LOCALIZED MEASUREMENT OF EMERGENT IMAGE FREQUENCIES BY GABOR WAVELETS

The measurement of instantaneous or locally-occurring signal frequencies is focal to a wide variety of variably-dimensioned signal processing applications. The task is particularly well-motivated for analyzing globally nonstationary, locally coherent signals having a significant fraction of their local spectral content concentrated over a narrow frequency range varying smoothly over time or space. The locally emergent frequencies characterizing such signals are manifested as, e.g., flow-like, granular, or oriented patterns in optical images, pure tonal components in music or whale song, and quasi-regular structures in three-dimensional micrographs of optical density. Measuring emergent signal frequencies requires spectral measurements accurate in both frequency and time or space, conflicting requirements that are shown to be balanced by a generalized uncertainty relationship. Such spectral measurements can be obtained from the responses of multiple wavelet-like channel filters that sample the signal spectrum, and that yield a locus of possible solutions for each locally emergent frequency. It is shown analytically that this locus of solutions is maximally localized in both space and frequency if the channel filters used are Gabor wavelets. A constrained solution is obtained by imposing a stabilizing term that develops naturally from the assumptions on the signal. The measurement of frequencies is then cast as an ill-posed extremum problem regularized by the stabilizing term, leading to an iterative constraint propagation algorithm. As a by-product, a measurement of the local image contrast is also obtained. The technique is demonstrated by application to a variety of two-dimensional textured images. A 3-D surface orientation-from-texture procedure further illustrates the method.