Image processing and low discrepancy sequences

It is well known that high-dimensional integrals can be solved with Monte Carlo algorithms. Recently, it was discovered that there is a relationship between low discrepancy sets and the efficient evaluation of higher-dimensional integrals. Theory suggests that for midsize dimensional problems, algorithms based on low discrepancy sets should out perform all other existing methods by an order of magnitude in terms of the number of sample points used to evaluate the integrals. We show that the field of image processing can potentially take advantage of specific properties of low discrepancy sets. To illustrate this, we applied the theory of low discrepancy sequences to some relatively simple image processing and computer vision related operations such as the estimation of gray level image statistics, fast location of objects in a binary image and the reconstruction of images from a sparse set of points. Our experiments show that compared to standard methods, the proposed new algorithms are faster and statistically more robust. Classical low discrepancy sets based on the Halton and Sobol' sequences were investigated thoroughly and showed promising results. The use of low discrepancy sequences in image processing for image characterization, understanding and object recognition is a novel and promising area for further investigation.