Building Nonredundant Adaptive Wavelets by Update Lifting 1

In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. Our scheme comprises an adaptive update lifting step and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two different update filters, and that this choice is triggered by the local gradient of the original signal. If the gradient is large (in some seminorm sense) it chooses one filter, if it is small the other. We derive necessary and sufficient conditions for the invertibility of such an adaptive system for various scenarios. Furthermore, we present some examples to illustrate our theoretical results.

[1]  A. Enis Çetin,et al.  Adaptive polyphase subband decomposition structures for image compression , 2000, IEEE Trans. Image Process..

[2]  Patrick L. Combettes,et al.  Convex Multiresolution Analysis , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  Jean-Christophe Pesquet,et al.  M-band nonlinear subband decompositions with perfect reconstruction , 1998, IEEE Trans. Image Process..

[4]  Jelena Kovacevic,et al.  Wavelets and Subband Coding , 2013, Prentice Hall Signal Processing Series.

[5]  Emmanuel J. Candès,et al.  The curvelet transform for image denoising , 2002, IEEE Trans. Image Process..

[6]  David L. Donoho,et al.  Orthonormal Ridgelets and Linear Singularities , 2000, SIAM J. Math. Anal..

[7]  Ricardo L. de Queiroz,et al.  Nonexpansive pyramid for image coding using a nonlinear filterbank , 1998, IEEE Trans. Image Process..

[8]  Richard G. Baraniuk,et al.  Nonlinear wavelet transforms for image coding via lifting , 2003, IEEE Trans. Image Process..

[9]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[10]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[11]  Henk J. A. M. Heijmans,et al.  Nonlinear multiresolution signal decomposition schemes. II. Morphological wavelets , 2000, IEEE Trans. Image Process..

[12]  M. Kunt,et al.  High compression image coding using an adaptive morphological subband decomposition , 1995, Proc. IEEE.

[13]  Henk J. A. M. Heijmans,et al.  Quantization of adaptive 2D wavelet decompositions , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[14]  Henk J. A. M. Heijmans,et al.  CONTENT-ADAPTIVE MULTIRESOLUTION ANALYSES , 2004 .

[15]  Luis F. Chaparro,et al.  Adaptive Morphological Representation of Signals: Polynomial and Wavelet Methods , 1997, Multidimens. Syst. Signal Process..

[16]  Richard Baraniuk,et al.  Lifting construction of non-linear wavelet transforms , 1998, Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380).

[17]  W. Sweldens.  The Lifting Scheme: A Custom - Design Construction of Biorthogonal Wavelets "Industrial Mathematics , 1996 .

[18]  Stéphane Mallat,et al.  Image compression with geometrical wavelets , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[19]  D. Donoho.  Wedgelets: nearly minimax estimation of edges , 1999 .

[20]  G. Piella,et al.  Adaptive lifting schemes with perfect reconstruction , 2002, IEEE Trans. Signal Process..

[21]  H. Heijmans,et al.  Adaptive update lifting with a decision rule based on derivative filters , 2002, IEEE Signal Processing Letters.

[22]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[23]  Robert D. Nowak,et al.  Adaptive wavelet transforms via lifting , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[24]  Henk J. A. M. Heijmans,et al.  Building adaptive 2D wavelet decompositions by update lifting , 2002, Proceedings. International Conference on Image Processing.

[25]  Wim Sweldens,et al.  The lifting scheme: a construction of second generation wavelets , 1998 .

[26]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing, 2nd Edition , 1999 .