Comptes Rendus
no. 12
Mathematical analysis/Theory of signals
Best bases for signal spaces

Presented by: Haïm Brézis

Yonathan Aflalo1; Haïm Brezis2, 3; Alfred Bruckstein1; Ron Kimmel1; Nir Sochen4
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, Volume 354 (2016) no. 12, pp. 1155-1167.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.10.002

Yonathan Aflalo 1; Haïm Brezis 2, 3; Alfred Bruckstein 1; Ron Kimmel 1; Nir Sochen 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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Yonathan Aflalo; Haïm Brezis; Alfred Bruckstein; Ron Kimmel; Nir Sochen. Best bases for signal spaces. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1155-1167. doi : 10.1016/j.crma.2016.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.002/

[1] Y. Aflalo; H. Brezis; R. Kimmel 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] H. Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Universitext Series, 2010

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

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

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