Discrete multi-dimensional linear transforms over arbitrarily shaped supports

In order to apply a multi-dimensional linear transform, over an arbitrarily shaped support, the usual practice is to fill out the support to a hypercube by zero padding. This does not however yield a satisfactory definition for transforms in two or more dimensions. The problem that we tackle is: how do we redefine the transform over an arbitrary shaped region suited to a given application? We present a novel iterative approach to define any multi-dimensional linear transform over an arbitrary shape given that we know its definition over a hypercube. 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 specific adaptation of the general definition of the transform which is suitable for compression or segmentation algorithms