Browsing by Author "Wells, Raymond O., Jr."

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    Extension theorems for solutions to over-determined systems of partial differential equations
    (1966) Krueger, Joe Elgin; Wells, Raymond O., Jr.
    Ehrenpreis has shown that, under suitable conditions, an infinitely differentiable solution to an overdetermined system of linear partial differentiable operators with constant coefficients has an extension across the bounded part of the complement of an annular region. It it shown in this thesis that a similar result holds for distribution solutions. Necessary and sufficient conditions for such extension are discussed and several theorems are also given on the extension problem for solutions to one differential operator.
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    Image processing via undecimated wavelet systems
    (2000) Zhang, Huipin; Wells, Raymond O., Jr.
    We have studied undecimated wavelet transforms and their applications in image denoising. Because of the redundancy of the undecimated wavelet transform, the inversion scheme which implements the Moore-Penrose inverse of the forward transform makes undecimated wavelet systems have excellent performance in signal denoising. We propose an image denoising algorithm that prunes the complete undecimated discrete wavelet packet binary tree to select the best basis. Since we believe discarding the small coefficients permits to choose the best basis from the set of coefficients that will really contribute to the reconstructed image, we propose to select the best basis based on the thresholded wavelet coefficients rather than the original ones. We also propose an exponential decay model for autocorrelations of undecimated wavelet coefficients of real-world images. This is a model that captures the dependency of wavelet coefficients within a scale. With this model we present a parametric solution for FIR Wiener filtering in the undecimated wavelet domain. The persistence property of wavelet coefficients indicates strong dependency across scales. To capture the persistence of UDWT we propose an extension of the wavelet-domain hidden Markov tree model (HMT). By introducing the concept of composite coefficient, we simplify the general coefficient graph to be a tree-structure graph which is very suitable for training to obtain the HMT model parameters. This Bayesian framework allows us to formulate the image denoising problem as computing the posterior estimate.
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    Mathematical modeling of seismic data: Multiscale noise removal and recognition algorithm
    (2000) Wang, Yuan; Wells, Raymond O., Jr.
    In this thesis, we present a novel artifact removal algorithm based on a seismic acquisition artifact data model, in which the artifact pattern is postulated to be additive to the actual geological signals. The algorithm can effectively remove artifacts that are highly correlated with the postulated pattern without affecting the spatial resolution. The main assumption of this algorithm is that the artifact is orthogonal to the underlying signal at certain scales of its DWT coefficients. The underlying signal is modeled as a sum of a short range dependent random noise and a deterministic signal. We prove that when the number of sampling points goes to infinity, the underlying signal and the artifact are asymptotically orthogonal to each other at a certain fine scale of its DWT coefficients. We name this algorithm as Wavelet Statistical Orthogonal Noise Reduction Scheme. We prove that with the WSO noise reduction scheme and for this noise model, when the number of sampling points goes to infinity, the average error between the evaluated artifact and the real artifact will go to zero and the probability of any non zero relative error between the estimated underlying signal and the real underlying signal will go to zero. Therefore, we know that the bias introduced from the orthogonality assumption will be reduced to zero as the number of sampling points goes to infinity. We use numerical experiments to verify that the above assumption is statistically true especially for large data sets, and we also use both synthetic and real seismic data sets to demonstrate the performance of this algorithm.
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    The mathematical theory and applications of biorthogonal Coifman wavelet systems
    (1996) Tian, Jun; Wells, Raymond O., Jr.
    In this thesis, we present a theoretical study of biorthogonal Coifman wavelet systems, a family of biorthogonal wavelet systems with vanishing moments equally distributed between scaling functions and wavelet functions. One key property of these wavelet systems is that they provide nice wavelet sampling approximation with exponential decay. Moreover they are compactly supported, symmetric, have growing smoothness with large degrees, and converge to the sinc wavelet system. Using a time domain design method, the exact formulas of the coefficients of biorthogonal Coifman wavelet systems of all degrees are obtained. An attractive feature behind it is that all the coefficients are dyadic rational, which means that we can implement a very fast multiplication-free discrete wavelet transform, which consists of only addition and shift operations, on digital computers. The transform coding performance of biorthogonal Coifman wavelet systems is quite comparable to other widely used wavelet systems. The orthogonal counterparts, orthogonal Coifman wavelet systems, are also discussed in this thesis. In addition we develop a new wavelet-based embedded image coding algorithm, the Wavelet-Difference-Reduction algorithm. Unlike zerotree type schemes which use spatial orientation tree structures to implicitly locate the significant wavelet transform coefficients, this new algorithm is a direct approach to find the positions of significant coefficients. It combines the discrete wavelet transform, differential coding, binary reduction, ordered bit plane transmission, and adaptive arithmetic coding. The encoding can be stopped at any point, which allows a target rate or distortion metric to be met exactly; the decoder can also terminate the decoding at any point, and produce a corresponding reconstruction image. Our algorithm provides a fully embedded code to successively approximate the original image source; thus it's well suited for progressive image transmission. It is very simple in its form (which will make the encoding and decoding very fast), and has a clear geometric structure, which enables us to process the image data in the compressed wavelet domain. The image coding results of it are quite competitive with almost all previous reported image compression algorithms on standard test images.