Mathematical Morphology in Computer Graphics, Scientific Visualization and Visual Exploration

Roerdink, Jos B. T. M.

doi:10.1007/978-3-642-21569-8_32

Cited by 8 publications

(2 citation statements)

References 44 publications

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“…Mathematical morphology is a theory for the analysis of spatial structures in images. It has evolved over decades to a very successful field in image processing, see, for example, [9][10][11] for an overview. There are two main building blocks of usual morphological operators.…”

Section: Introductionmentioning

confidence: 99%

Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation

Kahra

Breuß

2023

J Math Imaging Vis

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The basic filters in mathematical morphology are dilation and erosion. They are defined by a structuring element that is usually shifted pixel-wise over an image, together with a comparison process that takes place within the corresponding mask. This comparison is made in the grey value case by means of maximum or minimum formation. Hence, there is easy access to max-plus algebra and, by means of an algebra change, also to the theory of linear algebra. We show that an approximation of the maximum function forms a commutative semifield (with respect to multiplication) and corresponds to the maximum again in the limit case. In this way, we demonstrate a novel access to the logarithmic connection between the Fourier transform and the slope transformation. In addition, we prove that the dilation by means of a fast Fourier transform depends only on the size of the structuring element used. Moreover, we derive a bound above which the Fourier approximation yields results that are exact in terms of grey value quantisation.

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Section: Introductionmentioning

confidence: 99%

Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation

Kahra

Breuß

2023

J Math Imaging Vis

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“…However, most of these methods are limited in terms of shape, size or flatness of SE, or specific hardware that is needed, cf. [12], [14]- [17], [24].…”

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confidence: 99%

A Novel Approach for Computation of Morphological Operations Using the Number Theoretic Transform

Sridhar

Breuß

2022

Lecture Notes in Computer Science

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In this paper we consider the fundamental operations dilation and erosion of mathematical morphology. Many powerful image filtering operations are based on their combinations. We establish homomorphism between max-plus semiring of integers and subset of polynomials over the field of real numbers. This enables to reformulate the task of computing morphological dilation to that of computing sums and products of polynomials. Therefore, dilation and its dual operation erosion can be computed by convolution of discrete linear signals, which is efficiently accomplished using a Fast Fourier Transform technique. The novel method may deal with non-flat filters and incorporates no restrictions on shape or size of the structuring element, unlike many other fast methods in the field. In contrast to previous fast Fourier techniques it gives exact results and is not an approximation. The new method is in practice particularly suitable for filtering images with small tonal range or when employing large filter sizes. We explore the benefits by investigating an implementation on FPGA hardware. Several experiments demonstrate the exactness and efficiency of the proposed method.

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Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform

Kahra

Sridhar

Breuß

2021

Lecture Notes in Computer Science

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Mathematical Morphology in Computer Graphics, Scientific Visualization and Visual Exploration

Cited by 8 publications

References 44 publications

Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation

Properties of Morphological Dilation in Max-Plus and Plus-Prod Algebra in Connection with the Fourier Transformation

A Novel Approach for Computation of Morphological Operations Using the Number Theoretic Transform

Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform

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