Orthogonal discrete periodic Radon transform. Part I: theory and realization

Lun, Daniel Pak-Kong; Hsung, Tai-Chiu; Shen, Tak-Wai

doi:10.1016/s0165-1684(02)00498-x

Cited by 26 publications

(7 citation statements)

References 19 publications

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“…For the case , projections are required resulting in a certain amount of oversampling, which is easily and exactly corrected by dividing each coefficient for all slices by [13]. Orthogonal forms of the discrete FST also exist which do not require this sampling correction [14], [15]. The image can then be recovered from the projections by computing the 1-D DFT of these projections, placing these slices along the vectors and into 2-D DFT space directly (without interpolation) and computing the inverse 2-D DFT.…”

Section: A Discrete Fourier Slice Theoremmentioning

confidence: 99%

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

Chandra

Normand²,

Kingston

et al. 2014

IEEE Signal Process. Lett.

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International audienceThe discrete Fourier slice theorem is an important tool for signal processing, especially in the context of the exact reconstruction of an image from its projected views. This paper presents a digital reconstruction algorithm to recover a two dimensional (2-D) image from sets of discrete one dimensional (1-D) projected views. The proposed algorithm has the same computational complexity as the 2-D fast Fourier transform and remains robust to the addition of significant levels of noise. A mapping of discrete projections is constructed to allow aperiodic projections to be converted to projections that assume periodic image boundary conditions. Each remapped projection forms a 1-D slice of the 2-D Discrete Fourier Transform (DFT) that requires no interpolation. The discrete projection angles are selected so that the set of remapped 1-D slices exactly tile the 2-D DFT space. This permits direct and mathematically exact reconstruction of the image via the inverse DFT. The reconstructions are artefact free, except for projection inconsistencies that arise from any additive and remapped noise. We also present methods to generate compact sets of rational projection angles that exactly tile the 2-D DFT space. The improvement in noise suppression that comes with the reconstruction of larger sized images needs to be balanced against the corresponding increase in computation time

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Section: A Discrete Fourier Slice Theoremmentioning

confidence: 99%

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

Chandra

Normand²,

Kingston

et al. 2014

IEEE Signal Process. Lett.

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“…However, it has a high level of redundancy. The orthogonal discrete periodic radon transform (ODPRT) [5] is another transform that is based on linear congruences. MRT has been applied to image compression by making use of a set of unique MRT coefficients [6,7].…”

Section: Introductionmentioning

confidence: 99%

A New Integer-to-Integer Transform

Roy¹

2020

Number Theory and Its Applications

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This chapter presents a detailed analysis of an integer-to-integer transform that is closely related to the discrete Fourier transform, but that offers insights into signal structure that the DFT does not. The transform is analyzed for its underlying properties using concepts from number theory. Theorems are given along with proofs to help establish the salient features of the transform. Two kinds of redundancy exist in the transform. It is shown how redundancy implicit in the transform can be eliminated to obtain a simple form. Closed-form formulas for the forward and inverse transforms are presented.

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“…Black pixels lie on the centre of the wrapped lines, the values of these pixels are summed to give the value of one element in the transform. (a) Pixels on x ≡ 4y + 7 (mod 31) sum to give R4 (7). N is prime here so only one element on each row and column is sampled by a discrete wrapped line.…”

Section: Introductionmentioning

confidence: 99%

“…A non-redundant form of the DPRT with orthogonal bases was presented by Lun and Hsung and Shen in 2003 [7], it is termed the Orthogonal DPRT (ODPRT).…”

Section: Introductionmentioning

confidence: 99%

A Discrete Modulo N Projective Radon Transform for N × N Images

Kingston

Svalbe

2005

Lecture Notes in Computer Science

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Abstract. This paper presents a Discrete Radon Transform (DRT) based on congruent mathematics that applies to N × N arrays where N ∈ N. This definition incorporates and is a natural extension of the more restricted cases of the finite Radon transform [1] where N must be prime, the discrete periodic Radon transform [2] where N must be a power of 2, and the DRT over p n [3], where N must be a power of a single prime. The DRT exactly and invertibly maps a 2-D image to a set of 1-D projections of length N . Projections are found as the sum of the pixels centred on a parallel set of discrete lines. The image is assumed to be periodic and these lines wrap around the array under modulo N arithmetic. Properties of the continuous Radon transform are preserved in the DRT; a discrete form of the Fourier slice theorem applies, as does the convolution property. A formula is given to find the projection set required to be exactly invertible for arrays with N any composite number, as well as a means to determine the level of redundancy in sampling that is introduced on such composite arrays.

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Orthogonal discrete periodic Radon transform. Part I: theory and realization

Cited by 26 publications

References 19 publications

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

A New Integer-to-Integer Transform

A Discrete Modulo N Projective Radon Transform for N × N Images

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