Local Image Analysis By Gabor Wavelets
Considerable recent attention has focused on multi-scale signal processing paradigms that entail decompositions into elementary signals, or wavelets, that embody local spatial and/or temporal structures. While much research has been dedicated towards understanding the mathematical properties of these representations and methods for computing/inverting them, considerable effort is also being applied to exploiting them in a wide variety of problems, e.g., in the analysis and coding of two-dimensional images. spectrally localized basis. Measuring locally-occurring signal frequencies is focal in many applications, particularly those involving nonstationary coherent signals, where a significant fraction of the local signal spectrum is concentrated over a narrow frequency range varying smoothly over 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 image frequencies requires spectral measurements accurate in both frequency and space, conflicting requirements balanced by a generalized uncertainty relationship. Such spectral measurements can be obtained from the responses of multiple wavelet-like channel filters that sample the image 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, in the sense of a generalized p-uncertainty, when the channel filters are Gabor wavelets. A constrained solution is obtained by imposing a stabilizing term which develops naturally from the image model. 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. The efficacy of the technique is demonstrated by application to a variety of image processing tasks, including image texture segmentation, flow computation, local measurement of fractal image dimension, and the measurement of three-dimensional object shape from projected surface texture. We consider the restricted problem of measuring image frequencies on a spatio-