Open AccessProceedings Article10.1109/ICIP.2005.1530466

Exponential Radon transform inversion based on harmonic analysis of the Euclidean motion group

- 01 Jan 2005

- Vol. 3, pp 613-615

TL;DR: A new method based on harmonic analysis of the Euclidean motion group Fourier domain provides a deconvolution type inversion for the exponential Radon transform.

Abstract: This paper presents a new method for the exponential Radon transform inversion based on harmonic analysis of the Euclidean motion group (M(2)). The exponential Radon transform is modified to be formulated as a convolution over M(2). The convolution representation leads to a block diagonalization of the modified exponential Radon transform in the Euclidean motion group Fourier domain, which provides a deconvolution type inversion for the exponential Radon transform. Numerical examples are presented to show the viability of the proposed method.

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In this paper, the authors present a new derivation of the spherical harmonic decomposition of the projection slice theorem using harmonic analysis of the Euclidean motion group ( M ( N ) ).. Deconvolution using harmonic analysis of M ( N ) leads to spherical harmonic decomposition of the projection slice theorem.

Fig. 2. Details from reconstructed images.

Fig. 1. Reconstruction of the modified Shepp-Logan phantom using a. first method, b. second method, c. third method, and d. FBP. For the first method n = 0. For second and third methods, all ns that the M(2)-Fourier coefficients are computed and σ = 10−8 are used for reconstruction. Below and on the right are the plots of horizontal and vertical slices passing from the center of the original image (thin line) and the reconstructed image (bold line).

Citations

•Journal Article•10.1016/J.JMVA.2010.08.012

Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images

Martin Crane, +1 more

- 01 Feb 2011

- Journal of Multivariate Analysis

TL;DR: A nonparametric methodology for estimating the mean change for matched samples on a Lie group is developed, finding that for k>=5, a manifold of projective shapes of k-ads in 3D has the structure of a 3k-15 dimensional Lie group that is equivariantly embedded in a Euclidean space, therefore testing for mean change amounts to a one sample test for extrinsic means on this Lie group.

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•Journal Article•10.3934/IPI.2007.1.457

A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group

Can Evren Yarman, +1 more

- 01 Jul 2007

- Inverse Problems and Imaging

TL;DR: In this article, the authors proposed a new method for the exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane, which can be expressed as a convolution on the motion group.

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Deconvolution Over Groups in Image Reconstruction

Birsen Yazici, +1 more

- 01 Jan 2006

- Advances in Imaging and Electron Physics

TL;DR: In this article, a minimum mean square error (MMSE) solution for the deconvolution problems formulated over groups using the group representation theory and the concept of group stationarity is presented.

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Proceedings Article•10.1117/12.647807

An inversion method for the exponential radon transform based on the harmonic analysis of the Euclidean motion group

Can Evren Yarman, +1 more

- 02 Mar 2006

TL;DR: In this paper, the authors proposed a new method for exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane, which can be expressed as a convolution on the motion group.

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References

•Book

The Mathematics of Computerized Tomography

Frank Natterer

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TL;DR: In this paper, the Radon transform and related transforms have been studied for stability, sampling, resolution, and accuracy, and quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.

...read moreread less

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Journal Article•10.1109/TASSP.1979.1163232

Compensation of tissue absorption in emission tomography

S. Bellini, +3 more

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- IEEE Transactions on Acoustics, Speech, ...

TL;DR: In this paper, the authors presented a technique for overcoming the problem of tissue absorption in emission tomography by solving a linear partial-differential equation that links the derivatives of the spectrum with respect to absorption and spatial frequencies.

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Journal Article•10.1109/42.476106

A unified analysis of exact methods of inverting the 2-D exponential radon transform, with implications for noise control in SPECT

Charles E. Metz, +1 more

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- IEEE Transactions on Medical Imaging

TL;DR: An analysis of the 2-D Fourier transform of the modified sinogram reveals that all previously-proposed linear methods can be interpreted as special cases of a broad class of methods, and that each method in the class can be implemented, in principle, by any one of four distinct techniques.

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Journal Article•10.1109/42.3940

The circular harmonic transform for SPECT reconstruction and boundary conditions on the Fourier transform of the sinogram

W.G. Hawkins, +2 more

- 01 Jan 1988

- IEEE Transactions on Medical Imaging

TL;DR: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates.

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Journal Article•10.1088/0031-9155/34/3/003

Image reconstruction algorithm for single-photon-emission computed tomography with uniform attenuation

T Inouye, +2 more

- 01 Mar 1989

- Physics in Medicine and Biology

TL;DR: A new method is presented for obtaining an analytical solution to the image-reconstruction problem in single-photon-emission computed tomography by applying an analytical continuation process to the two-dimensional Fourier transform of the image, which is derived from the one- dimensional Fourier transforms of projection functions.

...read moreread less

Exponential Radon transform inversion based on harmonic analysis of the Euclidean motion group

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Citations

Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images

A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group

Deconvolution Over Groups in Image Reconstruction

An inversion method for the exponential radon transform based on the harmonic analysis of the Euclidean motion group

References

The Mathematics of Computerized Tomography

Compensation of tissue absorption in emission tomography

A unified analysis of exact methods of inverting the 2-D exponential radon transform, with implications for noise control in SPECT

The circular harmonic transform for SPECT reconstruction and boundary conditions on the Fourier transform of the sinogram

Image reconstruction algorithm for single-photon-emission computed tomography with uniform attenuation

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