1. A transform coder for encoding a multi-dimensional signal with an arbitrarily-shaped domain, the transform coder being configured to perform at least the steps of: obtaining the multi-dimensional signal; performing a domain adaptive transform on the signal that involves applying a filter comprising a convolution operator which is configured to process pixels near a boundary of, but not in an interior of, the arbitrarily-shaped domain, the convolution operator resulting in a weighted sum of values contained in the multi-dimensional signal; quantizing the resultant decomposition coefficients; and encoding and transmitting the quantized values over an information channel to a decoder for reconstruction of an approximated signal, wherein the domain adaptive transform is applied such that points external to the arbitrarily-shaped domain but within support of the filter (or filters) are included in the result of the convolution but are further multiplied (or re-weighted) by a set of additional weighting factors.