Discrete radon transform
TLDR
It is shown that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves.Abstract:
This paper describes the discrete Radon transform (DRT) and the exact inversion algorithm for it. Similar to the discrete Fourier transform (DFT), the DRT is defined for periodic vector-sequences and studied as a transform in its own right. Casting the forward transform as a matrix-vector multiplication, the key observation is that the matrix-although very large-has a block-circulant structure. This observation allows construction of fast direct and inverse transforms. Moreover, we show that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves. In fact, we describe not a single transform, but a class of transforms, representatives of which correspond in one way or another to discrete versions of the RT and its generalizations. An interesting observation is that the exact inversion algorithm cannot be obtained directly from Radon's inversion formula. Given the fact that the RT has no nontrivial one-dimensional analog, exact invertibility makes the DRT a useful tool geared specifically for multidimensional digital signal processing. Exact invertibility of the DRT, flexibility in its definition, and fast computational algorithm affect present applications and open possibilities for new ones. Some of these applications are discussed in the paper.read more
Citations
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Journal ArticleDOI
The finite ridgelet transform for image representation
Minh N. Do,Martin Vetterli +1 more
TL;DR: This work proposes an orthonormal version of the ridgelet transform for discrete and finite-size images and uses the finite Radon transform (FRAT) as a building block to overcome the periodization effect of a finite transform.
The Radon Transform - Theory and Implementation
TL;DR: In this article, the Radon and Hough transform is used for curve detection in digital images and for reconstruction of tomography images, and a new fast scheme for estimating curve parameters is presented.
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High‐resolution velocity gathers and offset space reconstruction
TL;DR: In this paper, a high-resolution procedure to reconstruct common-midpoint (CMP) gathers is presented, in which the target is the artifacts-free, aperture-compensated velocity gather.
Journal ArticleDOI
Deep Learning Techniques for Inverse Problems in Imaging
Gregory Ongie,Ajil Jalal,Christopher A. Metzler,Richard G. Baraniuk,Alexandros G. Dimakis,Rebecca Willett +5 more
TL;DR: A taxonomy that can be used to categorize different problems and reconstruction methods in deep neural networks and discusses the tradeoffs associated with these different reconstruction approaches, caveats and common failure modes.
Directional multiresolution image representations
TL;DR: This thesis focuses on the development of new "true" two-dimensional representations for images using a discrete framework that can lead to algorithmic implementations and a new family of block directional and orthonormal transforms based on the ridgelet idea.
References
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David Slepian,H. O. Pollak +1 more
TL;DR: In this paper, the authors apply the theory developed in the preceding paper to a number of questions about timelimited and bandlimited signals, and find the signals which do the best job of simultaneous time and frequency concentration.
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John Illingworth,Josef Kittler +1 more
TL;DR: This survey will provide a useful guide to quickly acquaint researchers with the main literature in this research area and it seems likely that the Hough transform will be an increasingly used technique.
Book
The Radon Transform and Some of Its Applications
TL;DR: In this article, the authors provide basic information about the properties of radon transform and provide guidance to literature related to transform, and are aimed at those with a basic undergraduate background in mathematics.
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Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case
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