A model-based expansion on interpolation for multiresolution sparse data

Abstract

This paper addresses the interpolation of sparse irregular data when these sparse data belong to diff erent scales. We propose an algorithm to iteratively approximate the intermediate values between irregularly sampled data, when a set of sparse values at coarser scales is known. This is possible if there is a characterized model for the multiresolution decomposition / reconstruction scheme of the dataset. Although the problem is ill-posed, and there are infi nite solutions, this approach gives an easy scheme to interpolate the values of a signal using all the information available at diff erent scales. This reconstruction method could be used as an extension on any interpolation. A simplifi ed one-dimensional case illustrates the explanation; the scheme is based on a fast dyadic wavelet transform and its inversion, using a fi lter bank analysis/synthesis implementation for the wavelet transforms model. This can be a basis method suitable for applied cases where there are sparse measures from diff erent instruments that are sensing the same scene simultaneously with several resolutions. Extensions of the method to sparse multiresolution data with higher dimensions (images or vector fi elds) also off er some promising preliminary results.