Abstract

Wavelets that are localized both in space and in spatial frequency are useful for vision modeling and image processing. For example, in image compression one would like localization in spatial frequency so that the small spatial filters are insensitive to low spatial frequencies to which the visual system is highly sensitive. One would also like spatially localized filters to reduce interference between adjacent features. Several investigators, including Gabor, asked what particular wavelet shape minimizes the joint space-spatial frequency uncertainty (the Heisenberg uncertainty). Consider the class of functions that are an nth-order polynomial times a Gaussian. Gabor proved that the nth-order Hermite polynomial produced the extremum of the Heisenberg uncertainty. He believed that the extremum he found was the minimum. We found that Gabor was partly right and partly wrong. A Hermite polynomial when multiplied by a Gaussian is indeed an extremum of the Heisenberg uncertainty for the class functions that are an nth-order polynomial times a Gaussian. The problem is that it maximizes rather than minimizes the uncertainty.

© 1991 Optical Society of America