A recursive approach to Multiscalar Data Interpolation of sparsely sampled sea surface measurements at different spatial resolutions

In many oceanographic studies there is a need to reconstruct a signal from a set of sparse measurements. 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 when there is an approximation to a model for the multiresolution decomposition/reconstruction scheme of the dataset. Although the problem is ill-posed, this approach gives an easy scheme to interpolate the values of a signal using all the information available at different resolutions. This reconstruction method could be used as an extension of any interpolation method to optimize the multiresolution sparse data fusion. A simplified one-dimensional case illustrates the explanation; it is an algorithm based on a recursive scheme of a fast dyadic wavelet transform and its inversion, using a filter 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 different instruments that are sensing the same scene simultaneously with several resolutions. Extensions of the method to sparse multiresolution dataset with higher dimensions (images or vector fields) also offer some promising preliminary results.