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Small Support Spline Riesz Wavelets in Low Dimensions

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Abstract

In Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006), a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in Han and Shen (J. Fourier Anal. Appl. 11:615–637, 2005). Motivated by these two papers, we develop in this article a general theory and a construction method to derive small support Riesz wavelets in low dimensions from refinable functions. In particular, we obtain small support spline Riesz wavelets from bivariate and trivariate box splines. Small support Riesz wavelets are desirable for developing efficient algorithms in various applications. For example, the short support Riesz wavelets from Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006) were used in a surface fitting algorithm of Johnson et al. (J. Approx. Theory 159:197–223, 2009), and the Riesz wavelet basis from the Loop scheme was used in a very efficient geometric mesh compression algorithm in Khodakovsky et al. (Proceedings of SIGGRAPH, 2000).

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References

  1. de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer, Berlin (1993)

    MATH  Google Scholar 

  2. Cai, J.-F., Chan, R.H., Shen, L., Shen, Z.: Restoration of chopped and nodded images by framelets. SIAM J. Sci. Comput. 30, 1205–1227 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cai, J.-F., Chan, R.H., Shen, L., Shen, Z.: Simultaneously inpainting in image and transformed domains. Numer. Math. 112, 509–533 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cai, J.-F., Chan, R.H., Shen, L., Shen, Z.: Convergence analysis of tight framelet approach for missing data recovery. Adv. Comput. Math. 31, 87–113 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cai, J.-F., Chan, R.H., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24, 131–149 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cai, J.-F., Ji, H., Liu, C., Shen, Z.: Blind motion deblurring using multiple images. J. Comput. Phys. 228(14), 5057–5071 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cai, J.-F., Ji, H., Liu, C., Shen, Z.: Blind motion deblurring from a single image using sparse approximation. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009)

    Google Scholar 

  8. Cai, J.-F., Osher, S., Shen, Z.: Linearized Bregman iteration for frame based image deblurring. SIAM J. Imaging Sci. 2, 226–252 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cai, J.-F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8, 337–369 (2009)

    Article  MathSciNet  Google Scholar 

  10. Chui, C.K., Wang, J.Z.: On compactly supported spline wavelets and a duality principle. Trans. Am. Math. Soc. 330, 903–915 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cohen, A., Daubechies, I.: A new technique to estimate the regularity of refinable functions. Rev. Mat. Iberoam. 12, 527–591 (1996)

    MATH  MathSciNet  Google Scholar 

  12. Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dong, B., Shen, Z.: Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22, 78–104 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Daubechies, I.: Ten Lectures on Wavelets. CBMS Series. SIAM, Philadelphia (1992)

    Google Scholar 

  15. Daubechies, I., Han, B., Ron, A., Framelets, Z. Shen: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Han, B.: On dual tight wavelet frames. Appl. Comput. Harmon. Anal. 4, 380–413 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Han, B.: Analysis and construction of optimal multivariate biorthogonal wavelets with compact support. SIAM J. Math. Anal. 31, 274–304 (1999/00)

    Article  MathSciNet  Google Scholar 

  18. Han, B.: Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets. J. Approx. Theory 110, 18–53 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Han, B.: Vector cascade algorithms and refinable function vectors in Sobolev spaces. J. Approx. Theory 124, 44–88 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Han, B.: Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl. 24, 693–714 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Han, B.: Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix. Adv. Comput. Math. 24, 375–403 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Han, B.: On a conjecture about MRA Riesz wavelet bases. Proc. Am. Math. Soc. 134, 1973–1983 (2006)

    Article  MATH  Google Scholar 

  23. Han, B.: Refinable functions and cascade algorithms in weighted spaces with Hölder continuous masks. SIAM J. Math. Anal. 40, 70–102 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Han, B.: Matrix extension with symmetry and applications to symmetric orthonormal complex M-wavelets. J. Fourier Anal. Appl. 15, 684–705 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Han, B., Jia, R.Q.: Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29, 1177–1199 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Han, B., Jia, R.Q.: Characterization of Riesz bases of wavelets generated from multiresolution analysis. Appl. Comput. Harmon. Anal. 27, 321–345 (2007)

    Article  MathSciNet  Google Scholar 

  27. Han, B., Kwon, S.G., Park, S.S.: Riesz multiwavelet bases. Appl. Comput. Harmon. Anal. 20, 161–183 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Han, B., Shen, Z.: Wavelets with short support. SIAM J. Math. Anal. 38, 530–556 (2006)

    Article  MathSciNet  Google Scholar 

  29. Han, B., Shen, Z.: Wavelets from the Loop scheme. J. Fourier Anal. Appl. 11, 615–637 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Han, B., Shen, Z.: Dual wavelet frames and Riesz bases in Sobolev spaces. Constr. Approx. 29, 369–406 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Li, S., Liu, Z.: Riesz multiwavelet bases generated by vector refinement equation. Sci. China Ser. A 52, 468–480 (2009)

    Article  MathSciNet  Google Scholar 

  32. Ji, H., Riemenschneider, S., Shen, Z.: Multivariate compactly supported fundamental refinable functions, duals and biorthogonal wavelets. Stud. Appl. Math. 102, 173–204 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ji, H., Shen, Z., Xu, Y.H.: Wavelet frame based method for scene reconstruction. J. Comput. Phys. 229, 2093–2108 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Jia, R.Q., Shen, Z.: Multiresolution and wavelets. Proc. Edinb. Math. Soc. 37, 271–300 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jia, R.Q., Wang, J.Z., Zhou, D.X.: Compactly supported wavelet bases for Sobolev spaces. Appl. Comput. Harmon. Anal. 15, 224–241 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. Johnson, M.J., Shen, Z.W., Xu, Y.H.: Scattered data reconstruction by regularization in B-spline and associated wavelet spaces. J. Approx. Theory 159, 197–223 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  37. Loop, C.: Smooth subdivision surfaces based on triangles. MSc thesis, University of Utah (1987)

  38. Lorentz, R., Oswald, P.: Criteria for hierarchical bases in Sobolev spaces. Appl. Comput. Harmon. Anal. 8, 32–85 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. Khodakovsky, A., Schröder, P., Sweldens, W.: Progressive geometry compression In: Proceedings of SIGGRAPH (2000)

    Google Scholar 

  40. Riemenschneider, S., Shen, Z.: Box splines, cardinal series, and wavelets. In: Chui, C.K. (ed.) Approximation Theory and Functional Analysis, pp. 133–149. Academic Press, New York (1991)

    Google Scholar 

  41. Riemenschneider, S., Shen, Z.: Wavelets and pre-wavelets in low dimensions. J. Approx. Theory 71, 18–38 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  42. Ron, A., Shen, Z.: Affine systems in L 2(ℝd): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ron, A., Shen, Z.: Affine systems in L 2(ℝd) II: dual systems. J. Fourier Anal. Appl. 3, 617–637 (1997)

    Article  MathSciNet  Google Scholar 

  44. Shen, Z.: Wavelet frames and image restorations. In: Bhatia, R. (ed.) Proceedings of the International Congress of Mathematicians, vol. IV, pp. 2834–2863. Hindustan Book Agency, Hyderabad (2010)

    Google Scholar 

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Correspondence to Qun Mo.

Additional information

Communicated by David Walnut.

Research of B. Han is supported in part by NSERC Canada under Grant RGP 228051. Research of Q. Mo is supported in part by NSF of China under Grants 10771090 and 10971189, the NSF of Zhejiang province under grant Y6090091, the doctoral program foundation of ministry of education of China under grant 20070335176. Research of Z. Shen is supported in part by several grants at National University of Singapore.

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Han, B., Mo, Q. & Shen, Z. Small Support Spline Riesz Wavelets in Low Dimensions. J Fourier Anal Appl 17, 535–566 (2011). https://doi.org/10.1007/s00041-010-9147-0

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  • DOI: https://doi.org/10.1007/s00041-010-9147-0

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