no. 12
Mathematical analysis/Theory of signals
Best bases for signal spaces
[Bases optimales pour des espaces de signaux]

Présenté par : Haïm Brézis

Aflalo, Yonathan1 ; Brezis, Haïm2, 3 ; Bruckstein, Alfred1 ; Kimmel, Ron1 ; Sochen, Nir4
1 Computer Science Department, Technion – I.I.T., 32000 Haifa, Israel
2 Department of Mathematics, Technion – I.I.T., 32000 Haifa, Israel
3 Department of Mathematics, Rutgers University, USA
4 Department of Applied Mathematics, Tel Aviv University, Tel Aviv 69978, Israel
Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1155-1167.

Dans cette note, nous abordons le problème de la recherche de bases orthonormales optimales en vue de représenter des signaux définis de façon, soit statistique, soit déterministe, ou selon une combinaison des deux. Dans tous les cas, nous montrons que ces bases proviennent de l'analyse spectrale d'une matrice hermitienne qui regroupe l'information émanant des signaux que l'on souhaite représenter. Nous prouvons aussi l'unicité de la base dans le cas discret.

We discuss the topic of selecting optimal orthonormal bases for representing classes of signals defined either through statistics or via some deterministic characterizations, or combinations of the two. In all cases, the best bases result from spectral analysis of a Hermitian matrix that summarizes the prior information we have on the signals we want to represent, achieving optimal progressive approximations. We also provide uniqueness proofs for the discrete cases.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.10.002
Aflalo, Yonathan 1 ; Brezis, Haïm 2, 3 ; Bruckstein, Alfred 1 ; Kimmel, Ron 1 ; Sochen, Nir 4

1 Computer Science Department, Technion – I.I.T., 32000 Haifa, Israel
2 Department of Mathematics, Technion – I.I.T., 32000 Haifa, Israel
3 Department of Mathematics, Rutgers University, USA
4 Department of Applied Mathematics, Tel Aviv University, Tel Aviv 69978, Israel 
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     title = {Best bases for signal spaces},
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     pages = {1155--1167},
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Aflalo, Yonathan; Brezis, Haïm; Bruckstein, Alfred; Kimmel, Ron; Sochen, Nir. Best bases for signal spaces. Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1155-1167. doi : 10.1016/j.crma.2016.10.002. http://www.numdam.org/articles/10.1016/j.crma.2016.10.002/

[1] Aflalo, Y.; Brezis, H.; Kimmel, R. On the optimality of shape and data representation in the spectral domain, SIAM J. Imaging Sci., Volume 8 (2015) no. 2, pp. 1141-1160

[2] Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Universitext Series, 2010

[3] Gammell, J.D.; Barfoot, T.D. The probability density function of a transformation-based hyperellipsoid sampling technique, 2014 | arXiv

[4] Mirsky, L. A trace inequality of John von Neumann, Monatsh. Math., Volume 79 (1975), pp. 303-306

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