Continuous and discrete wavelet transforms
SIAM Review
Sobolev characterization of solutions of dilation equations
SIAM Journal on Mathematical Analysis
Energy moments in time and frequency for two-scale difference equation solutions and wavelets
SIAM Journal on Mathematical Analysis
Inequalities of Littlewood-Paley type for frames and wavelets
SIAM Journal on Mathematical Analysis
Multiresolution analysis for surfaces of arbitrary topological type
Multiresolution analysis for surfaces of arbitrary topological type
Multiresolution analysis for surfaces of arbitrary topological type
ACM Transactions on Graphics (TOG)
Compactly supported tight affine spline frames in L2Rd
Mathematics of Computation
Wavelets for computer graphics: theory and applications
Wavelets for computer graphics: theory and applications
Wavelet Algorithms for High-Resolution Image Reconstruction
SIAM Journal on Scientific Computing
Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics - Special issue: Approximation theory, wavelets, and numerical analysis
Biorthogonal loop-subdivision wavelets
Computing - Geometric modelling dagstuhl 2002
Efficient wavelet construction with Catmull–Clark subdivision
The Visual Computer: International Journal of Computer Graphics
Restoration of Chopped and Nodded Images by Framelets
SIAM Journal on Scientific Computing
Tight wavelet frames for subdivision
Journal of Computational and Applied Mathematics
Advances in Computational Mathematics
√3-Subdivision-Based Biorthogonal Wavelets
IEEE Transactions on Visualization and Computer Graphics
A 4-point interpolatory subdivision scheme for curve design
Computer Aided Geometric Design
Image denoising using a tight frame
IEEE Transactions on Image Processing
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This paper is about the construction of univariate wavelet bi-frames with each framelet being symmetric. As bivariate filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices. For open surfaces, special multiresolution algorithms are designed to process boundary vertices. When the multiresolution algorithms derived from univariate wavelet bi-frames are used as the boundary algorithms, it is desired that not only the scaling functions but also all framelets be symmetric. In addition, the algorithms for curve/surface multiresolution processing should be given by templates so that they can be easily implemented. In this paper, first, by appropriately associating the lowpass and highpass outputs to the nodes of Z, we show that both biorthogonal wavelet multiresolution algorithms and bi-frame multiresolution algorithms can be represented by templates. Then, using the idea of the lifting scheme, we provide frame algorithms given by several iterative steps with each step represented by a symmetric template. Finally, with the given templates of algorithms, we obtain the corresponding filter banks and construct bi-frames based on their smoothness and vanishing moments. Two types of symmetric bi-frames are studied in this paper. In order to provide a clearer picture on the template-based procedure for bi-frame construction, in this paper we also consider the template-based construction of biorthogonal wavelets. The approach of the template-based bi-frame construction introduced in this paper can be extended easily to the construction of bivariate bi-frames with high symmetry for surface multiresolution processing.