A., Welk, M., DGMM 2024, pp. 325-337]. This is based on a method by Burgeth and Kleefeld [Burgeth, B., Kleefeld, A., ISMM 2013, pp. 243-254], who regard the colours as symmetric $2\times2$ matrices and compare them by means of the Loewner order in a bi-cone through different suprema. However, we will replace the supremum with the LogExp approximation for the maximum instead. This allows us to transfer the associativity of the dilation from the one-dimensional case to the higher-dimensional case. In addition, we will investigate the minimality property and specify a relaxation to ensure that our approach is continuously dependent on the input data.

Matrix-Valued LogSumExp Approximation for Colour Morphology

TL;DR

This paper presents an analysis of a novel approach that replaces the supremum within a morphological operation with the LogExp approximation of the maximum for matrix-valued colours and has the advantage of extending the associativity of dilation from the one-dimensional to the higher-dimensional case.

Abstract

Mathematical morphology is a part of image processing that uses a window that moves across the image to change certain pixels according to certain operations. The concepts of supremum and infimum play a crucial role here, but it proves challenging to define them generally for higher-dimensional data, such as colour representations. Numerous approaches have therefore been taken to solve this problem with certain compromises. In this paper we will analyse the construction of a new approach, which we have already presented experimentally in paper [Kahra, M., Breuß, M., Kleefeld, A., Welk, M., DGMM 2024, pp. 325-337]. This is based on a method by Burgeth and Kleefeld [Burgeth, B., Kleefeld, A., ISMM 2013, pp. 243-254], who regard the colours as symmetric matrices and compare them by means of the Loewner order in a bi-cone through different suprema. However, we will replace the supremum with the LogExp approximation for the maximum instead. This allows us to transfer the associativity of the dilation from the one-dimensional case to the higher-dimensional case. In addition, we will investigate the minimality property and specify a relaxation to ensure that our approach is continuously dependent on the input data.

Paper Structure

This paper contains 11 sections, 19 theorems, 158 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General Definitions
  3. Colour Morphology
  4. The Loewner Order and the Decision of a Minimiser
  5. Characterisation of the Log-Exp-Supremum
  6. LES for a unique largest Eigenvalue
  7. LES for non-unique largest Eigenvalues
  8. General Characterisation and Properties of the LES
  9. Minimality of the LES
  10. Relaxation of the LES
  11. Conclusion and future Work

Key Result

lemma 1

Let the conditions lambda1_unique and rotation be fulfilled according to spectral_decomposition. Then $\boldsymbol u_1 = (1,0)^\textup{T}$ represents the major eigenvector of $\boldsymbol E_m$ for $m \rightarrow \infty$.

Figures (3)

  • Figure 1: Application of different morphological operators on a $256 \times 256$ grey-scale image with a $5 \times 5$ SE. From left to right: Downscaled original image from TAMPERE17 noise-free image database imageDB, dilation, erosion, opening and closing.
  • Figure 2: The HCL bi-cone according to BurgethKleefeld.
  • Figure 3: Visualisation of the Loewner ordering in the HCL bi-cone for the three colour matrices $\boldsymbol{X}_i$, $i = 1, 2, 3$, with $\boldsymbol{X}_3 \geq_{\textup{L}} \boldsymbol{X}_1$ and $\boldsymbol{X}_3 \geq_{\textup{L}} \boldsymbol{X}_2$. From left to right: The cone representation of a yellow colour $\boldsymbol X_1$ and a cyan-blue colour $\boldsymbol X_2$, the base circles of $\boldsymbol X_1$ and $\boldsymbol X_2$ with a third base circle of a green colour $\boldsymbol X_3$ that encompasses both of them, the cone representation of all three colours.

Theorems & Definitions (49)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 39 more