$ l _{ p } $ -Multiresolution Analysis: How to Reduce Ringing and Sparsify the Error

We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the $l _{ p }$-sense (not restricted to the usual p=2), where p can take non-integer values. The underlying image model is specified using shift-invariant basis functions, such as B-splines. The solution is well-defined and determined by an iterative optimization algorithm based on digital filtering. Its convergence is accelerated by the use of first and second order derivatives. For p close to 1, we show that the ringing is reduced and that the histogram of the detail image is sparse as compared with the standard case, where p=2.