Tight wavelet frame construction and its application for image processing
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As a generalized wavelet function, a wavelet frame gives more rooms for different construction methods. In this dissertation, first we study two constructive methods for the locally supported tight wavelet frame for any given refinable function whose Laurent polynomial satisfies the QMF or the sub-QMF conditions in Rd. Those methods were introduced by Lai and St¨ockler. However, to apply the constructive method under the sub-QMF condition we need to factorize a nonnegative Laurent polynomial in the multivariate setting into an expression of a finite square sums of Laurent polynomials. We find an explicit finite square sum of Laurent polynomials that expresses the nonnegative Laurent polynomial associated with a 3-direction or 4-direction box spline for various degrees and smoothness. To facilitate the description of the construction of box spline tight wavelet frames, we start with B-spline tight wavelet frame construction. For B-splines we find the sum of squares form by using Fej´er-Riesz factorization theorem and construct tight wavelet frames. We also use the tensor product of B-splines to construct locally supported bivariate tight wavelet frames. Then we explain how to construct box spline tight wavelet frames using Lai and St¨ockler’s method. In the second part of dissertation, we apply some of our box spline tight wavelet frames for edge detection and image de-noising. We present a lot of images to compare favorably with other edge detection methods including orthonormal wavelet methods and six engineering methods from MATLAB Image Processing Toolbox. For image de-noising we provide with PSNR numbers for the comparison. Finally we study the construction of locally supported tight wavelet frame over bounded domains. The situation of the construction of locally supported tight wavelet frames over bounded domains is quite different from the construction we explained above. We introduce a simple approach and obtain B-spline tight wavelet frames and box spline tight wavelet frames over finite intervals and bounded domains.