# Efficient Fourier base fitting on masked or incomplete structured data

## TL;DR

This paper introduces a fast method for Fourier base fitting on incomplete data, useful for biomedical imaging and diagnostics.

## Contribution

An efficient Fourier base fitting method for masked or incomplete data with significant computational optimizations.

## Key findings

- Peak reconstruction errors in masked regions remained below 10% of the data range.
- Matrix assembly time was reduced by 75-fold in 3D cases using optimizations.
- SVD can be used for regularization but becomes computationally expensive with more Fourier modes.

## Abstract

Fourier base fitting for masked or incomplete structured data holds significant importance, for example in biomedical image data processing. However, data incompleteness destroys the simple unitary form of the Fourier transformation, necessitating the construction and solving of a linear system—a task that can suffer from poor conditioning and be computationally expensive. Despite its importance, suitable methodology addressing this challenge is not readily available.

In this study, we propose an efficient and fast Fourier base fitting method suitable for handling masked or incomplete structured data. The developed method can be used for processing multi-dimensional data, including smoothing and intra-/extrapolation, even when confronted with missing data.

The developed method was verified using 1D, 2D, and 3D benchmarks. Its application is demonstrated in the reconstruction of noisy and partially unreliable brain pulsation data in the context of the development of a biomarker for non-invasive craniospinal compliance monitoring and neurological disease diagnostics.

The study investigated the impact of different analytical and numerical performance improvement measures (e.g., term rearrangement, precomputation of recurring functions, vectorization) on computational complexity and speed. Quantitative evaluations on these benchmarks demonstrated that peak reconstruction errors in masked regions remained acceptable (i.e., below 10 % of the data range for all investigated benchmarks), while the proposed computational optimizations reduced matrix assembly time from 843 s to 11 s in 3D cases, demonstrating a 75-fold speed-up compared to unoptimized implementations. Singular value decomposition (SVD) can optionally be employed as part of the solving-step to provide regularization when needed. However, SVD quickly becomes the performance limiting in terms of computational complexity and resource cost, as the number of considered Fourier modes increases.