# Inverse Multiscale Discrete Radon Transform by Filtered Backprojection

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## Abstract

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## Featured Application

**The startup company Wooptix is using the inversion by filtered backprojection of the multiscale Radon transform to accelerate inverse problem solutions in the context of wavefront phase optical measurements and barcode detection.**

## Abstract

## 1. Introduction

#### 1.1. Multiscale DRT

**u**(x if what we are considering are the horizontal lines), is now split in two, since we are transforming bits that go from being considered ‘vertical bands’,

**v**, to being considered ‘slopes’, s. See Figure 1.

**v**is basically the forward DRT algorithm. It can be seen that it is performing just two sums to compute a new datum. The described outer loops account for the computational complexity: $O({N}^{2}logN)$.

#### 1.2. Inverse DRT

#### 1.3. Initial Idea

## 2. Proposed Method

#### 2.1. Extending Backprojection

#### 2.2. Characterization of the Impulse Responses

#### 2.2.1. Time Variant Impulse Responses

#### 2.3. Deconvolving Different Impulse Responses

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DRT | Discrete Radon transform |

PSNR | Peak Signal to Noise Ratio |

LTI | Linear time-invariant |

## References

- Radon, J. Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Akad. Wiss.
**1917**, 69, 262–277. [Google Scholar] - Kak, A.C.; Slaney, M. Principles of Computerized Tomographic Imaging; SIAM: University City, PA, USA, 2001. [Google Scholar]
- Bracewell, R. Strip Integration in Radio Astronomy. Aust. J. Phys.
**1956**, 9, 198–271. [Google Scholar] [CrossRef][Green Version] - Götz, W.; Druckmüller, H. A fast digital Radon transform—An efficient means for evaluating the Hough transform. Pattern Recognit.
**1996**, 29, 711–718. [Google Scholar] [CrossRef] - Brady, M.L. A fast discrete approximation algorithm for the Radon transform. SIAM J. Comput.
**1998**, 27, 107–119. [Google Scholar] [CrossRef] - Brandt, A.; Dym, J. Fast calculation of multiple line integrals. SIAM J. Sci. Comput.
**1999**, 20, 1417–1429. [Google Scholar] [CrossRef][Green Version] - Press, W.H. Discrete Radon transform has an exact, fast inverse and generalizes to operations other than sums along lines. Proc. Natl. Acad. Sci. USA
**2006**, 103, 19249–19254. [Google Scholar] [CrossRef] [PubMed][Green Version] - Marichal-Hernandez, J.G.; Luke, J.P.; Rosa, F.L.; Rodriguez-Ramos, J.M. Fast approximate 4-D/3-D discrete Radon transform for lightfield refocusing. J. Electron. Imaging
**2012**, 21, 023026-1. [Google Scholar] [CrossRef] - Averbuch, A.; Coifman, R.R.; Donoho, D.L.; Israeli, M.; Shkolnisky, Y. A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform. SIAM J. Sci. Comput.
**2008**, 30, 764–784. [Google Scholar] [CrossRef][Green Version] - Averbuch, A.; Coifman, R.R.; Donoho, D.L.; Israeli, M.; Shkolnisky, Y.; Sedelnikov, I. A Framework for Discrete Integral Transformations II—The 2D Discrete Radon Transform. SIAM J. Sci. Comput.
**2008**, 30, 785–803. [Google Scholar] [CrossRef] - Averbuch, A.; Shkolnisky, Y. 3D Fourier based discrete Radon transform. Appl. Comput. Harmon. Anal.
**2003**, 15, 33–69. [Google Scholar] [CrossRef][Green Version] - Shkolnisky, Y. Pseudo-Polar Fourier Transform Software. Available online: https://sites.google.com/site/yoelshkolnisky/software (accessed on 5 November 2020).
- Ragan-Kelley, J.; Adams, A.; Sharlet, D.; Barnes, C.; Paris, S.; Levoy, M.; Amarasinghe, S.; Durand, F. Halide: Decoupling Algorithms from Schedules for High-Performance Image Processing. Commun. ACM
**2017**, 61, 106–115. [Google Scholar] [CrossRef] - Marichal-Hernandez, J.G.; Cárdenes, Ó.G.; González, F.L.R.; Kim, D.H.; Rodríguez-Ramos, J.M. Three-dimensional multiscale discrete Radon and John transforms. Opt. Eng.
**2020**, 59, 1–23. [Google Scholar] [CrossRef] - Lüke, J.; Almunia, J.; Rosa, F. Framework for developing prototype bioacustic devices in aid of open sea Killer Whale protection. Bioacoust. Int. J. Anim. Sound Its Rec. Bioacoust.
**2011**, 20, 287–296. [Google Scholar] [CrossRef]

**Figure 1.**Depiction of partial transforms of multiscale DRT. As the stage increases, the vertical bands decrease, and the slopes become more numerous.

**Figure 2.**Depiction of extension of backprojected lines to a domain three times larger than input image size. See text for details.

**Figure 3.**Depiction of how initial DRT values map to a 4 times bigger DRT. A few dashed lines on the left are represented on Radon slope-displacement space on the right. On green, initially contiguous values in slope with incrementally decreasing displacement, map to slopes at distance four and with double displacement increments, on a 4 times bigger DRT, on blue. Moreover, the blue originally odd slopes get an additional increase of 3 in slope, and 2 in displacement that will be discussed later.

**Figure 8.**

**Top**row, input image and conventional backprojection.

**Bottom**row, extended backprojection.

**Figure 9.**Extended backprojection from previous figure, filtered assuming there is a unique impulse response.

**Figure 10.**

**Left**, first iteration.

**Right**, second iteration of deconvolution using $64+64$ discrete impulse responses on a ${256}^{2}$ image.

**Figure 11.**Images reconstructed with fewer than $N/4$ impulse responses, $N=256$.

**Top**row, from left to right, reconstructions for four, eight and sixteen impulse responses per dimension.

**Bottom**row, from left to right, 32 and 64 ($N/4$) impulse responses. Then, the original picture to compare with. The obtained PSNR from lesser to greater number of different impulse responses considered are: [24.97, 27.36, 30.98, 32.96, 33.08] dB.

**Figure 12.**

**Top**row, 8 horizontal half-impulse responses on a $32\times 32$ problem.

**Middle**row, there are only 4 different half-impulse responses that approximate the original ones. They were obtained with k-means clustering. Bottom row, absolute difference between original and approximated half-impulse responses enhanced five times, to reveal the otherwise imperceptible error.

**Figure 14.**Execution times of the proposed method compared with the best multiscale DRT and Fourier-slice inversion alternatives.

**Figure 15.**Images recovered after applying gaussian distributed noise of $5\%$ of the magnitude of transformed coefficients. From

**top**to

**bottom**, and

**left**to

**right**: Press’ method, ours, Shkolnisky’s, original image.

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## Share and Cite

**MDPI and ACS Style**

Marichal-Hernández, J.G.; Oliva-García, R.; Gómez-Cárdenes, Ó.; Rodríguez-Méndez, I.; Rodríguez-Ramos, J.M.
Inverse Multiscale Discrete Radon Transform by Filtered Backprojection. *Appl. Sci.* **2021**, *11*, 22.
https://doi.org/10.3390/app11010022

**AMA Style**

Marichal-Hernández JG, Oliva-García R, Gómez-Cárdenes Ó, Rodríguez-Méndez I, Rodríguez-Ramos JM.
Inverse Multiscale Discrete Radon Transform by Filtered Backprojection. *Applied Sciences*. 2021; 11(1):22.
https://doi.org/10.3390/app11010022

**Chicago/Turabian Style**

Marichal-Hernández, José G., Ricardo Oliva-García, Óscar Gómez-Cárdenes, Iván Rodríguez-Méndez, and José M. Rodríguez-Ramos.
2021. "Inverse Multiscale Discrete Radon Transform by Filtered Backprojection" *Applied Sciences* 11, no. 1: 22.
https://doi.org/10.3390/app11010022