- Research Article
- Open Access
Construction of Orthonormal Piecewise Polynomial Scaling and Wavelet Bases on Non-Equally Spaced Knots
EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 027427 (2007)
This paper investigates the mathematical framework of multiresolution analysis based on irregularly spaced knots sequence. Our presentation is based on the construction of nested nonuniform spline multiresolution spaces. From these spaces, we present the construction of orthonormal scaling and wavelet basis functions on bounded intervals. For any arbitrary degree of the spline function, we provide an explicit generalization allowing the construction of the scaling and wavelet bases on the nontraditional sequences. We show that the orthogonal decomposition is implemented using filter banks where the coefficients depend on the location of the knots on the sequence. Examples of orthonormal spline scaling and wavelet bases are provided. This approach can be used to interpolate irregularly sampled signals in an efficient way, by keeping the multiresolution approach.
Mallat S: A Wavelet Tour of Signal Processing. 2nd edition. Academic Press, San Diego, Calif, USA; 1999.
Vetterli M, Kovacevic J: Wavelets and Subband Coding. Prentice Hall, Englewood Cliffs, NJ, USA; 1995.
Chui CK (Ed): Wavelets: A Tutorial in Theory and Applications. Academic Press, San Diego, Calif, USA; 1993.
Rioul O, Duhamel P: Fast algorithms for wavelet transform computation. In Time-Frequency and Wavelets in Biomedical Signal Processing. Edited by: Akay M. Wiley-IEEE Press, New York, NY, USA; 1997:211-242. chapter 8
Buhmann MD, Micchelli CA: Spline prewavelets for non-uniform knots. Numerische Mathematik 1992,61(1):455-474. 10.1007/BF01385520
Daubechies I, Guskov I, Schröder P, Sweldens W: Wavelets on irregular point sets. Philosophical Transactions of the Royal Society of London. A 1999,357(1760):2397-2413. 10.1098/rsta.1999.0439
Daubechies I, Guskov I, Sweldens W: Commutation for irregular subdivision. Constructive Approximation 2001,17(4):479-514. 10.1007/s00365-001-0001-0
Lyche T, Mørken K, Quak E: Theory and Algorithms for non-uniform spline wavelets. In Multivariate Approximation and Applications. Edited by: Dyn N, Leviatan D, Levin D, Pinkus A. Cambridge University Press, Cambridge, UK; 2001:152-187.
De Boor C: A Practical Guide to Splines. revised edition. Springer, New York, NY, USA; 2001.
Farin G: Curves and Surfaces for CAGD. 5th edition. Morgan-Kaufmann, San Fransisco, Calif, USA; 2001.
Chihab N, Zergaïnoh A, Duhamel P, Astruc JP: The influence of the non-uniform spline basis on the approximation signal. Proceedings of 12th European Signal Processing Conference (EUSIPCO '04), September 2004, Vienna, Austria
Unser MA: Ten good reasons for using spline wavelets. Wavelet Applications in Signal and Image Processing V, July 1997, San Diego, Calif, USA, Proceedings of SPIE 3169: 422–431.
About this article
Cite this article
Zergaïnoh, A., Chihab, N. & Astruc, J.P. Construction of Orthonormal Piecewise Polynomial Scaling and Wavelet Bases on Non-Equally Spaced Knots. EURASIP J. Adv. Signal Process. 2007, 027427 (2007). https://doi.org/10.1155/2007/27427
- Information Technology
- Basis Function
- Quantum Information
- Bounded Interval
- Filter Bank