Improving Inverse Wavelet Transform by Compressive Sensing Decoding with Deconvolution

In this paper we propose an alternative decoding method for inverse wavelet transform when only partial coefficients are available. We have been inspired by the recently developed compressive sensing (CS) decoding, which is capable in recovering sparse signals from a few linear and non-adaptive measurements. Let x be a sparse signal with N entries and only K out of them are non-zero, and y be its approximation coefficients. Classic CS decoding such as l1-minimization can be applied to decode x from y, and it indeed provides better reconstruction of sparse signals than direct inverse transform, as demonstrated by our simulation results. When coefficients have been quantized, the performance of CS decoding decreases more severely compared with direct inverse transform, but still better than the latter once the signal is sparse enough.