Biorthogonal multivariate filter banks from centrally symmetric matrices. (English) Zbl 1079.65141
A general construction of vector-valued multivariate orthonormal filter banks having linear phase was proposed by Q. Chen, C. A. Micchelli, S. Peng, and Y. Xu [SIAM J. Matrix Anal. Appl. 25, No. 2, 517–531 (2003; Zbl 1043.42026)] by a matrix factorization approach and the use of block centrally symmetric matrices. In this paper the authors introduce a new method to generate block centrally symmetric matrices, and the structure of such matrices is completely resolved. These results are then used for the construction of filter banks with linear phase in the biorthogonal case.
Reviewer: Hermann Render (Logrono)
MSC:
| 65T60 | Numerical methods for wavelets |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
Keywords:
multivariate filter banks; matrix factorization; linear phase; biorthogonal; block centrally symmetric matricesCitations:
Zbl 1043.42026References:
| [1] | Chen, Q. H.; Micchelli, C. A.; Peng, S.; Xu, Y., Multivariate filter banks having a matrix factorization, SIAM J. Matrix Anal. Appl., 25, 517-531 (2003) · Zbl 1043.42026 |
| [2] | Dahmen, W.; Micchelli, C. A., Biorthogonal wavelet expansions, Constr. Approx., 13, 293-328 (1997) · Zbl 0882.65148 |
| [3] | Herrmann, O., On the approximation problem in nonrecursive digital filter design, IEEE Trans. Circuit Theory, CT-18, 411-413 (1971) |
| [4] | Jia, R. Q., Approximation properties of multivariate wavelets, Math. Comp., 67, 647-665 (1998) · Zbl 0889.41013 |
| [5] | Jia, R. Q.; Micchelli, C. A., Using the refinement equation for the construction of prewavelets V: extensibility of trigonometric polynomials, Computing, 48, 61-72 (1992) · Zbl 0765.65023 |
| [6] | Kovacevic, J.; Vetterli, M., Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for \(R^n\), IEEE Trans. Inf. Theory, 38, 533-555 (1992) |
| [7] | Micchelli, C. A., Using the refinement equation for the construction of pre-wavelets, Numer. Algorithms, 1, 75-116 (1991) · Zbl 0759.65005 |
| [8] | C.A. Micchelli, Using the refinement equation for the construction of pre-wavelets VI: Shift invariant subspaces, in: Approximation Theory, Spline Functions and Applications (Maratea, 1991), 213-222, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 356, Kluwer Acad. Publ., Dordrecht, 1992; C.A. Micchelli, Using the refinement equation for the construction of pre-wavelets VI: Shift invariant subspaces, in: Approximation Theory, Spline Functions and Applications (Maratea, 1991), 213-222, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 356, Kluwer Acad. Publ., Dordrecht, 1992 · Zbl 0790.42018 |
| [9] | Micchelli, C. A.; Xu, Y., Reconstruction and decomposition algorithms for biorthogonal multiwavelets, Multidimen. Syst. Signal Process., 8, 31-69 (1997) · Zbl 0872.42010 |
| [10] | Nguyen, T. Q.; Vaidyanathan, P. P., Two-channel perfect reconstruction FIR QMF structure which yield linear phase FIR analysis and synthesis filters, IEEE Trans. Acoust. Speech Signal Proc., 37, 676-690 (1989) |
| [11] | Vaidyanathan, P. P., Multivariate Systems and Filter Banks (1993.), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0784.93096 |
| [12] | Vaidyanathan, P. P., Quadrature mirror filter banks, M-band extensions and perfect reconstruction techniques, IEEE ASSP Mag., 4, 3, 4-20 (1987) |
| [13] | Vetterli, M., Filter banks allowing perfect reconstruction, Signal Process., 10, 3, 219-244 (1986) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
