Recovering wavelet coefficients from binary samples using fast transforms

TL;DR

This paper introduces a fast, memory-efficient algorithm for computing the change-of-basis matrix between Walsh-Hadamard samples and wavelet bases, significantly accelerating wavelet coefficient reconstruction in medical imaging.

Contribution

The authors develop an algorithm that reduces computational complexity from O(NM) to O(N log N) and storage from O(NM) to O(2^q), enabling faster wavelet coefficient recovery from binary samples.

Findings

Computes matrix-vector products in O(N log N) operations.

Reduces storage requirements to O(2^q).

Speeds up wavelet coefficient reconstruction in practice.

Abstract

Recovering a signal (function) from finitely many binary or Fourier samples is one of the core problems in modern medical imaging, and by now there exist a plethora of methods for recovering a signal from such samples. Examples of methods, which can utilise wavelet reconstruction, include generalised sampling, infinite-dimensional compressive sensing, the parameterised-background data-weak (PBDW) method etc. However, for any of these methods to be applied in practice, accurate and fast modelling of an $N \times M$ section of the infinite-dimensional change-of-basis matrix between the sampling basis (Fourier or Walsh-Hadamard samples) and the wavelet reconstruction basis is paramount. In this work, we derive an algorithm, which bypasses the $NM$ storage requirement and the $\mathcal{O}(NM)$ computational cost of matrix-vector multiplication with this matrix when using Walsh-Hadamard samples and wavelet reconstruction. The proposed algorithm computes the matrix-vector multiplication in $\mathcal{O}(N\log N)$ operations and has a storage requirement of $\mathcal{O}(2^q)$, where $N=2^{dq} M$, (usually $q \in \{1,2\}$) and $d=1,2$ is the dimension. As matrix-vector multiplications is the computational bottleneck for iterative algorithms used by the mentioned reconstruction methods, the proposed algorithm speeds up the reconstruction of wavelet coefficients from Walsh-Hadamard samples considerably.