Géométries de l’information et des matrices de...

Air Systems Division

Géométries de l’information et des matrices de covariances : application en RadarFrédéric BarbarescoPoitiers, 5 Mai 2010

Air Systems Division2

Acknowledgement

Marc Arnaudon, Équipe de probabilités, Laboratoire de Mathématiques et Applications (LMA),CNRS : UMR 6086, université de Poitiers Yang Le, PhD student (in collaboration with THALES),

université de Poitiers Léon Brillouin Seminar on « Geometric Science of

Information » launched in collaboration with : Frank Nielsen, LIX, Ecole Polytechnique Arshia Cont, IRCAM


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


Léon Brillouin Seminar Corpus : « Geometric Science of Information » Place : IRCAM (Arshia Cont), Paris Animation : F. Barbaresco (Thales), Arshia Cont (IRCAM), Frank Nielsen

(X/LIX) Laboratories : IRCAM (Arshia Cont, Gerard Assayag, Arnaud Dessein) Polytechnique (Frank Nielsen) 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) …http://www.informationgeometry.org/Seminar/seminarBrillouin.html


La SEE en quelques mots

Lieu de rencontres science-industrie-société Reconnue d’utilité publique (avantage fiscal) Environ 2000 adhérents et 5000 personnes impliquées Présence industrielle forte (~50%) 19 clubs techniques et 12 groupes régionaux Organise des conférences et journées d’études Suscite et attire des congrès internationaux sur le sol

français Membre institutionnel français de l’IFAC et de l’IFIP Remet des distinctions et médailles Produit 3 revues (REE, 3E.I, eSTA) Produit 3 monographies par an Présente sur le Web: http://www.see.asso.fr et groupe

LinkedIn SEE


Méthodologie Signal, Image,

Information, Décision Automatique et

automatisation industrielle Génie logiciel Systèmes informatiques

de confiance Systèmes complexes

Applications Systèmes électriques Transports terrestres Systèmes aérospatiaux Sécurité Globale Ingénierie biomédicale

Les Clubs Techniques

Physique Ondes et propagation Foudre

Technologie Métrologie Composants Télécommunications Systèmes radars, sonars

et radioélectriques Systèmes optroniques

d’observation et de surveillance Matériels électriques Systèmes industriels

+ discussions avec IFIP pour meilleure prise en compte de leurs activités


Publications SEE

Revues REE, Revue de l'Electricité et de l'Electronique 10 numéros par an. Couvre tous les domaines d’activité de la SEE: Electricité, Signal, Automation,

Electronique, Computing, Composants, Communications, etc. Première partie « nouvelles de nos métiers » et un dossier thématique Vente par abonnement ou au numéro. Archives consultables sur le site

3EI, Revue de l'Enseignement de l'Electrotechnique et de l'Electronique Industrielle Support à l’Education – cible les professeurs et les industriels des professions

concernées. La 3E.I développe des dossiers scientifiques, techniques et historiques. Vente par abonnement ou au numéro.

e-STA, Revue des Sciences et Technologies de l'Automatique Monographies SEE/CNRS Editions Bulletin SEE Informations sur la vie de l ’association et annonces et programmes des

événements.


MONOGRAPHIE SEE/CNRS EDITIONS

Monographies Gestion de la Complexité et de l’information dans les

grands systèmes critiques Sécurité globale Micro et nanoélectronique À venir: « Le traitement de l'Information en interaction

avec les mathématiques et la physique », et « Le développement du réseau de transport d’électricité en France »

Comment commander? www.see.asso.fr

Prix : 39 EUR + 5 EUR (frais de port)


Preamble : Radar Data Structure


Technology breakthroughs open new future for RADAR : Low Frequency Radar HFSWR Radar (HF Surface Wave Radar, HF Skywave Radar) VHF/UHF Radar (Alerteur,…)

Bi/Multi-static Radar : Passive Radar (A/D Radio, TNT, …) MIMO Radar (RIAS, …)

Bi-Band & Ultra-Wide-Band Radar Shared Aperture Antenna

Large deployable/deformable/conformal Antenna Modular Integrated Mast Multifunction/Multimission/Multi-Use Intelligent Radar « Cognitive & Software » Radar Full-Digital Radar Antenna « Waveform Diversity & Design » : coloured waveforms, sparse

sampling, Phase conjugation & Time Reversal

RENEWEL OF RADAR


Radar Antenna measures the EM Wave to extract: Reflectivity Information

Doppler Information

Polarimetric Information

Radar Signal

Doppler fouillis atmosphérique Doppler cible (vitesse)

Ellipse polarisation sur sphère de Poincaré

Doppler relatif cible (ISAR)


itffi

tf

r

cVr

rDoppler

eecRtuAtz

etuts

VfcV

cVf

Doppler

r

22

2

1/0

0

0

2.)( :Reception

).()( :Emission

.2/1

/2

ChristianDoppler

011

1

01

11011

cccc

ccccc

z

z

z

zER

n

n

nn

Time Covariance Matrix :Toeplitz Hermitian Positive Definite

HippolyteFizeau

Radar Antenna measures the EM Wave to extract: Reflectivity Information

Doppler Information

Polarimetric Information

Radar Signal

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

n

kktéreflectivi

n

zn

Rz

zZ

0

21 1


STAP (Space-Time Adaptive Processing) Principle

STAP Principle

Nml

ml

ml

ml

Ml

l

l

lt

z

zz

z

z

zz

Z2

1

2

1

, with

gate range lpulse m

element n

th

th

th

nmlz

jammerclutternoise

tt

t

RRRRVZVZER

nVZ

,


Doppler-Fizeau : Poincaré or Einstein Model ?

"L'histoire des Sciences montre que les progrès de la Science ont constamment été entravés par l'influence tyrannique de certaines

conceptions que l'on avait fini par considérer comme des dogmes.Pour cette raison, il convient de soumettre périodiquement à un examen très approfondi les principes que l'on a fini par admettre

sans plus les discuter."

Louis de Broglie - Nouvelles Perspectives en Microphysique.

We show that the image by the Lorentz Transformation of a spherical light wavefront, emitted by a moving source, is not a spherical light wavefront but an ellipsoidal light wavefront. Poincaré’s elongated ellipse is the direct geometrical representation of Poincaré’s relativity of simultaneity. Einstein’s circles are the direct geometrical representation of Einstein’s convention of synchronisation. Poincaré’s ellipse supposes another convention for the definition of space-time units involving that the Lorentz Transformation (LT) of an unit of length is directly proportional to the LT of an unit of time: This is Poincaré’s definition of isotropic elongated distance (with dilated travel time). The historical (polemical) problem of priorities is therefore scientifically solved because Einstein’s explicit kinematics and Poincaré’s implicit kinematics are not the same.

Special Relativity: Einstein’s Spherical Waves versus Poincaré’s Ellipsoidal Waves, Yves Pierseaux, Annales de la Fondation Louis de Broglie, Volume 30 no 3-4, 2005


Doppler Sagnac Experiment : 1913

GYROLASER

On ne saurait, à aucun point de vue, comparer cette expérience àcelle de M. Michelson. Celle-ci est du second ordre en fonction de

la vitesse de translation et son importance tient à ce qu’elle est venue mettre en évidence de manière aigüe la nécessité

d’introduire un cinématique nouvelle...

Paul Langevin

Sagnac non null result (1913) :

The main problem of rotating platform with Einstein’s kinematics is precisely Einstein’s invariance of one way speed of light, t+ = t− or t = 0 in the proper system.

This is the reason why Langevin solves the problem in the framework of General Relativity. In Poincaré’s relativistic kinematics we can have in the system of the source

t+ t− . With Poincaré’s elongated ellipse and Poincaré’s group with rotations [Reignier J.], we predict immediately (at the second order k) the experimentally measured difference of time.

2

4cωAγtt

2

2

02

2cvRtdx

cvdtt

R


New Geometric Foundation of Radar Processing


Non-Euclidean Geometry & Lie Group Theory

“On pourrait dire d’une manière paradoxale que le point de départ du non-euclidisme réside dans l’épuration d’une notion pure, dans la simplification d’une notion simple…Le groupe apporte la preuve d’une mathématique fermée sur elle-même. Sa découverte clos l’ère des conventions, plus ou moins indépendantes, plus ou moins cohérentes” Gaston BACHELARD, La Philosophie géométrique, in « Le

nouvelle Esprit scientifique », 1934


Objective

We propose new robust Radar algorithms based on : Geometry of Covariance Matrices Intrinsic Geometry of HPD(n) & SO(n) Information Geometry deduced from data statistical distributions

Mean/Barycenter/Center-of-Mass of sample covariance matrices Robust Median Estimation of sample covariance matrices in

inhomogeneous data Anisotropic Diffusion on a graph of covariance matrices

We apply this approach for : Doppler/Array Processing STAP Processing Phase Processing Polarimetry Processing


Robust Detector should be defined based on Geometry of Covariance matrices

Signal Processing is classically based on flat metric & normed space. The flat metric can still be used because the set of SPD matrices is convex but

vector space assumption can yield to degraded algorithms. the set of SPD matrices with flat metric is not a geodesically complete space

since the geodesic A+t(B-A) is not a positive matrix for all t

We propose to use geometry of symmetric cone This geometry in the framework of Information Geometry , takes into account

matrix variance

Flat Metric versus Information Geometry Metric

)()( ,

2

))((,2

ABtAtBABA

BABATrBABABA TF

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

2/12/1

)( ,

, with log

ABAAAtABAAABA

XXTrXXXBAAt

TFF


From Linear Algebra to Lie Group Theory & Metric Space

Since the mid-20th century, Radar Signal Processing is mainly based on : Euclidean Geometry : Euclidean geometry corresponds to

the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3. Normed Space : It is a theorem of P. Jordan and J. von

Neumann that each normed space satisfying parallelogram law is Euclidean. Therefore, the Euclidean spaces are the normed spaces satisfying the semi-parallelogram law. Linear Algebra : Linear Algebra is concerned with the

study of vector spaces and transformation functions represented by matrices. A fundamental role in linear algebra is played by the notions of linear combination, span, and linear independence.



Geometry of Metric Space : Maurice René Fréchet has introduced the entire concept of metric spaces and functional theory on this space, where a notion of distance (called a metric) between elements of the set is defined. Any normed vector space is a metric space by defining d(x, y) = ||y − x|| but not the contrary. Every metric space is a topological space. We are more interested by Jacques HadamardSpace, called complete CAT(0) space. CAT(k) spaces were named byMikhail Gromov and they are metric space with curvature that is bounded from above by k.

Geometry of Symmetric space: A symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. For Riemannian Symmetric space, these inversion are isometric geodesic symmetries. More generally for Lie theory a symmetric space is a homogeneous space G/H for a Lie group G such that the stabilizer H of a point is an open subgroup of the fixed point set of an involution of G. They were classified by Élie Cartan and Marcel Berger. More specifically, we are interested by Cartan–Hadamard-van-Mangoldt manifold, a Riemannian manifold that is complete and simply-connected, and has everywhere non-positive sectional curvature.



Geometry of Kähler Manifold : A Hermitian symmetric space is a Kähler manifold which is a Riemannian symmetric space. Kähler Geometry was introduced by Erik Kähler and this is a manifold with unitary structure satisfying an integrability condition and where Kähler metric is given by Hessian of real function called Kähler potential. Especially, the non-compact Hermitian symmetric spaces can be realized as bounded domains in complex vector spaces.

Geometry of Lie Groups : a Lie group is a group and a differentiable oriented manifold where there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity. We are interested by Geometry on Lie Groups.

Geometry of Symplectic Manifold and Siegel Space : In his book “Symplectic Geometry”, C.L. Siegel has defined distance between complex matrices whose the imaginary part is Positive Definite. Siegel Upper-Half Space is an extension for complex matrix of the Poincaré’s upper half space (this is also a symmetric space). The associate metric and distance is invariant under generalized Möbius transform. A particular case is space of Symmetric or Hermitian Positive Definite Matrices.



Geometry on Symmetric Cones : Symmetric Hermitian Space is equivalent to a tube domain, where we can apply Semi-Simple Lie Group Theory and Jordan Algebra. It could envisaged as a particular case of Bruhat-Tits space.

Geometry of Robust Statistic on abstract spaces & Fréchet-Karcher Mean or Median : Statistics on manifolds is a critical aspect of different fields of applied mathematic. Maurice René Frechet imprimatur has been appropriated within this domain. The Fréchet mean, alsoknown as the Karcher mean, is the point that minimizes the sum of the squared geodesic distance to each point in a set of points on a manifold. More generally, on a metric space, the Fréchet mean still be defined. Replacing square geodesic distance by geodesic distance, we can extend this approach to estimate a median in metric space (called Fermat-Weber point in Physic). We investigate the geometric median of a probability measure on a complete Riemannian manifold and prove the uniqueness. By regarding the Weiszfeld algorithm as a subgradient procedure, we introduce a sub-gradient algorithm to estimate the median and prove that this algorithm always converges without condition of the sign of curvatures.



Geometry of Information : Information Geometry has been introduced by C.R.Rao, and axiomatized by N. Chentsov, with same roots that the well-known Cramer-Rao bound (Cramer-Rao bound has been introduced by Maurice René Fréchet in 1939, published in 1943 and extended to multivariate case by Georges Darmois in 1945), allows to build a distance between statistical distributions that is invariant to non-singular parameterization transformations.


Geometry on Symmetric Cones Information Geometry & Geometry on Symmetric Cones

Close links with Carl Ludwig Siegel (1896-1981) works in Symplectic Geometry (Siegel nth upper-half space SHn) Geometry on Symmetric Cone and on Symmetric Space (Bruhat-Tits

Space, Cartan-Hadamard Manifold) have been studied in many fields


German Mathematical Works

H. Karcher, “Riemannian center of mass and mollifier smoothing”, Comm. Pure Applied Math., °30,pp.509-541,1977

E. Kähler, “Über eine bemerkenswerte Hermitesche Metrik”, Abh., Math. Sem. Hamburg

Univ., n°9, pp.173-186, 1933

K. T. Sturm, “Probability measures on metric spaces of nonpositive curvature”. In vol.: Heat kernels and analysis on manifolds, graphs, and metric spaces, Contemp. Math. 338 ,

357-390, 2003

C.L. Siegel, "Symplectic Geometry", Academic Press, New York, 1964

H.C.F. von Mangoldt, “Über diejenigen Punkte auf positiv gekrümmten flächen, welche die

eigenschaft haben, dass die von ihnen ausgehenden geodätischen Linien nie aufhören, kürzeste Linien zu sein“, J. Reine Angew.Math.,

n°91, pp.23-52, 1881


French Mathematical Works

M. Gromov,"Hyperbolic groups". Essays in group theory. Math. Sci. Res. Inst. Publ. 8. New York: Springer. pp. 75–263, 1987

M. Fréchet, “L’intégrale abstraite d’une fonction abstraite d’une variable abstraite et son application à la

moyenne d’un élément aléatoire de nature quelconque”, Revue Scientifique, pp. 483-512, 1944

J. Hadamard, « Les surfaces à courbures opposées et leurs lignes géodésiques », J. Math, sér. 5, 4, pp.27-73, 1898

E. Cartan, « Leçons sur la géométrie des espaces de Riemann », 2nd édition, revue et

augmentée. Paris, Gauthier-Villars, 1946

F. Bruhat and J. Tits, « Groupes réductifs sur un corps local », IHES, n°41, pp.5-251, 1972


« 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

iYXZZdYdZYTrdsSiegel with 112

Carl Ludwig siegel


Center of Mass : From Appolonius of Perga toElie Cartan



Elie Cartan & Center of Mass : Historically, von Mangoldt, E. Cartan & J. Hadamard proved that a simply connected surface of non-negative curvature is homeomorphic to the plane. For this kind of Manifold, any two points are joined by a unique geodesic, and we can define notion of center of mass and its uniqueness. German Hermann Karcher has introduced Riemannian center of mass, and proved that for negative curvature, the minimum is unique (it is named Karcher’s barycentre). The existence of a center of mass in the large for manifolds with non-positive curvature was proven and used by Elie Cartan back in the 1920’s. He used this to prove that maximal compact subgroups of Lie groups are always conjugate. The general case was employed by Eugenio Calabi in an unpublished note. This holds because a symmetric space of non-positive curvature is nothing but the quotient of a non-compact Lie group by one of its maximal compact subgroups. All these irreductible symmetric spaces have been classified by E. Cartan& M. Berger.


Center of Mass : from Appolonius of Perga to Elie Cartan In Euclidean space, the center of mass is defined for finite set of points

by arithmetic mean:

Appolonius of Perga was the first to discover that this point minimizes the function of distances:

This extends to general Riemannian manifolds. Elie Cartan has proved that the function :

is strictly convex (its restriction to any geodesic is strictly convex as a function of one variable), achieves a unique minimum at a point called the center of mass of A for the distribution da.

Miix ,...,1

M

iicenter x

Mx

1

1

M

ii

xcenter xxMinx

1arg

M

ii

xcenter xxdMinx

1

2 ),(arg

A

daamdmf ),(21: 2

E. Cartan, “Groupes simples clos et ouverts et géométrie Riemannienne”, J. Math Pures & Appl., n°8, pp.1-33, 1929


Center of Mass : from Elie Cartan to Herman Karcher Center of Mass is characterized by being the unique zero of the

gradient vector field:

where is the “exponential map” and is the tangent vector at of the geodesic from to :

Herman Karcher has introduced a gradient flow intrinsic on the Manifold that converges to the center of mass, called Karcher Barycenter :

In the discrete case, the center of mass for finite set of points is given by:

)(exp 1 A

m daaf

exp(.) )(exp 1 am

m m a mm Ta)(exp 1

)()0( with )(.exp)(1 nnnmnn mfmfttmn

Pushed by Normal Jacobi Field (Sum of Tangent vectors of

Geodesics)

Karcher Barycenter :

Normal Jacobi Field Equal to zero (Sum of Tangent vectors of

Geodesics = 0)

)(exp.exp1

11

M

iimmn xtm

nn

H. Karcher, “Riemannian Center of Mass and Mollifier Smoothing ”, Com. Pure & Applied Math. Vol30, 1977


Center of Mass : from Herman Karcher to Maurice Fréchet 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 probabilistic variable where lies on a manifold is introduced by Emery as an exponential barycenter :

In Classical Euclidean space, we recover classical definition ofExpectation 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


Laplace Median Definition : « valeur probable »

mxEMinmediandxxPmedian

mX 5.0).(0

Laplace has proved in 1774 that : let F be the cumulative distribution function of an (absolute continuous) random variable; the median is defined as the value μ such that F(μ) = 0.5. Laplace proved that μ is also the value minimizing the average of the absolute deviations, where the deviation between two values is their L1 (called also Manhattan) distance i.e., the absolute value of their differences. Laplace called this value "le milieu de probabilité" or "la valeur probable". The term median has been introduced by Cournot in l’Exposition de la théorie des chances in 1883.


Median/Fermat’s Point : From Laplace to Maurice Fréchet

Statistics on manifolds is a critical aspect of different fields of applied mathematic.

Center of Mass is not useful for robust statistic. Replacing L2 square geodesic distance by L1 geodesic distance, we can extend this approach to estimate a Median in metric space (called Fermat-Weber ‘s point in Physic). Fréchet studied Median statistic using

compared to :

Classically, in Euclidean space, Median point minimizes

or equivalently :

mxEMinmmmedian

2mxEMinmmmean

M

ii

xmedian xxdMinx

1),(arg

M

iii

xmedian xxxxMinx

1

/arg


Median/Fermat’s Point : From Fréchet to Wesfeld

More generally, on a metric space, the Fréchet mean still be defined. Replacing square geodesic distance by geodesic distance, we can extend this approach to estimate a median in metric space

For Riemannian extension :

We cannot directly extend the Karcher Flow to median computation in the discrete case :

because could vanish if

A m

m

MinA

daaahdaamdmh

)(exp)(exp),(

21:

1

1

)(exp

exp.exp

11

1

1

M

k km

kmmn x

xtm

n

n

n

)(exp 1km x

n

xm kn


Wesfeld Algorithm for Median in metric space

Yang Le & Marc Arnaudon have investigated the geometric median of a probability measure on a complete Riemannian manifold and prove the uniqueness. By regarding the Weiszfeld algorithm as a subgradient procedure, they have introduced a sub-gradient algorithm to estimate the median and prove that this algorithm always converges without condition of the sign of curvatures.

with

LE Yang, "Riemannian Median and its Estimation”, LMS Journal of Computation and Math., 17 Nov. 2009

nm n

n

nGk km

kmmn x

xtm

)(exp

exp.exp

1

1

1 nkm mxkG

n /


Information Geometry


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 :


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


Chentsov/Rao GeometryInformation Geometry and Kullback Divergence (Rao, 1945) Riemannian metric of Information is deduced from 2nd order Taylor

expansion of Kullback divergence :

3

,.).(

!21)/(),/( dOddgdxpxpK

jijiij

jijiij θ.θ

xpE.dxθ.θxpxpg

/log/log/

22

Link with Fisher Information Matrix Riemannian metric of Information is given by Fisher matrix I() :

jiijjiij θ

xp.θ

xpEnd g gI

/log/log)(a)(,

22 /log/log.. xpdExpdVardIdds T Cramer-Rao bound given by : 1ˆˆ

IE

T

)()()( 22 dswdsWw


Chentsov Information Geometry Information Geometry & Kullback divergence Chentsov (student of Smirnov) was the first to identify Kullback geometric

properties. He has proposed an analogue of Pythagoras theorem :

qpE.dx

xqxp.xpnd K(p,q)

.K(m,p)h D(m,p)(q,p) witD(m,q)D(m,p)then, D

.dξpqm).(qR or .dξ

pqm.: such that

ns p,q,m istributio and the d, a K-bowlrR(p,R),Let B

p

KKK

K

ln//log/a

2

0lnln

210

222

2

mq

p

Chentsov has named it « Asymmetric PythagorianGeometry ». Locally, this geometry could be identified as a symmetric Riemannian Geometry : Information Geometry


Example : Gaussian scalar distributionGaussian Law Fisher Information matrix is given by :

mIEI

T and )(ˆˆ with

2001

.)( 12

2

22

2

2

2

22

2.2.2..

ddmddmdIdds T

Fisher matrix induced the following differential metric :

.2

imz 1

iziz

22

22

1.8

dds

Poincaré model of hyperbolic Space :

Geometry ofGaussian LawIs Geometry of

Hyperbolic Poincare Space


Example : Gaussian scalar distribution Gaussian Law Metric :

If we set , we can integrate along one radial :

Homeomorphisme is used then to compute distance between two arbitrary points in the unit disk :

Distance between two Gaussian Law is then given by :

rrd

rdrds

11ln.2

1.8

2

22

)(0 and .1

)( .

je

and .2

1),( with

),(1),(1ln.2,,,

2

22112

izizimz

mmD

r


Information Geometry of Multivariate Gaussian PDF

Multivariate Gaussian Distribution of zero mean Riemannian Information metric is given by Rao Metric

T

ZRZn

ABTrBAAAA

RdRRdRRTrdseR

RZp

, and ,with

.. det1/

2

22/12/12121

Multivariate Gaussian of non zero mean Isometries are given by following homeomorphisms :

2112 dRRTrdmdmds T

),(x, with '

isometry ,',',222 RnGLRAadsdsds

RAAamARmRmn

TT

22

111

1then As

RR dsdsdRRRdRIRR Invariance by inversion

12

1121 ,, RRDRRD


Distance between Multivariate Gaussian Law of zero Mean From Rao Metric to Rao distance

For , Rao distance is given by extented eigenvalues :

0det...detwith

log..log,

122/1

122/1

1

1

222/112

2/1121

2

RRIRRR

RRRRRDn

kk

TABTrBA , with ,2

James Theorem If , then is driven by a Khi-2 law with n.(n+1)/2

degrees of freedom.21 RR

n

kk

n1

2log2

Swain Generalization (contraste function) Distance from contraste function C3 on [0,[ :

n

kkvRRD

121

2 )(, 1 , 0 if 0)( and2/1)('' , 0)1(')1(

v

vvv

)()( if ,, 121

221

2 vwRRDRRD wv

22/12/12 .. RdRRds

Rao metric

1

1.)1()log(n

nn

nIAA

21 mm


Information Metric & sphericity testSphericity Test Spherecity test based on Information Geometry is different of classical one :

0det

log21,

:0 with :

1

22

1

0

nin

n

i

inn

nn

nn

I

IDIHIH

n

i

n

iiinn

nn

iiopt n

ID1

2

1

22/1

1

log1log21,

Sphericity test based on Kullback divergence :

n

ii

n

iinnnopt

nnnn

nnIKtr

n

nntrIK

11log1log

21,1

logdetlog121,


Geometry of Covariance Matrix


Problem : « Barycentre of N HPD matrices »

Mean of N HPD matrices

Median of N HPD matrices (Fermat-Weber Point)

Matrix Mean/Median Barycenter : Optimisation criteria

N

kk

XXBXdXfX

1,arg)(arg minmin

2

N

kk

XXBXdXfX

1

22 ,arg)(arg minmin

VTBXfgrad X

N

kkX

1

12 exp)(

1

N

kk

AXBXdXfX

11 ,arg)(arg minmin

VT

BXdBXfgrad X

N

k k

kX

1

1

1 ,exp)(

nXn XfgradXn 21 .exp

NkkB 1

nXn XfgradXn 11 .exp


Fréchet-Karcher Center of Mass Gradient Flow Gradient flow on Manifold

Pushed by Normal Jacobi Field (Sum of Tangent vectors of

Geodesics)

Fréchet-Karcher Barycenter :

Normal Jacobi Field Equal to zero (Sum of Tangent vectors of

Geodesics = 0)


Information Geometry for Gaussian Multivariate Law of zero mean and Intrinsic Geometry of Hermitian Positive Definite matrices (particular case of Siegel Space) lead to same metric and distance

Information Geometry :

Geometry of Siegel Upper-Half Space :

Metric and Geometry of Covariance Matrices

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

iYXZZdYdZYTrdsSiegel with 112

nRYX 0

212nn dRRTrds

212nn dRRTrds

nn

nnnnn

RRTrn

nnn

RRE

mZmZR

eRRZp nn

ˆ and

.ˆwith

..)()/(1.ˆ1

*

2

.)/(ln)(

ji

nnij

ZpEg

0nm

with


Information Geometry is given by : Advantages based on Information Geometry : invariante metric by non singular parametrization change

Metric taking into account parameters statistics

« Good Metric » for Information Geometry Meaning

212nn dRRTrds

)()()( 22 dswdsWw

dRddIdds

IER

..).(.et

Rao-Cramer de borne : ˆ

12

1


Intrinsic metric is given by : Advantage based on Siegel Geometry : Siegel Space Isometries are given by quotient group

with symplectic group :

Only one metric is invariant by , Siegel metric :

« Good Metric » for Intrinsic Geometry Meaning

212nn dRRTrds

nSH nIRnSpRnPSp 2/),(),( ),( FnSp

1)(

DCZBAZZM

DCBA

M

nTT

TT

IBCDADBCA

FnSpDCBA

M

symmetric et ),(

),2(0

0 , /),2(),( RnSL

II

JJJMMFnGLMFnSpn

nT

)(ZM ZdYdZYTrdsSiegel

112 iYXZ avec

nRYX 0

212nn dRRTrds


Metric is given by : Associate distance is : Obtained by integration :

In general case, for Siegel Space :

From metric to Distance 212 dRRTrds

n

kkRRRRRd

1

222/112

2/1121

2 log..log,

0det 12 RR avec

0 with XSHiYXZ n

n

n

k k

kSiegel SHZZZZd

21

1

221

2 , with 11

log,

with 0.),(det 21 IZZR

12121

1212121,

ZZZZZZZZZZR


In case of Siegel Space We can observe that 2nd derivative of in is given by :

From Metric to Distance

ZZRZ ,1 ZZ 1

11112 .2/1)(2 YZddZYZZZdZZdZRD

111

1111,

ZZZZZZZZZZR

RDTrZddZYYTrds 2112 .2


Poincaré & Siegel Upper Half Plane & Disk

0et avec

*112

2

2

2

222

yiyxzdzdzyyds

ydz

ydydxds

- *1*1*2

22

22

11

1

dwwwdwwwds

w

dwds

),(et ),( avec

*112

CnHDPYCnHermXiYXZ

dZdZYYTrds

*1*1*2 11 dWWWdWWWTrds

izizw

1 iIZiIZW

Poincaré Upper Half Plane Poincaré Unit disk

Siegel Upper Half Plane Siegel Unit disk


Poincaré Upper-half plane


Hyperbolic Geometry : Poincaré, Klein, Minkowski


Poincaré Upper Half Plane & Disk in Art (Escher )


Hyperbolic Geometry in Art

Irène Rousseau


Siegel Upper Half 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


Covariance Matrix Mean


Covariance Matrix Mean

2

N

kk

XXBXdXfX

1

22 ,arg)(arg minmin

VTBXfgrad X

N

kkX

1

12 exp)( nXn XfgradX

n 21 .exp

2/12/1

2/12/12/12/11

2/12/1

)(exp

logexp

XeXV

XUXXXUVXX

X

X

2/1

log2/1

11

2/12/1

n

XBX

nn XeXX

N

knkn

with

2/12/12/12/12 log,log, XBXXBXBXd kkkwith


Geodesic between two matrices X and Y given by :

Mean is given by Karcher Barycenter :

X

Y

Geodesic and Symmetrized Geometric Mean

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

)( XYXXXXeXt tYXXt

2/1log2/1

2/1..2/1

2/12/12/12/11

2/12/1

2/12/1

),(exp

),(exp

logexp)(

XeXtv

XeXtv

XYXXXVVgradv

YXXtYXX

XvXtYXX

XYX

YX

YX

2/1/21 2/12/12/12/1 XYXXXYX

YX )1( and )0( 1,0with t


Gradient Flow on Manifold Gradient flow on Manifold

Tangent at X of geodesic from X to Bk

Sum of tangents at X of geodesics from X to

2/12/12/12/10

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

log)()(

2/12/1

XXBXXdt

tdXeXXXBXXt

ktk

k

XBXttkk

k

kkk

2/1

1

2/12/12/1

10 log )( XXBXXG

dttdG

N

kkX

N

kt

k

kX

NkkB 1


Karcher Barycenter & Jacobi Field

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


Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fixes the point p and reverses geodesics throughthat point. It follows that the derivative of the map at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M. M is said to be locally Riemannian symmetric if its geodesic

symmetries are in fact isometric, and (globally) Riemannian symmetric if in addition its geodesic symmetries are defined on all of M. A Riemannian Manifold (M,g) is symmetric when, for all point x

of M, there exist one isometry σx:MM such that : σx(x) = x ; dσx(x) = − Id.

This isometry σx is called involution at x.

Riemannan Symmetric Space


For considered space, geodesic between two matrices A et B is given by :

For Symmetric space of Symmetric or Hermitian Definite Positive Matrices, for each couple (A,B), there exist one bijective isometry such that :

and This isometry has a unique fixed point Z which is given by

barycenter of (A,B), the geometric mean previously defined :

with

And properties : and This different of classical approach in signal processing with

assumption of an euclidean vectorial normed space :

with and

Riemannian Symmetric Space

1,0 avec )( 2/1log.2/12/1 2/12/12/1 2/12/1

tAeAABAAAt BAAtt

),( BAG

),( BAG BA ),( ABG BA

ZXdXXGd BA ,2,),( )(X -1

),( BABAXG BA 2/12/12/12/12/1 ABAAABA

X- ),( BABAXG BA

2BABA

FBA

),( BABAG BA ),( IBAdG BA


Geodesic and Symmetrized Geometric Mean

),( AGB BA

BGA BA ),( BAXBAXG(A,B)

X

2

BABA

Normed Space : Isometry(classical signal processing approach)

),( AGB BA

BGA BA ),(

)(X -1),( BABAXG BA

X

2/1/21 2/12/12/1 ABAAABA

Metric space : Isometry

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

tABAAAt t

Normed Space : Geodesic

A)0(

B)1(

A)0(

B)1(

1,0)(

tABtAt

Metric space : Geodesic

Null curvature Negative curvature


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

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

2/12/12/12/12/12/11

))(()(

ABAAAXBAAXAA

BAAXAAXAABAAAXAXAABXXA

Symmetrized Geometric Mean Involved Mean is a Symmetric or Hermitian Positive Definite Matrix

given by :

This is a Karcher Barycenter, that minimize :

In case where matrices can commute, we recover classical geometric mean :

which is not symmetric positive definite An other way of computaion of « symmetrized »Geometric Mean is

given by solution of following Ricatti Equation :

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

22/12/122/12/1 loglog XBBXAAMin

X

2/1ABBA


Covariance Matrix Median


N

kk

XXBXdXfX

11 ,arg)(arg minmin

Covariance Matrix Median

2/12/1

2/12/12/12/11

2/12/1

)(exp

logexp

XeXV

XUXXXUVXX

X

X

0det and

loglogwith 1

22/12/1

2/1loglog

2/11

12/12/1

2/12/1

kn

M

iiFnkn

nXBXXBX

nn

BX

XBX

XeXX

N

k Fnkn

nkn

1

VT

BXdBXfgrad X

N

k k

kX

1

1

1 ,exp)( nXn XfgradX

n 11 .exp


Gradient Flow Convergence Problem Gradient Flow for Median Computation

Cannot be differentiate at point X : Convergence Problem In theory : regularization in neighborhood of dat sets (algorithm of

Huiling Le, but slow convergence) In practice : Very often, data sets are around median point : convergence Take an adpated iterative step

0det and

loglogwith 1

22/12/1

2/1loglog

2/11

12/12/1

2/12/1

kn

M

iiFnkn

nXBXXBX

nn

BX

XBX

XeXX

N

k Fnkn

nkn


Median sub-gradient algorithm Proof : See THALES PhD Thesis (Yang Le supervised by Marc

Arnaudon)

otherwise 1

allfor if : criteria Stopping

0det and

loglogwith

,1,loglog

1

22/12/1

2/12/1

2/12/1

2/12/11

NS

Bk, XS

BX

XBX

XBNkXBXXBXS

XeXX

Fn

knFn

kn

M

iiFnkn

nkk

Fnkn

nknn

nSS

nnn

Fn

n


Illustration of Time Covariance Matrices (Doppler= Raw Doppler

Spectrum

Mean Doppler Spectrum

Median Doppler Spectrum


Anisotropic Diffusion on a graph of covariance matrices


Diffusion Heat Fourier Equation on matrices graph Diffusion Fourier heat equation on 1D graph of scalar values In a normed vectorial space in 1D case, diffusion equation is

approximated by discret Laplacian :

with arithmetic mean : Discret Fourier heat equation can be written :

Based on analogy with Karcher Flow, diffusion equation on 1D graph of Symmetric or Hermitian matrices can be written with geometric mean between :

nnnnnn uu

xxuu

xuu

xtu

xu

tu

ˆ21

211

2

2

2/ˆ 11 nnn uuu

2,,,,2,12 with ˆ.).1(ˆ2

xtuuuu

xtuu tntntntntnn,t

tnA ,ˆ tntn AA ,1,1 ,

2/1

,1/21 2/1

,1,12/1,1

2/1,1,2

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog22/1

,1,

ˆ and x

t2with

ˆ2/1

,,2/1

,2

tntntntntntn

tntntntntntn

XXXx

t

tntn

XXXXXX

XXXXXXeXXtntntn

J. Fourier


Isotropic Diffusion based on Fourier Heat Equation

2

2

2

2

yu

xuu

tu


Diffusion Heat Fourier Equation on matrices graph Diffusion Equation on a 1D graph of matrices

Geometric Details : colinearity of tangent vectors for geodesics and between , and :

As in Euclidean case weighted mean is replaced by geodesic weighted barycenter :

2/1

,1/21 2/1

,1,12/1,1

2/1,1,2

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog22/1

,1,

ˆ and x

t2with

ˆ2/1

,,2/1

,2

tntntntntntn

tntntntntntn

XXXx

t

tntn

XXXXXX

XXXXXXeXXtntntn

)(1,

, tn

tn

XX )(,

,

ˆ tn

tn

XX tnX , 1, tnX tnX ,

ˆ

2/1,,

2/1,2

2/1,1,

2/1,

ˆlog2log

tntntntntntn XXXx

tXXX

tn

tntnXtnX

tn

tn

tn

tn XX

XX

XX TT

dd

xt

dd

,

,1,,

,

,

1,

, ˆ

0

ˆ

2

0

//)(

.2)(

yxyx )1(.

2/12/12/12/1 ABAAABA

J. Fourier


Diffusion Heat Fourier Equation on scalar 1D graph Diffusion Equation on a 1D graph of scalar values

J. Fourier

tnu ,

tntntn uuu ,12/1,1,ˆ

tnu ,1tnu ,1

1,0.).1(

vuvu

t

t+1

range

tnu ,tnu ,1 tnu ,1

tnu ,ˆ

1n,tu

time

tntntn uuu ,,1, ˆ

1

0

2ˆ and 2with

ˆ.).1(

,1,1,2

,,1

tntntn

tntnn,t

uuu

xt

uuu


Diffusion Heat Fourier Equation on matrices 1D graph Diffusion Equation on a 1D graph of matrices

2/1

,1/21 2/1

,1,12/1,1

2/1,1,2

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog22/1

,1,

ˆ and x

t2with

ˆ2/1

,,2/1

,2

tntntntntntn

tntntntntntn

XXXx

t

tntn

XXXXXX

XXXXXXeXXtntntn

J. Fourier

tnX ,

tntntn XXX ,12/1,1,ˆ

tnX ,1tnX ,1

1,0

2/1 2/12/12/1

ABAAABA

t

t+1

range

tnX ,tnX ,1 tnX ,1

tnX ,ˆ

1n,tX

time

tntntn XXX ,,1,ˆ

1

0


Diffusion Equation : Discret & continuous Diffusion Equation on 1D graph of Symmetric or Hermitian

Positive Definite Matrices :

To recover diffusion equation in continuous case, Campbell-Hausdorff equation should be used :

2/1

,1/21 2/1

,1,12/1,1

2/1,1,2

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog22/1

,1,

ˆ and x

t2with

ˆ2/1

,,2/1

,2

tntntntntntn

tntntntntntn

XXXx

t

tntn

XXXXXX

XXXXXXeXXtntntn

YXXYYXterms

XYYYXXYXYXee YX

,with 4

,,121,,

121,

21log


Diffusion Equation : Discret & continuous Discret Diffusion Equation by mean of Campbell-Hausdorff

Equation :

2/1

,,2/1

,2/1

,1,2/1

,2

2/1,,

2/1,2

2/1,1,

2/1,

2/1,,

2/1,

2/1,1,

2/1,

2/1,,

2/1,

2/1,1,

2/1,

ˆloglogˆloglog

ˆlog,log1

ˆlog2log

0ˆlog,log21

ˆloglog

0log2/1

,,2/1

,2/1

,1,2/1

,2/1

,,2/1

,2/1

,1,2/1

,

tntntntntntn

tntntntntntn

tntntntntntn

tntntntntntn

XXXXXXXXXXXX

XXXXXXx

XXXxt

XXX

XXXXXX

XXXXXX

eeIee tntntntntntntntntntntntn

2

2

2

2 log,log2

loglogx

Xt

Xdtx

Xt

X


Anisotropic Diffusion Equation Classical Anisotropic Diffusion : Diffusion Equation on a Manifold is given by Laplace-Beltrami

equation : Riemannian Metric :

Diffusion Equation :

1D isotropic case :

1D anisotropic case :such that :

2

222

xu

tudxds

Njiji

N

kjiji ggdxdxgds

1,,1

,2 with

Njiji

ji j

ij

i

ggxugg

xgtu

1,,1

, with )det(

)det(1

22

222 1 dxxududxds

2

1xug

xu

xu

xxu

tu

2/122/12

11


Anisotropic Diffusion Equation : Scalar 1D Case 1D anistropic case :

Discret anisotropic diffusion equation :

with

xu

xu

xxu

tu

2/122/12

11

xuu

xuu

xt

uu tntntn

tntntn

tntnn,t

,1,,

,,1,

,,1

.

2

,1,1, 1

xuu tntn

tn

2

,,1, 1

xuu tntn

tn

2

,1,, 1

xuu tntn

tn


Anisotropic Diffusion : scalar case

IsotropicDiffusion

AnisotropicDiffusion


Anisotropic Diffusion : scalar case

In Blue, signal during anisotropic diffusion

In Greenn, substraction of blue signal to original

signal(targets detection)


Anisotropic Diffusion : 1D Matrices Graph Case In 1D anisotropic Case, by analogy with scalar case, extension

for graph of matrices :

For distance , we use :

0detwith

loglog,

21

1

22/112

2/1121

XX

XXXXXdM

iiF

2/1

,1 2/1

,1,12/1,1

2/1,1

2/1,1

log2/1,1,

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog2/1,1,

2/1,1,1

2/1,1

2/1,,

2/1,

ˆwith

ˆ

tntntntntntnXXX

tntn

tntntntntntnXXX

tntn

XXXXXXeXX

XXXXXXeXXtntntn

tntntn

2/12,1,

,

2/12,,1

,

2/12,1,1

,2,,,

,,

,

,1 and

,1

,1,

. ,

xXXd

xXXd

xXXd

xt

tntntn

tntntn

tntntn

tntntn

tntn

tn


Diffusion Isotrope Simulation


Diffusion anisotrope Simulation


new processing chain in Doppler Processing domain


Rationale for Use of Diffusive CFAR : Low Altitude Threats New Operational Requests Detection of new low altitude threats (small and/or stealth, agile, asymmetric,…) Increase Reaction Time against “Ultra Critical threats”

Classical CFAR are not edge-preserving & poorly take into account Clutter statistic : probability of detection is not optimal close to clutter transitions

Clutter transitions are most threatening areas : crest-line & unmasking areas (threats : furtive helicopter pop-up with missile shooting, low altitude cruise missile & UAV, asymmetric threats, rockets/batteries,…)

Coast-Lines

Crest-Lines

Example of Naval radarradial in Littoral Area

UAV Heli ULA Cruise

Define Edge-preserving CFAR based on clutter statistic


Rationale for Use of Diffusive CFAR : Low Altitude Threats

BEHIND THISCREST LINE

…

Increase Survivability against Lethal Threatwith Limited Time Exposure behind Crest-Lines :

Improve Critical Reaction Time

…LETHAL THREAT

WITH LIMITEDTIME EXPOSURE

Classical Helicopter

Helicopter withLimited Exposure

Potential ThreatBehind each Crest-lines


Principle of Doppler CFAR based on “Matrix Mean”

If we can define distance between two HPD matrices and Mean of N HPD matrices : Detection is given by threshold of

distance between matrix of cell under test and « Matrix Mean » of the neighborhood

For short time series waveforms, Classical FFT or Doppler Filter Banks are not efficient, and suffer of the following drawbacks : Poor Doppler Resolution If Target Doppler is in between two

Doppler filters, detection is sub-optimal High intensity of Ground Clutter is not

limited to zero-Doppler filter but pollution is spread over all filters due to poor Filter-Banks Resolution & Doppler Filter side lobes.


DOPPLER MATRIX CFAR

Main Goal Improve Doppler Detection of small/stealth targets in inhomogeneous clutter

using Doppler Matrix CFAR based on Differential Geometry of Covariance Matrices space (CFAR Mean or CFAR median)

Main advantages Improve performances compared with classical FFT (or Doppler Filter Bank)

in case of Waveform Bursts with few number of pulses (<16) Regularized High Resolution Doppler Analysis Based on statistics of parameters (Cramer-Rao Bound via Fisher matrix) Robust Environment assessment based on Median

Improve detection in inhomogeneous clutter (e.g. : Low altitude target on Ground Clutter, small target on Sea clutter)

Improve detection for closely spaced targets

Possible Extensions Use for sparse regular sampling waveforms (for pulses interleaving) Use for STAP (Spatio-Temporal Adaptative Processing) on spatio-temporal

covariance matrices Use for DORT method (Time Reversal & retrodirective techniques)


Doppler Matrix CFAR by Matrix Mean

FFTI&Q

log|.|

Sliding Mean

>S

log|.|

Sliding Mean

>S

OR

I&QCovariance

Matrix

R

-

-

Robust

Matrices

Distance

Sliding Matrix Mean

>S

1B 2B kB 1kB NB


Doppler Matrix CFAR by Matrix Median or Anisotropic Diffusion

I&QCovariance

Matrix

R

Robust

Matrices

Distance

Anisotropic Diffusion

on 1D graph of matrices

>S

I&QCovariance

Matrix

R

Robust

Matrices

Distance

Sliding Matrix Median

>S



Doppler Matrix CFAR by Matrix or Complex Autoregressive Medians

I&QComplex

Autoregressive

Models (CAM)

Robust

CAM

Distance

Sliding CAM median

>S

Median of Complex Autoregressive Models

I&QCovariance

Matrix

R

Robust

Matrices

Distance


>S


Doppler CFAR by Matrices Mean

I&QCovariance

Matrix

R

Robust

Matrices

Distance

Sliding Matrix Mean

>S

2

N

kk

XXBXdXfX

1

22 ,arg)(arg minmin

VTBXfgrad X

N

kkX

1

12 exp)( nXn XfgradX

n 21 .exp

2/12/1

2/12/12/12/11

2/12/1

)(exp

logexp

XeXV

XUXXXUVXX

X

X

2/1

log2/1

11

2/12/1

n

XBX

nn XeXX

N

knkn

avec

2/12/12/12/12 log,log, XBXXBXBXd kkkwith


N

kk

XXBXdXfX

11 ,arg)(arg minmin

Matrix Median CFAR

I&QCovariance

Matrix

R

Robust

Matrices

Distance


>S

2/12/1

2/12/12/12/11

2/12/1

)(exp

logexp

XeXV

XUXXXUVXX

X

X

0det and

loglogwith 1

22/12/1

2/1loglog

2/11

12/12/1

2/12/1

kn

M

iiFnkn

nXBXXBX

nn

BX

XBX

XeXX

N

k Fnkn

nkn

1

VT

BXdBXfgrad X

N

k k

kX

1

1

1 ,exp)( nXn XfgradX

n 11 .exp



I&QCovariance

Matrix

R

Robust

Matrices

Distance


On 1D graph of matrices

>S

2/1

,1 2/1

,1,12/1,1

2/1,1

2/1,1

log2/1,1,

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog2/1,1,

2/1,1,1

2/1,1

2/1,,

2/1,

ˆwith

ˆ

tntntntntntnXXX

tntn

tntntntntntnXXX

tntn

XXXXXXeXX

XXXXXXeXXtntntn

tntntn


Matrix CFAR byComplex Autoregressive Model Median


Covariance Matric Toeplitz structure is not conserved by previous Gradient Flow. Need for Using Complex Autoregressive Model Parametrization Gradient Flow driving reflection coefficients to converge toward

CAR Median

Median of Complex Autoregressive Model

I&QComplex

Autoregressive

Model (CAR)

Robust

CAR

Distance

Sliding CAR Median

>S

Tnn P 110

)(

1

122

22

0

02

1)(.

n

ii

in

din

PdPnds

,arg1

min

N

kkmed d

with

And metric


Kähler Geometry Erich Kähler Geometry is an extension of classical

Riemannian Geometry for complex Manifold : Positive Definite Riemannian Form that defines Kähler metric is given

by :

Kähler condition : There exist locally a Kähler Potential function, (and Pluri-harmonic equivalents) such that :

Erich Kähler has prooved that Ricci tensor is then given by :

And scalar curvature :

n

ji

jiji dzdzgds

1,

2 .2

jiji zzg

2

ji

lkji zz

gR

detlog2

n

lklk

lk RgR1,

.


Complex Autoregressive ModelRadar Signal Model : Multivariate Gaussian Complex Circular Model :

Radar Model :

Complex Autoregressive Model :

Link with Issai Schur’s algorithm (1875-1941) [Alpay] D. Alpay, « Algorithme de Schur, espaces à noyau reproduisant et théorie des

systèmes », Panoramas et synthèse, n°6, Société Mathématique de France, 1998

nnnnnnn

RRTrn

nnn

RREmZmZR

eRRZp nn

ˆ and .ˆwith

..)()/(1.ˆ1

ki

kkkkT

nn

nnn

n

eiyxzzzZ

ZZER

m

with

Positive DefiniteHermitian Toeplitz

processmean zero 0

1

TNN

NNkknn

N

knkn

Nkn aaAbbEbzaz )()(

12

0,*

1

)( ... and with


Regularized Burg Algorithm for Robust CAR estimation Regularized Burg Algorithm (THALES Patent : F. Barbaresco, « Procédé et

dispositif de détermination du spectre de fréquence d’un signal », brevet n° 95 06983, Juin 1995)

)(.)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


Block Structure of Covariance Matrix and Partial Iwasawa Decomposition


Partial Iwasawa Decomposition

Partial Iwasawa decomposition : the components of a positive definite or semi-definite matrix in the Iwasawa coordinates : Iwasawa, K., « On some types of topological groups », Ann. Math. Vol.

50, n°3, pp. 507–558, 1949 Every n×n positive definite matrix G can be uniquely expressed

using its Iwasawa components as follows.

where W & V are HPD matrices of size k×k and m×m respectively By computing the matrix multiplications in previous equation, we

derive the following parametrization of positive semidefinite matrices :

ABBBAIXI

VW

Gm

k

where

000

VWXXWXWXW

G


Partial Iwasawa Decomposition

by using the Cholesky decomposition of , where A is a lower triangular matrix with non-negative diagonal elements, we can establish equivalence by defining B to be the matrix that satisfies the equation .

We derive the following parametrization for the Gram matrix :

If W is positive definite, then the Cholesky factor A is unique,is also unique, V=0 and therefore, the parametrization

in is unique

BA

BA

BBBAABAA

VWXXWXWXW

G

AAW

WXBA

AXB


Autoregressive Model & Structured Covariance Matrix

Covariance Matrix and its inverse can be expressed by mean of complex autoregressive model : Block Structure of Covariance Matrix :

1111111

1111

....

nnnnnn

nnnn AARA

AR

111

1111111

....

nnn

nnnnnnn RAR

RAARAR

*)()(111

121 .

00100100

where1

.0

and .1with VVAA

A nn

nnnnn


Non-Symmetric Square Root of Siegel Group If we consider Cholesky decomposition of covariance matrix :

Cholesky decomposition (Goldberg inversion algorithm) :

All distribution of n-dimensionnal variable is associated with Affine Group. It is the element such that its action on vector

Is transformed to random vector :

This representation of Affine Group elements could be considered as non symmetric square root of Siegel Group element :

2/11

2/1112/1

11

2n

1111

121

. and 01

1with

.

1.1..

nnnnn

n

nnnn

nnnnnnn

AW

AAAA

WWR

),0(~ nn INZ),(~ nnn ANX

XAZZA nnnn

1.

11.

01

12/11

2/111

1111

1

.1

nnnn

n

AAAA


Block matrix structure & extended eigen-values Matrix or that is

used for Rao Metric computation has same block structure

Invariance of Block structure :

This block structure to compute extended eigen-values :

2/1)1(1)2(2/1)1( .. nnn RRR 2/1)1()2(2/1)1( .. nnn RRR

)1(1

)2(1

1-n)1(1

)2(1

2/1)1(1

)1(11

111111

1112/1)1(1)2(2/1)1(

and ..with

....

..

n

nnnnnn-

nnnnnn

nnnnnnn

AARW

WWWW

RRR

111

1)(

11)()(

1,

)(

1

1)()1(

2)1(1)(

11)()()(

.....1

0.

..

nnnn

knnn

kn

k

nk

n

in

kn

i

ninn

knnn

kn

kn

WUIUXX

XWF


We can deduce an algorithm that could be parallelized according to CAR model order for each extended eigen-values :

111

1)(

11)()(

1,

)(

1

1)()1(

2)1(1)(

11)()()(

.....1

0.

..

nnnn

knnn

kn

k

nk

n

in

kn

i

ninn

knnn

kn

kn

WUIUXX

XWF

..

.1 )(1,

)(2/1

1

12)()1(

2)1(12)(

)(1,

)(

nk

nk

n

in

kn

i

ninn

knk

nk

XXXW

XX

1

12)1(

2)1(1

)1(

12)(1,

)(

1)(

1..

.1.

n

kn

k

nkn

nk

nnk

nk

nn XW

X

F

nnnnn

nn

nn

n Tr

)(1

)1(1

)(2

)(1

)1(1

)( ...0

1111

11.

nnnnn WW

TrTr

)()4( F

stricly increasing curve on each interval :

)3(1

)3(2

)3(3

4Tr

)4(1

Renormalisation at each iteration :

avec

)4(3

)4(2

)4(4

.A

1

1

n

1=k)(

1,

)(1)()(

1-n

1

1)()(

nk

nk

nk

n

n

k

nk

n

XXF

F

Interpretation in term of projection of CAR vector :

0

Block matrix structure & extended eigen-values


Information Geometry Metric for Complex Autoregressive Model


Kähler metric for Complex Autoregressive Model In the framework of Affine Information Geometry, Kähler

metric is given by Hessian of Entropy, considered as Kähler Potential Function :

Entropy of Multivariate Gaussian Law of Zero Mean :

Entropy can be parametrized by mean of reflection coefficients

enRRΦ logdetlog~

-RHΗΗΦg

jiij

et ~2

0

1

1

2 ..ln.1ln).(~ Penkn)(RΦn

kkn

n

kk

nnn

zn

P1

20

10

11

21

1 avec

.1

1

1

20

1

0

1 1detn

k

kn

kn

n

kknR

nn

nnnnn

RRTrn

nnn

RRE

mZmZR

eRRZp nn

ˆet

.ˆ avec

..)()/(1.ˆ1


Erich Kähler…

Entropy U=-logdet[R] of complex autoregressive model can be considered as a Kähler Potential function. If we use reflection coefficient for Entropy parametrization, we recover metric initially proposed by Erich Kähler.

« Kähler Erich, Mathematical Works », Edited by R. Berndt and O. Riemenschneider, Berlin, Walter de Gruyter, ix, 2003

Seminal Erich Kähler Paper, 1932, Hambourg


Kähler Metric : hyper-Abelian Case We recover Kähler metric proposed in his seminal paper

of 1932 : Kähler called this case, « hyper-Abelian » :

Other metric proposed by E. Kähler, called « hyper-fuchian » :

),(ln1ln.1

1

2 zzKzΦ D

n

kkk

,...1 1 / and

1),(kernelBregman

with1

1

2

nkzz

zzz : K

k

n

kkD

k

1 / with 1ln.1k

2

1

2n

k

n

kk zzzΦ


We define a « Doppler » metric in case of complex autoregressive case as Hessian of Kähler Potential function, where this potential is given by entropy in framework of affine Information geometry :

Kähler potential parametrized by :

Metric can be computed according to reflection coefficients :

10

1

1

2 ..ln.1ln).(~

enkn)(RΦn

kkn

Tnn

nTn

n P )()(1110

)(

20

2011

nPng 221

).(

i

ijij

ing

1

122

22

0

02

1)(.

n

ii

in

din

PdPnds

Kähler metric for Complex Autoregressive Model

TnP 110


Scalar Curvature of Complex Autoregressive Model

We use Ricci Tensor given by Erich Kähler in case of complex Manifold :

In Kähler Geometry, Ricci Tensor is given by :

In case of complex autoregressive model, we have :

We can compute negative scalar curvature :

ji

lkji zz

gR

detlog2

1,...,2for 1

2

12

22

20

11

nkR

PR

k

kllk

lk

lklk RgR

,.

n

n

j jnR

1

0 )(1.2


Autoregressive Model # Kähler-Einstein Metric The metric is not of Kähler-Einstein kind, but a close

« matricial » structure : A metric is called Kähler-Einstein metric if its Ricci tensor is

proportional to the metric :

In case of Kähler-Einstein metric, Kähler potential is solution of Monge-Ampère equation :

For complex autoregressive model, we have :

,....,2 where

)(1.2 with

1)(

1

0

)()(

indiagB

jnBTrRgBR

n

n

j

nij

nij

ji

ji

lklkji zz

kzz

gkgkR

2

0

2

00 .detlog

constant : with .

function cholomorphiPotentialKähler

with )det( 02

ψ : Φ :

eg klk


Median Algorithm on Reflection coefficients 1/2We have , where is a n uplet of reflection coefficients for all order and with signal power . We compute first , in R (classical method on scalar median).We compute then median of complex reflection coefficients : First, reflection coefficients are transformed from unit disk to upper half space with :

where Geodesic are circle orthogonal to real axis

1)()(0

)(1

)(1

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

nkkkn

kkk DRPP N ,...1

HDC :1

)(

)()()(

11

kj

kjk

jk

j iz

)( )(1)( kk Cz

)))log(),...,log(),(log(exp( )(0

)2(0

)1(00

Nmedian PPPmedianeP

)(1

)(1

)( ,, kn

kk

)(0

kP

c


Median Algorithm on Reflection coefficients 2/2We initiate algorithm with arbitrary point Z in H :

At step p, displacement is driven by gradient given by:

where Z is estimated and ck is circle origin of geodesic between Zand zk.

Integer p, initiated at 1, decreases if difference between to steps is lower than 1/p.After convergence, we compute value in unit disk :

N

kk

lp

l

kl

plk

lp

lplZ pcZ

cZzZsigneit1

)()(

)()()()()(

, /1)(

)Re(2 )()(

2)(2)()(

pl

kl

pl

klk

l ZzZz

c

TnZZZ )0(1

)0(1

)0(

with

)Im()Re()Im(

)Re( )()(

,

)(,)(

)(p

lplZ

plZp

lZZ

tt

Zc pl

)arg( )()()(

plZ

pl

pl cZ

pcZ

cZcZsigne

MkicZcZ

pl

pl

pl

pl

pl

Zp

l

Zp

l

Zp

lp

lZp

lZp

l/1

))(Re(2

1exp)(

)(

)()()()(

)()()()()1(

iZiZ)C(Zμ

PPPmedianePZCP

convl

convlconv

lmedianl

Nmedian

)(

)()(

)(0

)2(0

)1(00 )))log(),...,log(),(log(exp(

with ))(,(


Gradient Flow in Unit Disk

)( plZC

)1( plZC

)1()1(ll zC

)2()2(ll zC

)3()3(ll zC )4()4(

ll zC


Results on Reflection Coefficients

Mean in blue

Median in Black


« Doppler Matrix CFAR» by median autoregressive model

FFTI&Q

log|.|

Scalar CFAR

>S

log|.|

Scalar CFAR

>S

OU

I&QRegularized

AR

Model

-

-

Robust

Siegel

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

New Chain


Iterated median in Poincaré Disk for reflection coefficient

Disk isometries (conforme transformation : angles preservation) :

Algorithm :*

,

,*, )(

)(1

)(wzewez

zzwwzez

wi

iwi

w

*nmedian,n

nmedian,nmedian,n

*nk,n

nk,nk,n

l,n

m

lkk k,n

k,nnn

n

mm,,median,

wμwμ

μ

.wμwμ

hen μ,...,m tFor k

εμl/ with μμ

γw

as w n as longIterate on,...,μμ,...,μμ and μ

Init :

1

11

00

1

1

1

10010

n ofonOptimisati



12

3

0,median

30,3

10,1 20,2

00, median Problem is simplified : we start from All geodesics are radials and space is quasi-euclidean close to 0

Initialisation

00, median

Classical Approach of Median Flow New Paradigm of Dual Median Flow

Fixed

Points

Point Driven

By median flow

Points Driven

By Dual Median Flow

Fixed point



n,3

n,1n,2

nmedian,

Classical Median Flow

1,3 n

1,1 n

Dual Median 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


Regularized Burg Algorithm

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

11 , .

1

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

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

1

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

:Init .

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

MnIteratea

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


Iterated Mean in Poincaré Disk Algorithm :

*nmedian,n

nmedian,nmedian,n

*nk,n

nk,nk,n

l,n

m

lkk k,n

k,nk,nnn

n

mm,,median,

wμwμ

μ

.wμwμ

hen μ,...,m tfor k

εμl/ with μμ

μarcthγw

n until wIterate on,...,μμ,...,μμ et μ

tion :Initialisa

1

11

.

00

1

1

1

10010

n Optimise


Iterated Computation of median in Poincaré disk


Doppler smoothing by diffusion


Figures of Merit

COR curve

Red : Matric Doppler CFAR based on CAR median

Black : Classical FFT

Gain of

New processing

chain


Figure of Merit for closed targets

COR Curves

Red : Matrix Median CFAR

Black : Classical FFT

Gain of

New processing

Chain


Results

Normal conditions


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

taux

de

cibl

es d

étec

tés

taux de faus s es alarmes

algorithme médianFft


Results

Closely separated targets


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

taux

de

cibl

es d

étec

tés




Median versus Mean


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.7

0.75

0.8

0.85

0.9

0.95


taux

de

déte

ctio

n

médianmoyenne


Result

Traget close to clutter edges


0 0.01 0.02 0.03 0.04 0.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

taux

de

cibl

es d

étec

tés




Effet diffusion-1


0 0.01 0.02 0.03 0.04 0.05 0.06

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1


taux

de

déte

ctio

n

s ans diffus ionavec diffus ion

Pour l’algorithme médian:


Médian du coefficient de réflexion

COR curbes Cyan : Median in Poiincaré Disk Red : Median in Poincaré Half-Plane Blue : Mean in Poincaré Disk Yellow : Mean in Poincaré half-Plane Black : FFT

0 0.5 1 1.5 2 2.5 3

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

taux

de

cibl

es d

étec

tés



Médian du coefficient de réflexion

COR Curves (closely separated targets) Cyan : Median in Poiincaré Disk Red : Median in Poincaré Half-Plane Blue : Mean in Poincaré Disk Yellow : Mean in Poincaré half-Plane Black : FFT

0 1 2 3 4 5 6 7 8 9 10

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

taux

de

cibl

es d

étec

tés



Radar STAP Processing


STAP (Space-Time Adaptive Processing) Principle

STAP Principle

ttoutput ZSVSZwZ 2/12/1ˆ

Nml

ml

ml

ml

Ml

l

l

lt

z

zz

z

z

zz

Z2

1

2

1

, with

gate range lpulse m

element n

th

th

th

nmlz

jammerclutternoise

tt

t

RRRRVZVZER

nVZ

,


Secondary Data Covariance Matrix Estimation

Sample Data & CFAR Detector

ˆ 1VSw

1

ˆ011

2

tt

t

ZSZVSVZw


M

kkZkZRMS

1)()(ˆ.


Classical STAP Equation : Likelihood Ratio Test Joint Probability Density Function of primary & 2nd data set

set dataprimary fromVector Input Single :set Data 2nd fromVector Input Single :)(

tZkZ

M

ktt

M

ktt

i

MRRTr

Nti

M

ktt

kZkZVZVZM

R

i

kZkZZZM

R

i

R

kZkZEk , Re(R)π

,Z(M)),,Z(Zp

kZpZpMZ),Z(Zp

i

11

10

11

)()(1

1

: hypothesis noise-plus-signal , 1

)()(1

1

: hypothesis alone noise , 0

)()(with det11

)()(,,1

1


Classical STAP Equation : Likelihood Ratio Test Likelihood ratio test : The PDF of the inputs is maximized over all unknown

parameters separately for each of the two hypothesis The ratio of these maxima is the detection statistic The maximizing parameter values are the maxium likelihood

(ML) estimators of these parameters : the maximized PDFs are obtained by replacing the unknown parameters by their ML estimators The ML estimator of a covariance matrix is equal to the sample

covariance matrix

NRRTrRR

Re,Z(M)),,Z(ZpMax

iiML

iNtiR

1 ˆwith

det11


Maximum Likelihood Hermitian covariance matrix N.R. Goodman’s Theorem : Consider M independent

identically distributed N-variate complex Gaussian random variable , k=1,…,M as a sample of size N from a population with PDF . Let DH be the set of NxN Hermitian positive definite matrices. Over the domain DH the maximum likelihood estimator of the Hermitian covariance matrix is :

Proof :

)()(1ˆ

1kZkZ

MR

M

kZ

ZRZp /)(kZ

ZR̂ZR

M

kZZ

kZRkZ

MZ

NM

M

kZZ

kZRkZRMNML

eR

RkZpRMZZpL

M

kZ

1

1

)()(

1

)()(detlog)log(ln

det1/)(/)(),...,1( 1

1


Classical STAP Equation : Likelihood Ratio Test

Gerelarizd Likelihood Ratio Test :

We consider : (M times the sample covariance matrix of the secondary

data) that satisfy Wishart distribution

0

1

0

1

0

0

1

detdet

detdet

)(,),1(,

)(,),1(,

RMinR

RRMax

MZZZpMax

MZZZpMaxMax

tR

tR

M

kkZkZS

1)()(

VZSVZSRM

ZSZSRMZZSRM

ttN

ttN

tt

11

100

1detdet1

1detdet1)1(



Minimisation over :

Likelihood ratio is given by :

We put :

VSVZSV

VSVZSVVSVZSZ

ZSZVSVZSZVZSVZ

tttt

tttttt

1

212

1

111

11211

.

.Re2

1

1detdet

1

211

1

1

0

VSVZSV

ZSZ

ZSZRRMax

ttt

tt

10 with 1

11

1

00

000

11

21

η

ιZSZVSV

ZSV

tt

t



GLRT : Generalized Likelihood Ratio Test :

.ˆ11ˆ

ˆ

1ˆ

011

21

MZRZ

MVRV

ZRV

SM

R

tt

t

.ˆ11ˆ

ˆ

ˆˆ

011

2

1

MZRZ

MVRV

Zw

VRw

tt

t


Classical STAP Equation : AMF

AMF (Adaptive Matched Filter) Test :

1ˆ with ˆ

ˆ1

21

SM

RVRV

ZRV t

VRVZRV

VRVZRV

VRVZRV

Λ

VRVZRVΛMax

VRVZRVΛeΛ

,Z(M)),,Z(Zp

,Z(M)),,Z(ZpMax

,Z(M)),,Z(Zp,Z(M)),,Z(ZpMaxΛ

ttt

t

α

tZRZαVZRαVZ

t

tα

t

t

tttt

1

21

1

21

1

21

1

1

121*

0

1

0

1

.2log

logargˆ

Re2log

1

1

11

11



GLRT : Generalized Likelihood Ratio Test :

Adaptive Matched Filter (AMF) :

1

ˆ

)()(ˆ. with ˆ

011

21

1

tt

t

M

k

ZSZVSVZw

kZkZRMSVSw

ˆˆ

ˆ

)()(ˆ. with ˆ

01

21

1

wSwZw

kZkZRMSVSw

t

M

k

• CFAR Property

• better Pd at low SINR

• CFAR Property

• More robust to sidelobes targets



Optimum STAP interpretation

Optimum STAP :

Space-Time Steering Vector :

filter Matched : SZ

NoiseceInterferen whitten :

ˆ

ˆ

1/2

2/1

2/12/1

2/12/11

yV

ZSy

ZSVSZwZ

VSSVSw

output

t

ttoutput

)1(2

2)1(2

)1(2

22

)1(2

2

1

1

1

,

s

t

s

s

Nj

jNj

Nj

jj

Nj

j

e

ee

e

ee

e

e

V

mkkm

NNNN

N

N

zzEQ

QQQ

QQQQQQ

ZZER

tttt

t

t

with

21

22221

11211


Parametric Adaptive Matched Filter


Parametric Adaptive Matched Filter (PAMF) Parametric Adaptive Matched Filter (PAMF) methodology for STAP

and detection : This methodology is based on approximating the interference spectrum

with a multichannel autoregressive (AR) model of low order. a low-order AR model provides an accurate representation of

simulated and measured interference for a wide variety of system and scenario conditions leads to reduced computational requirements

The modeling fidelity is attained using a small fraction of the Reed-Brennan rule training data set, thus presenting reduced secondary data requirements

the method offers dramatic improvement in detection performance over the conventional adaptive matched filter (AMF)

the method offers dramatic improvement in detection performance over the conventional adaptive matched filter (AMF)

the PAMF provides significantly improved detection performance over the AMF using only a small fraction of the secondary data required bythe AMF

J. R. Román, M. Rangaswamy, D. W. Davis, Q. Zhang, B. Himed,and J. H. Michels., « Parameteric adaptive matched filter forairborne radar applications. IEEE Transactions Aerospace Electronic Systems, 36(2) :677–692, April 2000


Parametric Adaptive Matched Filter (PAMF)

Multichannel autoregressive (AR) model : Nuttall, A., « Multivariate linear predictive spectral analysis

employing weighted forward and backward averaging: A generalization of Burg’s algorithm. », Technical report TR-5501, Naval Underwater Systems Center, New London, CT, 1976 Strand, O. N., « Multichannel complex maximum entropy (auto-

regressive) spectral analysis », IEEE Transactions on Automatic Control, AC-22, n°4, pp 634-640, 1977

Rami KANHOUCHE, « Méthodes mathématiques en traitement du signal pour l’estimation spectrale », thèse de doctorat de l’ENS Cachan, 21 Décembre 2006 (tel-00136093, version 1 - 12 Mars 2007)


Parametric Adaptive Matched Filter (PAMF)

MF and AMF :

LDU decomposition :

M

k

tAMF kZkZRMS η

VSVZSV

Λ1

01

21

)()(ˆ. with

ηVRVZRV

Λ tMF

1

21

2

2/12/1

22/12/12/12/1 and

VRRVZRRV

ZRVR tt

diagramblock main thealong 10 matricesHermitian matrix diagonal-block a :

diagonalblock main thealong matrixIdentity JxJh matrix wit triangular-blockLower :

with

x

x

x

),...,N-(iCDCD

CA

ADAR

JJi

JNJN

JNJN


Parametric Adaptive Matched Filter (PAMF) the matrix coefficients of the nth-order multichannel

forward linear predictor for the process:

JNNNN

JNNN

J

J

J

IANANANAINANANA

IAAIA

I

A

)1()3()2()1(0)4()3()2(

00)1()2(000)1(0000

1111

222

22

1

1

1

12

11

1

0000000000

ND

DD

D

)1(),...,1(),( nnn AnAnA


Parametric Adaptive Matched Filter (PAMF) MF based on LDU decomposition :

the multichannel element vectors are given by :

12/12/111 ADDAR VADVAD

ZADVADΛ

t

MF 12/112/1

12/112/1

tMF ZA

VAu

uDuD

DuDΛ

1

1

2/12/1

2/12/1

with

nIAknZkADnDn

NnknZkAn

Jnn

ktnnn

n

ktn

)0( with

)()()()(

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

0

2/12/1

0

TT

tTt

Ttt

Jtt

NZZZZ

NnCnZZ

)1()1()0(

data theof series time1,...,1,0/)(:


Parametric Adaptive Matched Filter (PAMF) the multichannel element vectors are given by :

Parametric Matched Filter by approximation to the MF with a simplified structure : retain only the vector sequence for the filter of order P (1P N-1) :

filter ingock whitenspatial)bl(or ation transform whiteningspatial a is step This instant. each timeat

dimension spatial thealong elements eduncorrelat generates :

matrix Identity theismatrix covariance its : )(matrix covarianceor with error vect predictionnth theis : )(

2/1

n

J

n

D

InDn

covarianceerror predictor theand tscoefficien

predictionlinear order -Pth : ,1,...,1,0),(

1,...,1,0 and )0(with

)()()()(0

2/12/1

P

J

P

ktPP

DPNkkA

PNnIA

PknZkADnDn


Parametric Adaptive Matched Filter (PAMF) PMF & PAMF schemes:

1,...,1,0 and )0(with

)()()()(

)()()()(

0

2/12/1

0

2/12/1

PNnIA

PknVkADnuDn

PknZkADnDn

J

P

kPP

P

ktPP

1

0

21

0

)()(

)()(

PN

n

PN

nPMF

nn

nnΛ

The inverse of an AR model is an MA model : the MA whitening

filter is implemented as a multichannel tapped delay line


Parametric Adaptive Matched Filter (PAMF) Multivariate Autoregressive Model and Identification

Algorithms : A complex-valued Gaussian-distributed zero-mean stationary,

vector random process is a multichannel AR process of order P, denoted as AR(P), if it satisfies a relation of the form :

For stability, all the system poles must lie inside the unit circle in the complex plane. The multivariable poles of an AR(P) system are the

multivariable zeros of its inverse MA(P) system.

)(nd

sequence noise white: ,0~)()0( and ,parametersmatrix JxJ

luedcomplex va constant, : ,...,2,1/)(

filter whitening: )()()(0

P

J

P

k

DCNnIA

PkkA

kndkAn


Multivariate Burg AlgorithmENS Cachan PhD thesis


Multivariate Burg Algorithm MV-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

*11

*11

11

12

1121

11

kekeEP

kekeEP

kekeEP

kekekekekekeA

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

nn

nn

nn

nn

nn

nnnn

nn

nn

Siegelnn

nn

nn DiskAIAA

11

*11

11


Multivariate Burg Algorithm Open Question : How to estimate Robust Multivariate

Burg Estimation in case in Inhomogeneous area : Regularization of Multivariate Burg (avoid to estimate AR order) Median MV-AR Model estimation by Fréchet Barycenter

See explanation done in Doppler Processing Case


Sample Covariance Matrix Estimation by Fréchet-Karcher Barycenter


Inhomogeneous Clutter

Sample Covariance Matrix estimation in inhomogeneous clutter will be the first drawback of these methods due to :Clutter Amplitude variationDoppler Spectral broadening variationClutter EdgesTargets in secondary data


Fréchet-Karcher exponential Barycenter Fréchet définition of Expectation as center of mass or

barycenter :

Karcher’s result :

Karcher’s flow :

)()(,

argˆ

2 dPRTdTG

TGMinRER

Z

TZFréchetFréchetZ

)( , of space tangent : with

)()(exp 1

Z

ZT

RT

TdPRTgradG

)()0(with )(.exp)(

TgradGTgradGtt T


Fréchet-Karcher Barycenter : Discrete Case Fréchet définition of Expectation as center of mass or

barycenter :

Karcher’s result :

Karcher’s flow :

M

k

TZFréchetFréchetZ

kZkZTdTG

TGMinRER

1

2 )()(,

argˆ

)()( , of space tangent : with

)()(exp1

1

kZkZT

TkZkZTgradGM

kT

2/1)()(log

2/11

1

2/12/1

)()0( with )(.exp)(

n

TkZkZTt

nn

T

TeTT

TgradGTgradGttM

knn

M

kkZkZ

MT

1)()(1ˆ


Iterated Median in Siegel Disk


Mostow Decomposition Mostow Theorem :

Every matrix of can be decomposed :

whereis unitaryis real antisymmetricis real symmetrix

Can be deduce from

Lemma : Let and two positive definite hermitian matrices, 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 M

SSS eMeM 1*


Mostow Decomposition Lemma : Let and two positive definite hermitian

matrices, there exist a unique positive definite hermitian matrix such that :

Proof :is unique hermitian positive definite square root of

Observe that is geodesic center of and for symmetric space of hermitian positive definite matrices

A B

X BXAX

2/12/12/12/12/1

2/12/12/12/12/1

2/12/122/12/1

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

ABAAAX

BAAXAA

BAAXAA

BAAXAAAXABXAX

2/1A A

X 1A B


Mostow Decomposition Corollary : if is Hermitian Positive Definite, there exist

a unique real symmetric matrix such that :

Proof :Positive Definite Hermitian Matrix, and with

same property. From previous Lemma, there exist a unique hermitian positive definite matrix such that :

Exponential providing an homeomorphism between symmetric et positive definite symmetric spaces, we have to proof that is positive definite

MSSS eMeM 1*

M *M 1M

XXXMM 1*

X

XXXXMMXMXM

MXXMXMXMM

*1*

*1**

1*1*1**1****

because


MMPPPPPPS

PPPPPeePeP SSS

avec log.2/1

:corrolary Lemma2/12/12/1*2/12/1

2/12/12/1*2/12/1212*

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

avec log21

:y injectivit lleexponentie 2

iASeMeU


Mostow Decomposition Lemma :

Matrix is Hermitian Positive Definite and verify .There exist one particular real antisymmetric

such that : Proof :

SS PeeY IYY * A

iAeY 2

IeePeePeeYYePePePePeeYY

SSSSSSSS

SSSS

212*212*

**

But

0et 2

/**1 *

SiASHHHeYYe

eYHHDPYHH

H


Siegel Isometry To find an isometry of Siegel Disk such that

, We determine an isometry of Siegel half-plane such that , and we use Cayley transform :

Isometric Automorphism of Siegel Half-Plane :

Isometric Automorphism of Siegel Disk :

nSD)(XfW

0)( WfW)ˆ(ˆ

ˆ XfW

nSH iIWfW )ˆ(ˆˆ

)XΦ(X ˆ 1ˆˆˆ

iIXiIX)XΦ(X

iIWfYXYY

M

LL

IXI

Mf

YLSHiYX(W)ΦW

W

W

n

)ˆ(ˆ0

00

0 matrice de ˆation transformLa

et ˆ

ˆ2/1

2/12/1

1

ˆ

2/11

0 avec

matrice de ation transformLa 1

Wf

iIIiII

C

CMCf

W

-W

1)(

DCZBAZZf

DCBA

M


Iterated Computation of Median in Siegel disk

median,nGmedian,n

k,nGk,n

Fnk

m

lkk

nknn

nknknknknknknknknk

S

nk

SS

nkiA

nkk,nSiA

nknk

Fn

mm,,median,

iii

m

WfW

Wf W,...,m k

εHl/ HγG

WWPPPPPPS

ith eWeeWeUHeeUW

Gn

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

iIZiIZW,...,mi

,...,ZZ

n

n

nknknknknknk

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

,,,,,,


Metric for SO(n) Groupmean, median and diffusion


Pulse Compression in Radar : LFM

BT

BTR sin)(


Low Signal to noise ratio

Matched Filter for LFM

SignalWithoutNoise

After MatchedFilter

WithNoise

Remenber :With non Matched filter


Phase Code : Barker code

Barker code of length 13 is longer code, composé with only 0 ou de 1, that maximises, by autocorrélation, the ratio main peak versus secundary peak. This code is made of 13 digit :

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


2 4 6 8 10 12-2

-1.5

-1

-0.5

0

0.5

1

1.5

-800 -600 -400 -200 0 200 400 600 800-200

0

200

400

600

800

1000

1200

1400

1600

1800


Distance & geodesic on SO(n) Compact Group SO(n) :

Distance between SO(n) matrices :

Geodesic between SO(n) matrices :

Tangent of geodesic :

Constraints on Karcher barycentre :

XXtraceXRRRRd TFF

T avec log, 2121

21log1211)( RRttT T

eRRRRt

1detet /)()( RIRRnGLRnSO T

2110

log)( RRRdt

td T

t

0log0log)(111 0

N

kk

TN

kk

TN

k t

k RRRRRdt

td

k


Barycentre and median on SO(n) Karcher Barycentre computation :

Median Computation

N

kk

Tn RX

nnn eXX 1

log1

1

FkTnn

RXRX

n

nn RXkSeXX nSk FkTn

kTn

log/ with loglog1

1


Diffusion Heat Fourier Equation on matrices graph Diffusion Fourier heat equation on 1D graph of scalar values In a normed vectorial space in 1D case, diffusion equation is

approximated by discret Laplacian :

with arithmetic mean : Discret Fourier heat equation can be written :

Based on analogy with Karcher Flow, diffusion equation on 1D graph of Symmetric or Hermitian matrices can be written with geometric mean between :

nnnnnn uu

xxuu

xuu

xtu

xu

tu

ˆ21

211

2

2

2/ˆ 11 nnn uuu

2,,,,2,12 with ˆ.).1(ˆ2

xtuuuu

xtuu tntntntntnn,t

J. Fourier

/21

,1,1,1,2

,,,

ˆlog2

,1,

ˆet x

t2 avec

ˆ,,2

tnT

tntntn

tnT

tntn

XXx

t

tntn

XXXX

XXXeXXtn

Ttn

tnX ,ˆ tntn XX ,1,1 ,


Diffusion Heat Fourier Equation on matrices graph Diffusion Equation on a 1D graph of matrices

Geometric Details : colinearity of tangent vectors for geodesics and between , and :

As in Euclidean case weighted mean is replaced by geodesic weighted barycenter :

J. Fourier

)(1,

, tn

tn

XX )(,

,

ˆ tn

tn

XX tnX , 1, tnX tnX ,

ˆ

tnT

tntnT

tn XXx

tXX ,,21,,ˆlog2log

tn

tntnXtnX

tn

tn

tn

tn XX

XX

XX TT

dd

xt

dd

,

,1,,

,

,

1,

, ˆ

0

ˆ

2

0

//)(

.2)(

yxyx )1(.

BAABA T

/21

,1,1,1,2

,,,

ˆlog2

,1,

ˆet x

t2 avec

ˆ,,2

tnT

tntntn

tnT

tntn

XXx

t

tntn

XXXX

XXXeXXtn

Ttn


Diffusion Heat Fourier Equation on scalar 1D graph Diffusion Equation on a 1D graph of scalar values

J. Fourier

tnu ,

tntntn uuu ,12/1,1,ˆ

tnu ,1tnu ,1

1,0.).1(

vuvu

t

t+1

range

tnu ,tnu ,1 tnu ,1

tnu ,ˆ

1n,tu

time

tntntn uuu ,,1, ˆ

1

0

2ˆ and 2with

ˆ.).1(

,1,1,2

,,1

tntntn

tntnn,t

uuu

xt

uuu


Diffusion Heat Fourier Equation on matrices 1D graph Diffusion Equation on a 1D graph of matrices

J. Fourier

tnX ,

tntntn XXX ,12/1,1,ˆ

tnX ,1tnX ,1

t

t+1

range

tnX ,tnX ,1 tnX ,1

tnX ,ˆ

1n,tX

time

tntntn XXX ,,1,ˆ

1

0

1,0

BAABA T

/21

,1,1,1,2

,,,

ˆlog2

,1,

ˆet x

t2 avec

ˆ,,2

tnT

tntntn

tnT

tntn

XXx

t

tntn

XXXX

XXXeXXtn

Ttn


Anisotropic Diffusion on graph of SO(n) matrices In1D case, Anisotropic diffusion is given by :

For distance , we take :

2/12,1,

,

2/12,,1

,

2/12,1,1

,2,,,

,,

,

,1,1,1

log,1,

,,,ˆlog

,1,

,1et

,1

,1,

. ,

ˆwith

ˆ

,1,1

,,

xXXd

xXXd

xXXd

xt

XXXeXX

XXXeXX

tntntn

tntntn

tntntn

tntntn

tntn

tn

tnT

tntnXX

tntn

tnT

tntnXX

tntn

tnT

tn

tnT

tn

XXTraceX

XXXXd

T

F

T

with

log, 2121


Iterated Computation of SO(2) matrices Mean

Iterated Algorithm:


0 0.2 0.4 0.6 0.8

-1

-0.9

-0.8

-0.7

-0.6

-0.5


Iterated Computation of SO(2) matrices Median

Iterated Algorithm :


0.2 0.4 0.6 0.8 1

-0.95

-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

Non unicity


Diffusion



Results


0 100 200 300 400 500 600 700-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Mean: Median: Anisotropic Diffusion:

Raw signal


Extension to POLARIMETRIC DIFFUSIVE CFAR


Unit Sphere Poincaré Model of Polarimetric Data

2;

2

4;

4

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


Noisy Data

Geometric Diffusion of Polarimetric Data

Beltrami Flow

Noisy Data (Blue) and Data after diffusion (Green) (,)


Questions

« L’eau ronge les montagnes et comble les vallées. Si elle le

pouvait, elle réduirait le monde à une sphère parfaite »

Léonard de Vinci [Cod. Atl. 185 V]

Géométries de l’information et des matrices de...

Documents

Transcript of Géométries de l’information et des matrices de...