# Central and Periodic Multi-Scale Discrete Radon Transforms
## 8 Conclusions  
In this work, two new variations to discrete Radon transform, with a multi-scale approach, have been introduced. Both reduce the number of computations of conventional DRT, almost by half. Depending on the deployment platform, this translates into a halving of execution times, compared to the fastest alternative available, to the best of our knowledge. The methods benefit greatly when ported to parallel languages and GPUs. The central DRT is of interest as a substitute to conventional DRT whenever a forward Radon transform is required, but not its inverse. The other useful condition is that, for the field of application, lines at the corners can mostly be ignored. The usefulness of periodic DRT may seem reduced, but it is exactly what was required, in order to accelerate the computation of the discrete Curvelet transform. Figure 21: From left to right: detailed of images, recovered with Curvelet inverse transform after thresholding at 100%, 85%, and 75% of the accumulated magnitude of coefficients, obtained from Curvelet forward transform, which employs periodic DRT as founding tool for computing ridgelets. 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. The search for deepening the knowledge of discrete Radon transforms will be continued in the future. Some of the objectives that remain open are:  
* 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 smartphones. The following are available online at [[https://www.mdpi.com/article/10.3390/app112210606/s1](https://www.mdpi.com/article/10.3390/app112210606/s1)]([https://www.mdpi.com/article/10.3390/app112210606/s1](https://www.mdpi.com/article/10.3390/app112210606/s1)): Video S1: A screen recording of a demonstration mobile phone application that uses the proposed central DRT to detect, in real-time, the most prominent rectangle seen by the device's camera. Conceptualization, [PERSON] and [PERSON]; methodology, [PERSON] and [PERSON]; software, [PERSON], [PERSON], and [PERSON]; validation, [PERSON]; formal analysis, [PERSON] and [PERSON]; visualization, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON], [PERSON], and [PERSON]; project administration, [PERSON]. All authors have read and agreed to the published version of the manuscript. This research was partially funded by 'Estrategia de Especializacion inteligente de Canarias RIS-3', Government of the Canary Islands, and European Regional Development Fund, under grant number ProID2020010066, as well as \"Promotion of open dissemination of research excellence results\", University of La Laguna, and the Council of Economy, Knowledge and Employment of the Government of the Canary Islands. [PERSON]-H. was partially supported by Wooptix S.L., through 'Convenio de colaboracion Wooptix-Universidad de la Laguna, 2019-2020'. Not applicable. Data Availability Statement: Data sharing is not applicable to this article. The images in the experiments are the result of applying the provided algorithms, and are available on request from the corresponding author. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. The following abbreviations are used in this manuscript: