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 be a sparse signal with
entries and only out of them are non-zero, and be its approximation coefficients.
Classic CS decoding such as ????-minimization can be applied to decode from , and it
indeed provides better reconstruction of sparse signals than direct inverse transform, as
demonstrated by our simulation results in Figure 1. 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.