Abstract
We present in this paper several approaches for performing nonlinear multiscale transforms to an image. Both redundant and non-redundant transformations are discussed. The median based multiscale transforms and some of their applications are detailed. Finally we show how a multiscale vision model can be used to decompose an image, and how several transformations can be combined in order to benefit of the advantages of each of them.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. H. Adelson, E. Simoncelli, and R. Hingorani. Optimal image addition using the wavelet transform. SPIE Visual Communication and Image Processing II, 845:50–58, 1987.
M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies. Image coding using wavelet transform. IEEE Transactions on Image Processing, 1(2):205–220, 1992.
A. Bijaoui and F. Rué. A multiscale vision model adapted to astronomical images. Signal Processing, 46:229–243, 1995.
F. Bonnarel, P. Fernique, F. Genova, J. G. Bartlett, O. Bienaymé, D. Egret, J. Florsch, H. Ziaeepour, and M. Louys. ALADIN: A reference tool for identification of astronomical sources. Astronomical Data Analysis Software and Systems VIII, ASP Conference Series, Vol. 172. Ed. David M. Mehringer, Raymond L. Plante, and Douglas A. Roberts. ISBN: 1-886733-94-5 (1999), p. 229., 8:229–232, 1999.
E. J. Breen, R. J. Jones, and H. Talbot. Mathematical morphology: A useful set of tools for image analysis. Statistics and Computing journal, 10:105–120, 2000.
C. S. Burrus, R. A. Gopinath, and H. Guo. Introduction to Wavelets and Wavelet Transforms. Prentice Hall, 1998.
P. J. Burt and A. E. Adelson. The Laplacian pyramid as a compact image code. IEEE Tranactions on Communications, 31:532–540, 1983.
R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo. Wavelet transforms that map integers to integers. Appl. Comput. Harmon. Anal., 5(3):332–369, 1998.
E. J. Candès. Harmonic analysis of neural netwoks. Applied and Computational Harmonic Analysis, 6:197–218, 1999.
E. J. Candès and D. Donoho. Curvelets. Technical report, Statistics, Stanford University, 1999.
E. J. Candès and D. Donoho. Ridgelets: the key to high dimensional intermittency? Phil. trans; R. Soc. Lond. A, 357:2495–2509, 1999.
S. S. Chen, D. L. Donoho, and M. A. Saunder. Atomic decomposition by basis pursuit. SIAM J. Math. Anal., 20(1):33–61, 1998.
C. H. Chui. Wavelet Analysis and Its Applications. Academic Press, 1992.
R. Claypoole, G. M. Davis, W. Sweldens, and R. Baraniuk. Nonlinear wavelet transforms for image coding via lifting. IEEE Transactions on Image Processing, 2000. submitted.
R. L. Claypoole, R. G. Baraniuk, and R. D. Nowak. Lifting construction of nonlinear wavelet transforms. In IEEE International Conference IEEE-SP Time-Frequency and Time-Scale Analysis, pages 49–52, 1998.
A. Cohen, I. Daubechies, and J. C. Feauveau. Biorthogonal bases of compactly supported wavelets. Communications in Pure and Applied Mathematics, 45:485–560, 1992.
I. Daubechies. Time-frequency localization operators: A geometric phase space approach. IEEE Transactions on Information Theory, 34(1):605–612, 1988.
I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992.
I. Daubechies, I. Guskov, P. Schröder, and W. Sweldens. Wavelets on irregular point sets. Phil. Trans. R. Soc. Lond. A, To be published.
I. Daubechies and W. Sweldens. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl., 4(3):245–267, 1998.
D. L. Donoho. Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data. Proceedings of Symposia in Applied Mathematics, 47, 1993.
D. L. Donoho. Nonlinear pyramid transforms based on median-interpolation. SIAM J. Math. Anal., 60(4):1137–1156, 2000.
O. Egger and W. Li. Very low bit rate image coding using morphological operators and adaptive decompositions. In IEEE International Conference ICIP-94 on Image Processing, volume 2, pages 326–330, 1994.
24. D. A. F. Florêncio and R. W. Schafer. Perfect reconstructing nonlinear filter banks. In IEEE International Conference ICASSP-96 on Acoustics, Speech, and Signal Processing, volume 3, pages 1814–1817, 1996.
J. Goutsias and H. J. A. M. Heijmans. Nonlinear multiresolution signal decomposition schemes. part 1: Morphological pyramids. IEEE Transactions on Image Processing, 9(11):1862–1876, 2000.
F. J. Hampson and J.-C. Pesquet. A nonlinear subband decomposition with perfect reconstruction. In IEEE International Conference ICASSP-96 on Acoustics, Speech, and Signal Processing, volume 3, pages 1523–1526, 1996.
H. J. A. M. Heijmans and J. Goutsias. Multiresolution signal decomposition schemes. part 2: Morphological wavelets. IEEE Transactions on Image Processing, 9(11):1897–1913, 2000.
L. Huang and A. Bijaoui. Astronomical image data compression by morphological skeleton transformation. Experimental Astronomy, 1:311–327, 1991.
X. Huo. Sparse Image Representation via Combined Transforms. PhD thesis, Stanford Univesity, August 1999.
M. Louys, J.-L. Starck, S. Mei, F. Bonnarel, and F. Murtagh. Astronomical image compression. Astronomy and Astrophysics, Suppl. Ser., 136:579–590, 1999.
S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, 1998.
S. Mallat and F. Falzon. Analysis of low bit rate image transform coding. IEEE Transactions on Signal Processing, 46(4):1027–42, 1998.
S. G. Mallat. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674–693, 1989.
S. G. Mallat and Z. Zhang. Atomic decomposition by basis pursuit. IEEE Transactions on Signal Processing, 41(12):3397–3415, 1993.
G. Matheron. Elements pour une théorie des milieux poreux. Masson, Paris, 1967.
G. Matheron. Random Sets and Integral Geometry. Willey, New York, 1975.
F. Murtagh, J. L. Starck, and M. Louys. Very high quality image compression based on noise modeling. International Journal of Imaging Systems and Technology, 9:38–45, 1998.
A. Said and W. A. Pearlman. A new, fast, and efficient image codec based on set partitioning in hierarchival trees. IEEE Trans. on Circ. and Syst. for Video Tech., 6(3):243–250, 1996.
Peter Schröder and Wim Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. Computer Graphics Proceedings (SIGGRAPH 95), pages 161–172, 1995.
J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.
J. M. Shapiro. Embedded image coding using zerotrees of wavelet coefficients. IEEE Transactions on Signal Processing, 41(12):3445–3462, 1993.
J. L. Starck, A. Bijaoui, I. Vatchanov, and F. Murtagh. A combined approach for object detection and deconvolution. Astronomy and Astrophysics, Suppl. Ser., 147:139–149, 2000.
J. L. Starck, E. Candès, and D. L. Donoho. The curvelet transform for image denoising. IEEE Transactions on Image Processing, 2001. submitted.
J. L. Starck and F. Murtagh. Multiscale entropy filtering. Signal Processing, 76(2):147–165, 1999.
J. L. Starck, F. Murtagh, and A. Bijaoui. Image Processing and Data Analysis: The Multiscale Approach. Cambridge University Press, Cambridge (GB), 1998.
J. L. Starck, F. Murtagh, and R. Gastaud. A new entropy measure based on the wavelet transform and noise modeling. Special Issue on Multirate Systems, Filter Banks, Wavelets, and Applications of IEEE Transactions on CAS II, 45(8), 1998.
J. L. Starck, F. Murtagh, B. Pirenne, and M. Albrecht. Astronomical image compression based on noise suppression. Publications of the Astronomical Society of the Pacific, 108:446–455, 1996.
G. Strang and T. Nguyen. Wavelet and Filter Banks. Wellesley-Cambridge Press, Box 812060, Wellesley MA 02181, fax 617-253-4358, 1996.
W. Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal., 29(2):511–546, 1997.
W. Sweldens and P. Schröder. Building your own wavelets at home. In Wavelets in Computer Graphics, pages 15–87. ACM SIGGRAPH Course notes, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Starck, JL. (2002). Nonlinear Multiscale Transforms. In: Barth, T.J., Chan, T., Haimes, R. (eds) Multiscale and Multiresolution Methods. Lecture Notes in Computational Science and Engineering, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56205-1_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-56205-1_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42420-8
Online ISBN: 978-3-642-56205-1
eBook Packages: Springer Book Archive