Fonction symétrique | EPFL Graph Search

En mathématiques, une fonction symétrique est une fonction invariante par permutation de ses variables. Le cas le plus fréquent est celui d'une fonction polynomiale symétrique, donnée par un polynôme symétrique. Définition Une fonction f(x_1,\ldots,x_n) en n variables est symétrique si pour toute permutation s de l'ensemble d'indices {1, … ,n}, l'égalité suivante est vérifiée : :f(x_1,\dots,x_n)=f(x_{s(1)},\dots,x_{s(n)}). Pour n = 1, toute fonction est symétrique. Pour n = 2, la fonction (x,y)\mapsto\mathrm e^{xy} est symétrique, alors que la fonction x\operatorname e^y ne l'est pas. Une équation f(x_1,\ldots,x_n)=0 est une équation symétrique lorsque la fonction f(x_1,\ldots,x_n) est symétrique. Exemples Les fonctions :f(x_1,x_2) = x_1 + x_2 et f(x_1,x_2) = x_1x_2 sont symétriques. Le discriminant en trois variables :f(x_1,x_2,x_3)

Generalized Biorthogonal Daubechies Wavelets

Thierry Blu, Michaël Unser, Cédric René Jean Vonesch

We propose a generalization of the Cohen-Daubechies-Feauveau (CDF) and 9⁄7 biorthogonal wavelet families. This is done within the framework of non-stationary multiresolution analysis, which involves a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. We consider a dual pair of such multiresolutions, where the scaling functions at a given scale are mutually biorthogonal with respect to translation. Also, they must have the shortest-possible support while reproducing a given set of exponential polynomials. This constitutes a generalization of the standard polynomial reproduction property. The corresponding refinement filters are derived from the ones that were studied by Dyn et al. in the framework of non-stationary subdivision schemes. By using different factorizations of these filters, we obtain a general family of compactly supported dual wavelet bases of

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. In particular, if the exponential parameters are all zero, one retrieves the standard CDF B-spline wavelets and the 9⁄7 wavelets. Our generalized description yields equivalent constructions for E-spline wavelets. A fast filterbank implementation of the corresponding wavelet transform follows naturally; it is similar to Mallat's algorithm, except that the filters are now scale-dependent. This new scheme offers high flexibility and is tunable to the spectral characteristics of a wide class of signals. In particular, it is possible to obtain symmetric basis functions that are well-suited for image processing.

SPIE2005

Multiwavelet-Like Bases for High Quality Image Interpolation

Thierry Blu, Michaël Unser

We present a simple but generalized interpolation method for digital images that uses multiwavelet-like basis functions. Most of interpolation methods uses only one symmetric basis function; for example, standard and shifted piecewise-linear interpolations use the “hat” function only. The proposed method uses q different multiwavelet-like basis functions. The basis functions can be dissymmetric but should preserve the “partition of unity” property for high-quality signal interpolation. The scheme of decomposition and reconstruction of signals by the proposed basis functions can be implemented in a filterbank form using separable IIR implementation. An important property of the proposed scheme is that the prefilters for decomposition can be implemented by FIR filters. Recall that the shifted-linear interpolation requires IIR prefiltering, but we find a new configuration which reaches almost the same quality with the shifted-linear interpolation, while requiring FIR prefiltering only. Moreover, the present basis functions can be explicitly formulated in time-domain, although most of (multi-)wavelets don't have a time-domain formula. We specify an optimum configuration of interpolation parameters for image interpolation, and validate the proposed method by computing PSNR of the difference between multi-rotated images and their original version.

SPIE2003

Generalized Biorthogonal Daubechies Wavelets

Thierry Blu, Michaël Unser, Cédric René Jean Vonesch

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SPIE2005