Certainty & Uncertainty in Filter Bank Design Methodology
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Transcript of 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
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
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 '
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
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())
Unitary irreducible representation:
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 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 2D Affine Group B = (s11,s12,s21,s22,b1,b2)| all
Unitary irreducible representation:
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 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 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 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 G Group (Torresani)
Unitary irreducible representation:
The Generators:
The Solution: x
ixTx
ixxxT bi
a
)(,)( 2
abxe
axbaaU xi a 111)(),),((
1)(,,
|,),(12aaba
baaB
xiiii exx ab 21
)(
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
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 spaces- modulation 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)(
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
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 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 solution
Conclusions & Future WorkConclusions & Future Work
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