Numerical harmonic analysis on the hyperbolic plane

Fridman, Buma L.; Kuchment, Peter; Lancaster, Kirk E.; Lissianoi, Serguei; Mogilevsky, M.I.; Ma, Daowei; Пономарев, И. Н.; Papanicolaou, Vassilis G.

doi:10.1080/00036810008840889

Cited by 4 publications

(5 citation statements)

References 2 publications

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“…There are some applications of this theory to image processing [19,41] and to other problems in computational science. Fridman [12] discusses a Fourier transform for the hyperbolic group, the 3-dimensional subgroup of the Möbius group that fixes the unit disc. However, the theory appears not to have been developed to the point where it can be used as effectively as the standard Fourier invariants.…”

Section: Invariants Of Imagesmentioning

confidence: 99%

Möbius Invariants of Shapes and Images

Marsland

McLachlan²

2016

SIGMA

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Abstract. Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the Möbius group PSL(2, C), which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known Möbius invariants, and then develop an algorithm by which shapes can be recognised that is Möbius-and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a Möbius-invariant signature of grey-scale images.

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Section: Invariants Of Imagesmentioning

confidence: 99%

Möbius Invariants of Shapes and Images

Marsland

McLachlan²

2016

SIGMA

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“…In particular, the papers mentioned above contain explicit inversion formulas for the hyperbolic geodesic Radon transform. The formula obtained in [92] was numerically implemented in [48] and works as nicely and stably as the standard inversions of the regular Radon transform 8 .…”

Section: Reconstruction Algorithms and The Hyperbolic Integral Geometrymentioning

confidence: 99%

“…These Fourier transforms were numerically implemented in [48]. However, the deconvolution part is the one that makes the whole problem extremely unstable.…”

Section: Reconstruction Algorithms and The Hyperbolic Integral Geometrymentioning

confidence: 99%

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Generalized transforms of Radon type and their applications

Kuchment¹

2006

The Radon Transform, Inverse Problems, and Tomography

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These notes represent an extended version of the contents of the third lecture delivered at the AMS Short Course "Radon Transform and Applications to Inverse Problems" in Atlanta in January 2005. They contain a brief description of properties of some generalized Radon transforms arising in inverse problems. Here by generalized Radon transforms we mean transforms that involve integrations over curved surfaces and/or weighted integrations. Such transformations arise in many areas, e.g. in Single Photon Emission Tomography (SPECT), Electrical Impedance Tomography (EIT) thermoacoustic Tomography (TAT), and other areas. 1 Numerous other reasons to study this transform are known, e.g. Radar and Sonar imaging, approximation theory, PDEs, potential theory, complex analysis, etc. [2, 93]. Although in dimensions higher than two one should probably use the word "spherical" rather than "circular," we will use for simplicity the latter. This should not create any confusion.

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“…• The MRI technique [17,18,74] involves integration over hyperplanes the standard Radon transform in 3D [30,42,43]. • Electrical impedance tomography [11,15] somewhat unexpectedly leads to integrations along horocycles and geodesics in the hyperbolic plane [8,9,26,27,49,60]. • A range of contemporary novel techniques, such as thermoacoustic/photoacoustic tomography (TAT/PAT) and synthetic aperture radar (SAR) [6,13,14,56,57,59,61] lead to integrations over circles/spheres, the circular/spherical Radon transform [22].…”

Section: Introductionmentioning

confidence: 99%

Compton camera imaging and the cone transform: a brief overview

2018

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While most of Radon transform applications to imaging involve integrations over smooth sub-manifolds of the ambient space, lately important situations have appeared where the integration surfaces are conical. Three of such applications are single scatter optical tomography, Compton camera medical imaging, and homeland security. In spite of the similar surfaces of integration, the data and the inverse problems associated with these modalities differ significantly. In this article, we present a brief overview of the mathematics arising in Compton camera imaging. In particular, the emphasis is made on the overdetermined data and flexible geometry of the detectors. For the detailed results, as well as other approaches (e.g. smaller-dimensional data or restricted geometry of detectors) the reader is directed to the relevant publications.Only a brief description and some references are provided for the single scatter optical tomography.

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Numerical harmonic analysis on the hyperbolic plane

Cited by 4 publications

References 2 publications

Möbius Invariants of Shapes and Images

Möbius Invariants of Shapes and Images

Generalized transforms of Radon type and their applications

Compton camera imaging and the cone transform: a brief overview

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