Digital Signal Processing Reference
In-Depth Information
Itoh [
65
] has recently defined a map from complete Riemannian manifold of
negative curvature to its boundary in term of its Poisson kernel, as Douady and Earle
map, to investigate geometry of the pull-back metric of the Fisher information metric
by this map based on a paper of Friedrich [
55
] that studied the space of probability
measure with respect to the Fisher information metric.
More recently, Airault et al. [
4
] have extended study of Brownian motion on the
diffeomorphism group of the circle to Brownian motion on Jordan curves in
C
based
on a Douady-Earle type conformal extension of vector fields on the circle to the disk.
The aim of one of Malliavin's projects was the construction of natural measures on
infinite dimensional spaces, like Brownian measures on the diffeomorphism group
of the circle, on the space of univalent functions of the unit disk and on the space
of Jordan curves in the complex plane. He understood that unitarizing measures
for representations of Virasoro algebra can be approached as invariant measures of
Brownian motion on the diffeomorphism group with a certain drift defined in terms
of a Kähler potential [
2
,
3
,
82
,
83
].
9.20 Conclusion
Fréchet median with Information Geometry and Geometry of
HPD
(
n
) matrices is
a new tool for Radar Signal Processing that could improve drastically performance
and robustness of classical methods, in Doppler processing and in STAP. Obviously,
these approaches could be extended to Array Processing and Polar Data Processing in
the same way on respectively spatial covariance matrix and Polar covariance matrix.
Future works will be dedicated to deepen close relations of Information Geometry
with Lagrange Symplectic Geometry and Geometric Quantization.
I would like to give many thanks to all member of Brillouin seminar for inter-
esting discussion, hosted in IRCAM by Arshia Cont, since 2009. I am especially
very graceful for Le YANG under supervision of Marc Arnaudon, who has proven
consistency and convergence of all these algorithms with rigorous developments and
generalizations [
120
].
In 1943 seminal Fréchet's paper [
52
], where he introduced for the first time what
it is called nowadays “Cramer-Rao Bound”, we can read at the bottom of first page
Le contenu de ce mémoire a formé une partie de notre cours de statistique mathéma-
tique de l'Institut Henri Poincaré pendant l'hiver 1939
-
1940
. With help of Cédric
Villani, new Intitut Henri Poincaré director, we have looked for this Fréchet Lecture
in IHP without success. I have recently visited in Archive of French Academy of
Science, quai Conti, the “Fonds Fréchet” that are made of 28 boxes with all origi-
nal manuscripts, papers and works of Fréchet. For the time being, I have had only
time to read papers of Box 16 where Fréchet study statistics of human profiles and
cranes with computing resources of first IHP Computation Center. I hope to have
enough time soon to explore all these “Fonds Fréchet” to find this historical Lec-
ture. More recently, Emery gave me the advice to look for Fréchet's document in
Pantheon-Sorbonne university Archive.
