# Inverse Multiscale Discrete Radon Transform by Filtered Backprojection
## 4 Discussion  
Our method is compared in this section with two alternative algorithms that compute the inverse Radon transform. One represents Fourier-based methods that we will name after the author of the software package: [PERSON] (1996); [PERSON] (1996); [PERSON] (1996); another represents the only method currently available for inversion of the multiscale DRT: that due to Press Press (1996). These methods are compared with ours in terms of processing time, for the same quality target of 30 dB PSNR, and on the same machine, a PC with an intel 19-9900K processor running at 3.60 GHz, with implementations as optimal as possible. Performance was measured in the same conditions for all algorithms and averaged for several executions.  
Our method will be measured with the number of impulse responses \(K=N/16\). For the image of reference, Figure 11, the method from [PERSON] requires 6 iterations to get the same level of quality. [PERSON]'s method needs 5 iterations to obtain a similar PSNR.  
Our extended backprojection algorithm was implemented in Halide language ([PERSON], 1996), and the convolutions in the Fourier domain in OpenCV, compiled specifically to the selected processor using intrinsics functions. Convolutions were computed in parallel using the 8 cores-16 threads of that CPU. The code provided by [PERSON] (1996) is parallelized and uses effectively the 8 CPU cores. [PERSON] method was implemented by us in Python using Numpy to use optimized, parallel, matrix-vector functions, and Cython when needed, obtaining C compiled code to avoid memory copies in Python.  
Figure 14 shows the comparison between the different algorithms. For every image side size in the comparison, from 64 to 2048, our proposed method is systematically faster than the other two methods. The speedup is approximately \(3x\) in comparison with the  
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 13: Execution times for \(N=256\) using different number of impulse responses: 4 to 64.  
second fastest method. But depending on the size and method with which we confront it, the speedup can be more than ten times greater.  
Figure 15 shows how the different methods react when there is noise in the transformed domain. The recovery that our proposed method achieves coincides completely with [PERSON] inversion. Although visually there is a difference in contrast with [PERSON]'s method, the PSNR tell us the fidelity of reconstruction falls in the same proportion (from 30 dB to 15 dB) when we add a similar quantity of gaussian noise in the transformed domain.  
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**: [PERSON] method, ours, [PERSON]’s, original image.