基于结构容许覆盖的框架构造

数学物理学报 ›› 2011, Vol. 31 ›› Issue (3): 670-677.

基于结构容许覆盖的框架构造

刘国军^1,2, 冯象初¹, 张伟斌¹

1.西安电子科技大学理学院西安 750071|2.宁夏大学数学计算机学院银川 750021

收稿日期:2009-07-05 修回日期:2010-07-13 出版日期:2011-06-25 发布日期:2011-06-25
基金资助:
国家自然科学基金(60872138, 61001156)资助

Construction of Frames Based on Structured Admissible Covering

LIU Guo-Jun^1,2, FENG Xiang-Chu¹, ZHANG Wei-Bin¹

1.Department of Applied Mathematics, Xidian University, Xi'an |710071|2.School of Mathematics and Computer Science, Ningxia University, Yinchuan |750021

Received:2009-07-05 Revised:2010-07-13 Online:2011-06-25 Published:2011-06-25
Supported by:
国家自然科学基金(60872138, 61001156)资助

摘要/Abstract

摘要：

利用一般的结构容许覆盖理论, 借助曲线波型框架的构造流程, 得到了频域空间R²(ξ)中的脊波型覆盖Q={Q_T}_T∈T, 这里仿射变换T扮演着平移, 膨胀和调制的作用. 此外, 通过联合正交脊波和所构造的脊波型覆盖构造了一些新的脊波型框架,即脊波型紧框架, 对偶脊波型框架, 以及单尺度对偶脊波型框架. 与Candes的脊波紧框架相比, 除了框架本身, 其对偶框架也具有显式表达式.

关键词: 框架, 结构容许覆盖, 脊波, 曲线波

Abstract:

Using the general theory of the structured admissible covering, a Ridgelet-type covering Q={Q_T}_T∈T of the frequency space R²(ξ) is analogously introduced along with the constructed flow of Curvelet-type frame. The affine transformation T acts on the roles of dilation, translation, and modulation. Most of all, some different Ridgelet-type frames are given, such as Ridgelet-type tight frame, Dual Ridgelet-type frame, and Monoscale Dual Ridgelet-type frame, by combining the orthonomal Ridgelet with the constructed covering. Compared with Candes's Ridgelet tight frame, besides the analysis frames, their dual frames also have an explicit form.

Key words: Frame, Structured admissible covering, Ridgelet, Curvelet

中图分类号:

42C15

刘国军, 冯象初, 张伟斌. 基于结构容许覆盖的框架构造[J]. 数学物理学报, 2011, 31(3): 670-677.

LIU Guo-Jun, FENG Xiang-Chu, ZHANG Wei-Bin. Construction of Frames Based on Structured Admissible Covering[J]. Acta mathematica scientia,Series A, 2011, 31(3): 670-677.

参考文献

[1] Duffin R J, Schaeffer A C. A class of nonharmonic Fourier series. Transactions of the American Mathematical Society, 1952, 72(2): 341--366

[2] Wiley R G. Recovery of band-limited signals from unequally spaced samples. IEEE Transactions on Communications, 1978, 26: 135--137

[3] Daubechies I. The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information Theory, 1990, 36(5): 961--1005

[4] Daubechies I, Grossman A. Frames in the Bargmann space of entire functions. Communications on Pure and Applied Mathematics, 1988, 41(2): 151--164

[5] Daubechies I, Grossman A, Meyer Y. Painless nonorthogonal expansions. Journal of Mathematical Physics, 1986, 27(5): 1271--1283

[6] Mallat S, Zhong S. Characterization of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1992, 14(7): 710--732

[7] Donoho D L, Johnstone I M. Ideal spatial adaptation via wavelet shrinkage. Biometrika, 1994, 81(3): 425--455

[8] Candes E J. Harmonic analysis of neural networks. Applied and Computational Harmonic Analysis, 1999, 6(2): 197--218

[9] Donoho D L. Orthonormal ridgelet and straight singularities. SIAM Journal on Mathematical Analysis, 2000, 31(5): 1062--1099

[10] Candes E J. Monoscale Ridgelet for the Representation of Images with Edges. California: Stanford University, 1999: 1--26

[11] Candes E J, Donoho D L. Curvelets-a Surprisingly Effective Nonadaptive Representation for Objects with Edges//Schumaker L L ed. Saint-Malo Proceedings. Nashville: Vanderbilt University Press, 1999: 1--10

[12] Flesia A G, Hel-Or H, Averbuch A, et al. Digital Implementation of Ridgelet Packets//Stoeckler J, Welland G V ed. Beyond Wavelets. New York: Academic Press, 2001: 1--34

[13] Tan S, Jiao L C. Ridgelet bi-frame. Applied and Computational Harmonic Analysis, 2006, 20(3): 391--402

[14] Tan S, Zhang X R, Jiao L C. Dual Ridgelet Frame Constructed Using Biorthonormal Wavelet Basis. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing. Philadelphia: PA, 2005: 997--1000

[15] Bai J, Feng X C. Digital ridgelet reconstruction based on local dual frame. Science in China, 2005, 48(6): 782--794

[16] Candes E J, Donoho D L. New tight frames of curvelets and optimal representations of objects with piecewise singularities.
Communications on Pure and Applied Mathematics, 2004, 57(2): 219--266

[17] 黄永东, 程正兴. α带小波紧框架的显式构造方法. 数学物理学报, 2007, 27(1): 7--18

[18] 肖雪梅, 朱玉灿. Banach空间中框架的对偶原理. 数学物理学报, 2009, 29(1): 94--102

[19] Borup L, Nielsen M. Frame decomposition of decomposition spaces. Journal of Fourier Analysis and Applications, 2007, 13(1): 39--70

[20] Triebel H. Theory of Function Spaces. Monographs in Mathematics 78. Basel: Birkhauser Verlag, 1983