An Approach to Adaptive Quadratic Structuring Functions Based on the Local Structure Tensor

Landström, Anders

doi:10.1007/978-3-319-18720-4_61

Cited by 2 publications

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“…More recently, Landström [35] has considered space-adaptive generalisations. Table 2 gives a compact representation of the morphological examples that we have discussed.…”

Section: Dilation and Erosion Scale-spacesmentioning

confidence: 99%

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Schmidt

Weickert

2016

J Math Imaging Vis

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It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramér-Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton-Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, α-scale-spaces, summed α-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.

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“…More recently, Landström [35] has considered space-adaptive generalisations. Table 2 gives a compact representation of the morphological examples that we have discussed.…”

Section: Dilation and Erosion Scale-spacesmentioning

confidence: 99%

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Schmidt

Weickert

2016

J Math Imaging Vis

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“…A partial differential equation (PDE) approach was proposed [30] to resolve that issue; the parameter of the model being linked with the intrinsic image features. More recently, this was achieved from the local structure tensor by defining elliptic [31] or quadratic [32] SF. For a survey on adaptive MM, please see [33, 34].…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous morphological PDEs for robust and adaptive image shock filters

Diop¹,

Angulo

2020

IET image process

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Classical morphological filters suffer from well performing in a noisy environment, and intrinsic image structures are not taken into account. The authors propose here an alternative to overcome such weaknesses, by properly using robust shock filters and inhomogeneity. Thus, they obtain multiscale morphological operators by using image edge functions as local weights in inhomogeneous Hamiltonians in classical multiscale dilations/erosions formulated with partial differential equations (PDEs). They provide the equivalent sup–inf‐based formulations, and derive sharpening/enhancement methods. In addition, they establish the PDE associated with the asymptotical iterations of the proposed robust and adaptive filters. The good behaviours of the proposed sup–inf and PDE‐based methods are illustrated on synthetic, greyscale, and colour images; results are analysed both qualitatively and quantitatively.

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An Approach to Adaptive Quadratic Structuring Functions Based on the Local Structure Tensor

Cited by 2 publications

References 20 publications

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

Inhomogeneous morphological PDEs for robust and adaptive image shock filters

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