Harmonic Analysis on SL(2,C) with Applications in...

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Harmonic Analysis on SL(2,C) with Applications in Cognitive Vision Systems Jacek Turski University of Houston-DTN ESI05: Special Semester on Modern Methods of Time-Frequency Analysis: 4 th Workshop on Noncommutative Computational Harmonic Analysis. The E. Schrödinger International Institute of Mathematical Physics, Vienna, July 5, 2005

Transcript of Harmonic Analysis on SL(2,C) with Applications in...

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Harmonic Analysis on SL(2,C) with Applications in Cognitive Vision

Systems

Jacek TurskiUniversity of Houston-DTN

ESI05: Special Semester on Modern Methods of Time-Frequency Analysis: 4th Workshop on Noncommutative Computational Harmonic Analysis. The E. Schrödinger International Institute of Mathematical Physics, Vienna, July 5, 2005

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Outline

I. Mathematical Formulation1. The Conformal Camera2. Analysis on SL(2,C)3. Unitary Representations of SL(2,C)4. Projective Fourier Analysis (Noncompact Picture)

II. Applications5. Talk given at the Boston University

III. Additional Topics6. 2D-Shape Analysis by Mumford & Sharon7. PSL(2,C) in Modern Complex Analysis, Geometry and Sciences

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I. Mathematical Formulation

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1. The Conformal Camera

j � x 1 , x 2 , x 3 � � x 3 � ix 1

x 2

SL � 2 , � � � SU � 2 � ASU � 2 �

k � z � a z� bbz�a

, k � SU�2�

x2�1

Image Projective Transform. Tgf�z� � f�g�1 � z�, g � PSL�2,��

PSL�2,�� � SL�2,��/��I�

The Central Projection on the Image Plane

h � z � d �1z�cd

, h � NA

Polar Factor.:

The Quotient:

A �d 0

0 d�1|d � �� , N �

1 0

c 1|c � �

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����� �����

The Riemann Sphere � � � � ���

Aut��� � PSL�2,��

�0,0,0�

j|S 2 �y1 ,y2 ,y3� �y3�iy1

y2

�y1 , y2 , y 3 �

� �y3�iy1

y2

Stereographic Projection�3

S�0,1,0�2

j|S 2 �0,0,0� � �

Image Plane �: x 2 � 1

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� � � g � � �d��cb��a

z1 � 0 has slope �

����� �����

The Comple x Line P�� 2 � � P�

1

Proj. Transform. PSL�2,� �

� 2 � � z 1 z 2 � � � z 1 � x 2 � iy , z 2 � x 3 � ix 1 � Embedding of the Camera:

� 2 Image Plane �: z1 � 1

slope �

slope � �

� �

z1

z2z1�

z2�

� gz1

z2

�az1 � bz2

cz1 � dz2

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2. Analysis on SL(2,C)

���� ������������ SL�2,�� � NMANDerived from SL�2,�� � NB �pB where p � 0 1

�1 0and B � MAN is the Borel subgroup of SL�2,��

������� ������������ SL�2,�� � SU�2�ANg � a b

c d� SL�2,�� in the form g � kan with

k � SU�2�, a � A and n � N

M �e i� 0

0 e�i�

Important subgroups of SL�2,��: SU�2�, A, N, N � 1 c0 1

|c � � and

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��� � ������ �������� �� G:�G : G ��� such that drg � �G dlg

SL�2,�� is unimodular: �G � 1

B � SL�2,�� is not: �Ba b

0 a�1� |a|4

SL�2,� � is semisimple : it is reductive (stable under conjugate transpose)and has finite center.

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3. Unitary Representations of SL(2,C)

A unitary representation � of G (a group homomorphism � : G � U�V�to unitary operators, such that g � ���g�v,w is continuous) is a discreteseries if it has a non-zero coefficient ���g�v,w that is square-integrable.

A semisimple G has discrete series if it has a compact Cartan subgroup(a maximal abelian subgroup).

SL�2,��’s maximal abelian, MA � ��, is noncompact; SL�2,�� doesnot have a discrete series.

SL�2,�� has two series of unitary representations: the principal seriesand complementary series. They do not have square-integrable representations.

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- Inducing the Principal Series

The process of induction produces representations of G from the representationsof a closed sugroups P of G.

For SL�2,�� the principal series is induced from the Borel subgroup B � MANas follows:

The action ���g�F�x� � F�g�1 x� is initially taken on the space

V� � �F � C�SL�2,���|F�gb� � ��1 �b��B �b��1/2 F�b��

and then completed in the in the norm ||F||SU�2�2 � �

SU�2�|F�k�2 dk .

(by Iwasawa decomposition F is determined by F|SU�2�). Here

��b� � � l�m�� s�a� � a|a |

l|a| is; l � �, s � �.

where b �a b

0 a �1�

a/|a | 0

0 |a |/a

|a | 0

0 |a |�1

1 b/a

0 1.

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- Noncompact Picture of Induced Represent.

The restriction of F to N, which is one-to-one by Gauss decomposition. Here

V� � �F � C�SL�2,���|F�nb� � ��1�b��B�b��1/2F�n��.

Also, ||F||SU�2�2 � ||F||

N2 .

Explicitly, if g � a bc d

� SL�2,��, n � 1 z0 1

then, the action is

���g�F�n� � F�g�1n� � F�n�g�1n�m�g�1n�a�g�1n�n�

� |�bz � d|�is�2 �bz�d|�bz�d|

�lF

1 0az�c�bz�d

1

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- The Principal Series on L2(C)

� Because F in V� satisfies F�gman� � |�|�is�2 �|�|

�lF�g�, it is

N-invariant. Therefore, it is a function on SL�2,��/N ��2

� F � V� if and only if F �z1�z2

� �m � nF z1z2

� � � |�|is�2 �|�|

kF z1

z2� �

where m � �1/2��k � is� � 1 and m � �1/2���k � is� � 1

Thus, F z1z2

� � � |z1 |is�2�z1 /|z1 |�k��z2 /z1 � where ��z� F 1z� �

��� ��������� ������ �� SL�2,��

����z� � |��z � �|�is�2 ��z��|��z��|

�k�

�z����z�� , � � L2���

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- Compact Picture of Induced Represent.

The restriction of F to SU�2�. The dense space is

V� � �F � C�SU�2��|F�km� � ��m��1F�k��

with the norm ||F||SU�2�. The restriction is one-to-one by Iwasawa decomposition.

The action is is complicated

���g�F�k� � F�g�1k� � �s�a ��g�1k���1�B��g�1k�a ��g�1k�n��1/2

�l��g�1k���1F��g�1k��

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4. Projective Fourier Analysis (Noncompact Picture)

D

gD

g � �

Image Plane

z1

z2 � 2

Note: PFT is the standard Fourier integral in log-polar coord. �u,�� given by� � eu�i�

� Write h� 1z� �� f�z� and extend by h �

�z � |�|�1 f��z�

along the complex lines. Then, h�g�1 z1z2

� �� � f�g�1 � z�

� Define F and � in V� as follows:

F z1z2

� � � i2�|�|�is �

|�|

�kh �z1

�z2d�d� , ��z� � F 1

z� �

� Taking � � �z, we can write ��z� � |z|is�1�z/|z|�k f �s, k�

��� ���������� ������� ��������� �����

f �s, k� � i

2� f���|�|�is�1 �

|�|

�kd�d�

��� ������� ���

f�z� � 1�2��2 �k���

� � f �s, k�|z|is�1 z

|z|

kds Some Borel Characters z

|z|

k|z|is

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The usual Plancherel’s theorem in log-polar coordinates gives its projective counterpart:

i2�|f�z�|2dzd z � 1

�2��2 �k���� �|

f �s,k�|2ds

The Convolution in the noncompact picture is defined on the subgroup MA as follows:

f1 � f2�z� � i2� f1���f2�g�1 � z� d�d �

|�|2d�d� , g � ��1/2 0

0 �1/2e�i�/2 00 e i�/2

and � � �ei�

The we can prove The Convolution Property: f1 � f2�k, s� � f 1�k, s�

f 2�k, s�

- Plancherel’s and Convolution Theorems

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Summary.

� The inverse PFT gives decomposition in terms of the characters�l,s � |z|is�z/|z|�l of B with the projective Fourier transform as thecoefficients.

� The Gauss decomposition SL�2,�� � NB implies that B exhauststhe projective part of SL�2,�� as N represents translations.

� Note that all unitary representations of the group SL�2,�� are infinite asopposed to the fact that all finite irreducible unitary representations of Bare 1D.

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II.Applications

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���������������� ������� ���������������� ������� ���������������� ������� ���������������� ������� ���������������� ������� ���������������� ������� ���������������� ������� ���������������� �������

���������� ��� ���� ���������� ��� ���� ���������� ��� ���� ���������� ��� ���� ���������� ��� ���� ���������� ��� ���� ���������� ��� ���� ���������� ��� ����

���������������� ��������������������������������� ��������������������������������� ��������������������������������� ��������������������������������� ��������������������������������� ��������������������������������� ��������������������������������� �����������������

99thth International Conference on Cognitive and Neural Systems International Conference on Cognitive and Neural Systems

Boston University, May 18Boston University, May 18--21, 200521, 2005

5. Talk given at the Boston University

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Outline

A. The Brain Visual pathway1. Biological Facts2. Local V1 Topography3. On Global Topography4. Cyclopean Symmetry

B. On Binocular Image Processing5. Mathematical Background6. The Conformal Camera7. Cortical Image Processing8. Binocular Vision

- Cortical Images- Horopter

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A. The Brain Visual pathway

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1. Biological Facts

��������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ������

50% of the primary visual cortex (V1) is used toprocess input from the foveal region of 4% of theretinal area

��������������������������������������������������������������������������������

Image is transmitted from the retina to the visual cortex along the visual pathway in a precise,although changing, retinotopic arrangement, resulting in many maps of features superimposed in visual cortex (topography, orientation, eye dominance, motion direction, etc).

-20 º

-40 º

-60 º

-80 º

20 º 40 º

60 º

80º

0 º

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Ganglion cells with neural fibers

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2. Local V1 Topography

• With the ganglion density (away from the central part of the fovea) and a uniform V1 packing call density ,

• gives the coordinate mapping

• which defines the local V1 topography

c��r�rdrd� � cdud�

�k lnr, �� k lnz

z � k lnz

��r� � r2

� u � k lnr

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After the work of Eric Schwartz and his group:

(1) The mapping is an accepted approximation of the V1

topographic structure.

(2) A better model represents a full topography in terms of the mapping

w � k ln z�az�b

w � k ln�z � a�

3. On Global Topography

The parameter “a” removes singularity but also destroys the symmetry of the log.

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Author: Herman Gomes

4. Cyclopean Symmetry

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B. On Binocular Image Processing

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5. Mathematical Background

We constructed Computational Harmonic Analysis of the group SL(2,C) -- the group that provides image projective transformations in the Conformal Camera. Remarkably, the resulting image representation is also well adapted to retinotopic mapping of the brain’s visual pathway.

��� ���������� ������� ��������� �����

f �s, k� � i

2� f���|�|�is�1 �

|�|

�kd�d�

��� ������� ���

f�z� � 1�2��2 �k���

� � f �s, k�|z|is�1 z

|z|

kds

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The conformal camera reduces the projective degrees of freedom to a minimal set of image projective transformations.

6. The Conformal Camera

Image plane

S�0,1,0�2

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7. Cortical Image Processing

PFT is the standard FT in log-polar coordinates:

We discretize PFT to compute it in log-polar coordinates by FFT

�u, �� where u � ln r

Nonuniform Log-polar Sampling

– Retinal Image

Inverse DPFT in Log-polar Coord.

– Cortical Image

f �s, k� � �

0

2�/L �ln ra

ln rb g�u,��e�i�us��k�dud�

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The Discrete PFT and Its Inverse (Log-polar)

Projective “covariance” of PFT

fm ,n� � 1

MNk�0

M�1

�l�0

N�1

� f k,le

�u m,n�

ei2�u m,n� k/Tei�m,n

�lL

f m,n � �k�0

M�1 �l�0N�1 fk,le

uk e �i2�umk/T e�i2��nlL

fk,l �1

MN �m�0M�1�n�0

N�1 f m,ne�uk e i2�umk/Tei2��nlL

um ,n� � i�m ,n

� � lnzm ,n� � ln�g�1 � zm ,n�

- DPFT

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8. Binocular Vision--the conformal cameras model eyes

Binocular Disparity

Cortical Projection of the Right Visual Field

Right Eye

Left Eye

Right Hemisphere

Left Hemisphere

V2 & V3

V1

V1

V2 & V3

Fixationpoint

LGN

LGN

Fovea

Fovea

Cortical Projection of the Left Visual Field

Right Visual Field

Left Visual Field

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- Cortical Images

Cortical Image from The Left Eye

Cortical Image from The Right EyeThe Projection onto the Right Eye Retina

The Projection onto the Left Eye Retina

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- Horopter

Theorem. For binocular system with the conformal cameras, the horopters in the visual plane are conics, closely matching the empirical horopters.

Cortical Image of the Right Visual Field

V2 &V3

V1

V1

V2 & V3

Fixation point

M

Horopter = Zero-Disparity Curve

Cortical Image of the Left Visual Field

With retinas acting as if they were spheres, the geometric horopters are circles, known as the Vieth-Muller circles.

Empirical horopters resemble conics

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Conclusions

Because:

1. The camera with silicon retina sensors produces the image similar to the topographic image in V1,

2. The line singularity most likely exists in the fovea (the split theory*),

The head–eye–visual cortex integrated vision system makes possible an efficient, biologically realistic computational approach for binocular vision designs in robotic systems

* M. Lavidor and V. Walsh, The nature of foveal representation. Nature Reviews: Neuroscience 5, 729-735, 2004

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III. Additional Topics

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6. 2D-Shape Analysis [SM]

[SM] Sharon, E. and Mumford, D. 2D-Shape Analysis using Conformal mapping, IEEE on Conference CVPR Vol. 2, 2004.

Ingredients

� The manifold of shapes S � �simple closed smooth curves in���

S denotes S modulo scaling and translation

� The group Diff�S1� � �diffeomorphisms : S1 S1�� The quotient space Diff�S1�/PSL�2,��.

The Idea

� An intriguing metric space Diff�S1�/PSL�2,�� of 2D shapes comesvia the theory of Teichmüller spaces. In this space, every simple closedcurve (a “shape”) is represented by a ‘fingerprint’ which is an elementof Diff�S1�/PSL�2,��.

The Resulting Classification of 2D shapes is PSL�2,�� invariant!

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� From Shapes to Diffeomorphisms. Let �� � � and�� � � c � S1 � ���. Further, for � S, let � � Int�� �

and � � Ext�� � ���.By The Riemann mapping theorem, there exist conformal maps

�� : �� �, unique up to any m � PSL�2,���� : �� �; ����� � �, ��

� ��� � 0

� The shape fingerprint of : � � Diff�S1 �; � � ���1 � � � : S1 S1

� It gives a bijective map fromS to Diff�S1 �/PSL�2,��, that is,

S Diff�S1 �/PSL�2,��.

- A brief review of [SM] paper

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� From Diffeomorphisms to Shapes: Welding. Given PSL�2,�� in Diff�S1 �/PSL�2,��,choose any � in PSL�2,��. Construct an abstract Riemann surface by welding

�� and � � using � to identified their boundaries. The result must be �. (Fact 3 due to Bers)

Fact 3. For every surface Y � � 3 diffeomorphic to S2 , there is conformal map : � Yup to any m � PSL�2,��.

� It gives a curve , and two conformal mappings�� : �� � and �� : �� �, such that��

�1 � �� |S1 � �, which is unique up to anym � PSL�2,��.

� The shape via welding:

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� A complex structure. If f : �1 �2 is a biholomorphic mappingbetween Riemann surfaces, then they are holomorphically or conformallyequivalent, �1 �2 .

� Uniformization Theorem for Simply-Connected Riemann Surfaces.Up to conformal equivalence, there exists three simply-connected Riemann surfaces:

1. �; Aut��� � PSL�2,��2. �; Aut��� �Aff�1,�� (affine Möbius transformations z � dz � c)3. � � �z � �; |z| � 1� � � �z � �; Im�z� � 0�;

Aut��� � PSU�1, 1� Aut��� � PSL�2,��.

� Uniformization Theorem for Riemann Surfaces.Every Riemann surface� is conformally equivalent to�\�, where �, (the universal cover of��, is one of

�, � or � and the covering group � is a subgroup ofAut��� admitting a freely discontinuous action on�.

7. PSL(2, C) in Modern Complex Analysis, Geometry and Sciences

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� Fact 1. Two Riemann surfaces are conformally equivalent � 1 � 2 if and only ifthey have the same universal cover � and their covering groups are conjugate in Aut���:

there exists a g � Aut��� such that g� 1 g�1 � � 2.

� Example 1. Since every element of PSL�2,�� fixes some point of� , no proper

subgroup of Aut��� can act freely discontinuously on � , the only Riemann surface with

universal cover � is � itself.

� Fact 2. If the universal covering space of� is � , then� is conformally equivalent to� , � � or a torus � 2 . The respective covering groups are: �e�, � and � � � .

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� Example 2. Consider a lattice in�:

���1 ,�2� � �m�1 � n�2 |m, n � �,� k � ��� � � �.

Now, let hk : z � z � � k �k � 1, 2� be generators of ���1,�2�. Since for

m �1 0

c d� Aut��� we have: mhkm�1�z� � z � d� k. Thus, similar lattices

���1 ,�2� and ���1,�2� are conjugate in Aut���.

Taking d � 1/�1 ,

���1 ,�2� is conjugate to ��1,��.

Since the lattices are identical if and only if ��1 ,�2� and ��1� ,�2

� � are related by anelement of SL�2,��, we conclude that �/��1,�� and �/���1,��� are conformallyequivalent if and only if for � � �2 /�1 and �� � �2

� /�1� , assumed �, �� � �,

�� � m���; m � PSL�2,��.

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� Teichmüller space. In the last example (g � 1): Two surfaces �/��1,�� �/��1, ���are conformally equivalent if and only if �� � m���, m � PSL�2,��, �, �� � �.Thus we can think of �/PSL�2,�� as representing the set of all complex structureswhich are imposed on a surface of genus 1. �/PSL�2,�� is called the moduli spaceof a torus and the group PSL�2,�� is referred to as the modular group.

� More generally, for each integer g � 0, let Rg be the set of all conformal equivalenceclasses of compact Riemann surface of genus g, called space of moduli of genus g.We have seen that R1 can be identified with �/PSL�2,��.

� Teichmüller space Tg is a metric space, homeomorphic to � 6g�6 , admitting adiscontinuous group g of isometries such that Rg � Tg/ g . Thus, for g � 1,T1 is the upper half-space �, the metric is the hyperbolic metric, 1 is the modulargroup PSL�2,��. However, note that T1 � � is not a covering space ofR1 � �/PSL�2,��.

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� The universal Teichmüller space T�1� (introduced by Bers) is the simplestTeichmüller space that contains Teichmüller spaces as complex submanifolds.It contains an infinite-dimensional complex manifold Diff�S1�/PSL�2,�� asa complex submanifold

� Diff�S1�/PSL�2,�� plays an important role in the string theory basedon Kähler geometry and in 2D-shape analysis via the theory of Teichmüllerspaces (Mumford & Sharon)

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Conclusions. The group PSL�2,�� is inherently relevant to:

(1) Classical Geometry: The (nonmetric) complex projective line ��, PSL�2,��� isan umbrella for metric geometries: elliptic, parabolic and hyperbolic

(2) Modern Complex Analysis: Modular and Teichmüller spaces

(3) Modern Geometry: Geometrization Theory of 3-manifolds (Thurston)

(4) Classical Physics: Special Theory of Relativity: SL�2,�� is the double cover ofSO�1,3�, the group of Lorentz transformations (Penrose&Rindler)

(5) Modern Physics: String Theory

(6) Computational Vision: Human-like Vision System (Turski) and 2D ShapeAnalysis (Mumford&Sharon)