Certainty & Uncertainty in Filter Bank Design Methodology

Certainty & Uncertainty in

Filter Bank Design Methodology

Chen SagivChen Sagiv

Joint work with:

Nir Sochen

Yehoshua Zeevi

Peter Maass & Dirk Lorenz

Stephan Dahlke

The Motivation: Maximal Accuracy Minimal Uncertainty

location

frequency

scale

orientation

Image features

The Motivation

The Motivation Signal & Image Processing applications call for:Signal & Image Processing applications call for:

““optimal” mother-wavelet optimal” mother-wavelet ““optimal” filter bankoptimal” filter bank

Possible criteria for optimality: Possible criteria for optimality: The “optimal” mother-wavelet provides The “optimal” mother-wavelet provides

maximal accuracy - minimal uncertainty maximal accuracy - minimal uncertainty The “optimal” filter bank constitutes a tight The “optimal” filter bank constitutes a tight

frameframe

The quest for the “optimal” functionThe quest for the “optimal” function

The many faces of the Uncertainty PrincipleThe many faces of the Uncertainty Principle A case study: The Weyl-Heisenberg GroupA case study: The Weyl-Heisenberg Group Previous StudiesPrevious Studies The 2D Affine GroupThe 2D Affine Group The 1D Affine Weyl-Heisenberg GroupThe 1D Affine Weyl-Heisenberg Group

The GThe G SolutionSolutionThe The -modulation spaces solution-modulation spaces solutionA Gabor-wavelet flavored solutionA Gabor-wavelet flavored solution

Conclusions & Future WorkConclusions & Future Work

The Uncertainty Principle – Quantum Mechanics View

““The more precisely the The more precisely the position is determined, the position is determined, the

less precisely the less precisely the momentum is known in momentum is known in

this instant, and vice this instant, and vice versa.versa.” ”

Heisenberg, Heisenberg, uncertainty paperuncertainty paper

1927 1927

Werner Heisenberg1927

The Uncertainty Principle – Signal Processing View

Fourier Transform of Signal

Signal

There is There is nono such thing as such thing as instantaneous frequencyinstantaneous frequency

The Short-Time Fourier Transform (STFT)The Short-Time Fourier Transform (STFT)

The Uncertainty Principle – Signal Processing View

The Gaussian-modulated The Gaussian-modulated complex exponentials: complex exponentials: GaborGabor functions achieve functions achieve maximal accuracy – maximal accuracy – minimal uncertainty minimal uncertainty

Dennis Gabor 1969

21* t

The Uncertainty Principle – Harmonic Analysis View

S, T are self-adjoint operators.

< P > Ψ = < PΨ , Ψ > : mean of the action of operator P

[S,T]=ST-TS commutator

Then the following holds:S Ψ * T Ψ 0.5 * | < [S,T] > Ψ |

Minimizers of the joint uncertainty The inequality turns into equality iff there

exists i such that:

( S - < S > ) = ( T - < T > )

is the minimizer of the uncertainty principle





Windowed Fourier Transform

Weyl-Heisenberg Group G = (w,b) | b,w Group Law: (w,b) ° (w’,b’) = (w+w’,b+b’)Unitary irreducible representation

The windowed Fourier Transform: )()(),( bxxbU e xi

R

iwx

H

dxebxxf

fbwUbwfWFT

)()(

,),(),(

The Weyl-Heisenberg Group: Generators

Commutation Relation

dxdixbwU

bix

xxbwUix

bb

b

T

T

00

00

,

,

)(),()(

)(),()(

iTTb ,

The minimizer of the 1D Weyl-Heisenberg Group

From the constraint for equality, we obtain the following ODE:

with a solution:

bbww TT

22 )()(wi

b xix ecex

bw ix '





1D Wavelet Transform 1D Affine Group

A = {(a,b) | a,b , a 0}

Group Law:(a,b) ° (a’,b’) = (aa’,ab’+b)

Unitary irreducible representation:

a

bxΨa

1(x)b)ΨU(a,

1D Wavelet Transform 1D Affine GroupMinimizer of uncertainty (Dahlke & Maass):

ab iixcx 21

)()(

Real Imaginary

B = (a,b,)| a+, b 2, SO(2)Group Law:

(a,b,) ° (a’,b’, ’) = (aa’,a b’+b, +’) (x,y) = (x cos() – ysin(), x sin() + ycos())


a

bya

bxa

yxbaU 21 ,||

1),(),,(

No non-zero minimizer

2D Wavelet Transform 2D Similitude Group

Solution 1: Dahlke & Maass Adding elements of the enveloping algebra.Adding elements of the enveloping algebra.

Considering: TConsidering: T, T, Taa, ,

A possible solution is the Mexican hat function: A possible solution is the Mexican hat function:

(r)= [2-2(r)= [2-2rr22]exp(- ]exp(- rr22 ) . ) .

2221 bbb TTT

Solution 2: Ali, Antoine, Gazeau

[T[Taa , T , Tb1b1] & [T] & [T , T , Tb2 b2 ]]

[T[Taa , T , Tb2b2] & [T] & [T , T , Tb1 b1 ]]

Define a new operator: Define a new operator: Find a minimizer for: [TFind a minimizer for: [Taa , T , T ] and [T] and [T , T, T++/2 /2 ]]

with respect to a fixed direction with respect to a fixed direction ..

)sin()cos(21

bb TTT

Ali, Antoine, Gazeau

The 1D solution in Fourier The 1D solution in Fourier Space:Space: Cauchy Wavelets : Cauchy Wavelets : (()= c )= c ss exp(- exp(- ))

The 2D solution in Fourier The 2D solution in Fourier Space:Space:(k)= c |k|(k)= c |k|ss exp(- exp(- k kxx),),

s > 0, s > 0, > 0, k > 0, kx x > 0> 0

Solution in the time domain

Solution in the spatial domain

in

crea

ses

s increases

in

crea

ses

s increases





The 2D Affine Group B = (s11,s12,s21,s22,b1,b2)| all


21 bybxSDyxbSU ,),(),(

2221

1211

ssss

S 12212211 ssssD

“Solution” #1: Going back to SIM(2) Adapting the solutions of Adapting the solutions of Dahlke & Maass and Ali, Dahlke & Maass and Ali,

Antoine, GazeauAntoine, Gazeau: : Total orientation: Total orientation: TT = Ts= Ts12 12 – Ts– Ts21 21

Total Scale: Total Scale: TTscale scale = Ts= Ts11 11 + Ts+ Ts22 22

“Solution” #2: Subspace Solution]Ts11,Ts12],[Ts11,Ts21,[

]Ts11,Tb1] ,[Ts12,Tb2[

]Ts22,Ts21],[Ts22,Ts12,[

]Ts22,Tb2] ,[Ts21,Tb1[

i, s.t. yiii bbs exyx 21115.0,

12

1

b





The Gabor-Wavelet Transform:

B = (, a, b)| , a+, b 2

Problem: This representation is not square integrableProblem: This representation is not square integrableSolution: work with quotients. Solution: work with quotients.

a

bxea

xbawU iwx

||1)(),,(

Gabor Wavelets Transform AWH Group

R

dxea

bxxfa

bawfGWT aik

)(1),,(





The G Group (Torresani)


The Generators:

The Solution: x

ixTx

ixxxT bi

a

)(,)( 2

abxe

axbaaU xi a 111)(),),((

1)(,,

|,),(12aaba

baaB

xiiii exx ab 21

)(





Modern Frame Theory in Banach Spaces (Feichtinger & Grochenig)

A group G in a Hilbert space HA group G in a Hilbert space H An associated generalized integral transformAn associated generalized integral transform The Coorbit-Spaces (LThe Coorbit-Spaces (Lpp space) space) Discretization of the representationDiscretization of the representation Frames Frames

Example:Example:The Euclidean Plane and the Weyl-Heisenberg & The Euclidean Plane and the Weyl-Heisenberg & Wavelets framesWavelets frames

Generalization of the Feichtinger/Grochenig theory to quotient spaces(Dahlke, Fornasier, Rauhut, Steidel, Teschke)

Coorbit Spaces associated with Affine Group Coorbit Spaces associated with Affine Group Besov SpacesBesov Spaces

Coorbit Spaces associated with WH Group Coorbit Spaces associated with WH Group Modulation SpacesModulation Spaces

Coorbit Spaces associated with Affine WH Group Coorbit Spaces associated with Affine WH Group - modulation spacesmodulation spaces

The 1D AWH group w.r.t. the -modulation spaces The section: a = The section: a = (((( leads to the leads to the

representation:representation:

We select: a = We select: a = (((( = ( 1 + = ( 1 + ‖‖‖‖p p ((--

The representation is then given by: The representation is then given by:

))(()()()(,, )(21

bxexbU bxi

)(||1||1)()(,, )(2 bxexbU bxi

The selection of the section

The 1D AWH group w.r.t. the -modulation spaces The infinitesimal generators:The infinitesimal generators:

The solution obtained is:The solution obtained is:

)()()()()(

2 xxxixTxxT

dxd

dxd

b

ixibii

exx 221

1)(





Possible Solution: Gabor-Wavelet What about the representation:

where: (a) = 1/aThe Generators are then:

)()(),),(( bxaeaxbaaU ikax

xiexT

xixkxexT

ikxb

iikxa

)(

)( 2

Numerical Solution:



The GThe G SolutionSolutionThe The -modulation spaces solution-modulation spaces solution


Summary Minimizers for the Minimizers for the AffineAffine Group in Group in 2D2D Minimizers for the Minimizers for the Affine Weyl-HeisenbergAffine Weyl-Heisenberg group in group in 1D1D

Inerpolating between Fourier and Wavelet Transforms Inerpolating between Fourier and Wavelet Transforms using using -modulation spaces-modulation spaces Obtaining the uncertainty minimizers in a Obtaining the uncertainty minimizers in a constrained constrained environmentenvironment

Future Work

http://www.tau.ac.il/~chensagi

Our motivation: Gabor Space Active Contours

Sochen, Kimmel & Malladi

-150 -100 -50 0 50 100 150-150

-100

-50

0

50

100

150

u [hz]

v [h

z]

In the frequency domain deltateta=pi/15,L=0.4, a=1.685

The Uncertainty Principle for G

The ODE:

The Solution:

where

)()()()()(2

)( xxxiex

xxixxixkxe b

ikxa

ikx

)()( xsex

dtteixxxikxs

ikt

ba

log.log)( 50

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Certainty & Uncertainty in Filter Bank Design Methodology