Certainty & Uncertainty in Filter Bank Design Methodology

Certainty & Uncertainty in Filter Bank Design Methodology Chen Sagiv

Joint work with: Nir Sochen Yehoshua Zeevi Peter Maass & Dirk Lorenz Stephan Dahlke

The Motivation: Maximal Accuracy Minimal Uncertainty scale location frequency orientation

The Motivation Image features

The Motivation • Signal & Image Processing applications call for: • “optimal” mother-wavelet • “optimal” filter bank • Possible criteria for optimality: • The “optimal” mother-wavelet provides maximal accuracy - minimal uncertainty • The “optimal” filter bank constitutes a tight frame

The quest for the “optimal” function • The many faces of the Uncertainty Principle • A case study: The Weyl-Heisenberg Group • Previous Studies • The 2D Affine Group • The 1D Affine Weyl-Heisenberg Group • The G Solution • The -modulation spaces solution • A Gabor-wavelet flavored solution • Conclusions & Future Work

The Uncertainty Principle – Quantum Mechanics View • “The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” Heisenberg, uncertainty paper 1927 Werner Heisenberg 1927

The Uncertainty Principle – Signal Processing View Signal Fourier Transform of Signal

There is no such thing as instantaneous frequency

The Short-Time Fourier Transform (STFT)

Dennis Gabor 1969 The Uncertainty Principle – Signal Processing View • The Gaussian-modulated complex exponentials: Gabor functions achieve maximal accuracy – minimal uncertainty

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: SΨ * TΨ  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

G = (w,b) | b,w   • Group Law: (w,b)° (w’,b’) =(w+w’,b+b’) • Unitary irreducible representation • The windowed Fourier Transform: Windowed Fourier Transform Weyl-Heisenberg Group

The Weyl-Heisenberg Group: Generators Commutation Relation

The minimizer of the 1D Weyl-Heisenberg Group • From the constraint for equality, we obtain the following ODE: • with a solution:

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:

1D Wavelet Transform 1D Affine Group • Minimizer of uncertainty (Dahlke & Maass): Imaginary Real

2D Wavelet Transform 2D Similitude Group • 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: No non-zero minimizer

Solution 1: Dahlke & Maass • Adding elements of the enveloping algebra. • Considering: T, Ta, • A possible solution is the Mexican hat function: • (r)= [2-2r2]exp(- r2 ) .

Solution 2: Ali, Antoine, Gazeau • [Ta , Tb1] & [T , Tb2 ] [Ta , Tb2] & [T , Tb1 ] • Define a new operator: • Find a minimizer for: [Ta , T ] and [T , T+/2 ]with respect to a fixed direction .

s increases Ali, Antoine, Gazeau • The 1D solution in Fourier Space: • Cauchy Wavelets : ()= c s exp(- ) • The 2D solution in Fourier Space: (k)= c |k|s exp(- kx), s > 0,  > 0, kx > 0  increases Solution in the time domain s increases  increases Solution in the spatial domain

The 2D Affine Group • B = (s11,s12,s21,s22,b1,b2)| all  • Unitary irreducible representation:

“Solution” #1: Going back to SIM(2) • Adapting the solutions of Dahlke & Maass and Ali, Antoine, Gazeau: • Total orientation: T = Ts12 – Ts21 • Total Scale: Tscale = Ts11 + Ts22

“Solution” #2: Subspace Solution [Ts11,Ts12],[Ts11,Ts21], [ Ts11,Tb1], [Ts12,Tb2] [Ts22,Ts21],[Ts22,Ts12], [ Ts22,Tb2], [Ts21,Tb1]   i, s.t.

Gabor Wavelets Transform AWH Group • The Gabor-Wavelet Transform: • B = (, a, b)| , a+, b 2 • Problem: This representation is not square integrable • Solution: work with quotients.

The G Group(Torresani) • Unitary irreducible representation: • The Generators: • The Solution:

Modern Frame Theory in Banach Spaces (Feichtinger & Grochenig) • A group G in a Hilbert space H • An associated generalized integral transform • The Coorbit-Spaces (Lp space) • Discretization of the representation Frames • Example: • The Euclidean Plane and the Weyl-Heisenberg & Wavelets frames

Generalization of the Feichtinger/Grochenig theory to quotient spaces(Dahlke, Fornasier, Rauhut, Steidel, Teschke) • Coorbit Spaces associated with Affine Group  Besov Spaces • Coorbit Spaces associated with WH Group  Modulation Spaces • Coorbit Spaces associated with Affine WH Group  - modulation spaces

The 1D AWH group w.r.t. the -modulation spaces • The section: a =  (( leads to the representation: • We select: a =  (( = ( 1 + ‖‖p (- • The representation is then given by:

The selection of the section

The 1D AWH group w.r.t. the -modulation spaces • The infinitesimal generators: • The solution obtained is:

Possible Solution: Gabor-Wavelet • What about the representation: • where: (a) = 1/a • The Generators are then:

Numerical Solution:

The quest for the “optimal” function • The many faces of the Uncertainty Principle • A case study: The Weyl-Heisenberg Group • Previous Studies • The 2D Affine Group • The 1D Affine Weyl-Heisenberg Group • The G Solution • The -modulation spaces solution • Conclusions & Future Work

Summary • Minimizers for the Affine Group in 2D • Minimizers for the Affine Weyl-Heisenberg group in 1D • Inerpolating between Fourier and Wavelet Transforms using -modulation spaces • Obtaining the uncertainty minimizers in a constrained environment Future Work

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

Our motivation: Gabor Space Active Contours Sochen, Kimmel & Malladi

The Uncertainty Principle for G • The ODE: • The Solution: where

Certainty & Uncertainty in Filter Bank Design Methodology