Optimally Localized Wavelets and Smoothing Kernels

It is well-known that the Gaussian functions and, more generally, their modulations-translations (the Gabor functions) have the unique property of being optimally localized in space and frequency in the sense of Heisenberg’s uncertainty principle. In this thesis, we address the construction of complex wavelets modeled on the Gabor function, and smoothing kernels based on the Gaussian. We proceed by relaxing the exact form of the Gaussian and Gabor function, and by approximating them using spline functions. In particular, we construct a family of spline wavelets, termed Gabor-like wavelets, which provide arbitrary close approximations of the Gabor function. On the other hand, we introduce a family of compactly supported box splines to approximate the Gaussian, both isotropic and anisotropic. The attractive feature of these spline wavelets and kernels is that we are able to develop fast and efficient algorithms for implementing the associated transforms. The Gabor-like wavelet is obtained within the framework of multiresolution analysis by combining Hilbert transform pairs of B-spline wavelets. To begin with, we provide a rigorous understanding of why the Hilbert transform goes well with wavelets. We show that at the heart of this is the characteristic vanishing-moment property of wavelets and certain fundamental invariances of the Hilbert transform. The former allows us to ensure that the Hilbert transform (which is non-local) of a localized wavelet is again well-localized provided that it has sufficient number of vanishing moments, while the latter allows us to seamlessly integrate it into the multiresolution framework of wavelets. Guided by these facts, we formulate a general recipe for constructing a pair of wavelet bases that form a Hilbert transform pair. Using this recipe, we are able to identify a pair of B-spline wavelets that are related through the Hilbert transform. We show that the complex wavelet derived from this pair converges to a Gabor function as its order gets large. We next extend the construction to higher dimensions using the directional Hilbert transform and tensor-products wavelets. This results in a system of complex wavelets that closely resemble the directional Gabor functions. We develop an efficient numerical algorithm for implementing the associated complex wavelet transforms on finite periodic data. We next identify the complete family of transforms which share the fundamental invariances of the Hilbert transform. Based on this family of transforms and its particular properties, we are able to provide an amplitude-phase interpretation of the signal representation associated with the Gabor-like wavelet transform. This allows us to understand the significance of the amplitude and phase information associated with the transform. As an application, we develop a coarse-to-fine stereo-matching algorithm that does dynamic programming on the sub-sampled Gabor-like wavelet pyramid instead of the raw pixel intensities. The crucial feature of our pyramid was that it provides near translationinvariance at the cost of moderate redundancy. The translation-invariance is absolutely essential for encoding the local spatial translations between the stereo pair. Based on the particular Gabor-like form of our wavelets, we also provided a mathematical explanation