Fourier Volume Rendering

Often, 3D objects are defined by a 3D cloud of data points, each point with its own density. Take, for example, a magnetic resonance image (MRI) of a person\'s head. The output is a cube of numbers indicating the density at each point. It takes a lot of processing to project this 3D volume data onto a 2D screen. Each data point has to be looked at and projected separately. Fourier volume rendering speeds up the rendering for a particular kind of projection, the Radon transform. This projection looks like an X-ray. The speedup comes from the Fast Fourier Transform (FFT). The Radon transform and the Fourier transform are good friends, and happen to be linked. The Radon transform of a volume can be efficiently extracted from the Fourier transform of the volume. The result is a change in algorithmic complexity from O(N^3) to O(N^2 logN). This technique was invented and applied to regular (rectangular) grid volume data some time ago. We hope to extend it to non-regular grids.