Central and Periodic Multi-Scale Discrete Radon Transforms
Abstract
:Featured Application
Abstract
1. Introduction
1.1. Radon Transform, Use, and Algorithms for Its Computation
1.2. Conventional Discrete Radon Transform
1.3. Founding Idea on Periodic DRT
1.4. Structure of this Paper
2. Conventional 2D Discrete Radon Transform
2.1. Forward Multi-Scale Discrete Radon Transform
Discrete Lines in Conventional DRT
2.2. Full Sinogram Construction
- The black filled circle will be considered our {0, 0} reference point.
- The reference point will be joined by discrete lines, with the empty circles on the opposite extreme; so, those empty circles depict the plurality of the slopes to be considered.
- The reference and the slopes are at the extremes of an axis marked with a coarse black line. This axis is coincident with the discrete line with null displacement and slope.
- For the maximal slope, the reference point will be joined with the red circle. This line of maximal slope is one of the diagonals of the plane and has, as normal, the vectors depicted with dashed arrows. Those can be interpreted as positives or negatives.
- The rest of discrete lines, those corresponding to the ascents from 1 to , have, as a finishing vertex of their normals, the black dots that describe arches close to the axis, where crosses are depicted.
- Those crosses represent the displacements that must be considered.
2.3. Radon Adjoint Transform
2.4. Radon Inverse Transform
3. Central DRT
3.1. Discrete Lines in Central DRT
3.2. Memory Footprint at Partial Transforms
3.3. Central DRT Algorithm
- N data in column S, descending from row index , are the result of adding N data in column , descending from row index , and N data in column , descending from row index .
- N data in column , descending from row index , are the result of adding N data in column , descending from row index , and N data in column , descending from row index .
4. Periodic DRT
5. Results and Discussion
5.1. Time Measurements and Quality of Reconstructions
5.2. Comparison of Central and Periodic DRT
5.3. Adjoint Operators of Central and Periodic DRT
5.4. Convergence of Inversion Algorithm
6. Example of Central DRT Application
Implementation on Mobile Devices
7. Example of Periodic DRT Application
8. Conclusions
- A fast inversion algorithm for central DRT.
- The manner to operate on non-square domains.
- An algorithm to simultaneously compute more than one quadrant at a time.
- The efficient implementation of the whole Curvelet pair of transforms on smart-phones.
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
ARGB | Alpha Red Green Blue |
CPU | Central Processing Unit |
DRT | Discrete Radon Transform |
FFT | Fast Fourier Transform |
FFTW | Fast Fourier Tranform in the West [31] |
FMT | Fast Mojette Transform |
fps | Frames per Second |
GPU | Graphical Processing Unit |
SFF | Shape From Focus |
SIMD | Single Instruction Multiple Data |
SoC | System on Chip |
PPFRT | Pseudo-Polar Fourier Radon Transform |
Appendix A
Algorithm A1 Compute the central DRT of a quadrant |
Input: Image f(x, y), consisting of data. Output: Central Radon transform of f, (s, d), consisting of data.
|
Algorithm A2 Compute the periodic DRT of a quadrant |
Input: Image f(x, y), consisting of data. Output: Periodic Radon transform of f, (s, d), consisting of data.
|
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Central DRT Times Model & SoC | Resolution | Time in ms |
---|---|---|
Snapdragon™ 821 (Adreno™ 530) | 41.9 | |
14.2 | ||
Snapdragon™ 835 (Adreno™ 540) | 35.4 | |
12.3 | ||
Snapdragon™ 845 (Adreno™ 630) | 25.4 | |
5.9 |
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Gómez-Cárdenes, Ó.; Marichal-Hernández, J.G.; Phillip Lüke, J.; Rodríguez-Ramos, J.M. Central and Periodic Multi-Scale Discrete Radon Transforms. Appl. Sci. 2021, 11, 10606. https://doi.org/10.3390/app112210606
Gómez-Cárdenes Ó, Marichal-Hernández JG, Phillip Lüke J, Rodríguez-Ramos JM. Central and Periodic Multi-Scale Discrete Radon Transforms. Applied Sciences. 2021; 11(22):10606. https://doi.org/10.3390/app112210606
Chicago/Turabian StyleGómez-Cárdenes, Óscar, José G. Marichal-Hernández, Jonas Phillip Lüke, and José M. Rodríguez-Ramos. 2021. "Central and Periodic Multi-Scale Discrete Radon Transforms" Applied Sciences 11, no. 22: 10606. https://doi.org/10.3390/app112210606