2D Fourier Transform on a non-rectangular space

[tanus5]

2D Fourier Transform on a non-rectangular area

Is it possible to perform a Fourier transform on a shape instead of a rectangular region? To be specific I am attempting to make a linear zoom function that doesn't produce any pixelation and that mimics natural blur that occurs with distance. After sketching the problem in Blender it seems that the best approach would be to utilize a frustum from an [STRIKE]ovular[/STRIKE] elliptical cone, generate a Fourier transform from an [STRIKE]oval-bound[/STRIKE] ellipse-bound section of the image, and extend the transform with additional null high frequencies to produce multiple pixels from a single pixel (or strip off high frequencies for zoom-out). Finally I would apply an inverse transform. This operation would be performed on a pixel-by-pixel basis for each pixel of the final image. As good as this may sound I'm not familiar with a Fourier transform on anything other than rectangular area. Can anyone suggest a good place to start?

Note: Computing power is not an issue for this problem as the system will be parellelized across 4096 cores at ~ 1mhz each for a total of nearly 4 teraflops of computing power using 2x 280X Radeon GPU's.

 

[tanus5]

After a good amount of Googling it seems that I'm looking for a variation of the EDFT algorithm (Elliptic Discrete Fourier Transform). I have not yet been able to find a comprehensive explanation of an EDFT algorithm. What I need is a comprehensive explanation of an elliptic Fourier transform and an example EDFT implementation. All I've found so far is general information that this algorithm isn't very popular which is probably why I haven't been able to find a good explanation of it. It appears that the actual EDFT loops around the perimeter and what I'm looking for is something that loops from the perimeter to the center. For use with other shapes other than the ellipse it seems that a 3D transform is needed where the third dimension is not spatial but a fuzzy (0-1) alpha value indicating the percentage of the pixel which contains the shape.