On the Axioms of Scale Space Theory

Duits, Remco; Florack, Luc; Graaf, J. de; Romeny, Bart ter Haar

doi:10.1023/b:jmiv.0000024043.96722.aa

Cited by 114 publications

(105 citation statements)

References 28 publications

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“…It is in fact an element of the jet-space of the traditional Gaussian scale-space. It is interesting to note the similarity with the scale-space generated by the complex diffusion operator [11], as well as the study of α-scale-spaces [5,6] and the Poisson Scale-Space [7].…”

Section: The Integral H(t) = T Tmentioning

confidence: 99%

The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

2006

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Abstract. The uncertainty principle is a fundamental concept in the context of signal and image processing, just as much as it has been in the framework of physics and more recently in harmonic analysis. Uncertainty principles can be derived by using a group theoretic approach. This approach yields also a formalism for finding functions which are the minimizers of the uncertainty principles. A general theorem which associates an uncertainty principle with a pair of self-adjoint operators is used in finding the minimizers of the uncertainty related to various groups. This study is concerned with the uncertainty principle in the context of the Weyl-Heisenberg, the SIM(2), the Affine and the Affine-WeylHeisenberg groups. We explore the relationship between the two-dimensional affine group and the SIM(2) group in terms of the uncertainty minimizers. The uncertainty principle is also extended to the Affine-Weyl-Heisenberg group in one dimension. Possible minimizers related to these groups are also presented and the scale-space properties of some of the minimizers are explored.

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Section: The Integral H(t) = T Tmentioning

confidence: 99%

The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

2006

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“…Often identified with Gaussian low-pass filtering, a rigorous analysis of underlying scale-space axioms [53] has led to the discovery of the Poisson scale-space [18] and more general α scale-spaces [8].…”

Section: Introductionmentioning

confidence: 99%

Adaptive Filtering Using Channel Representations

Felsberg

2011

Computational Imaging and Vision

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This review article gives an overview on adaptive filtering methods based on channel representations. The framework of channel representations and its relation to density estimation is introduced. The fast and accurate scheme of virtual shift decoding is introduced and applied in several variants of channel smoothing:• channel smoothing with alpha-synthesis for improving stability of edge-enhancing filtering • orientation adaptive channel smoothing with applications to coherence-enhancing filtering • channel smoothing using graph-cuts for improving filtering results at corners • channel-coded feature maps (CCFMs) which lead to a significant speed-up of channel averaging • CCFM-based smoothing based on optimal parameters derived from a novel uncertainty relationFor each method, an algorithmic description and some examples of results are provided, together with discussions and references of the original papers. Cross connections to other articles in this volume are given where appropriate.

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“…Other papers in the general area include [1,2,3,4,6,7,12,13,14,15,18,21,22,23,24], and we refer to these for additional background, and for fundamentals in the theory.…”

Section: Introductionmentioning

confidence: 99%

Ergodic scales in fractal measures

Jørgensen

2011

Math. Comp.

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Abstract. We will consider a family of fractal measures on the real line R which are fixed, in the sense of Hutchinson, under a finite family of contractive affine mappings. The maps are chosen such as to leave gaps on R. Hence they have fractal dimension strictly less than 1. The middle-third Cantor construction is one example. Depending on the gaps and the scaling factor, it is known that the corresponding Hilbert space L 2 (μ) exhibits strikingly different properties. In this paper we show that when μ is fixed in a certain class, there are positive integers p such that multiplication by p modulo 1 induces an ergodic automorphism on the measure space (support(μ), μ).

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On the Axioms of Scale Space Theory

Cited by 114 publications

References 28 publications

The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

Adaptive Filtering Using Channel Representations

Ergodic scales in fractal measures

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