Discrete multi-dimensional linear transforms over arbitrarily shaped supports

Inorder to apply a multi-dimensional linear transform over an arbitrarily shaped support, the usual practice is to ll out the support to a hypercube by zero padding. This does not however yield a satisfactory de nition for transforms in two or more dimensions. The problem that we tackle is: how do we rede ne the transform over an arbitrary shaped region suited to a given application? We present a novel iterative approach to de ne any multi-dimensional linear transform over an arbitrary shape given that we know its de nition over a hyper-cube. The proposed solution is (1) extensible to all possible shapes of support (whether connected or unconnected) (2) adaptable to the needs of a particular application. We also present results for the Fourier Transform, for a speci c adaptation of the general de nition of the transform which is suitable for compression or segmentation algorithms.