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A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform

2008
*
SIAM Journal on Scientific Computing
*

*The*algorithm relies on a

*discrete*Fourier slice theorem, which associates

*the*

*discrete*

*Radon*

*transform*with

*the*pseudo-polar Fourier

*transform*[14] . ...

*The*

*discrete*2D definition of

*the*

*Radon*

*transform*is shown to be geometrically faithful as

*the*lines used for summation exhibit no wraparound effects. ...

*The*

*fast*algorithms for

*the*pseudo-polar Fourier

*transform*provide

*fast*forward and inverse algorithms for

*the*

*discrete*

*Radon*

*transform*. ...

##
###
Page 4858 of Mathematical Reviews Vol. , Issue 2003f
[page]

2003
*
Mathematical Reviews
*

(FIN-TUT-SG; Tampere

*Fast*quantum -D Fourier and*Radon**transforms*. ...*The*algorithms are based on*the*decomposition of*the**transform*into a product of an n-dimensional*discrete**Radon**transform*and a family of parallel one-dimensional*transforms*. ...##
###
3-D Discrete Analytical Ridgelet Transform

2006
*
IEEE Transactions on Image Processing
*

This

doi:10.1109/tip.2006.881936
pmid:17153944
fatcat:u5v7waxsjfaobiqelzi7e5yjh4
*transform*uses*the*Fourier strategy for*the*computation of*the*associated 3-D*discrete**Radon**transform*. ...*The*innovative step is*the*definition of a*discrete*3-D*transform*with*the**discrete*analytical geometry*theory*by*the*construction of 3-D*discrete*analytical lines in*the*Fourier domain. ... Principles of*the*3-D*Discrete**Radon**Transform**The**discretization*of*the*3-D*Radon**transform*is*the*extension of*the*2-D*discrete**Radon**transform*method: 1) Compute*the*3-D*Discrete*Fourier*Transform*of ...##
###
Discrete analytical Ridgelet transform

2004
*
Signal Processing
*

This

doi:10.1016/j.sigpro.2004.07.009
fatcat:zu7hwjxnkrck7bv6jl52nc5tia
*transform*uses*the*Fourier strategy for*the*computation of*the*associated 3-D*discrete**Radon**transform*. ...*The*innovative step is*the*definition of a*discrete*3-D*transform*with*the**discrete*analytical geometry*theory*by*the*construction of 3-D*discrete*analytical lines in*the*Fourier domain. ... Principles of*the*3-D*Discrete**Radon**Transform**The**discretization*of*the*3-D*Radon**transform*is*the*extension of*the*2-D*discrete**Radon**transform*method: 1) Compute*the*3-D*Discrete*Fourier*Transform*of ...##
###
The Radon transform on SO(3): a Fourier slice theorem and numerical inversion

2008
*
Inverse Problems
*

Based on a Fourier slice theorem

doi:10.1088/0266-5611/24/2/025011
fatcat:3oq2efczhvgvpjctiakr3acjia
*the**discrete*inverse*Radon**transform*of a function sampled on*the*product space S 2 × S 2 of two two-dimensional spheres is determined as*the*solution of a minimization ... This communication presents a novel approach to*the*numerical inversion of*the*one-dimensional*Radon**transform*on SO(3). ... Acknowledgments*The*authors gratefully acknowledge financial support by Deutsche Forschungsgemeinschaft, grant "high resolution texture analysis" (PR 331/11, SCHA 465/15) and thank*the*referees and*the*...##
###
Compressed sensing electron tomography

2013
*
Ultramicroscopy
*

Still demonstrates

doi:10.1016/j.ultramic.2013.03.019
pmid:23834932
fatcat:im76pmc4fndsdjhujzao7emyee
*the*feasibility of undersampled recovery, which is evidently of interest for some tomography applications. ... Experimental results For phantoms with*the*nanoparticle statistical properties, CS-ET recovery is markedly better than alternative methods. ... ET*Radon*data can be numerically*transformed*into 2D Fourier samples on a polar grid. From a computational perspective, these are non-uniform*discrete*Fourier*transform*(NDFT) samples. ...##
###
Fast numerical inversion of the attenuated Radon transform with full and partial measurements

2004
*
Inverse Problems
*

*The*method is based on an extension of

*the*

*fast*slant stack algorithm developed for

*the*

*Radon*

*transform*. ... We propose a numerical method to simulate and invert

*the*two-dimensional attenuated

*Radon*

*transform*(AtRT) from full (360 • ) or partial (180 • ) measurements. ...

*The*first author also acknowledges support from an Alfred P Sloan fellowship. ...

##
###
Irregular Sampling Of The Radon Transform Of Bandlimited Functions

2013
*
Zenodo
*

Publication in

doi:10.5281/zenodo.54450
fatcat:sdbvioek5rcylpij5gytoqjxk4
*the*conference proceedings of SampTA, Bremen, Germany, 2013 ... SAMPLING THEOREMS FOR*THE**RADON**TRANSFORM**The*developments from*the*previous section allow us to apply*the**theory*of bandlimited functions to*the*sampled*Radon**transform*. ... Bandlimitedness conditions also appear implicitly in reconstruction techniques based on*discretizations*of*the*inverse*Radon**transform*. ...##
###
Irregular Sampling of the Radon Transform of Bandlimited Functions
[article]

2013
*
arXiv
*
pre-print

First, we prove that

arXiv:1307.1151v1
fatcat:uaknufdkeve2hawistcqzrv3nu
*the**Radon**transform*is a continuous L2-operator for certain classes of bandlimited signals. ... We provide conditions for exact reconstruction of a bandlimited function from irregular polar samples of its*Radon**transform*. ... SAMPLING THEOREMS FOR*THE**RADON**TRANSFORM**The*developments from*the*previous section allow us to apply*the**theory*of bandlimited functions to*the*sampled*Radon**transform*. ...##
###
Page 2581 of Mathematical Reviews Vol. , Issue 92e
[page]

1992
*
Mathematical Reviews
*

over a finite field is initiated chiefly by

*the*possible applications of these*transforms*to problems of coding*theory*, combinatorics, and other branches of*discrete*mathematics. ... Summary: “We present a*fast*algorithm which evaluates a*discrete*Laplace*transform*with N points at M arbitrarily distributed points with C(N+M) work, where C depends only on*the*precision 44 INTEGRAL ...##
###
Application of Radon Transform for Fast Solution of Boundary Value Problems for Elliptic PDE in Domains With Complicated Geometry
[chapter]

2010
*
Matrix Methods: Theory, Algorithms and Applications
*

Here we extend

doi:10.1142/9789812836021_0030
fatcat:2yzqk73l65ginihnqhbeelsi3e
*the*proposed approach and construct another version of GRmethod based on application of*the*direct*Radon**transform*[*Radon*(1917) ] to*the*PDE [Sigurdur (1999); Gelfand and Shapiro (1955) ... GRmethod consists in application of*the**Radon**transform*directly to*the*PDE and in reduction PDE to assemblage of Ordinary Differential Equations (ODE). ... Acknowledgement Author acknowledge to VIEP BUAP, SEP and CONACYT Mexico for*the*support of*the*part of this investigation in*the*frame of*the*Projects No GRA-EXC08-*I*and No CB-2006-1-57479. ...##
###
Computational complexity lower bounds of certain discrete Radon transform approximations
[article]

2018
*
arXiv
*
pre-print

For

arXiv:1801.01054v1
fatcat:a4s7vky6pfgbnimskdymbtukq4
*the*computational model where only additions are allowed,*the*Ω(n^2 n) lower bound on operations count with respect to image size n× n is obtained for two types of*the**discrete**Radon**transform*implementations ... :*the**fast*Hough*transform*and a generic strip pattern class which includes*the*classical Hough*transform*, implying*the**fast*Hough*transform*algorithm asymptotic optimality. ... , Igor Polyakov and Alexey Savchik for verifying*the*proofs and locating a few misprints, and Andrey Gladkov for help with*the*FHT pictures. ...##
###
Page 44 of Mathematical Reviews Vol. , Issue 88k
[page]

1988
*
Mathematical Reviews
*

Summary: “We describe

*the**discrete**Radon**transform*(DRT) and*the*exact inversion algorithm for it. ... This observation allows construction of*fast*direct and inverse*transforms*. ...##
###
Binary Shape Normalization Using the Radon Transform
[chapter]

2003
*
Lecture Notes in Computer Science
*

This paper presents a novel approach to normalize binary shapes which is based on

doi:10.1007/978-3-540-39966-7_17
fatcat:wxiso37e6bfv3djxla6vbjcgnu
*the**Radon**transform*.*The*key idea of*the*paper is an original adaptation of*the**Radon**transform*. ...*The*binary shape is projected in*Radon*space for different levels of*the*(3-4) distance*transform*. ... cary over to*the**discrete**Radon**transform*only approximately, due to errors of*discretization*. ...##
###
Central and Periodic Multi-Scale Discrete Radon Transforms

2021
*
Applied Sciences
*

*The*multi-scale

*discrete*

*Radon*

*transform*(DRT) calculates, with linearithmic complexity,

*the*summation of pixels, through a set of

*discrete*lines, covering all possible slopes and intercepts in an image ... Central DRT can provide almost a 2× speedup over conventional DRT, probably becoming

*the*faster

*Radon*

*transform*algorithm available, at

*the*cost of ignoring 15% of

*the*summations in

*the*corners. ...

*The*details on

*the*formulation of

*the*

*discrete*

*Radon*

*transform*can be found there. ...

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