In this thesis, we study subdivision schemes, biorthogonal wavelets and wavelet-based image compression. Subdivision schemes are important in computer aided geometric design to generate curves and high dimensional surfaces. First, we shall characterize the Lp convergence of any subdivision scheme with a finitely supported refinement mask in multivariate case. Then we shall study the error behaviour of any subdivision scheme if there is a round-off of its refinement mask. Next, we study a special kind of subdivision schemes—interpolatory subdivision schemes. We shall analyze the optimal properties, such as sum rules of an interpolatory refinement mask and the smoothness of its associated refinable function, of any interpolatory subdivision scheme. A general construction of optimal interpolatory subdivision schemes is presented. Next, we shall study biorthogonal multivariate wavelets since there is a well known close relation between interpolatory subdivision schemes and biorthogonal wavelets. We shall study the optimal approximation and smoothness properties of any biorthogonal wavelet. More importantly, a general and easy way (CBC algorithm) is presented to construct multivariate biorthogonal wavelets. As an example, a family of optimal bivariate biorthogonal wavelets is given. Finally, we try to apply our results in image compression and a 2-D wavelet transform C++ program is established which can use a library of bivariate biorthogonal wavelet filters.
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机译：在本文中，我们研究了细分方案，双正交小波和基于小波的图像压缩。细分方案对于计算机辅助几何设计以生成曲线和高维曲面非常重要。首先，在多变量情况下，我们将用有限支持的精炼掩码来刻画任何细分方案的 L p 收敛性。然后，如果对细分方案进行了精修，则我们将研究任何细分方案的错误行为。接下来，我们研究一种特殊的细分方案-内插细分方案。我们将分析任何插值细分方案的最佳属性，例如插值精炼蒙版的求和规则及其关联的可提炼函数的平滑度。提出了最佳插值细分方案的一般构造。接下来，我们将研究双正交多元小波，因为插值细分方案与双正交小波之间存在众所周知的紧密关系。我们将研究任何双正交小波的最佳逼近度和平滑度。更重要的是，提出了一种通用且简便的方法（CBC算法）来构造多元双正交小波。例如，给出了一个最佳的双变量双正交小波族。最后，我们尝试将结果应用于图像压缩，并建立了一个二维小波变换C ++程序，该程序可以使用双变量双正交小波滤波器库。
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