| The traditional two-dimensional B-spline tight frames, which are the tensor product of one-dimensional B-spline wavelets, can only offer two directions of horizontal and vertical. While Non-tensor product wavelet may contain more than two directions. Note that the bivariate B-spline denoted by B(x,y) in the spline space S21(Δmn(2)) is defined on a four-directional mesh, which can contain the informations at four directions. In this thesis, we verify that the bivariate B-spline function satisfies the assumptions of the Unitary Extension Principe (UEP), and give the explicit expression of the bracket product relevant with B(x, y) and its bounds. Then according to the UEP. we construct a non-product B-spline tight frame by using the bivariate B-spline. In addition, we obtain the masks of the B-spline tight frame. From the experiment of the decomposition of the image with eight directions, we can know that the non-tensor product has two more directions than the tensor product tight frame. The experiments of image denoising and deblurring show that the non-tensor product tight frame is better than the tensor product of the linear B-spline tight frame. The thesis is organized as follows:The first chapter introduces the problems of constructions of the bivariate B-spline wavelets tight frame, and the research about B-spline wavelets tight frame at home and abroad.The second chapter introduces the basic knowledge about the tight frame and the spline space S21(Δmn(2)), and gives the corresponding recursive formula about masks of the univariate B-spline wavelet tight frame.The third chapter verifies that the bivariate B-spline function, in the spline space S21(Δmn(2)), satisfies the assumptions of UEP. Then according to the UEP, we construct a non-tensor product B-spline tight frame.The fourth chapter presents the numerical examples of image processing, image denoising, edge detection and image deblurring, with three different B-spline tight frames. |
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