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Applications Radarde la Géométrie de l'information

associée aux matrices de covariances : traitements spatio-temporels

Frédéric BARBARESCOThales Air Operations

Domaine Surface Radar, Direction TechniqueDépartement Développements Avancés


Radar Applications ofInformation Geometry

based on covariance matrices : Space-Time Processing

Frédéric BARBARESCOThales Air Operations

Domaine Surface Radar, Direction TechniqueDépartement Développements Avancés


Radar Signal Processing :Doppler Processing

4 /4 / Doppler Sensors : Radar, Sonar & Lidar

Radar (RAdio Detection And Ranging) Technology

107 Years old (C. Hülsmeyer, 1904)

Sonar (SOund Navigation And Ranging)

(piezoelectric effect) 95 years old (Paul Langevin & Constantin Chilowski, 1916)

Lidar (LIght Detection And Ranging)Technology

(laser) 51 years old (T. Maiman, 1960)

5 /5 / All these sensors use Doppler-Fizeau Effects

Woldemar Voigt (1850 - 1919)

Armand Hippolyte Louis Fizeau (1819 – 1896)

Christian Andreas Doppler (1803- 1853)

freq Radial Velocity(Doppler Spectrum Mean)

Var(freq) Turbulence (Doppler Spectrum Width)

6 /6 / Radar Processing based on Covariance Matrix

Doppler Processing(Time Covariance Matrix)

Antenna Processing(Space Covariance Matrix)

STAP Processing(Space-Time Covariance Matrix)

Polarimetric Radar Processing(Polarimetry Covariance Matrix)

7 /7 /


Military Air Defense Application

Detection of slow/stealth targets in inhomogeneous Clutter

New requirements in Air Defense to detect low altitude or surface targets at low elevation. Target Doppler is very close to fluctuating Clutter Doppler (Ground & Sea Clutters)

Detection of asymmetric & stealth targets in Ground Clutter Microlight Airplane General Aviation UAV & Micro UAV Micro Helicopter

Detection of small targets in Sea Clutter Wooden & inflatable canoe Jetski Unmanned Boat Naval Micro Helicopter Periscope

Detection of low RCS targetsDetection of teneous Doppler SignalIncrease Range & Reactivity

8 /8 /


Civil Air Traffic Control Application

Monitoring of turbulences: New Requirement in ATC (SESAR)

In Turbulences, signal is no longer characterized by Doppler velocity Mean but by Doppler Spectrum Width & “shape”

Atmospheric Air turbulences Eddy Dissipation Rate

Turbulent Kinetic Energy

Airplane Wake-Vortex turbulences (A380, B747-8) Circulation

Decay Rate

Windshear in Final Approach Headwind

Crosswind

Improve Safety by mitigating weather hazardsIncrease Capacity by reducing safe separations

9 /9 /


Challenges of Doppler Radar Processing: Detection on Ground

Detection of asymmetric & stealth targets in Inhomogenesous Ground Clutter

Classical Doppler Filter Banks (or FFT) are not efficient with very short bursts (<16 pulses) : Low Resolution of Doppler Filters with short Bursts (Low

sidelobes / high loss, wide filter)

If Target Doppler is between two Doppler filters, energy is spread on adjacent filters. Gain between 2 filters is lower thangain at filter center ("Straddling loss")

Ground Clutter Energy is not limited to zero-Doppler filter but pollution is spread over all filters due to poor Filter-Banks Resolution & Doppler side lobes in case of very short Bursts.

Filter 0 Filter 1 Filter 7Filter 2 Filtre 6Filter 3 Filter 4 Filter 5

Pollution of all Doppler Filters in case of Burst with low number of pulses

0.5 0.6 0.7 0.8 0.9 1e normalisee (V/Va)

0 0.1 0.2 0.3 0.4 0.50

0

0

0

0

0

0

0

Frequence normalisee

S/N gain [dB]

0-10-20-30-40-50-60 V/ Vamb

10 /10 /


Challenges of Doppler Radar Processing:Detection in Sea

Detection of slow targets in Sea Clutter

Sea Clutter is highly inhomogeneous Doppler fluctuation

Time/space Fluctuation

Sea Clutter is dependant of Sea current

Surface wind

fetch

Bathymetry

Sea Clutter is corrupted by Spikes due to breaking waves

“Moutonement”

Offshore SeaDoppler Spectrum

Close to the Shore Sea Doppler Spectrum

(Breaking waves)

11 /11 /


Challenges of Doppler Radar Processing: Atm. turbulences

Monitoring of atmospheric turbulences

Air turbulence is characterized by Spread of fluctuating speeds

This composition of different Doppler speeds in Radar cells generates a Widen Doppler Spectrum

Speed variance of Doppler Spectrum Width are related to 2 measures of turbulence : EDR: Eddy Dissipation Rate

TBE: Turbulent Kinetic Energy

12 /12 /

Monitoring of Wake-Vortex Turbulences

Wake vortex generate to contra-rotative roll-up spirals

Mean speed depends on cross-wind Wake-Vortex has spiral geometry with

increasing speed in the core and decreasing speed outside the core

Wake-Vortex Speed and structure depend on Wake-Vortex age/decay phase

Wake-Vortex Strength is characterized by Circulation in m2/s

Monitoring of Windshear

Inversion of speed in range or in altitude Microburst in the same radar cell


Challenges of Doppler Radar Processing: Wake Turbulences

Wake-Vortex Doppler Spectrum

Wind-ShearDoppler Spectrum


Probability Metrics: Information Geometry

& Optimal Transport

14 /14 /


Probability Metrics: Distance between 2 random variables

Question in Probability and Statistic: Could we define distance between 2 random variables ?

Or equivalently, could we define distance between 2 probability densities of these 2 random variables ?

)1(

)1(1

)1(2

)1(1

)1(

n

n

zz

zz

Z

)2(

)2(1

)2(2

)2(1

)2(

n

n

zz

zz

Z ???, )2()1( ZZd

)1()1( /Zp )2()2( /Zp ???, )2()1( d

15 /15 / 2 Definitions of Probability Metric

What is the good geometry : geometry with best properties for our applications in Radar Processing

Maurice René Fréchet

1957 (CRAS)Fréchet-Levy Distance

Optimal Transport Theory

1939 (IHP Lecture)Fréchet Bound (Cramer-Rao)

Information Geometry

1948 (Annales de l’IHP)Les éléments aléatoires de nature quelconque

dans un espace distanciéExtension of Probability/Statistic in abstract space

1928

16 /16 / Information Geometry: Fréchet-Darmois-Cramer-Rao Bound

Information Geometry :

Fisher Information Matrix and Fréchet-Darmois-Cramer-Rao Bound

19451943

(1939 IHP Lecture)

1ˆˆ

IE

FDCR Bound

*

2

, /ln)(

jiji

XpEI

Fisher Information Matrix R.A. Fisher

17 /17 / Information Geometry : Rao-Chentsov metric

Information Geometry :

Rao-Chentsov Metric defined with Kullback divergence

Kulback-Leibler Divergence :

Rao-Chentsov Metric (invariance by parameter changes)

jijiji ddgdIddXpXpKds

,

*,

2 )(/,/

jiji Ig , , )(

1945Calyampudi Radhakrishna Rao Nikolai Nikolaevich Chentsov

1964 (axiomatisation)

dxxqxpxpeEESupqpK qp

//ln/ln),(

18 /18 / Dual Information GeometryDual Coordinates systems & Potential functions

Potential Functions are Dual and related by Legendre transformation :

enΗ)(ηΗηΗΦ

nTrΘΨ

mmmη,ΗΗ

Σ)m,(Σθ,ΘΘ

T

T

T

2log2detlog21log2~~)log(2detlog22~~

,~2~

scoordinate Dual

1111

1112

11

ΘΗΦ

θηΦ

Η

ΘΨ

ηθΨ

~

~

and~

~

TT ΘΗTrH,Θ

ΨH,ΘΦ

~~with

~~~~

pEΗΦ log~~ Entropy

ji

*ij

jiij ΗΗ

ΦgΘΘΨg

~ and

~ 22

Hessians are convexe and define Riemannian metrics :

19 /19 / Combinatorial/Variational Foundation of Kullback Divergence

Combinatorial Fundation of Kullback Divergence Kullback Divergence can be naturally introduced by combinatorial elements and

stirling formula :

)(log),(

eEESupqpK qp

in

M

i i

ni

MMM nqNqqnnnP

i

1121 !

!,...,/,...,,

iq

M

ii Nn

1 Nnp i

i

n when ..2..! nenn nn

),(log.log11

qpKqppP

NLim

M

i i

iiMN

Let multinomial Law of N elements spread on M levels

with priors , and

Sirling formula gives :

Variational Foundation of Kullback Divergence Donsker and Varadhan have proposed a variational definition of Kullback divergence :

20 /20 / Kullback Divergence & VARADHAN’s Variational Approach

)(log),(

eEESupqpK qp

Donsker and Varadhan have proposed a variational definition of Kullback divergence :

),()1ln(),()()()(ln

)()(ln)()(ln)(

)()(ln)( :Consider

qpKqpKqpq

qppeEE

qp

qp

0)()(ln)()(ln)(),(

)()()(with

)()(ln)(

)(ln)(ln)(

)(

)(

qppeEEqpK

eqeqq

qqp

eEeEeEE

qp

qpqp

This proves that the supremum over all is no smaller than the divergence

Using the divergence inequality,

Link with « Large Deviation Theory » & Fenchel-Legendre Transform which gives that logarithm of generating function are dual to Kullback Divergence :

)(

(.)

)(

(.)

)(

log)(),(

)(log)()(),(

),()()()(log

xVqp

V

xV

V

p

xV

eEVESupqpK

dxxqedxxpxVSupqpK

qpKdxxpxVSupdxxqe

21 /21 / Optimal Transport Theory: Wasserstein distance

Wasserstein distance

Framework of optimal transport theory

Particular cases of Wasserstein distance

Case n = 1 : Monge-Kantorovich-Rubinstein distance

Case n=2 : Fréchet distance

)(,)(,,,

),(),(,

/1

/1

),(

YlawXlawYXdEInfW

yxdyxdInfW

nnn

n

nn

YXEInfQPdWYX

,

1 ,),(

2

,2 ,),( YXEInfQPdW

YX

Cédric Villani, Field Medal 2010, Book on « Optimal Transport

Theory : Old & New »

22 /22 / Optimal Transport Theory : Fréchet-Wasserstein distance

Fréchet distance

In 1957, Maurice René Fréchet has introduced a distance, based on Paul Levy’s paper (particular case of Wasserstein distance)

dxdyyxHyxYXEInfGFdYX

),()(, 22

,

2

23 /23 / Optimal Transport Theory : Fréchet-Wasserstein distance

Fréchet Distance

Fréchet’s paper from 1957 in CRAS :

Paul Levy’s letter to Maurice Fréchet (2nd of April 1958) … J’ai ainsi pu apprécier ce que vous aviez fait, en prenant comme point de départ de votre

mémoire ce que vous appelez ma première définition de la distance de deux lois de probabilité(en fait ce n’était pas la première). Vous l’avez d’ailleurs généralisée, en ce sens que je ne l’avais associée qu’à une de vos définitions de deux variables aléatoires. Et j’ai beaucoup admirécomment avec votre quatrième définition, vous arrivez à faire quelque chose de maniable d’une idée qui pour moi était surtout théorique, vu la difficulté de déterminer le minimum de la distance de deux variables aléatoires ayant les répartitions marginales données.

24 /24 / 4th definition of Fréchet distance

4th Fréchet Distance

Extreme Fréchet Copulas

4th Fréchet’s Distance

)(),(),(with

),(,

1

122

yGxFMinyxH

dxdyyxHyxGFd

)(),(),(

0,1)()(),(with

),(),(),(

1

0

01

yGxFMinyxHyGxFMaxyxH

yxHyxHyxH

25 /25 / Application for Multivariate Circular Gaussian Law of 0 mean

Model : Multivariate Circular Gaussian Law of zero mean

Rao-Chentsov(-Siegel) Metric & distance

Fréchet-Levy(-Wasserstein) distance

RRE

mZmZRe

RRmZp RRTr

n ˆ

ˆ with

det1,/

1ˆ

0m

22/12/1212

FdRRRdRRTrds

Case with

0det...detwith

ln..log,

2/12/11

222/12/12

XYXYX

n

kkFXYXYX

RRIRRR

RRRRRd

2/12/12/12 .2, XYXYXYX RRRTrRTrRTrRRd

26 /26 / Properties of each geometry

Information Geometry

Geodesic

Space with sectional Negative Curvature

Invariance by parameters changes

Wasserstein Geometry

Geodesic

Space with sectional Positive Curvature

YXYX

Xt

XYXXXRRRt

X

RRRRRRRRRReRt XYX

)2/1( and )1( , )0()( 2/12/12/12/12/1log2/1 2/12/1

12/12/12/12/12/1

,

,,)(

with

.)1(.)1(

YXXXYXXYX

YXkYYXkt

RRRRRRRD

DtItRDtItR

Existence and Unicity of

Barycenter only proved in case of

negative curvature by Elie

Cartan during 20’s

27 /27 / Information Geometry : Multivariate Gaussian Case, m=0

Information Geometry for Multicariate Gaussian Laws(cas m=0)

Rao-Chentsov distance

Particular case of Carl-Ludwig Siegel case (in the framework of symplectic geometry)

Siegel Metric

Invariant by all automorphisms of SHn

Particular Case :

0det...det avec

ln..log,

2/12/11

222/12/12

XYXYX

n

kkXYXYX

RRIRRR

RRRRRd

0Im/),( Y(Z)CnSymiYXZSH n

iYXZZdYdZYTrdsSiegel avec ... 112

nTT IBCDADCZBAZZM avec )( 1

2120dRRTrds

RYX

C.L. Siegel

28 /28 / Example : Monovariate Gauss-Laplace LawGauss-Laplace Law

Fisher Information Matrix for Gaussian Law :

mIEI

T and )(ˆˆ with

2001

.)( 12

2

22

2

2

2

22

2.2.2..

ddmddmdIdds T

Rao-Chentsov Metric from Information Geometry

H. Poincaré

.2

imz 1

iziz

22

22

1.8

dds

This metric is the Poincaré metric (model of hyperbolic geometry)

)*2()1(

)2()1()2()1(

2

)2()1(

)2()1(

22112

1),(with

),(1),(1log.2,,,

mmd

29 /29 / Poincaré Space in Art (Escher, Irène Rousseau )

30 /30 / Poincaré & Siegel Upper Half Plane & Disk

*1*1*2

22

22

11

1

dwwwdwwwds

w

dwds

izizw

1 iIZiIZW

Poincaré Upper Half Plane Poincaré Unit disk

Siegel Upper Half Plane Siegel Unit disk

),( and ),(with

112

CnHPDYCnHermXiYXZ

ZddZYYTrds

0 and avec

*112

2

2

2

222

yiyxzdzdzyyds

ydz

ydydxds

dWWWIdWWWITrds 112

31 /31 / Siegel Space

0Im/),( Y(Z)CnSymiYXZSH n

Siegel Upper Half Space

X

0Y

ZdYdZYTrds 112

n

iiiZZd

1

221

2 1/1log,

kkk YiXZ .

1k

2k

212 dRRTraceds

kkkk RNWRiZ ,0 if .

1k

2k

n

kkRRd

1

221

2 log,

12121

1212121,

ZZZZZZZZZZR

0...det 2/112

2/11 IRRR

0.,det 21 IZZR

L.K. Hua

C.L. Siegel

F. Berezin

32 /32 /Information Geometry based on Poincaré’s hyperbolic geometry

259 Letters between Mittag-Leffler & Poincaré

• « Acta Mathematica » was founded by Gösta Mittag-Leffler in 1882• Henri Poincaré has published paper on Fuchsian Group in first volume of Acta Mathematica: Henri Poincaré (1882) "Théorie des Groupes Fuchsiens", Acta Mathematica v.1, p.1.

33 /33 /Information Geometry based on Poincaré’s hyperbolic geometry

• Henri Poincaré (1882) "Théorie des Groupes Fuchsiens", Acta Mathematica v.1, p.1., 1882 published by Mittag-Leffler

34 /34 /

« At one point Siegel thought that too many unnecessary things were being published, so he decided not to publish anything at all »George PolyaThe Polya Picture Album, Encounters of a Mathematician, Birkäuser

Carl Ludwig SiegelWith George Polya

Carl Ludwig siegel

35 /35 / Information Geometry : Multivariate Gaussian Case, m=0

Information Geometry for Multivariate Gaussian Laws(cas m=0)

Geodesic :

Properties of this space

Symetric Space as studied by Elie Cartan : Existence of bijective geodesic isometry

Bruhat-Tits Space : semi-parallelogram inequality

Cartan-Hadamard Space (Complete, simply connected withnegative sectional curvature Manifold)

10 with ),(.))(,( ,tRRdttRd YXX

YXYX

Xt

XYXXXRRRt

X

RRRRRRRRRReRt XYX

)2/1( and )1( , )0()( 2/12/12/12/12/1log2/1 2/12/1

2/12/12/12/12/11-),( avec )(X ABAAABABABAXG BA

Xx)d(x,x)d(x,xd(x,z)),xd(x

xzxx

224

que tel ,2

22

122

21

21

36 /36 /

x1x

2x

z

U

VU VU V

Information Geometry : Multivariate Gaussian Case, m=0

Symetric Space as studied by Elie Cartan : Existence bijective geodesic isometry

Bruhat-Tits Space : semi-parallelogram inequality

2222 22 VUVUVU

),( AGB BA

BGA BA ),( )(X-1),( BABAXG BA

X

2/1/212/12/12/1 ABAAABA

1,0)( 2/12/12/12/1

tABAAAt t

A)0(

B)1(

22

21

2221 224 )d(x,x)d(x,xd(x,z)),xd(x

E. J. Cartan

M. Berger


Information Geometry & Fréchet-Karcher

Gradient Flow

38 /38 /

For detection of Slow & Stealth/Small Target in inhomogeneous clutter, we need simultaneously :

High Doppler Resolution with short Bursts Robust CFAR in inhomogeneous clutter & closely separated targets

Proposed Solution : OS-HR-Doppler-CFAR

Avoid drawbacks of Doppler Filters / FFT in case of short bursts Take advantages of Robust Ordered Statistic of OS-CFAR (Ordered

Statistic CFAR, Median-CFAR)

Challenges to define OS-HR-Doppler-CFAR :

Can we order « Doppler Spectrums » : NO there is no total order of covariance matrices R1>R2>…>Rn

There is only Partial « Lowner » order : R1>R2 if and only if R1-R2 Positive Definite

Can we define « Median » of « Doppler Spectrums » : YES !!! In a « Metric Space », the median is defined as the point that minimizes the « geodesic »

distance to each point (compared to the mean that minimizes the square distance to each point)

We can define a deterministic or stochastic gradient flow that converges fastly to « median spectrum » (Modified Karcher Flow : THALES Patent)

Transport Optimal : barycentre de Fréchet-WassersteinOS-HD-Doppler-CFAR

39 /39 /

In Rn, the center of mass is defined for finite set of points

Arithmetic mean:

This point minimizes the function of distances:

The median (Fermat-Weber Point) minimizes :

M

ii

xcenter xxdMinx

1

2 ),(arg

1x

2x

3x

4x

5x

M

ii

xcenter xxMinx

1arg

M

iicenter x

Mx

1

1

Miix ,...,1

M

iicenter x

Mx

1

1

M

ii

xcenter xxdMinx

1

2 ),(arg

Center of Mass : Arithmetic Mean and Median in Rn

M

ii

xmedian xxdMinx

1),(arg

1x

2x

3x

4x

5x

M

ii

i

xmedian

xx

xxMinx1

arg

mxEMinmmmedian 2mxEMinm

mmean

M

ii

xmedian xxdMinx

1),(arg

40 /40 /

Median Mean

x1

x2

x3

x1

x2

x3

Median Mean

x1

x2

x3

x1

x2

x3

x’2 x’2

outliers outliers

MEDIAN MEAN

Sensitivity to outliers : Median versus Mean

41 /41 / Extension of barycenter in metric space

Right Triangle

h2=a2+b2

Naive Mean of N Right Triangles :

Let N Right triangles :

« Arithmetic » Mean is not a Right triangle

Solution : Fréchet Mean (Center of Mass)

Consider the surface h2=a2+b2 in Space of coordinates (a,b,h)

Fréchet Mean/Barycenter/Center-of-Mass

2221 with ,, iii

Niiii bahhba

222

111

,,1,1,1

BAH

HBAhN

bN

aN

N

ii

N

ii

N

ii

22221

2

,,

/,, surfaceon defined (.,.)

,,,,,arg,,

bahhbad

HBAhbadMinHBA

geodesic

N

iiiigeodesicHBA

a

b

h

One Right Triangle could be represented by 1 point on

surface h2=a2+b2

42 /42 / Cartan Center of Mass and Karcher Flow

Cartan Center of Mass

Elie Cartan has proved that the following functional :

is strictly convexe and has only one minimum (center of mass of A for distribution da) for a manifold of negative curvature

Karcher Flow

Hermann Karcher has proved the convergence of the following flow to the Center of Mass :

E. J. Cartan

H. Karcher

A

daamdmf ),(: 2

)()0( avec )(.exp)(1 nnnnmnnn mfmfttmn

)(exp 1 A

m daaf

43 /43 / probability on a manifold : Emery’s exponential barycenter

Maurice René Fréchet, inventor of Cramer-Rao bound in 1939, has also introduced the entire concept of Metric Spaces Geometry and functional theory on this space (any normed vector space is a metric space by defining but not the contrary). On this base, Fréchet has then extended probability in abstract spaces.

In this framework, expectation of an abstract probabilisticvariable where lies on a manifold is introduced by Emery as an exponential barycenter :

In Classical Euclidean space, we recover classical definition of Expectation E[.] :

xyyxd ),(M. R. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace

distancié”, Annales de l’Institut Henri Poincaré, n°10, pp.215-310, 1948

xgEb )(xg x

0)(exp 1 dxPxgM

b

nn RX

Rp

n dxxpxgdxPxgxgEpqqRqp )()()()()()(exp, 1

M. Emery & G. Mokobodzki, “Sur le barycentre d’une probabilité sur une variété”, Séminaire de Proba. XXV, Lectures note in Math. 1485, pp.220-233, Springer, 1991

44 /44 / Mean & Median of N matrices HPD(n)

Mean of N Hermitian Positive Definite Matrices HPD(n)

Solution given by Karcher Flow with Information Geometry metric

Median (Fermat-Weber Point) of N matrices HPD(n)

PhD Yang Le supervised by Marc Arnaudon (Univ. Poitiers/Thales)

N

kk

N

kk

Xmoyenne XBXBXdMinArgX

1

22/12/1

1

2 ..log,

2/1

log2/1

11

2/12/1

n

XBX

nn XeXX

N

knkn

N

kk

N

kk

Xmediane XBXBXdMinArgX

1

2/12/1

1..log,

knnn

XBXXBX

nn BXkSXeXX nSk nkn

nkn

/ with 2/1loglog

2/11

2/12/1

2/12/1

45 /45 / Median on a Manifold: Karcher-Cartan-Fréchet

Gradient flow on Surface/Manifold

Gradient Flow :Pushed by Sum of Normalized Tangent vectors of

Geodesics

Fréchet-Karcher Barycenter : Sum of Normalized Tangent vectors

of Geodesics is equal to zero

A m

m

MinA

daaahdaamdmh

)(exp)(exp),(

21:

1

1

Sum of Normalized Tangent Vectors

)(exp

exp.exp

11

1

1

M

k km

kmmn x

xtm

n

n

n

Compute point on the surface In the direction of sum of Normalized

Tangent Vector

46 /46 / Mean : Karcher Barycenter

2/1log2/12/1 2/12/12/1 2/12/1

)( XeXXXBXXt XBXttkk

k

2/12/12/12/10 log)( XXBXX

dttd

ktk

N

k

N

kkt

k XXBXXdt

td1

2/1

1

2/12/12/10 0log)(

1B 2B

3B

4B

5B

6B

47 /47 / Comparison of Mean & Median Doppler Spectrum

Raw Doppler Spectrum

Mean Doppler Spectrum

Median Doppler Spectrum

Range axis

Doppler axis

No preservation of discontinuities

Perturbation by outlier

48 /48 /

Diffusion Fourier Equation on 1D graph (scalar case) Approximation par un Laplacien discret :

with arithmetic mean of adjacent points :

Dicrete Fourier Heat Equation for Scalar Values in 1D :

By Analogy, we can define Fourier Heat Equation in 1D grapf of HDP(n) matrices :

nnnnnn uu

xxuu

xuu

xtu

xu

tu

ˆ21

211

2

2

2/ˆ 11 nnn uuuJ. Fourier

Other approach to smooth spectrum : Fourier Heat Equation

2,,1,.2 with ˆ.).1(x

tuuu tntntn

tntntntntntntntn

tntntntntntnXXX

tntn

XXXXXXXX

XXXXXXeXX tntntn

,12/1,12/1

,12/12/1

,1,12/1,1

2/1,1,

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog2/1,1,

ˆwith

ˆ2/1,,

2/1,

49 /49 / Isotropic Diffusion of Doppler Spectrum

50 /50 / Anisotropic Diffusion of Doppler Spectrum

51 /51 /

Geodesic between matrix P and Geodesic between Q & R :

The geodesic projection is a contraction :

Geodesic Projection

Q

R

PP

Q

2/1

2/12/1 2/12/12/12/12/1,)( PPQRQQQPPtstts

s

QPdistQPdist MM ,)(),(

M : geodesically convex closed submanifold of

Hadamard Manifold SPdistP

MSM ,minarg)(

M(P)

M(Q)

P

Q


Optimal Transport Theory&

Gradient Flow

53 /53 / Optimal Transport : Fréchet-Wasserstein distance

Wasserstein Distance for Multivariate Gaussian Laws

Fréchet-Wasserstein distance

Proof If we set

Solution is given by:

YXYX RTrRTrRTrYEXEYXE

YXYXTrEYXE

,22

2

2)()(

0

Y

XW R

RR

YX

W

2/12/12/12/12/1

0..

1

XXYXX

RRRRRRRTrSup

YX

2/12/12/1 ...2 XYX RRRTrTr

XRRRRRY XXYXX ... 2/12/12/12/12/1

54 /54 / Optimal Transport : Fréchet-Wasserstein distance

Wasserstein Distance for Multivariate Gaussian Laws

Fréchet-Wasserstein distance

Geodesic If we set :

Optimal Transport : transport from to

2/12/12/122 .2,,, XYXYXYXYYXX RRRTrRTrRTrmmRmRmd

12/12/12/12/12/1,

YXXXYXXYX RRRRRRRD

YXkYYXkt

XYt

DtItRDtItR

mtmtm

,,)(

)(

.)1(.)1(

.)1(

)1()1()0()0(2)()()()(2 ,,,).(,,, RmRmWstRmRmW ttss

YYk RmNI ,, # YY RmN , XX RmN ,

XYYX mmyDyx ,)(

XYYXY mmvDmvv ,2

1)(

YYYXXx myRmymxRmx 11

55 /55 / Optimal Transport : Fréchet-Wasserstein Distance

Characteristics of this space (case m=0)

Wasserstein metric :

Tangent Space and Exponential Map :

Properties of this space

Alexandrov space

space Positive Sectional Curvature (geodesiquely convexe and simply connected)

YRXTrYXg YRN Y..),(),0(

)(,,0 ,0).(exp tRRN YYRNXt

XtItRXtItR kYktRY.)1(.)1()(,

VTYRN ,0

).(exp ,0 XtYRN

YRRN ,0

2222 )1(),0().1()(,.)0(,).1()(, dtttdtdttd

56 /56 / Optimal Transport : Fréchet-Wasserstein Barycenter

Wasserstein Barycenter for Multivariate Gaussian Laws(Case m=0)

Fréchet-Wasserstein Barycenter

Solution of Fréchet-Wasserstein Barycenter for N Multivariate Gaussian Laws of zero meams Constraint :

Iterative Solution Iterative solution (convergence for d=2, d>3 convergence conditionaly to the

initiation)

2

,2

1

22 ),( with ),( YXEInfWWInf

YX

N

kk

NkkRN 1,0

RRRRN

kk

1

2/12/12/1

with 2/12/1

1

2/1)(2)()1(ii

N

k

nk

nn RKKKKK


Stationnary Signal &Toeplitz Constraint on

Covariance Matrix :Gradient Flow through Partial

Iwasawa & CAR Model

58 /58 /

Covariances matrices are structured matrices that verify the following constraints : Toeplitz Structure (for stationary signal) :

Hermitian structure :

Positive Definite Structure :

How to built a flow that preserves the Toeplitz structure ?

0121

*10

*212

*101

*1

*2

*10

0121

*10

*212

*101

*1

*2

*10

rrrrrrrrr

rrrrrrr

rrrrrrrrr

rrrrrrr

R

n

n

n

n

n

1nR

conjuguate and d transpose: with nn RR

00det such that ,0, nnnn λ.IRZRZCZ

Additional Constraint : Toeplitz Structure

kknn rzzEn * ,

If you remember first example on « Right Triangle », in this case Pythagore constraint is replaced by HPD & Toeplitz constraints

59 /59 / Preservation of Toeplitz Structure

We use Partial Iwasawa Decomposition = Complex AR model

Information Geometry metric (metric = Hessian of Entropy)

1111

121

.1

.1..nnnn

nnnnnnn AAA

AWWR

11

21 .1

nnn

1011)(

nn

nn

Aμ

AA *)( .VJV

Tnn P 110

)(

)*()(

2 ~n

jn

iijg

100P

..ln.1ln).(.logdetlog,~0

1

1

210 PenkneR)P(R

n

kknn

with

1

122

22

0

0)()(2

1)(.

n

ii

inij

nn

din

PdPndgdds

TNN

NNkknn

N

knkn

Nkn aaAbbEbzaz )()(

12

0,*

1

)( and with

K. Iwasawa

E. Kähler

S. Bergman

1t with coefficien reflection : kk

60 /60 / Complex Autoregressive Model : Regularized Burg Algorithm

Regularized Burg Algorithm (THALES Patent)

)(.)1()( )1(.)()(

11 , .

1

).()2.( with ..2)1()(1

...2)1().(2 to1For : (n) tep .

1

)(.1ech.) nb. : (N 1 , )()()(f

:tion Initialisa .

1*

1

11

)(

)*1()1()(

)(0

221

)(

1

1

0

2)1()(21

21

1

1

1

)1()1()(*11

)0(0

1

20

00

kfkbkbkbkfkf

a

,...,n-k=aaa

a

nkakbkf

nN

aakbkfnN

MnSa

kzN

P

,...,N k=kzkbk

nnnn

nnnn

nn

n

nknn

nk

nk

n

nkN

nk

n

k

nk

nknn

N

nk

n

k

nkn

nk

nknn

n

N

k

61 /61 / Median AR model Through Median Reflection Coefficients k

n,3

n,1n,2

nmedian,

Classical Karcher Flow in Unit Disk

1,3 n

1,1 n

Duale Karcher Flow

εμl/ avec μμ

γw l,n

m

lkk k,n

k,nnn

1

nnmedian w1,

*nk,n

nk,nk,n .wμ

wμμ

11

1,2 n

*nmedian,n

nmedian,nmedian,n wμ

wμμ

11

62 /62 / Median AR model Through Median Reflection Coefficients k

63 /63 / Alternative to Karcher Flow : Arnaudon Stochastic Flow

Random selection at each iteration of 1 point in the

disk + evolution along the geodesic

nnrand

nnrandnn γw

),(

),(

)(exp

exp.exp

)(1

)(1

1nrandm

nrandmnmn x

xtm

n

n

n

M. Arnaudon, C. Dombry, A.Phan, L.Yang, ”Stochastic algorithms for computing means of probability measures”, http://hal.archives-ouvertes.fr/hal-00540623

64 /64 /

nnrand

nnrandnn γw

),(

),(

)(exp

exp.exp

)(1

)(1

1nrandm

nrandmnmn x

xtm

n

n

n

Arnaudon Stochastic Flow

65 /65 / OS-HR-Doppler-CFAR

FFTI&Q

log|.|

Scalar OS CFAR

>S

log|.|

Scalar OS CFAR

>S

OU

I&QRegularized

AR

Model

-

-

Robust

Rao

Distance

Median CFAR on

>S

CfenêtreTFAiimiiP

,,1,0 ,...,,

2

1*

,,

,,

2

,0

,0,,1,0,,1,0 ].[1

arglog,...,,,,...,,

m

n mediannjn

mediannjn

j

medianmedianmmedianmedianimii μ

thn)(mP

PmPPdist

Classical

Chain

OS-HRD

CFAR

66 /66 / Figures of Merit (sea clutter)

COR curve

Red : OS-HR-DOPPLER-CFAR based on CAR median

Black : Classical Doppler Filter & OS-CFAR

Gain of

New processing

chain

First tests on real recorded

sea clutter

67 /67 / Figure of Merit for Closely Separated Targets (sea clutter)

COR Curves

Gain of

New processing

Chain

Red : OS-HR-DOPPLER-CFAR based on CAR median

Black : Classical Doppler Filter & OS-CFAR

First tests on real recorded

sea clutter

68 /68 /In Progress: tests on real Air Defense Radar Ground Clutter records

Doppler Filters

High ResolutionDoppler Filters

Waveform 1 Waveform 2

69 /69 /

Robustness is given by : Metric & Distance between Doppler Spectrums take into account variances of

estimation (Rao’s metric of Information Geometry is given by « Fisher Information Matrix » used in Cramer-Rao Bound)

HR Doppler estimation is « Regularized » (Regularized Burg Algorithm) because with short bursts, we cannot estimate AR model order or dimension of « signal space » (e.g. MUSIC)

Detector is not based on whitening filters (sub-optimal because filter weights have high variances with short bursts) but on robust distance

Median is a basic tool of Robust statistic : low sensitivity to outliers (Fréchethas studies main advantages of “Median” compared to “Mean”)

Jointly Robust Doppler Estimation Robust Distance

Robust Statistics Robust Detector

ROBUSTNESS of OS-HRD-CFAR


Information Geometry for turbulences monitoring

71 /71 /

THOMASSET Cyrille : 01/04/2011

Monitoring of Atmospheric Turbulences

Distance between AR model of order 1 and Regularized AR model of maximum order

Laminar Flow

Turbulent Flow2

1

2 11

ln21)(

n

k k

kknS

72 /72 /

STAR2000 (I&Q)

Upgrade of STAR2000 ATC Radar Weather Channel

Rain Rate

Atmospheric Turbulence

Wind Radial Speed

73 /73 / WakeWake--Vortex MonitoringVortex Monitoring

74 /74 /

Wake Vortex Roll-up(Arrival)

DepartureArrival

East Configuration

WakeWake--Vortex Monitoring : Vortex Monitoring : ArrivalArrival

75 /75 /

Wake Vortex Roll-up(Departure)

DepartureArrival

West Configuration

WakeWake--Vortex Monitoring : Vortex Monitoring : DepartureDeparture

76 /76 / Opportunity Trials : ATC PSR Radar records of A380

T T+ 4s T+ 8s

0.5 km

A380 AirplaneRange ~ 4 km

HR Doppler Entropy

77 /77 / BOR-A Installation : 06/05/2011

78 /78 / BOR-A First tests


SPACE-TIME COVARIANCE MATRIX MEDIAN

COMPUTATION: STAP PROCESSING

80 /80 / Extension to STAP : Toeplitz-Block-Toeplitz Matrices

Covariance Matrix

Space or Time Covariance Matrix

Space-Time Covariance Matrix

Entropy Hessian metric

Karcher/Fréchet barycenter/Median

Partial Iwasawa Decomposition

Anistropic Fourier Diffusion on graph

Median in Poincaré’s disc

Hermitian Positive Definite Matrix Group : HPD(n) nICnSpCnPSp 2/),(),(

Symplectic Quotient Group :

Toeplitz structure

Multi-variate Entropy Hessian metric

Partial Iwasawa Decomposition

Median in Siegel’s disk

Block-Toeplitz structure

Complex Autoregressive

Polar decomposition in Poincaré’s disc

Mostow decomposition in Siegel’s disc

Maslov-Leray Index

INFORMATION GEOMETRY

LIE GROUP & RIEMANNIAN SYMMETRIC SPACES GEOMETRIES

K-Mean via G-PCA in SDn

K-Median in SD

Geodesic PCA (G-PCA)

enRRΦ logdetlog~

ji

jiRΦg

~2

Robust Space-Time Processing


Previous results can be extented to Block-ToeplitzMatrices :

0

,

01

1

01

10

1, ~~

RRRR

RRRR

RRRRR

Rn

nnp

n

n

np

*1~

n

n

R

RVR

000

000

p

p

p

JJ

J

Vwith


From Burg-like parameterization, we can deduced thisinversion of Toeplitz-Block-Toeplitz matrix :

nnnnpnn

nnnnp AARA

AR

....

1,

11,

npnnp

npnnnpnnnp RAR

RAARAR

,,

,,1

1, ....

p

pn

p

pnnp

nn

p

n

nn

n

-n

nn

nnn

IJAJ

JAJ

AA

A

AA

RαAA

*11

*11

1

11

01

011

1

.0

and

,.1with


Kähler potential defined by Hessian of multi-channel/Multi-variate entropy :

npji

npnpnp

RHessg

csteµRTrcsteRRΦ

,

,,, logdetlog~

0

1

1, det..log.detlog).(~ RenAAIknR

n

k

kk

kknnp

1

1

1120

10

2 )(.n

k

kk

kk

kkn

kk

kk

kkn dAAAIdAAAITrkndRRTrnds

84 /84 / Multi-Channel Burg Algorithm

Multi-Channel Burg Algorithm :

)()(

)()(

)1()(

with

)()()()()1()(2

2

)()()( with )()1()(

)1()()(

and

0010

01100

with )()()(

)()()(

,,...,,0,,...,,,...,,

1

111

11

1*11

*

00*111

111

0

0

*

0

*11

*12

*11

11

12

1121

11

kkEP

kkEP

kkEP

kkkkkkA

JJPPJJPPAJJPPTrMin

kZkkkJJAkk

kAkk

IAJlnkJZkJAk

lkZkAk

IJJAJJAJJAAAAAAAA

bn

bn

bn

fn

fn

fn

bn

fn

fbn

nN

k

bn

bn

nN

k

fn

fn

nN

k

bn

fn

nn

fn

bn

fbTn

fbn

nn

bn

fn

A

bff

nnn

bn

fn

bn

nn

fn

fn

nn

l

nl

bn

n

l

nl

fn

nnn

nn

nn

nn

nnnn

nn

nn

Siegelnnn

nn

nn DiskAIAA

111

11

11 .

85 /85 / Mostow Decomposition & Berger Fibration

Georges Daniel Mostow

(Yale University & US Academy of Sciences)

Marcel Berger

(IHES & French Academy of Sciences)

Mostows decomposition may be found in Georges Giraud’s paper of

1921

86 /86 / Mostow Decomposition (& Berger Fibration)

Mostow Theorem :

Every matrix of can be decomposed :

whereis unitaryis real antisymmetricis real symmetrix

Can be deduce from

Lemma : Let and two positive definite hermitianmatrices, there exist a unique positive definite hermitian matrix such that :

Corollary : if is Hermitian Positive Definite, there exist a unique real symmetric matrix such that :

M CnGL ,SiAeUeM

UAS

A B

X BXAX

MS

SS eMeM 1*

87 /87 /

MMPPPPPPS

PPPPPeePeP SSS

with log.21

:corrolary Lemma

2/12/12/1*2/12/1

2/12/12/1*2/12/12212*

Mostow Decomposition

Mostow Theorem :

All matrix of can be decomposed in :

is unitary, is real antisymmetric réelle and is real symmetric

Proof :

M CnGL ,SiAeUeM

U A S

SS

SSiASSSiAS

SiASSiA

ePePeeeeeeeeP

eeeMMPeUeM

212*

2222*

2

MMPPeei

A

Peee

SS

SSiA

with log21

:y injectivit lleexponentie 2

iASeMeU

88 /88 / Siegel Disc Automorphism

Automorphism of Siegel Disc given by :

All automorphisms given by :

Distance given by:

Inverse automorphism given by :

nSD

2/100

100

2/100)(

0ZZIZZIZZZZIZZ

tZn UZUZCnUUSDAut )()(/,),(

0

2/100

2/100

01

01

100

2/100

2/100

with

)()(0

ZZIZZIG

ZGIZGZ

ZZIZZZZIZZIG

Z

)(Φ1)(Φ1

log21,,,

WW

WZdSDWZZ

Zn

89 /89 / Iterated Computation of Median in Siegel disk

2/12/1

1111

1

2/112/11

1

,1

,

,,,2/1

,2/12/1

,*,

2/1,

2/1,,

2,

2,,,,

10010

11

)()(

then1For

with

with log.2/1

: w

until on Iterate

0

: 1For

Plane-Half Siegelin

,,,,,,

nnmedian,nnn

nnmedian,nGmedian,nmedian,nGmedian,n

nnk,nnnk,nnnk,n

k,nGk,n

Fnk

m

lkk

nknn

nknknknknknknknknk

S

nk

SS

nkiA

nkk,nSiA

nknk

Fn

mm,,median,

iii

m

GGIWGGIG

GGIGGWWWW

GGIWGIGWGGIW

W W,...,m k

εHl/ HγG

WWPPPPPPS

ith eWeeWeUHeeUW

Gn

,...,WW,...,WW et Wtion :Initialisa

iIZiIZW,...,mi

,...,ZZ

n

n

n

nknknknknknk


Extensionfor Polarimetric Data

91 /91 / STOKES VECTORS

)Im(2)(2

*123

*122

22

211

22

210

2

1

zzszzRes

zzszzs

zz

E

92 /92 / POLARIMETRY : POINCARE UNIT SPHERE

2;

2

2

arctan1

2

ss

2

arctan22

21

3

sss

2sin2sin2cos2cos2cos

Im2Re2

0

0

0

0

*12

*12

22

21

22

21

3

2

1

0

ssss

zzzz

zzzz

ssss

S

23

22

21

20 ssss

4;

4

93 /93 / Extension : Arnaudon Flow for Polar Data

Arnaudon Stochastic Flow on Sphere


Conclusion

95 /95 /


Conclusion

What are the benefits of Information Geometry

High Doppler Resolution Property Improve detection of slow targets Increase monitoring accuracy of turbulences

Robustness Property very short Burst (few number of pulses) with “Median” to be robust to outliers and clutter edges With metric that take into account parameters correlations/variances

Future Extension

For Robust STAP Robust Estimation of Secundary Data Covariance Matrix

For Polarimetric Data Processing Stochastic Flow on Unit Poincaré Sphere

Other actors

EADS, MBDA, ONERA in Europe MIT LL in US NUDT & BIT (China), Australian labs

96 /96 / QUESTIONS ?

Mosaïque d’Irène Rousseau(Le disque de Poincaré)

More Information :. Séminaire Léon Brillouin sur les « Sciences géométriques de l’information »http://www.informationgeometry.org/Seminar/seminarBrillouin.html. Séminaire Franco‐Indien CEFIPRA/THALES/Ecole‐Polytechnique « Matrix Information Geometries »http://www.informationgeometry.org/MIG/http://www.informationgeometry.org/MIG/MIG‐proceedings.pdfhttp://www.lix.polytechnique.fr/~schwander/resources/mig/slides/


GENERAL THEORY OFCOMPLEX SYMMETRIC

SPACES

98 /98 / Classical Symmetric Bounded Domains

Symmetric bounded domains in Cn are particular cases of symmetric spaces of noncompact type.

Elie Cartan has proved that there is :

4 types of classical symmetric bounded domains 2 exceptional types (group of motion E6 and E7)

Classical Symmetric Bounded Domains (extension of Poincaré Disk)

021

1ZZ

:such that row 1 and columnsn with matricesComplex : IV Type

porder of matrices symmetric-skewComplex : III Type

porder of matrices symmetricComplex : II Type

columns q and rows p with matricesComplex : I Type):(

MatrixComplex r rectangula:

2

t

ZZZZ

Ω

Ω

Ω

ΩconjugatetransposedIZZ

Z

t

IVn

IIIp

IIp

Ip,q

99 /99 / Kernel function of Symmetric Bounded Domains

Luo-Geng Hua has computed the kernel functions for all classical domains :

Particular case (p=q=n=1) : Poincaré Unit Disk

domain theof volumeeuclidean theis where

, : IV Typefor 2ZZ11,

1 , : III Type

1 , : II Type

, : I Type

for det1,

-**t*

*

nΩZWWWWZK

pΩ

pΩ

qpΩ

ZWIWZK

IVn

IIIp

IIp

Ip,q

2*

*

*11111

11,

1/

zwwzK

zzCzΩΩΩΩ IVIIIIII,

100 /100 / Analytic automorphisms of Bounded Domains

Groups of analytic automorphisms of these domains are locally isomorphic to the group of matrices which preservefollowing forms:

n

tIVn

p

p

p

ptIIIp

p

p

p

ptIIp

p

pIp,q

II

HHAHAHAHAΩ

II

LI

IHLALAHAHAΩ

II

KI

IHKAKAHAHAΩ

AI

IHHAHAΩ

00

,, , : IV Type

00

,0

0,, , : III Type

00

,0

0,, , : II Type

1det,0

0, , : I Type

2*

*

*

*

122211211

2221

1211

AZAAZAgZ

AAAA

A

101 /101 / Analytic automorphisms of Bounded Domains

All classical domains are circular following from Cartan’s general theory, and the point 0 is distinguished for the potential :

Berezin quantization is based on the construction of the Hilbert Space of functions analytic in :

ZZI

KZZKZZ detlog0,0

,ln,*

*

ZgZZgjZZKZgjzgjgZgZK

ZZdK

ZZKhc

ZZdK

ZZKZgZfhcgf

h

h

),( with ),(),(),(),(

),()0,0(),()(

),()0,0(),()()()(,

***

*/1*

1

*/1*

102 /102 / Berezin Quantification of Poincaré Unit Disk

The most elementary example of Berezian quantification is, in the case of complex dimension 1, given by the Poincaré unit Disk with volume element :

Map from path on D to automorphy factor :

*22 )1.(2/1 dzdzzi

*

2

*

2

**2

22**

1

)()(

)(lnRe2)(1ln)( : potentialKähler

1 where with )1,1(

/)1,1(1/

zzzF

zzgzF

zFazbgzFzzF

baabba

gSUg

SSUzCzD

)()(

1ln1ln))0(()0(

1*

*1

2

1

21*1*22

gFgFabba

g

babgFabgba

103 /103 /

Extension for Siegel Unit Disk :

)()(

lnRe2)())((lndetlog)( : potentialKähler

)(

0 where

00

with and with

/

**

**

1**

*

*

**

ZFZgFZBAtraceZFZgF

ZZItraceZZIzFAZBBAZZg

BAABIBBAA

II

JJJggBBBA

g

IZZZSD

t

t

t

n

Berezin Quantification of Siegel Unit Disk

BBItraceBBIgFABg lndetln))0(()0( 1*

104 /104 / Cartan Decomposition on Poincaré Unit Disk

Lemma of Cartan for radial coordinates in Poincaré Disk :

)(ln21ln)(

.)(

1.)2coth(..)2(8

)()()(

)()()()(

and 0

0with

:ion DecompositCartan

1 where)( and with )1,1(

1/

2

2

2

22

22222

21*)(

22****

chzzF

shdshdds

ethabzshebchea

tchtshtshtch

de

eu

udug

baazbbazzg

abba

gSUg

zzD

LB

ii

i

i

i

105 /105 / Iwasawa Decomposition on Poincaré Unit Disk

Iwasawa coordinates in Poincaré Disk :

eu

euisheb

euichea

Ntchtshtshtch

DK

ii

CCgCghNDKhg

baazbbazzg

abba

gSUg

zzD

i

i

with

22/

22/

101

and )()()()(

, 2/cos2/sin2/sin2/cos

11

21 , )( with : Dec. Iwasawa

1 where)( and with )1,1(

1/

22/

22/

1

22****

106 /106 / Hua-Cartan Decomposition on Siegel Unit Disk

Lemma of Hua for radial coordinates in Siegel Disk (Hua-Cartan) :

)()()(diag ,Let

)(with

0)()(0

exp)()()()(

)(

)(

norder of matricescomplex unitary and exist there0

00

0 , )(

)()()()()()()()(

0 with

21

210

20

21*

212

2

2

00

00

0

*0

*

00

00**

210

210

1121

n

t

n

t

t

t

n

n

nnn

thththdiagPZZeigenABPPUUABZ

diagZ

ZZ

ABBA

VBUB

VAUA

VUV

VABBA

UU

gnSpABBA

g

shshshdiagBchchchdiagA

107 /107 / Iwasawa Decomposition on Siegel Unit Disk

Iwasawa coordinates in Siegel Disk :

SBAiBUB

SBAiAUA

ABBA

KANhSiISi

SiSiINh

ABBA

Ah

SISI

NN

eediageediag

BABA

A

CU

UC

UUUUiUUiUU

gUU

Ughg

iIIiII

CCgCghABBA

g

AZBBAZg(Z)IZZZSD

n

n

n

2121

with

.2/.2/

.2/.2/)( ,)(

norder ofmatrix real , 0

/

00

00

00

21norder unitary ,

00

)(/

21 with )( ,

and /

0001

0001

*1

*1

11

00

00

00

00

*1

**

**

*

1**

1**

1

1

108 /108 / Iwasawa/Cartan Coordinates on Siegel Unit Disk

Iwasawa/Cartan coordinates relation in Siegel Disk :

1

000000

111*

0001

0001

0

*0

**

0000

00

00~

00

00

~2

~2

with

2121

: Iwasawa

:Cartan

~with

, .2/.2/

.2/.2/

SiBAASiBABH

PUUHUUABZ

SBAiBUB

SBAiAUA

VBUB

VAUA

ABBA

g

BASSBA

ABBA

MMABBA

SiISiSiSiI

M

tt

t

t

SSS

109 /109 / Special Berezin Coordinates

For every symmetric Riemannian space, there exist a dual space being compact. The isometry groups of all the compact symmetric spaces are described by block matrices (the action of the group in terms of special coordinates is described by the same formula as the action of the group of motions of the dual domain).

Berezin coordinates for Siegel domain :

Remark :

00

with , :ly equivalentor

, 1**

II

LLLI

ABBA

ABBA

t

t

t

BBItraceBBIgFABg lndetln))0(()0( 1*

IiIiII

CCC

AWAAWAWAAAA

21 with :Isometry

)(

1

122211211

2221

1211

110 /110 / Invariant metric in Special Berezin Coordinates

Let M be a classical complex compact symmetric space. The invariant volume and invariant metric in terms of special Berezin coordinates have the form :

Link with : For arbitrary Kählerian homogeneous space, the logarithm of the density for the invariant measure is the potential of the metric

WWIWWF

WWWWFgdWdWgds

WWdWWFWWd nL

det),( where

,ln with

,,,

*

*

*2

,

*,

2

***


Information surSéminaire Inter-disciplinaire

Léon Nicolas Brillouin sur « les Sciences Géométriques

de l’Information »

112 /112 / Séminaire Léon Brillouin

Corpus thématique : « Science géométrique de l’information »

Laboratoire d’accueil : IRCAM (Arshia Cont), Paris Animation : Arshia Cont (IRCAM), Frank Nielsen (Sony Research), F.

Barbaresco (Thales) Les laboratoires

IRCAM (Arshia Cont, Gerard Assayag, Arnaud Dessein) Polytechnique (Schwanger) Mines ParisTech (Pierre Rouchon, Silvere Bonnabel, Jesus Angulo) Telecom ParisTech (Hugues Randriam) SUPELEC (Mérouane Debbah, Romain Couillet) UTT Troyes (Hichem Snoussi) Univ. Poitiers (Marc Arnaudon, Le Yang) Univ. Montpellier (Michel Boyom, Paul Bryand) Observation de Nice (Cédric Richard) INRIA (Jean-Paul Zolesio, Rama Cont, Xavier Pennec) Thales (Frédéric Barbaresco, Jean-Francois Marcotorchino, François Gosselin) …

113 /113 / Séminaire Léon Brillouin « Science géométrique de l’information »

« La Science et la Théorie de l’Information », L. Brillouin, 1956

« Théorie scientifique de l’information d’une part,

mais aussi application de la théorie de l’information

à des problèmes de science pure. »

Théorie & géométrie de l’information

Probabilité & statistique

Géométrie Riemannienne (symétrique, Kählerien, métrique)

Géométrie & Groupes de Lie

Géométrie symplectique

Géométrie complexeGéométrie

discrète

Géométrie algébrique & arithmétique

Géométrie non commutative

Physique Statistique Thermo-

dynamique & Géométrie (Souriau,

Ruppeiner)

Science géométrique

de l’information

Physique quantique

114 /114 /

Site web: http://www.informationgeometry.org/Seminar/seminarBrillouin.html

9 Juin 2011 : Session « stochastic geometry & information geometry »

New Perspectives in Stochastic Geometry par Wilfrid S Kendall (univ. De Warwick, UK, [email protected] )

Pecularities of the q-exponential function in phi-exponential functions par Asuka Takatsu (IHES et Univ. De Nagoya, [email protected] )

5 Juillet 2011 : Session Digiteo & EPFL

Minimisation de l’Entropie Géométrique par Alfred Hero (Chaire DIGITEO lab, [email protected] )

Christophe Vignat (Ecole Polytechnique de Lausanne)

Octobre 2011 : Session « Entropies et Divergences » (29 Avril)

Estimateurs statistiques par minimisation des -divergences duales par Michel Broniatowski (Univ. UPMC/Paris-6, [email protected] )

Novembre 2011 : Session Franco-Italienne

Algebraic and Geometric Methods in Statistics par Paolo Gibilisco (univ. De Rome, [email protected] )

Learning the Fréchet Mean over the Manifold of Covariance Matrices par Simone Fiori (UniversitàPolitecnica delle Marche, [email protected] )

Decembre 2011 : Session historique (date à définir)

Œuvre de Léon Brillouin par Rémy Mosseri (UPMC Paris-6, [email protected] ) Léon Brillouin et les débuts de la théorie de l’information par Philippe Jacquet (INRIA & X/LIX,

[email protected] ) Le concept d’Entropie en Physique par Roger Balian (CEA, [email protected] )

Séminaire Brillouin : sciences géométriques de l’information

115 /115 / Projet PEPS

PEPS « Géométrie de l’Information »

Titre long du Projet : La géométrie de l’information : un cadre général nouveau pour l’analyse et le traitement de

signaux multiformes

Titre court du Projet : InfoGeo

Mots clés : Géométrie de l’information, analyse et traitement du signal, audio, image, radar,

télécommunications, mathématiques appliquées.

Résumé du Projet : Ce projet centré autour de la géométrie de l’information cherche à regrouper des acteurs de

différentes disciplines : audio,image, radar, télécommunications, mathématiques appliquées. L’objectif est d’amener les outils de la géométrie de l’information vers un nouveau champ d’applications : l’analyse et le traitement des signaux multiformes, pour lequel nous souhaitons formuler un cadre théorique générique. Afin de réunir les multiples compétences nécessaires, l’IRCAM, le LIX et Thales se sont associés et ont récemment initié un groupe de travail pluridisciplinaire autour de la géométrie de l’information avec des manifestations scientifiques prévues pour 2011. Le projet présenté permettra de soutenir le lancement de ce groupe de travail et de le consolider en étendant le réseau d’acteurs nationaux impliqués.

Coordinateur : Arshia cont (IRCAM) Equipes participantes : F. Nielsen (Sony Research), F. Barbaresco (Thales),

A. Dessein (IRCAM)

116 /116 / Séminaire Franco-Indien « Matrix Information Geometries »

« Matrix & Information Geometries », MIG’11

Lieu : Ecole Polytechnique et Thales Research & Technology Date : 23 au 25 Février 2011 Site web : http://www.informationgeometry.org/MIG/ Résumés : http://www.informationgeometry.org/MIG/MIG-proceedings.pdf Slides: http://www.lix.polytechnique.fr/~schwander/resources/mig/slides/ Photos : http://www.lix.polytechnique.fr/~schwander/resources/mig/pictures/ Financement :

CEFRIPA CEFRIPA (http://www.cefipra.org) : Programme bilatéral de coopération scientifique entre

l'Inde et la France financé respectivement par : le Département Science et Technologie (DST) du Ministère indien de la science et de la

technologie la Direction générale de la coopération internationale et du développement du Ministère

français des affaires étrangères.

Laboratoires indiens : Dehli Indian Institute of Statistics (Prof. Rajendra Bhatia) 2 IITs : Guwahati (Prof. Amit Kumar Mishra), Kharagpur Sociétés indiennes : The BuG Design, Honeywell Technology de Bangalore

Laboratoires français : Membres du séminaire Brillouin


R. Bhatia, « Positive Definite Matrices », Princeton university Press, 2007

TABLE OF CONTENTS:

Preface vii

Chapter 1: Positive Matrices 1

1.1 Characterizations 1

1.2 Some Basic Theorems 5

1.3 Block Matrices 12

1.4 Norm of the Schur Product 16

1.5 Monotonicity and Convexity 18

1.6 Supplementary Results and Exercises 23

1.7 Notes and References 29

Chapter 2: Positive Linear Maps 35

2.1 Representations 35

2.2 Positive Maps 36

2.3 Some Basic Properties of Positive Maps 38

2.4 Some Applications 43

2.5 Three Questions 46

2.6 Positive Maps on Operator Systems 49



Chapter 3: Completely Positive Means 65

3.1 Some Basic Theorems 66

3.2 Exercises 72

3.3 Schwarz Inequalities 73

3.4 Positive Completions and Schur Products 76

3.5 The Numerical Radius 81



Chapter 4: Matrix Means 101

4.1 The Harmonic Mean and the Geometric Mean 103

4.2 Some Monotonicity and Convexity Theorems 111

4.3 Some Inequalities for Quantum Entropy 114

4.4 Furuta's Inequality 125



Chapter 5: Positive Definite Functions 141

5.1 Basic Properties 141

5.2 Examples 144

5.3 Loewner Matrices 153

5.4 Norm Inequalities for Means 160

5.5 Theorems of Herglotz and Bochner 165



Chapter 6: Geometry of Positive Matrices 201

6.1 The Riemannian Metric 201

6.2 The Metric Space Pn 210

6.3 Center of Mass and Geometric Mean 215

6.4 Related Inequalities 222



121 /121 / GDR Maths & Entreprises

122 /122 / Mini-Symposia SMAI 2011

Mini-Symposia pour Congrès SMAI 2011, 5e Biennale Française des Mathématiques Appliquées et Industrielles Lieu : Guidel, Bretagne Date : 23 au 27 Mai 2011 Site web : http://smai.emath.fr/smai2011/index.php Thème : Géométrie de l’Information

Thème : La géométrie de l'information est un thème émergeant qui consiste à traiter des phénomènes aléatoires en regardant la géométrie différentielle des espaces de probabilités correspondants. Les applications sont nombreuses en machine learning, théorie de l'information, traitement du signal et aussi géométrie différentielle. Il s'agit ici de donner une introduction aux techniques et aux motivations de la géométrie de l'information. Ce thème intéresse tout autant des mathématiciens purs qu'appliqués, des informaticiens et des industriels.

Programme : Arshia Cont (IRCAM) Frank Nielsen (Ecole Polytechnique, Sony) Xavier Pennec (INRIA, Sophia) Frédéric Barbaresco (THALES)

123 /123 / GRETSI’11 : session spéciale SS2

Session spéciale SS2 « Science géométrique de l'information », au XXIIIème colloque GRETSI 2011

Lieu : Bordeaux Date : 5 au 8 Septembre 2011 Site web : http://www.gretsi2011.org/sessions-speciales.html Programme :

Frédéric Barbaresco (THALES), « Science géométrique de l’Information : Géométrie des matrices de covariance, espace métrique de Fréchet et domaines bornés homogènes de Siegel »

Olivier SCHWANDER (Ecole Polytechnique), Frank NIELSEN (Sony), « Simplification de modèles de mélange issus d’estimateur par noyau »

Silvère Bonnabel (Ecole des Mines de Paris), « Convergence des méthodes de gradient stochastique sur les variétés riemanniennes »

Arnaud Dessein, Arshia Cont (IRCAM), « Segmentation statistique de flux audio en temps-réel dans le cadre de la géométrie de l’information »

V. Devlaminck (Université de Lille), « Modèles sous-jacents à certaines techniques d’interpolation géodésique dans l'espace des matrices de cohérence en optique de polarisation »

Pierre Formont (Supelec), Frédéric Pascal (Sondra), Jean-Philippe Ovarlez (ONERA), Gabriel Vasile(INPG Grenoble), Laurent Ferro-Famil (Université de Rennes), « Apport de la géométrie de l'information pour la classication d'images SAR polarimétriques

Xavier Pennec (INRIA), Marco Lorenzi (IRCCS, Italie), « Which parallel transport for the statistical analysis of longitudinal deformations? »

Hichem Snoussi (UTT Troyes), « Filtrage particulaire sur les vari´et´es riemanniennes » Marc Arnaudon (Université de Poitiers), Yang Le, « Algorithmes stochastiques pour calculer les p-

moyennes de mesures de probabilité et géométrie des matrices de covariance Toeplitz »

124 /124 / Sessions spéciales IRS’11

2 Sessions spéciales « New Generation of Advanced Radar Processing based on Information Geometry », au 12ème IRS 2011, International Radar Symposium

Lieu : Leipzig, Allemagne Date : 7-9 Septembre 2011 Site web : http://www.irs-2011.de Programme :

F. Barbaresco (THALES) : Geometric Radar Processing based on Fréchet Distance : Information Geometry versus Optimal Transport Theory

M. Frasca (MBDA Italy) : Optimal Cramer-Rao estimators for dimensions greater than two F. Opitz (EADS Gmbh) : Differential Geometry and Applications to Signal Processing and Tracking J.P. Ovarlez (ONERA), P. Formont (SONDRA), F. Pascal (SONDRA), G. Vasile (GIPSA) and L. Ferro-

Famil (Université de Rennes) : Contribution of Information Geometry for Polarimetric SAR Classification in Heterogeneous Areas

M. Arnaudon (Université de Poitiers), Le YANG & F. Barbaresco : Stochastic algorithms for computingp-means of probability measures, geometry of radar Toeplitz covariance matrices and applications to HR Doppler processing

Y. Cheng (NUDT, Chine), X. Wang, M. Morelande & Bill Moran (Université de Melbourne) : Bearings-Only Sensor Trajectory Optimization Using Accumulative Information

Y. Cheng (NUDT, Chine), X. Wang, M. Morelande & Bill Moran (Université de Melbourne) : Information Resolution of Joint Detection-Tracking Systems

F. Barbaresco (THALES) : Robust Statistical Radar Processing in Fréchet Metric Space: OS-HDR-CFAR and OS-STAP Processing in Siegel Homogeneous Bounded Domains

125 /125 / IGAIA’12 à Paris

Projet d’organisation du 4th IGAIA’12 à Paris

Organisateur : séminaire Brillouin (A. Dessein & A. Cont / IRCAM) Lieu : IRCAM, Paris Pour info sur 3rd IGAIA’10 :

http://www.mis.mpg.de/calendar/conferences/2010/infgeo.html

http://www.mis.mpg.de/calendar/conferences/2010/infgeo/slides.html

126 /126 / Informations générales : Livre

Géométrie Hessienne & travaux de Koszul

Soutenance de thèse de P. Byande (Prof. Boyom), 7. Déc. 2011 Livre de Hirohiko Shima, « Geometry of Hessian Structures », world

Scientific Publishing 2007

127 /127 / G. Pistone : Algebraic and Geometric Methods in Statistics

applications radar de la géométrie de l'information...

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