Introduction to Gabor Analysis - univie.ac.at · Radu Frunza Introduction to Gabor Analysis. Introduction to Gabor Analysis Radu Frunza Fourier Transform and Operators Gabor Frames

Introductionto GaborAnalysis

Radu Frunza

FourierTransformand Operators

Gabor Frames

Lattices

WindowFunctions

Gabor FramesRevised

Openquestions

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Introduction to Gabor AnalysisTheoretical and Computational Aspects

Radu Frunza

Numerical Harmonic Analysis Groupunder the supervision of

Prof. Dr. Hans Georg Feichtinger

30 Oct 2012

Radu Frunza Introduction to Gabor Analysis


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Outline

1 Fourier Transform and Operators

2 Gabor Frames

3 Lattices

4 Window Functions

5 Gabor Frames Revised

6 Open questions

7 Final Mention



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Fourier Transform and OperatorsDFT/ iDFT

• Discrete Fourier TransformGiven an input signal f of length L, the DFT of f is avector F of the same length L, with elements:

F (k) :=

L∑n=1

f(n)e−2πi(k−1)(n−1)/L, 1 ≤ k ≤ L (1.1)

• Inverse Discrete Fourier TransformThe inverse DFT is given by:

f(n) =1

L

L∑k=1

F (k)e2πi(k−1)(n−1)/L, 1 ≤ n ≤ L (1.2)



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Fourier Transform and OperatorsFundamental Operators

• TranslationFor x ∈ ZL we define the translation operator Tx by:

(Txf)(t) := f(mod(t− x, L)) (1.3)

• ModulationFor ω ∈ ZL we define the modulation operator Mω by:

(Mωf)(t) := e2πiωt/Lf(t) (1.4)

• Note:Tx is also called a Time shift, and Mω a Frequency shift.TxMω and MωTx are called Time-Frequency shifts.



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Fourier Transform and OperatorsTranslation, Modulation

• In MATLAB

• Tx = rot(eye(L),−x)• Mω = diag(exp(2 ∗ pi ∗ i ∗ ω ∗ (1 : L)/L))

Figure: Translation Figure: Modulation



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Fourier Transform and OperatorsFundamental Operators II

• InvolutionThe involution ∗ is defined by:

f∗(x) := f(−x) (1.5)

• ConvolutionThe convolution of two vectors f, g of length L is thevector f ∗ g defined by:

(f ∗ g)(x) :=

L∑y=1

f(y)g(x− y) (1.6)



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Fourier Transform and OperatorsConvolution

Figure: Random signal Figure: Result after convolution



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Fourier Transform and OperatorsFundamental Operators III

• Inner ProductThe inner product of two vectors f and g of length L isdefined as:

〈f, g〉 :=

L∑k=1

f(k)g(k) (1.7)



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Fourier Transform and OperatorsShort Time Fourier Transform

• STFTThe discrete STFT of a vector f with respect to a windowvector g is given as:

(Vgf)(x, ω) := 〈f,MωTxg〉 , ∀x, ω ∈ ZL (1.8)

• SpectrogramThe spectrogram of f with respect to a window function gsatisfying ‖g‖2 = 1 is defined to be:

SPECgf(x, ω) := |Vgf(x, ω)|2 (1.9)

• Note(s):The spectrogram non-negative, covariant andenergy-preserving.



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Fourier Transform and OperatorsSpectrograms

Figure: Spectrogram 1 Figure: Spectrogram 2



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Gabor FramesFrames

• DefinitionA finite sequence {g1, g2, ..., gN} of vectors gj of length Lis a frame if ∃A,B > 0 such that:

A

L∑k=1

|f(k)|2 ≤N∑j=1

| 〈f, gj〉 |2 ≤ BL∑k=1

|f(k)|2, (2.1)

for all vectors f of length L.

• Note(s):Any two constants A,B satisfying (2.1) are calledframe bounds.We must have N > L elements in the sequence {gj}j=1,N

in order to obtain a frame.



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Gabor FramesAnalysis Operator

• DefinitionFor any subset {gj : j ∈ J} the analysis operator C isgiven by

Cf = {〈f, gj〉 : j ∈ J} (2.2)

• Note(s):C is also called coefficient operator.In MATLAB: C = [g1; g2; . . . gJ ].



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Gabor FramesSynthesis Operator

• DefinitionThe synthesis operator D is defined for a finite sequencec = (cj)j∈J by

Dc =∑j∈J

cjgj (2.3)

• Note(s):D is also called reconstruction operator.In MATLAB: D = C ′.



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Gabor FramesFrame Operator

• DefinitionThe frame operator S is defined by

Sf =∑j∈J〈f, gj〉 gj (2.4)

• Note(s):In MATLAB: S = C ′C = DD′.



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Gabor FramesDual Frame

• DefinitionIf {gj : j ∈ J} is a frame with frame bounds A,B > 0,then

{S−1gj : j ∈ J

}is a frame with frame bounds

B−1, A−1 > 0, the so-called dual frame.

• Note(s):Every f has non-orthogonal expansions:

f =∑j∈J

⟨f, S−1gj

⟩gj =

∑j∈J〈f, gj〉S−1gj (2.5)



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LatticesDefinition

• DefinitionA lattice Λ ⊆ Z2

L is a subgroup of Z2L. Any lattice can be

written in the form Λ = AZ2L, where A is an invertible

2× 2-matrix over ZL. Lattices in Z2L can be described as:

Λ = {(x, y) ∈ Z2L|(x, y) = (ak + dl, ck + bl), (k, l) ∈ Z2

L}(3.1)

with a, b, c, d ∈ N and

A =

(a dc b

)(3.2)



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LatticesSeparability

• SeparableA lattice Λ = aZL × bZL E Z2

L, for a, b ∈ N is called aseparable or product lattice. If Λ is separable then thereexists a generating matrix of Λ which is diagonal:

A =

(a 00 b

)(3.3)

• Non-separableThe generating matrix of any (non-separable) lattice canbe expressed in the form:

A =

(a 0s b

), (3.4)

where s ∈ N.



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LatticesSpyc

Figure: Separable lattice Figure: Quincux lattice



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LatticesIndexing and generating lattices

• Matrix representation

• Index storage

• Parameter representation

• L,a,b,s• LTFAT∗ parameters: L,a,M,lt

• Generating lattices

• Separable lattices• Non-separable lattices

∗Information on LTFAT tollboxRadu Frunza Introduction to Gabor Analysis


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LatticesSubgroups

• Counting ProblemFor a given size N and redundancy red, how many latticesare there?

• SeparableSay that the divisors of N ∗ red are {p1, p2, ..., pn}. Thenumber of separable lattices of redundancy red is ≤ n.

• ExamplesFor L = 480, red = 1.5, there are 10 separable lattices.For L = 480, red = 2, there are 20 separable lattices.



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LatticesSubgroups II

• Non-separable

Say N =∏j p

ajj and N × red = K =

∏j p

bjj . Clearly

0 ≤ bj ≤ 2aj ,∀j. Then we have:

sK(N) =

n∏j=1

spbjj

(pajj ) =

n∏j=1

{pb+1−1p−1 , 0 ≤ b ≤ a

p2a−b+1−1p−1 , a ≤ b ≤ 2a

• ExamplesFor L = 480, red = 1.5, there are 176 non-separablelattices.For L = 480, red = 2, there are 724 non-separablelattices.



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LatticesAdjoint Lattice

• NotationFor λ = (x, ω) ∈ Z2

L we typically write

π(λ) = TxMω, λ = (x, ω) (3.5)

• DefinitionFor any lattice Λ ∈ Z2

L we define the adjoint lattice Λo as

Λo ={λo ∈ Z2

L : π(λo)π(λ) = π(λ)π(λo),∀λ ∈ Λ}(3.6)

• ApplicationsJanssen representation/test



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LatticesSpyc II

Figure: Λ and Λo



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Window FunctionsGaussian window

• DefinitionLet

ϕ(x) = 21/4e−πx2

(4.1)

be the Gaussian on R.

• Properties

• Normalized:

‖ϕ‖2 = 1 (4.2)

• Minimizes the uncertainty principle/ provides the optimalresolution in the time-frequency plane.



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Window FunctionsModified Gaussians

• Stretched Gaussian windows

• Stretching operators• In practice: ϕα(x)

• Rotated Gaussian windows

• Rotation operators• Via Hermite∗ functions

∗Eigenfunctions of the Fourier transformRadu Frunza Introduction to Gabor Analysis


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Window FunctionsModified Gaussians II

Figure: Stretched Gaussian Figure: Stretched and RotatedGaussian



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Window FunctionsDual Window

• DefinitionIf {gj : j ∈ J} is a frame with frame operator S, thenγ = S−1g is called the canonical dual window.

Figure: Dual for a = 20, b = 16 Figure: Dual for a = 10, b = 32



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Gabor Frames RevisedClarifications

Schematically the time-frequency analysis of a signal consists ofthree distinct steps:

• A. Analysis: Given a signal or image f , its STFT Vgfwith respect to a suitable window is computed.

• B. Processing: Vgf(x, ω) is transformed into some newfunction F (x, ω).

• C. Synthesis: The processed signal or image is thenreconstructed using the modified inversion formula:

h =∑λ∈Λ

F (λ)π(λ)γ (5.1)

with respect to a suitable synthesis window γ



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Gabor Frames RevisedConcrete example - Digital image compression

• Method• Given an input 2D signal( image) f , decompose it into

simple parts( RGB/Y CbCr).• For each component, compute a 2D DFT.• Truncate the higher frequencies.• Reconstruct.

• Example



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Gabor Frames RevisedQuality Criteria - Frame Bounds

• Recall the definition of a frameA finite sequence {g1, g2, ..., gN} of vectors gj of length Lis a frame if ∃A,B > 0 such that:

A

L∑k=1

|f(k)|2 ≤N∑j=1

| 〈f, gj〉 |2 ≤ BL∑k=1

|f(k)|2,

for all vectors f of length L.

• Quality of a frameThe ratio B/A is a good indicator of how well a vectorcan be reconstructed using (2.5).



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Gabor Frames RevisedQuality Criteria - CN of Frame Operator

• Frame operatorThe Frame operator plays a central role in the synthesis ofthe signal, either via (2.5) or in computing the canonicaldual window for (5.1). Either way, the inverse of the frameoperator is required in the reconstruction.

• Condition numberThe condition number∗ is a good indicator of theinvertibility of the Frame operator.

∗2-normRadu Frunza Introduction to Gabor Analysis


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Gabor Frames RevisedGeometric Criteria

• DescriptionHow to arrange 2-dimensional balls in R2 in the mosteconomical way.

• Packing problemEach point of R2 may not belong to more than one ball.

• Covering problemEach point of R2 belong to at least one ball.

• Lattice Packing/CoveringA packing/covering is a lattice packing/covering for Λ ifit is of the form (Bx(0) + λ)λ∈Λ

• CriteriaLet r be the radius for the densest packing and R be theradius for the thinnest covering.Then q = vol(BR)/vol(Br) is an indicator of the qualityof the system G(Λ, g0).



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Gabor Frames RevisedPacking/Covering I

Figure: Circles, Sep. Lattice



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Gabor Frames RevisedPacking/Covering II

Figure: Circles, Hex. Lattice



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Gabor Frames RevisedPacking/Covering III

Figure: Circles, Sep. Lattice Figure: Ellipses, Sep. Lattice



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Gabor Frames RevisedQuality Criteria - Janssen Test

• DefinitionGiven a lattice Λ and a window function g, the Janssensum is defined as:

s(Λ, g) :=∑λ∈Λo

Vgg(λ) (5.2)

• ExplanationIf g is properly normalized, i.e. ‖g‖ = 1, then Vgg(0) = 1.If s(Λ, g) < 2, the system has good properties.



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Gabor Frames RevisedOther Quality Criteria

• CN of the dual window

• S0∗ norm of the dual window

• Composite quality criteria

∗1-norm of the STFTRadu Frunza Introduction to Gabor Analysis


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Gabor Frames RevisedEnd note

• Equivalence of Quality criteria?

Figure: Comparison of criteria



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Open questions

• Optimal Quality Criteria?

• Efficient computation of dual windows?

• Efficiency of non-separable lattices?



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I would like to thank:

• Prof. Dr. Hans Georg Feichtinger

• Dr. Maurice de Gosson

• Christoph Wiesmeyr

• Markus Faulhuber

• Peter Sondegaard

• Everyone in the audience: Thank you!



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References

• Hans G. Feichtinger, Ole Christensen, Stephan Paukner. Gabor Analysis for Imaging. In ”Handbookof Mathematical Methods in Imaging”, Springer Berlin, Vol.3 p.1271-1307, 2011.

• K. Grochenig. Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. BirkhauserBoston, Boston, MA, 2001.

• J.H. Conway, N.J.A. Sloane. Sphere Packings, Lattices and Groups, Springer, 1999.

• Chuanming Zong. Simultaneous Packing and Covering in the Euclidean Plane, Monatsh. Math. 134,247-255, 2002.

• Hans G. Feichtinger, Thomas Strohmer, Ole Christensen. A Group-theoretical Approach to GaborAnalysis. Opt. Eng., Vol.34 p.1697–1704, 1995.

• H. G. Feichtinger, N. Kaiblinger. 2D-Gabor analysis based on 1D algorithms. In Proc. OEAGM-97(Hallstatt, Austria), 1997.

• H. G. Feichtinger, W. Kozek, P. Prinz, and T. Strohmer. On multidimensional non-separable Gaborexpansions. In Proc. SPIE: Wavelet Applications in Signal and Image Processing IV, August 1996.

• Stephan Paukner. Foundations of Gabor Analysis for Image Processing, Nov 2007.

• A.J.E.M. Janssen. Gabor representation of generalized functions. J. Math. Anal. Appl., 83:377-394,October 1981.

• D. Gabor. Theory of communication. J. IEE, 93(26):429-457, 1946.


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Introduction to Gabor Analysis - univie.ac.at · Radu Frunza Introduction to Gabor Analysis. Introduction to Gabor Analysis Radu Frunza Fourier Transform and Operators Gabor Frames