On the Reconstruction of Wavelet-Sparse Signals From Partial Fourier Information

The problem of reconstructing a wavelet-sparse signal from its partial Fourier information has received a lot of attention since the emergence of compressive sensing (CS). The latest theory within the CS framework analyzes the local coherence between the Fourier and wavelet bases, and recover the signal from frequencies randomly selected according to a variable density profile. Unlike these developments, we adopt a new approach that does not need to analyze the (local) coherence. We show that the problem can be tackled by recovering the wavelet coefficients from the finest to the coarse scale, and only a small set of frequencies are needed to recover the coefficients exactly. As long as the scaling function satisfies a mild condition, the reconstruction is exact. Moreover the frequency set can be deterministically pre-selected and does not need to change even if the wavelet basis changes.