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arXiv:2107.05524

Quantum Radon Transform and Its Application

Guangsheng Ma, Hongbo Li, Jiman Zhao

Published Jul 12, 2021• 0 citations
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Abstract

This paper extends the Radon transform, a classical image processing tool for fast tomography and denoising, to the quantum computing platform. A new kind of periodic discrete Radon transform (PDRT), called quantum Radon transform (QRT), is proposed. The QRT has a quantum implementation that is exponentially faster than the classical Radon transform. Based on the QRT, we design an efficient quantum image denoising algorithm. The simulation results show that QRT preserves the good denoising capability as in the classical PDRT. Also, a quantum algorithm for interpolation-based discrete Radon transform (IDRT) is proposed, which can be used for fast line detection. Both the quantum extension of IDRT and the line detection algorithm can provide polynomial speedups over the classical counterparts.

DOI
10.48550/arxiv.2107.05524

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This paper extends the Radon transform, a classical image processing tool for fast tomography and denoising, to the quantum computing platform. A new kind of periodic discrete Radon transform (PDRT), called quantum Radon transform (QRT), is proposed. The QRT has a quantum implementation that is exponentially faster than the classical Radon transform. Based on the QRT, we design an efficient quantum image denoising algorithm. The simulation results show that QRT preserves the good denoising capability as in the classical PDRT. Also, a quantum algorithm for interpolation-based discrete Radon transform (IDRT) is proposed, which can be used for fast line detection. Both the quantum extension of IDRT and the line detection algorithm can provide polynomial speedups over the classical counterparts.
This article extends the Radon transform, a classical image-processing tool for fast tomography and denoising, to the quantum computing platform. A new kind of periodic discrete Radon transform (PDRT), called the quantum periodic discrete Radon transform (QPRT), is proposed. The quantum implementation of QPRT based on the amplitude encoding method is exponentially faster than the classical PDRT. We design an efficient quantum image denoising algorithm using QPRT. The simulation results show that QPRT preserves good denoising capability as in the classical PDRT. Also, a quantum algorithm for IDRT is proposed, which can be used for fast line detection. Both the quantum extension of IDRT and the line detection algorithm can provide polynomial speedups over the classical counterparts in certain cases.
The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length NNN, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any ddd-dimensional Hilbert transform in depth O(dlog⁡N)O(d\log N)O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
The Radon transform constitutes the conventional tool for tomosynthesis, i.e., the composition of cross-sections of an object from its projections. It is actually a version of the Fourier Transform, which is accompanied by the appropriate digital high pass filters for correct distribution of energy among the reconstructed frequency components. The Radon transform and its inverse are employed in their 2D and 3D versions, respectively, and the whole procedure is verified by the a priori known cross-sections to be reconstructed (known fandom). Usually, 3D medical image cubes, which are to be reconstructed, require powerful computational tools since the 2D projections are of high-resolution containing millions of pixels. Although the 3D FFT is very fast, the large number of projections will result in a 3D spectrum of very large dimensions. Inverting this spectrum with the inverse 3D FFT is extremely time consuming. In this work, the implementation of the 2D Radon transform using the 2D Quantum Fourier Transform is analytically presented. Simultaneously, its inverse version is realized by means of the Quantum inverse 3D FFT. For this purpose, a review of the necessary quantum computational units is presented for the implementation of the quantum 3D FFT and simultaneously simple examples of tomosynthesis are given by means of the quantum version of the 2D Radon transform and its inverse 3D counterpart. The whole procedure of the quantum tomosynthesis is analytically described.
Quantum computing is an emerging field of technology that utilizes unique properties such as the superposition principle and quantum entanglement, potentially enabling the development of techniques that are drastically faster or even the only feasible solutions compared to classical approaches. Thus, it promises enormous gains in several domains, including the processing of n-dimensional images, such as those produced by X-ray computed tomography. However, the latter is subject to physical effects that primarily include artifacts and quantum noise. Noise, in particular, is an unavoidable issue that needs to be addressed when processing X-ray data. Therefore, this work considers the denoising of 3D volumetric data using quantum computing. An adaptation of a mathematical denoising model was transformed to be suitable for quantum implementation, and preliminary experiments demonstrated its functionality with two-dimensional images. Here, we first lift this model to be applicable to 3D images. Next, we provide an implementation in form of quantum circuits and consider issues of a practical implementation occurring specifically in 3D. The final implementation is executed on a real X-ray computed tomography dataset, showing that proper denoising can be performed on quantum devices, yet current technological limitations inhibit the application to large datasets.
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Fast classical and quantum algorithms are introduced for a wide class of nonseparable nD discrete unitary K-transforms (DKT) KNn {\cal K}_{N^n } KNn​ .