No Access

MULTISELECTIVE PYRAMIDAL DECOMPOSITION OF IMAGES: WAVELETS WITH ADAPTIVE ANGULAR SELECTIVITY

Laboratoire de Télécommunications et Télédétection (TELE), Université Catholique de Louvain, Place du Levant 2, B-1348 Louvain-la-Neuve, Belgium

Corresponding author.

Search for more papers by this author

and

JEAN-PIERRE ANTOINE

Institut de Physique Théorique (FYMA), Université Catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

Search for more papers by this author

https://doi.org/10.1142/S0219691307002051Cited by:20 (Source: Crossref)

Abstract

Many techniques have been devised these last ten years to add an appropriate directionality concept in decompositions of images from the specific transformations of a small set of atomic functions. Let us mention, for instance, works on directional wavelets, steerable filters, dual-tree wavelet transform, curvelets, wave atoms, ridgelet packets, etc. In general, features that are best represented are straight lines or smooth curves as those defining contours of objects (e.g. in curvelets processing) or oriented textures (e.g. wave atoms, ridgelet packets). However, real images present also a set of details less oriented and more isotropic, like corners, spots, texture components, etc. This paper develops an adaptive representation for all these image elements, ranging from highly directional ones to fully isotropic ones. This new tool can indeed be tuned relatively to these image features by decomposing them into a Littlewood–Paley frame of directional wavelets with variable angular selectivity. Within such a decomposition, 2D wavelets inherit some particularities of the biorthogonal circular multiresolution framework in their angular behavior. Our method can therefore be seen as an angular multiselectivity analysis of images. Two applications of the proposed method are given at the end of the paper, namely, in the fields of image denoising and N-term nonlinear approximation.

Keywords:

AMSC: 42C40, 65T60, 68U10, 94A08, 62H35

References

J.-P. Antoineet al., Signal Process. 31, 241 (1993), DOI: 10.1016/0165-1684(93)90085-O. Crossref, Web of Science, Google Scholar
J.-P. Antoine and R. Murenzi, Signal Process. 52, 259 (1996), DOI: 10.1016/0165-1684(96)00065-5. Crossref, Web of Science, Google Scholar
J.-P. Antoine, R. Murenzi and P. Vandergheynst, Int. J. Imag. Syst. Tech. 7, 152 (1996). Crossref, Web of Science, Google Scholar
J.-P. Antoine, R. Murenzi and P. Vandergheynst, Appl. Comput. Harmon. Anal. 6, 314 (1999), DOI: 10.1006/acha.1998.0255. Crossref, Web of Science, Google Scholar
A. Karasaridis and E. Simoncelli, A filter design technique for steerable pyramid image transforms, Int. Conf. Acoustics Speech and Signal Processing (1996) p. 4. Google Scholar
E. P. Simoncelli, Bayesian Inference in Wavelet Based Models, eds. P. Müller and B. Vidakovic (Springer-Verlag, New York, 1999) pp. 291–308. Crossref, Google Scholar
M. N. Doet al., On the compression of two-dimensional piecewise smooth functions, IEEE Int. Conf. on Image Proc. — ICIP '01 (2001) pp. 14–17. Google Scholar
E. Candès and D. Donoho, Curve and Surface Fitting, eds. L. L. Schumakeret al. (Vanderbilt University, 1999) pp. 105–120. Google Scholar
E. Candèset al., SIAM Multiscale Model. Simul. 5, 861 (2006). Crossref, Web of Science, Google Scholar
L. Demanet and L. Ying, Wave atoms and sparsity of oscillatory patterns, preprint (June 2006) . Google Scholar
A. G. Flesia et al. , Beyond Wavelets , eds. J. Stoeckler and G. V. Welland ( Academic Press , 2002 ) . Google Scholar
R. R. Coifman and M. V. Wickerhauser, IEEE Trans. Inform. Theory 32, 712 (1992). Google Scholar
J.-P. Antoine et al. , Two-Dimensional Wavelets and Their Relatives ( Cambridge University Press , Cambridge, UK , 2004 ) . Crossref, Google Scholar
M. Holschneider, J. Math. Phys. 31, 39 (1990), DOI: 10.1063/1.528825. Crossref, Web of Science, Google Scholar
I. Daubechies , Ten Lectures on Wavelets ( SIAM , Philadelphia, PA , 1992 ) . Crossref, Google Scholar
B. Torrésani , Analyse Continue par Ondelettes ( InterÉditions/CNRS Éditions , Paris , 1995 ) . Crossref, Google Scholar
M. Unser, A. Aldroubi and M. Eden, IEEE Trans. Signal Process. 41, 821 (1993), DOI: 10.1109/78.193220. Crossref, Web of Science, Google Scholar
A. Cohen, I. Daubechies and J. Feauveau, Commun. Pure Appl. Math. 45, 485 (1992), DOI: 10.1002/cpa.3160450502. Crossref, Web of Science, Google Scholar
D. L. Donoho, IEEE Trans. Inform. Theory 41, 613 (1995), DOI: 10.1109/18.382009. Crossref, Web of Science, Google Scholar
J. Starck, E. Candès and D. Donoho, IEEE Trans. Image Process. 11, 670 (2002), DOI: 10.1109/TIP.2002.1014998. Crossref, Web of Science, Google Scholar
S. Haykin, Communication Systems, 4th edn. (New York, Wiley, 2000). Google Scholar
L. Jacques, Ondelettes, Repères et Couronne Solaire, PhD thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (2004) . Google Scholar
S. Mallat , A Wavelet Tour of Signal Processing ( Academic Press , San Diego , 1998 ) . Google Scholar
N. G. Kingsbury, Appl. Comput. Harmon. Anal. 3, 234 (2001). Google Scholar
I. W. Selesnick, R. G. Baraniuk and N. Kingsbury, IEEE Signal Process. Mag. 22(6), 123 (2005), DOI: 10.1109/MSP.2005.1550194. Crossref, Web of Science, Google Scholar
Caroline Chaux, IEEE Trans. Image Process. 15(8), 2397 (2006), DOI: 10.1109/TIP.2006.875178. Crossref, Web of Science, Google Scholar