Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T06:58:53.208Z Has data issue: false hasContentIssue false
  • English
  • Français

The Projection Method for Multidimensional Framelet and WaveletAnalysis

Published online by Cambridge University Press:  17 July 2014

B. Han
Get access

Abstract

The projection method is a simple way of constructing functions and filters byintegrating multidimensional functions and filters along parallel superplanes in the spacedomain. Equivalently expressed in the frequency domain, the projection method constructs anew function by simply taking a cross-section of the Fourier transform of amultidimensional function. The projection method is linked to several areas such as boxsplines in approximation theory and the projection-slice theorem in image processing. Inthis paper, we shall systematically study and discuss the projection method in the area ofmultidimensional framelet and wavelet analysis. We shall see that the projection methodnot only provides a painless way for constructing new wavelets and framelets but also is auseful analysis tool for studying various optimal properties of multidimensional refinablefunctions and filters. Using the projection method, we shall explicitly and easilyconstruct a tight framelet filter bank from every box spline filter having at least orderone sum rule. As we shall see in this paper, the projection method is particularlysuitable to be applied to frequency-based nonhomogeneous framelets and wavelets in anydimensions, and the periodization technique is a special case of the projection method forobtaining periodic wavelets and framelets from wavelets and framelets on Euclidean spaces.

Keywords

Projection methodwavelets and frameletstight framelets from box splinesdual framelet filter banksinterpolatory filtersorthonormal filters, frequency-based dual frameletsnonhomogeneous and homogeneous affine systemsFourier transform
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 83 - 110
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

C. de Boor, K. Höllig, S. Riemenschneider. Box splines. Series in Appl. Math. Sci. vol. 98. Springer-Verlag, New York, 1993.
C. K. Chui. An introduction to wavelets. Academic Press, Inc., Boston, MA, 1992.
Chui, C. K., He, W. J., Stöckler, J.. Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal., 13 (2002), 224–262. CrossRefGoogle Scholar
I. Daubechies. Ten Lectures on Wavelets. CBMS-NSF Series, 61, SIAM, Philadelphia, 1992.
Daubechies, I., Han, B.. Pairs of dual wavelet frames from any two refinable functions. Constr. Approx., 20 (2004), 325–352. CrossRefGoogle Scholar
Daubechies, I., Han, B., Ron, A., Shen, Z.. Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal., 14 (2003), 1–46. CrossRefGoogle Scholar
Ehler, M.. On multivariate compactly supported bi-frames. J. Fourier Anal. Appl., 13 (2007), 511–532. CrossRefGoogle Scholar
Han, B.. Properties of discrete framelet transforms. Math. Model. Nat. Phenom., 8 (2013), 18–47. CrossRefGoogle Scholar
Han, B.. Nonhomgeneous wavelet systems in high dimensions. Appl. Comput. Harmon. Anal., 32 (2012), 169–196. CrossRefGoogle Scholar
Han, B.. Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space. Appl. Comput. Harmon. Anal., 29 (2010), 330–353. CrossRefGoogle Scholar
Han, B.. Construction of wavelets and framelets by the projection method. Int. J. Appl. Math. Appl., 1 (2008), 1–40. Google Scholar
B. Han. The projection method in wavelet analysis. in Splines and Wavelets: Athens 2005, G. Chen and M.J. Lai eds., (2006), 202–225.
Han, B.. Symmetric multivariate orthogonal refinable functions. Appl. Comput. Harmon. Anal., 17 (2004), 277–292. CrossRefGoogle Scholar
Han, B.. Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl., 24 (2003), 693–714. CrossRefGoogle Scholar
Han, B.. Vector cascade algorithms and refinable function vectors in Sobolev spaces. J. Approx. Theory, 124 (2003), 44–88. CrossRefGoogle Scholar
Han, B.. Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. J. Comput. Appl. Math., 155 (2003), 43–67. CrossRefGoogle Scholar
Han, B.. Projectable multivariate refinable functions and biorthogonal wavelets. Appl. Comput. Harmon. Anal., 13 (2002), 89–102. CrossRefGoogle Scholar
Han, B.. Symmetry property and construction of wavelets with a general dilation matrix. Linear Algebra Appl., 353, (2002), 207–225. CrossRefGoogle Scholar
Han, B.. Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets. J. Approx. Theory, 110 (2001), 18–53. CrossRefGoogle Scholar
Han, B.. Analysis and construction of optimal multivariate biorthogonal wavelets with compact support. SIAM J. Math. Anal., 31 (2000), 274–304. CrossRefGoogle Scholar
Han, B.. On dual wavelet tight frames. Appl. Comput. Harmon. Anal., 4 (1997), 380–413. CrossRefGoogle Scholar
B. Han. Wavelets. M.Sc. thesis at the Institute of Mathematics, Chinese Academy of Sciences, China, 1994.
Han, B., Jia, R. Q.. Optimal C2 two-dimensional interpolatory ternary subdivision schemes with two-ring stencils. Math. Comp., 75 (2006), 1287–1308. CrossRefGoogle Scholar
Han, B., Jia, R. Q.. Optimal interpolatory subdivision schemes in multidimensional spaces. SIAM J. Numer. Anal., 36, (1998), 105–124. CrossRefGoogle Scholar
Han, B., Mo, Q.. Analysis of optimal bivariate refinable Hermite interpolants. Commun. Pure Appl. Anal., 6 (2007), 689–718. Google Scholar
Jia, R. Q.. Approximation properties of multivariate wavelets. Math. Comp., 67 (1998), 647–665. CrossRefGoogle Scholar
Lai, M. J., Stöckler, J.. Construction of multivariate compactly supported tight wavelet frames. Appl. Comput. Harmon. Anal., 21 (2006), 324–348. CrossRefGoogle Scholar
J. Liao. New interpolatory subdivision schemes in computer graphics. M.Sc. thesis at the University of Alberta, Canada, 2004.
Y. Meyer. Wavelets and operators. Cambridge University Press, Cambridge, 1992.
Ron, A., Shen, Z.. Affine systems in L2(ℝd): the analysis of the analysis operator. J. Funct. Anal., 148 (1997), 408–447. CrossRefGoogle Scholar