Celebratio Mathematica

A. Cohen, I. Daubechies, and J.-C. Feauveau: “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45 : 5 (1992), pp. 485–560. MR 1162365 Zbl 0776.42020 article

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.

Albert Cohen	Related
Jean-Christophe Feauveau	Related

A. Cohen and I. Daubechies: “A stability criterion for biorthogonal wavelet bases and their related subband coding scheme,” Duke Math. J. 68 : 2 (November 1992), pp. 313–335. MR 1191564 Zbl 0784.42022 article

A. Cohen and I. Daubechies: “Orthonormal bases of compactly supported wavelets, III: Better frequency resolution,” SIAM J. Math. Anal. 24 : 2 (1993), pp. 520–527. Part I was published in Commun. Pure Appl. Math. 41:7 (1988). MR 1205540 Zbl 0792.42019 article

A. Cohen, I. Daubechies, B. Jawerth, and P. Vial: “Multiresolution analysis, wavelets and fast algorithms on an interval,” C. R. Acad. Sci., Paris, Sér. I 316 : 5 (1993), pp. 417–421. MR 1209259 Zbl 0768.42015 article

Albert Cohen	Related
Björn David Jawerth	Related
Pierre Vial	Related

A. Cohen and I. Daubechies: “Non-separable bidimensional wavelet bases,” Rev. Mat. Iberoam. 9 : 1 (1993), pp. 51–137. MR 1216125 Zbl 0792.42021 article

We build orthonormal and biorthogonal wavelet bases of \( L^2(\mathbb{R}^2) \) with dilation matrices of determinant 2. As for the one dimensional case, our construction uses a scaling function which solves a two-scale difference equation associated to a FIR filter. Our wavelets are generated from a single compactly supported mother function. However, the regularity of these functions cannot be derived by the same approach as the one dimensional case. We review existing techniques to evaluate the regularity of wavelets, and we introduce new methods which allow to estimate the smoothness of non-separable wavelets and scaling funtions in the most general situations. We illustrate these with several examples.

A. Cohen and I. Daubechies: “On the instability of arbitrary biorthogonal wavelet packets,” SIAM J. Math. Anal. 24 : 5 (1993), pp. 1340–1354. MR 1234020 Zbl 0792.42020 article

A. Cohen, I. Daubechies, and P. Vial: “Wavelets on the interval and fast wavelet transforms,” Appl. Comput. Harmon. Anal. 1 : 1 (1993), pp. 54–81. MR 1256527 Zbl 0795.42018 article

Albert Cohen	Related
Pierre Vial	Related

A. Cohen, I. Daubechies, and A. Ron: “How smooth is the smoothest function in a given refinable space?,” Appl. Comput. Harmon. Anal. 3 : 1 (January 1996), pp. 87–89. MR 1374399 Zbl 0901.46024 article

Albert Cohen	Related
Amos Ron	Related

A. Cohen and I. Daubechies: “A new technique to estimate the regularity of refinable functions,” Rev. Mat. Iberoam. 12 : 2 (1996), pp. 527–591. MR 1402677 Zbl 0879.65102 article

A. Cohen, I. Daubechies, and G. Plonka: “Regularity of refinable function vectors,” J. Fourier Anal. Appl. 3 : 3 (1997), pp. 295–324. MR 1448340 Zbl 0914.42025 article

Albert Cohen	Related
Gerlind Plonka-Hoch	Related

A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore: “Tree approximation and optimal encoding,” Appl. Comput. Harmon. Anal. 11 : 2 (2001), pp. 192–226. MR 1848303 Zbl 0992.65151 article

Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual \( n \)-term approximation. We show that the restrictions of tree approximation cost little in terms of rates of approximation. We then use that result to design encoders for compression. These encoders are universal (they apply to general functions) and progressive (increasing accuracy is obtained by sending bit stream increments). We show optimality of the encoders in the sense that they provide upper estimates for the Kolmogorov entropy of Besov balls.

Albert Cohen	Related
Ronald Alvin DeVore	Related
Wolfgang A. Dahmen	Related

A. Cohen, I. Daubechies, O. G. Guleryuz, and M. T. Orchard: “On the importance of combining wavelet-based nonlinear approximation with coding strategies,” IEEE Trans. Inform. Theory 48 : 7 (2002), pp. 1895–1921. MR 1929999 Zbl 1061.94004 article

This paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory. Contrasting linear and nonlinear approximation spaces, we show that there are interesting classes of functions/random processes which are much more compactly represented by wavelet-based nonlinear approximation. These classes include locally smooth signals that have singularities, and provide a model for many signals encountered in practice, in particular for images. However, we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate-distortion sense. Based on this observation, we construct compression techniques and formulate the family of functions/stochastic processes for which they provide efficient descriptions in a rate-distortion sense. We show that this family invariably leads to Besov spaces, yielding a natural relationship among Besov smoothness, linear/nonlinear approximation order, and compression performance in a rate-distortion sense. The designed compression techniques show similarities to modern high-performance transform codecs, allowing us to establish relevant rate-distortion estimates and identify performance limits.

Albert Cohen	Related
Onur G. Guleryuz	Related
Michael T. Orchard	Related

A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore: “Harmonic analysis of the space BV,” Rev. Mat. Iberoamericana 19 : 1 (2003), pp. 235–263. MR 1993422 Zbl 1044.42028 article

Albert Cohen	Related
Ronald Alvin DeVore	Related
Wolfgang A. Dahmen	Related

A. Cohen, I. Daubechies, R. DeVore, G. Kerkyacharian, and D. Picard: “Capturing ridge functions in high dimensions from point queries,” Constr. Approx. 35 : 2 (April 2012), pp. 225–243. MR 2891227 Zbl 1318.62286 article

Constructing a good approximation to a function of many variables suffers from the “curse of dimensionality”. Namely, functions on \( \mathbb{R}^N \) with smoothness of order \( s \) can in general be captured with accuracy at most \( O(n^{-s/N}) \) using linear spaces or nonlinear manifolds of dimension \( n \). If \( N \) is large and \( s \) is not, then \( n \) has to be chosen inordinately large for good accuracy. The large value of \( N \) often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on \( N \) variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function \[ f(x) = g(a\cdot x) \] when both \( a\in\mathbb{R}^N \) and \( g\in C[0,1] \) are unknown. We establish estimates on how well \( f \) can be approximated using these point queries under the assumptions that \( g\in C^s[0,1] \). We also study the role of sparsity or compressibility of \( a \) in such query problems.

Ronald Alvin DeVore	Related
Dominique B. Picard	Related
Albert Cohen	Related
Gérard Kerkyacharian	Related

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