Progressive wavelet correlation using Fourier methods

This paper derives a multiresolution analysis technique for performing correlations on wavelet representations of images. The technique maps the images into the wavelet-frequency domain to take advantage of high-speed correlation in the frequency domain. It builds on Vaidyanathan's (1993) wavelet correlation theorem, which shows that subsamples of correlations of two signals can be obtained from a sum of correlations of subbands of wavelet representations of those signals. Our algorithm produces the correlations at lowest resolution by applying the convolution theorem to subband correlations. A new multiresolution technique fills in the missing correlation data by incrementally inverting the wavelet transform and refining the Fourier transform. When applied to JPEG representations of data, the lowest resolution correlations can he performed directly on the JPEG images to produce 1/64th of the correlation points. Each of three incremental steps quadruple the number of correlation points, and the process can be halted at any point if the intermediate results indicate that the correlation will not find a match.