Recovering Wavelet Coefficients from Binary Samples Using Fast Transforms

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