Central and Periodic Multi-Scale Discrete Radon Transforms

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 f...

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Autores principales:	Óscar Gómez-Cárdenes, José G. Marichal-Hernández, Jonas Phillip Lüke, José M. Rodríguez-Ramos
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	discrete Radon transform DRT numerical transforms curvelets SFF pruning Technology T Engineering (General). Civil engineering (General) TA1-2040 Biology (General) QH301-705.5 Physics QC1-999 Chemistry QD1-999
Acceso en línea:	https://doaj.org/article/7a58b12650174ef4bdcab9c87319d975
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spelling	oai:doaj.org-article:7a58b12650174ef4bdcab9c87319d9752021-11-25T16:32:57ZCentral and Periodic Multi-Scale Discrete Radon Transforms10.3390/app1122106062076-3417https://doaj.org/article/7a58b12650174ef4bdcab9c87319d9752021-11-01T00:00:00Zhttps://www.mdpi.com/2076-3417/11/22/10606https://doaj.org/toc/2076-3417The 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>×</mo><mn>4</mn><mi>N</mi></mrow></semantics></math></inline-formula>, instead of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mi>N</mi><mo>×</mo><mn>4</mn><mi>N</mi></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo></mrow></semantics></math></inline-formula> speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners.Óscar Gómez-CárdenesJosé G. Marichal-HernándezJonas Phillip LükeJosé M. Rodríguez-RamosMDPI AGarticlediscrete Radon transformDRTnumerical transformscurveletsSFFpruningTechnologyTEngineering (General). Civil engineering (General)TA1-2040Biology (General)QH301-705.5PhysicsQC1-999ChemistryQD1-999ENApplied Sciences, Vol 11, Iss 10606, p 10606 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	discrete Radon transform DRT numerical transforms curvelets SFF pruning Technology T Engineering (General). Civil engineering (General) TA1-2040 Biology (General) QH301-705.5 Physics QC1-999 Chemistry QD1-999
spellingShingle	discrete Radon transform DRT numerical transforms curvelets SFF pruning Technology T Engineering (General). Civil engineering (General) TA1-2040 Biology (General) QH301-705.5 Physics QC1-999 Chemistry QD1-999 Óscar Gómez-Cárdenes José G. Marichal-Hernández Jonas Phillip Lüke José M. Rodríguez-Ramos Central and Periodic Multi-Scale Discrete Radon Transforms
description	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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>×</mo><mn>4</mn><mi>N</mi></mrow></semantics></math></inline-formula>, instead of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mi>N</mi><mo>×</mo><mn>4</mn><mi>N</mi></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo></mrow></semantics></math></inline-formula> speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners.
format	article
author	Óscar Gómez-Cárdenes José G. Marichal-Hernández Jonas Phillip Lüke José M. Rodríguez-Ramos
author_facet	Óscar Gómez-Cárdenes José G. Marichal-Hernández Jonas Phillip Lüke José M. Rodríguez-Ramos
author_sort	Óscar Gómez-Cárdenes
title	Central and Periodic Multi-Scale Discrete Radon Transforms
title_short	Central and Periodic Multi-Scale Discrete Radon Transforms
title_full	Central and Periodic Multi-Scale Discrete Radon Transforms
title_fullStr	Central and Periodic Multi-Scale Discrete Radon Transforms
title_full_unstemmed	Central and Periodic Multi-Scale Discrete Radon Transforms
title_sort	central and periodic multi-scale discrete radon transforms
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/7a58b12650174ef4bdcab9c87319d975
work_keys_str_mv	AT oscargomezcardenes centralandperiodicmultiscalediscreteradontransforms AT josegmarichalhernandez centralandperiodicmultiscalediscreteradontransforms AT jonasphillipluke centralandperiodicmultiscalediscreteradontransforms AT josemrodriguezramos centralandperiodicmultiscalediscreteradontransforms
_version_	1718413146062848000

Central and Periodic Multi-Scale Discrete Radon Transforms

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