Three-dimensional multiscale discrete Radon and John transforms

Marichal-Hernández, José G.; Gómez-Cárdenes, Óscar; Rosa, F.; Kim, Do-Hyung; Rodríguez-Ramos, José Manuel

doi:10.1117/1.oe.59.9.093104

Cited by 4 publications

(7 citation statements)

References 27 publications

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“…Generalizations of the Brady-Yong algorithm for summations over lines and planes in three-dimensional images were first published by T.-K. Wu and M. Brady in 1998 [68], but this publication in the proceedings of a symposium on image processing and character recognition went unnoticed for a long time. Both algorithms have been repeatedly reinvented [26,69].…”

Section: Discussionmentioning

confidence: 99%

“…In the second loop (lines [19][20][21][22][23][24][25][26][27], matching subpatterns are from the list. The index table (ind) is supported, allowing each pattern to find its subpattern from the shortened list.…”

Section: Endmentioning

confidence: 99%

“…For somewhat different reasons, FHT is used in X-ray computed tomography [1,[24][25][26]. With significant computing resources involved, the desired reconstruction resolution parameters have not yet been achieved, so the volumes of processed data are constantly growing, which, when using asymptotically inefficient algorithms, sooner or later leads to a resource shortage.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a Fast Hough/Radon Transform as a Compact Summation Scheme over Digital Straight Line Segments

Nikolaev,

Ershov,

Kroshnin

et al. 2023

Mathematics

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The Hough transform, interpreted as the discretization of the Radon transform, is a widely used tool in image processing and machine vision. The primary way to speed it up is to employ the Brady–Yong algorithm. However, the accuracy of the straight line discretization utilized in this algorithm is limited. In this study, we propose a novel algorithm called ASD2 that offers fast computation of the Hough transform for images of arbitrary sizes. Our approach adopts a computation scheme similar to the Brady–Yong algorithm but incorporates the best possible line discretization for improved accuracy. By employing the Method of Four Russians, we demonstrate that for an image of size n×n where n=8q and q∈N, the computational complexity of the ASD2 algorithm is O(n8/3) when summing over O(n2) digital straight line segments.

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Section: Discussionmentioning

confidence: 99%

Section: Endmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a Fast Hough/Radon Transform as a Compact Summation Scheme over Digital Straight Line Segments

Nikolaev,

Ershov,

Kroshnin

et al. 2023

Mathematics

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“…In two recent papers, we have extended the conventional, multi-scale DRT to perform summation on planes and lines, in a three-dimensional volume [41]; we proposed an advantageous backward transform, based on filtered back-projection [42], again for the conventional DRT. We think both proposed central and periodic DRT will be equally extensible to three dimensions, and, in the case of periodic DRT, the filtered back-projection approach also seems feasible.…”

Section: Discussionmentioning

confidence: 99%

Central and Periodic Multi-Scale Discrete Radon Transforms

et al. 2021

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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, exclusively with integer arithmetic operations. An inversion algorithm exists and is exact and fast, in spite of being iterative. In this work, the DRT forward and backward pair is evolved to propose two faster algorithms: central DRT, which computes only the central portion of intercepts; and periodic DRT, which computes the line integrals on the periodic extension of the input. Both have an output of size N×4N, instead of 3N×4N, as in the original algorithm. Periodic DRT is proven to have a fast inversion, whereas central DRT does not. An interesting application of periodic DRT is its use as building a block of discrete curvelet transform. 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.

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“…We plan to extend this method to Radon inversion in more than two dimensions [14] and as a tool for blind acoustic source localization [15].…”

Section: Discussionmentioning

confidence: 99%

Inverse Multiscale Discrete Radon Transform by Filtered Backprojection

et al. 2020

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The Radon transform is a valuable tool in inverse problems such as the ones present in electromagnetic imaging. Up to now the inversion of the multiscale discrete Radon transform has been only possible by iterative numerical methods while the continuous Radon transform is usually tackled with the filtered backprojection approach. In this study, we will show, for the first time, that the multiscale discrete version of Radon transform can as well be inverted with filtered backprojection, and by doing so, we will achieve the fastest implementation until now of bidimensional discrete Radon inversion. Moreover, the proposed method allows the sacrifice of accuracy for further acceleration. It is a well-conditioned inversion that exhibits a resistance against noise similar to that of iterative methods.

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Three-dimensional multiscale discrete Radon and John transforms

Cited by 4 publications

References 27 publications

On a Fast Hough/Radon Transform as a Compact Summation Scheme over Digital Straight Line Segments

On a Fast Hough/Radon Transform as a Compact Summation Scheme over Digital Straight Line Segments

Central and Periodic Multi-Scale Discrete Radon Transforms

Inverse Multiscale Discrete Radon Transform by Filtered Backprojection

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