Optimized filter functions for filtered back projection reconstructions

TL;DR

This paper presents an analytical method to optimize filter functions in filtered back projection for CT, improving reconstruction quality without needing training data, supported by theoretical derivations and numerical experiments.

Contribution

It derives a closed-form filter function that minimizes reconstruction error in FBP, applicable to both infinite and finite measurement scenarios, without requiring training datasets.

Findings

Optimized filter functions improve CT reconstruction quality.

The derived filters are simple, closed-form, and easy to implement.

Numerical experiments confirm the effectiveness of the proposed filters.

Abstract

The method of filtered back projection (FBP) is a widely used reconstruction technique in X-ray computerized tomography (CT), which is particularly important in clinical diagnostics. To reduce scanning times and radiation doses in medical CT settings, enhancing the reconstruction quality of the FBP method is essential. To this end, this paper focuses on analytically optimizing the applied filter function. We derive a formula for the filter that minimizes the expected squared $\mathrm{L}^2$-norm of the difference between the FBP reconstruction, given infinite noisy measurement data, and the true target function. Additionally, we adapt our derived filter to the case of finitely many measurements. The resulting filter functions have a closed-form representation, do not require a training dataset, and, thus, provide an easy-to-implement, out-of-the-box solution. Our theoretical findings are supported by numerical experiments based on both simulated and real CT data.