Construction of uniformly symmetric bi-frames based on multiresolution template algorithm

ABSTRACT In this paper, we focus on the construction of wavelet bi-frames with three generators and uniform symmetry which play an important role in the applications for curve multiresolution processing. We first establish some templates of the decomposition and reconstruction algorithm corresponding to wavelet bi-framelet by several iterative steps. These templates allow parameterization of wavelet bi-framelet filter banks with three symmetric generators. Then the parameters in the filter can be determined by solving systems of nonlinear equations related to the sum rules for the refinement mask and the vanishing moments of framelets, which led to the wavelet bi-frames with desired properties. Some examples of uniform symmetry bi-frames with three generators are presented to illustrate efficient of our methods.

[1]  Qingtang Jiang,et al.  Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing , 2010, J. Comput. Appl. Math..

[2]  Bin Dong,et al.  MRA-based wavelet frames and applications , 2013 .

[3]  I. Daubechies,et al.  The Canonical Dual Frame of a Wavelet Frame , 2002 .

[4]  Qingtang Jiang,et al.  Biorthogonal Wavelets with Six-fold axial Symmetry for Hexagonal Data and Triangle Surface Multiresolution Processing , 2011, Int. J. Wavelets Multiresolution Inf. Process..

[5]  Qingtang Jiang,et al.  Highly symmetric bi-frames for triangle surface multiresolution processing , 2011 .

[6]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[7]  Bin Han,et al.  Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function , 2002, SIAM J. Matrix Anal. Appl..

[8]  B. Han,et al.  Pairs of Dual Wavelet Frames from Any Two Refinable Functions , 2004 .

[9]  Zuowei Shen,et al.  Dual Wavelet Frames and Riesz Bases in Sobolev Spaces , 2009 .

[10]  Qingtang Jiang.  Correspondence between frame shrinkage and high-order nonlinear diffusion , 2012 .

[11]  Martin Bertram,et al.  Biorthogonal Loop-Subdivision Wavelets , 2004, Computing.

[12]  Zuowei Shen,et al.  PSEUDO-SPLINES, WAVELETS AND FRAMELETS , 2007 .

[13]  Qingtang Jiang,et al.  Orthogonal and Biorthogonal $\sqrt 3$-Refinement Wavelets for Hexagonal Data Processing , 2009, IEEE Transactions on Signal Processing.

[14]  Bin Han,et al.  The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets , 2009, Math. Comput..

[15]  M. Ehler,et al.  Applied and Computational Harmonic Analysis , 2015 .

[16]  M. Ehler.  On Multivariate Compactly Supported Bi-Frames , 2007 .

[17]  Raymond H. Chan,et al.  Restoration of Chopped and Nodded Images by Framelets , 2008, SIAM J. Sci. Comput..

[18]  Qingtang Jiang,et al.  Triangular √3-subdivision schemes: the regular case , 2003 .

[19]  B. Han.  On Dual Wavelet Tight Frames , 1995 .

[20]  Jian-Feng Cai,et al.  A framelet-based image inpainting algorithm , 2008 .

[21]  Kaihuai Qin,et al.  √3-Subdivision-Based Biorthogonal Wavelets , 2007, IEEE Transactions on Visualization and Computer Graphics.

[22]  Qingtang Jiang,et al.  Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing , 2011, Adv. Comput. Math..

[23]  C. Chui,et al.  Compactly supported tight and sibling frames with maximum vanishing moments , 2001 .

[24]  Bin Han,et al.  Algorithm for constructing symmetric dual framelet filter banks , 2014, Math. Comput..

[25]  Kai Tang,et al.  Efficient wavelet construction with Catmull–Clark subdivision , 2006, The Visual Computer.

[26]  Qingtang Jiang,et al.  Highly symmetric 3-refinement Bi-frames for surface multiresolution processing , 2017 .

[27]  Bin Dong,et al.  Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond , 2017, Multiscale Model. Simul..

[28]  Zuowei Shen.  Wavelet Frames and Image Restorations , 2011 .

[29]  Jian-Feng Cai,et al.  Split Bregman Methods and Frame Based Image Restoration , 2009, Multiscale Model. Simul..

[30]  Bernd Hamann,et al.  Generalized B-spline subdivision-surface wavelets for geometry compression , 2004, IEEE Transactions on Visualization and Computer Graphics.

[31]  Qingtang Jiang,et al.  Wavelet bi-frames with uniform symmetry for curve multiresolution processing , 2011, J. Comput. Appl. Math..

[32]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

[33]  A. Ron,et al.  Affine systems inL2 (ℝd) II: Dual systems , 1997 .

[34]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

[35]  W. Sweldens.  The Lifting Scheme: A Custom - Design Construction of Biorthogonal Wavelets "Industrial Mathematics , 1996 .

[36]  B. Han.  DUAL MULTIWAVELET FRAMES WITH HIGH BALANCING ORDER AND COMPACT FAST FRAME TRANSFORM , 2008 .