Optimally Localized Wavelets and Smoothing Kernels
K.N. Chaudhury
Swiss Federal Institute of Technology Lausanne, EPFL Thesis no. 4968 (2011), 239 p., February 16, 2011.
It is wellknown that the Gaussian functions and, more generally, their modulationstranslations (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 Gaborlike 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 Gaborlike wavelet is obtained within the framework of multiresolution analysis by combining Hilbert transform pairs of Bspline 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 vanishingmoment property of wavelets and certain fundamental invariances of the Hilbert transform. The former allows us to ensure that the Hilbert transform (which is nonlocal) of a localized wavelet is again welllocalized 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 Bspline 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 tensorproducts 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 amplitudephase interpretation of the signal representation associated with the Gaborlike 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 coarsetofine stereomatching algorithm that does dynamic programming on the subsampled Gaborlike 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 translationinvariance is absolutely essential for encoding the local spatial translations between the stereo pair. Based on the particular Gaborlike form of our wavelets, we also provided a mathematical explanation of the near translationinvariance of our pyramid. From a computational standpoint, we show that a significant reduction of the run time is achieved in comparison with the standard dynamic programming algorithm.
In the second half of the thesis, we introduce a particular bivariate box spline termed the radiallyuniform box spline. As an application of the Central Limit Theorem, we show that it converges to a Gaussian as its order gets large. For a fixed order, we show how the parameters of the box spline can be tuned to approximate a fixed anisotropic Gaussian. In particular, we develop a simple rootfinding algorithm for controlling the anisotropy of the elliptical box splines.
We next investigate the efficient realization of spacevariant (or nonconvolution) Gaussian filters using these box splines. The realization of even the simple convolution Gaussian filter is known to be computationally challenging, particularly when the size of Gaussian is large. The number of computations required per pixel for a direct implementation of the filter scales linearly with the size of the filter. We demonstrated that it is possible to filter an image with Gaussianlike box splines of varying size, elongation and orientation using a fixed number of computations per pixel (constanttime implementation). The associated algorithm is easy to implement and uses simple preintegrations and local finitedifferences.
As an application of the Gaussianlike box splines and the associated filtering algorithm, we develop two algorithms for spacevariant filtering. The first of these is inspired by anisotropic Gaussian diffusion. The spacevariance in this case is in terms of the size, elongation, and orientation of the box splines, which are controlled using the local image features. The other scheme is based on a spacevariant form of the Gaussian bilateral filter. The spatial adaptability in this case was in terms of the size of the spatial Gaussian filter. The highlight is that we are able to develop a constanttime algorithm for implementing the bilateral filter by approximating the variable spatial filter using isotropic box splines, and by approximating the fixed range filter (locally) using a class of shiftable kernels. We demonstrate their usage by developing smoothing algorithms for signaladaptive denoising of images. As an application in a different direction, we develop box spline filters resembling the LaplacianofGaussian. Using this particular detector, and by appropriately modifying the basic filtering algorithm, we develop a fast templatematching algorithm for the detection of bright cells and nuclei in fluorescence images.

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AUTHOR="Chaudhury, K.N.",
TITLE="Optimally Localized Wavelets and Smoothing Kernels",
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YEAR="2011",
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©
2011
K.N. Chaudhury.
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