Inverse Multiscale Discrete Radon Transform by Filtered Backprojection

Marichal-Hernández, José G.; Oliva-García, Ricardo; Gómez-Cárdenes, Óscar; Rodríguez-Méndez, Iván; Rodríguez-Ramos, José Manuel

doi:10.3390/app11010022

Cited by 6 publications

(7 citation statements)

References 15 publications

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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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Section: Discussionmentioning

confidence: 99%

Central and Periodic Multi-Scale Discrete Radon Transforms

et al. 2021

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“…There are two published methods that deal with the inverse or backward discrete Radon transform. 10,11 Both use as intermediate step the adjoint transform, also called backprojection. While in other transforms it is possible to return from the transformed domain to the initial domain with an algorithm of the same complexity as the one used for the direct path; in the discrete Radon transform, where summations of the initial data have to be undone, this is not the case.…”

Section: Inversion Methodsmentioning

confidence: 99%

“…On the other hand, the method proposed by Marichal et al 11 expands the sinogram to backproject to an image size that is 3 times bigger than the original. This expanded backprojection can then be cleaned by deconvolving with the point spread functions, PSFs, of the direct+adjoint combined operator.…”

Section: Inversion Methodsmentioning

confidence: 99%

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A fast but ill-conditioned formal inverse to Radon transforms in 2D and 3D

Oliva-García

Marichal-Hernández²,

Gómez-Cárdenes³

et al. 2022

Real-Time Image Processing and Deep Learning 2022

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We present a formal inversion of the multiscale discrete Radon trasform, valid both for 2D and 3D. With the transformed data from just one of the four quadrants of the direct 2D Radon transform, or one of the twelve dodecants, in case of 3D Radon transform, we can invert exactly and directly, with no iterations, the whole domain. The computational complexity of the proposed algorithms will be O(N log N). With N the total size of the problem, either square or cubic. But this inverse transforms are extremely ill conditioned, so the presence of noise in the transformed domain turns them useless. Still we present both algorithms, and characterize its weakness against noise.

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“…The mathematical Radon transform is an integral transform that converts a function f defined on a plane into a function Rf defined on a two-dimensional space of lines on the plane, the value of which in a specific line is equal to the curvilinear integral of this function over this line [6].…”

Section: Radon Transformmentioning

confidence: 99%

Development of an Algorithm and Software System for Facing Panels Accounting on Production Lines

Ivanov,

Bilous,

Botvinnikov

et al. 2023

ACPS

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This paper aims to develop and implement an algorithm and an automated software system for the auto- matic accounting process of external facing panels during transportation on line conveyors. The method described in this paper is designed to simplify the process of production and accounting of wall-facing panels. This method can also serve as a model for implementing other manufacturers. The developed algorithm consists of the following steps: obtaining a video stream in real-time or from a file and its targeted processing and determining the number of moving objects of interest. The software accounting system created based on the developed algorithm analyzes the video data and stores all the necessary results and settings in the data- base. The software system can adapt to the accounting requirements of other types of similar products in other areas.

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Inverse Multiscale Discrete Radon Transform by Filtered Backprojection

Cited by 6 publications

References 15 publications

Central and Periodic Multi-Scale Discrete Radon Transforms

Central and Periodic Multi-Scale Discrete Radon Transforms

A fast but ill-conditioned formal inverse to Radon transforms in 2D and 3D

Development of an Algorithm and Software System for Facing Panels Accounting on Production Lines

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