Complexity of Radon transforms

TL;DR

This paper investigates the inherent complexity of Radon transforms in image reconstruction, analyzing Fourier slice theorem implications, and proposes methods to generate and utilize this complexity for improved reconstruction accuracy.

Contribution

It introduces new insights into the complexity of Radon transforms, analyzes singular points via Fourier slice theorem, and proposes methods including hybrid functions to enhance reconstruction processes.

Findings

Complexity of Radon transforms affects inverse reconstruction accuracy.

Analysis of singular points reveals sources of transform complexity.

Hybrid functions naturally introduce necessary complexity for better results.

Abstract

For the reconstruction problem, the universal representation of inverse Radon transforms implies the needed complexity of the direct Radon transforms which leads to the additional contributions. In the standard theory of generalized functions, if the outset (origin) function which generates the Radon image is a pure-real function, as a rule, the complexity of Radon transforms becomes in question. In the paper, we discuss the Fourier slice theorem analyzing the degenerated (singular) points as possible sources of the complexity. We also demonstrate the different methods to generate the needed complexity on the intermediate stage of calculations. Besides, we show that the introduction of the hybrid (Wigner-like) function ensures naturally the corresponding complexity. The discussed complexity provides not only the additional contribution to the inverse Radon transforms, but also it makes an essential impact on the reconstruction and optimization procedures within the frame of the incorrect problems. The presented methods can be effectively used for the practical tasks of reconstruction problems.