A generalization of matrix-valued Multiresolution analysis (MMRA) to matrix-valued frames, and the constructions of matrix-valued frames are considered and characterized. A matrix-valued frame Multiresolution analysis is defined in this paper. We provide necessary and sufficient conditions for constructing matrix-valued frames and Riesz bases of translates, and give the calculation method of matrix-valued dual Riesz basis. These conclusions are useful in providing theoretical basis for constructing matrix-valued frames and Riesz basis.

Wavelets and radial basis functions (RBF) have ubiquitously proved very successful to solve different forms of partial differential equations (PDE) using shifted basis functions, and as with the other meshless methods, they have been extensively used in scattered data interpolation. The current paper proposes a framework that successfully reconciles RBF and adaptive wavelet method to solve the Perona-Malik equation in terms of locally shifted functions. We take advantage of the scaling functions that span Multiresolution subspaces to provide resilient grid comprising centers. At the next step, the derivatives are computed and summed over these local feature collocations to generate the solution. We discuss the stability of the solution and depict how convergence could be granted in this context. Finally, the numerical results are provided to illustrate the accuracy and efficiency of the proposed method.

We present a new method for representation of Multiresolution graphic models. The considered models are curves, images and surfaces. The method is based on a discrete approach to the theory of wavelets. Multiresolution has various applications to computer graphics and pattern recognition, such as image compression and smoothing, graphics model processing, and scanning transformations. We concentrate on the Chaikin"s subdivision method for generating smooth models. This method makes use of an optimal projection of a Multiresolution model onto a space having a lower Multiresolution level. The optimization is made locally and discretely and results in biorthogonal wavelets. We demonstrate the effectiveness of the decomposition and reconstruction processes of our method through various examples.

A general concept of wavelets and Multiresolution analysis on Sobolev space over local fields of positive characteristic (Hs(K)) is given. A characterization of the scaling functions associated with an MRA is also given