Abstract
To construct a very smooth nonseparable multiscaling function, we impose polynomial approximation order 2 and add new conditions on the polyphase highpass filters. We work with a dilation matrix generating quincunx lattices, and fix the index set. Other imposed conditions are orthogonal filter bank and balancing. We construct a smooth, compactly supported multiscaling function and multiwavelet, and test the system on a noisy image with good results.
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References
Daubechies, I.: Ten lectures on wavelets. Society for Industrial and Applied Mathematics (1992)
Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, London (1999)
Cohen, A., Daubechies, I.: Non-separable bidimensional wavelet bases. Revista Matematica Iberoamericana 9, 51–137 (1993)
Kovacevic, J., Vetterli, M.: Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n. IEEE Trans. Inf. Theor. 38, 533–555 (1992)
Lawton, W., Lee, S., Shen, Z.: Stability and orthonormality of multivariate refinable functions. SIAM J. Math. Anal. 28, 999–1014 (1997)
Karoui, A., Vaillancourt, R.: Nonseparable biorthogonal wavelet bases of \(L^2(\Re ^n)\). CRM Proceedings and Lecture Notes American Math. Society 18, 135–151 (1999)
Ji, H., Riemenschneider, S., Shen, Z.: Multivariate compactly supported fundamental refinable functions, duals and biorthogonal wavelets. Studies in Applied Mathematics (to appear)
Strela, V., Heller, P., Strang, G., Topiwala, P., Heil, C.: The application of multiwavelet filterbanks to image processing 8, 548–563 (1999)
Plonka, G., Strela, V.: Construction of multiscaling functions with approximation and symmetry. SIAM Journal of Mathematical Analysis 29, 481–510 (1998)
Wajcer, D., Stanhill, D., Zeevi, Y.: Two-dimensional nonseparable multiwavelet transform and its application. In: Proc. IEEE-SP Intern. Symp. Time-Frequency and Time-Scale Analysis, pp. 61–64. IEEE Computer Society Press, Los Alamitos (1998)
Tay, D., Kingsbury, N.: Design of nonseparable 3-d filter banks wavelet bases using transformations of variables. IEE VISP 143, 51–61 (1996)
Ruedin, A.: Nonseparable orthogonal multiwavelets with 2 and 3 vanishing moments on the quincunx grid. Proc. SPIE Wavelet Appl. Signal Image Proc. VII 3813, 455–466 (1999)
Ruedin, A.M.C.: Balanced nonseparable orthogonal multiwavelets with two and three vanishing moments on the quincunx grid. Wavelet Appl. Signal Image Proc. VIII, Proc. SPIE 4119, 519–527 (2000)
Ruedin, A.M.C.: Construction of nonseparable multiwavelets for nonlinear image compression. Eurasip J. of Applied Signal Proc. 2002(1), 73–79 (2002)
Ruedin, A.: A nonseparable multiwavelet for edge detection. Wavelet Appl. Signal Image Proc. X, Proc. SPIE 5207, 700–709 (2003)
Ruedin, A.: Estimating the joint spectral radius of a nonseparable multiwavelet. In: IEEE Proc. XXIII Int. Conf. SCCC, pp. 109–115. IEEE Computer Society Press, Los Alamitos (2003)
Ruedin, A.M.C.: Dilation matrices for nonseparable bidimensional wavelets. In: Blanc-Talon, J., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2006. LNCS, vol. 4179, pp. 91–102. Springer, Heidelberg (2006)
Ron, A.: Smooth refinable functions provide good approximation orders. SIAM J. Math. Anal. 28, 731–748 (1997)
Cabrelli, C., Heil, C., Molter, U.: Accuracy of lattice translates of several multidimensional refinable functions. J. of Approximation Theory 95, 5–52 (1998)
Lebrun, J., Vetterli, M.: Balanced multiwavelets: Theory and design. IEEE Transactions on Signal Processing 46, 1119–1125 (1998)
Selesnick, I.: Balanced multiwavelet bases based on symmetric fir filters. Proceedings SPIE, Wavelet Applications in Signal Processing VII 3813, 122–131 (1999)
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Ruedin, A. (2007). Polyphase Filter and Polynomial Reproduction Conditions for the Construction of Smooth Bidimensional Multiwavelets. In: Blanc-Talon, J., Philips, W., Popescu, D., Scheunders, P. (eds) Advanced Concepts for Intelligent Vision Systems. ACIVS 2007. Lecture Notes in Computer Science, vol 4678. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74607-2_20
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DOI: https://doi.org/10.1007/978-3-540-74607-2_20
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