Recovering wavelet coefficients from binary samples using fast transforms
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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 × 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 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 𝒪(Nlog N) operations and has a storage requirement of 𝒪(2^q), where N=2^dq M, (usually q ∈{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.
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