Signal recovery and wavelet reproducing kernels

A class of signal recovery problems in which the degrading operator is linear shift-invariant and of low-pass filter type is studied. This low-pass filter, H, can be viewed as a scaling filter for which there exists an associated high-pass filter, G. H and G correspond to a discrete wavelet transform (if H is regular) or an octave-band filter bank transform (if H is not regular). The signal recovery problem can be reformulated as finding missing data at the finest scale. The wavelet reproducing equation then plays a fundamental role in determining a unique and stable recovery. It is shown that this approach is closely related to unconstrained and constrained least-squares techniques used in signal recovery. From the regularization point of view, a midband consistency function subject to a smoothness constraint is minimized. G arises naturally as a regularizing operator