Sampling and Approximation of Bandlimited Volumetric Data
This is an incremental theoretical contribution for signal processing and scientific computing, providing a method for approximating 3D functions from grid samples.
The paper presents an approximation scheme for bandlimited volumetric data using generalized prolate spheroidal wavefunctions, with a truncation rule and error bound, but no concrete numerical results are provided.
We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is based on expanding the given function in the basis of generalized prolate spheroidal wavefunctions, with the expansion coefficients given by weighted dot products between the samples of the function and the samples of the basis functions. As numerical implementations require all expansions to be finite, we present a truncation rule for the expansions. Finally, we derive a bound on the overall approximation error in terms of the assumed space/frequency concentration.