Isotropic Multiresolution Analysis: Theory and Applicationssaurabh/Banff-March-2009.pdfTitle...

Isotropic Multiresolution Analysis: Theory andApplications

Saurabh JainDepartment of Mathematics

University of Houston

March 17th 2009Banff International Research Station

Workshop on “Frames from first principles.”

S. Jain Isotropic Multiresolution Analysis March 17 2009 1 / 29

Collaborators & Acknowledgements

This work has been performed in collaboration with Manos Papadakis(UH), Juan R. Romero (U.Puerto Rico), Simon K. Alexander (UH),Shikha Baid (formerly UH).

We gratefully acknowledge the contributions and previous related work byRobert Azencott (UH and ENS) and Bernhard G. Bodmann (UH).

Outline

MotivationSeismic ImagingBiomedical Imaging and Segmentation

Isotropic Multiresolution AnalysisDefinitionCharacterizations in the theory of IMRAExamples of IMRAExtension Principles

Mathematical Framework for SegmentationFeature spaces and feature mapsSteerability

Example from Seismic Imaging

(a) (b)

Figure: Vertical cross-sections of volumes produced using radial (left) andnon-radial (right) filters. (Courtesy: Total E&P, USA)

Example from Biomedical Imaging

(a) 2D slice from 3D µCT x-raydata

(b) Slice from IntravascularUltra Sound data

Figure: Examples of medical 3D data sets.

Segmentation flowchart

analysis

clustering Segmentat ion

Stack intoFeature Vectors

Problems with tensor product basis

‘Barbara’ DB8 decomposition

‘Barbara’ DB8 high pass

IMRA high pass DB8 high pass

Problems with tensor product basis (Zoom)

IMRA high pass DB8 high pass

Outline

Notation and Preliminaries

We say that Ω ⊂ Rd is radial if for all O ∈ SO(d), OΩ = Ω.

A d × d matrix is expansive if it has integer entries and if all of itseigenvalues have absolute value greater than 1.

An expansive matrix is radially expansive if M = aO, for some fixeda > 1 and a matrix O ∈ SO(d).

A unitary dilation operator with respect to M is defined byDMf (x) = | det M|1/2f (Mx).

For y ∈ Rd , the (unitary) shift operator Ty is defined byTy f (x) = f (x − y).

A function f ∈ L2(Rd) is said to be isotropic if there exists a y ∈ Rd

and a radial function g ∈ L2(Rd) such that f = Tyg .

Definition

An IMRA is a sequence Vjj∈Z of closed subspaces of L2(Rd) satisfyingthe following conditions:

∀j ∈ Z, Vj ⊂ Vj+1,

(DM)jV0 = Vj ,

∪j∈ZVj is dense in L2(Rd),

∩j∈ZVj = 0,

V0 is invariant under translations by integers,

V0 is invariant under all rotations.

If P0 is the orthogonal projection onto V0, then

OP0 = P0O for all O ∈ SO(d),

where O is the unitary operator given by Of (x) := f (OT x)

Definition

∀j ∈ Z, Vj ⊂ Vj+1,

(DM)jV0 = Vj ,

∩j∈ZVj = 0,V0 is invariant under translations by integers,

Definition

∀j ∈ Z, Vj ⊂ Vj+1,

(DM)jV0 = Vj ,

∩j∈ZVj = 0,V0 is invariant under translations by integers,

Characterization of V0

TheoremLet V be an invariant subspace of L2(Rd) under the action of thetranslation group induced by Zd . Then V remains invariant under allrotations if and only if V = PWΩ for some radial measurable subset Ω ofRd .

Characterization of IMRAs

TheoremLet M be a radially expansive matrix and C := M∗. A sequence Vjj∈Zis an IMRA with respect to M if and only if Vj = PWC jΩ, where Ω isradial and satisfies

1 Ω ⊂ CΩ.

2 The set-theoretic complement of ∪∞j=1CjΩ is null.

3 limj→∞ |C−jΩ| = 0.

Moreover the only singly generated IMRAs are precisely Vj = PWC jΩ,where Ω is a radial subset of Td satisfying (1), (2) and (3).

Characterization of isotropic refinable functions

DefinitionA function φ in L2(Rd) is called refinable with respect to dilations inducedby an expansive matrix M, if there exists an H ∈ L∞(Td) such that

φ(M∗ξ) = H(ξ)φ(ξ), for a.e. ξ ∈ Rd .

The function H is called the low-pass filter or mask corresponding to φ.

TheoremIf φ ∈ L2(Rd) is a refinable function which is also isotropic and

limξ→0

φ(ξ) = L 6= 0,

then φ ∈ PWρ/(ρ+1), where ρ is the dilation factor of M.

Characterization of isotropic refinable functions

DefinitionA function φ in L2(Rd) is called refinable with respect to dilations inducedby an expansive matrix M, if there exists an H ∈ L∞(Td) such that

φ(M∗ξ) = H(ξ)φ(ξ), for a.e. ξ ∈ Rd .

The function H is called the low-pass filter or mask corresponding to φ.

TheoremIf φ ∈ L2(Rd) is a refinable function which is also isotropic and

limξ→0

φ(ξ) = L 6= 0,

then φ ∈ PWρ/(ρ+1), where ρ is the dilation factor of M.

Consequences

Corollary

There are no isotropic, refinable functions that are compactly supportedin the spatial domain.

Corollary

If the Fourier transform of φ is supported outside the ball of radius 1/2centered at the origin then the MRA generated by φ is not an IMRA.

Consequences

Corollary

There are no isotropic, refinable functions that are compactly supportedin the spatial domain.

Corollary

If the Fourier transform of φ is supported outside the ball of radius 1/2centered at the origin then the MRA generated by φ is not an IMRA.

Examples of IMRA

Example

The sequence of closed subspaces Vj = PW2jB(0,ρ), for any ρ > 0 andj ∈ Z is an IMRA.

Example

Let B(0, r , s) denote the (2-dimensional) annulus centered at the originhaving inner radius r and outer radius s.

Define the sets A =∞⋃

n=1B(0, rn, 2

n−1), with rn = 2n−1 − (1/16)n, and

B = B(0, 1/2).Define Ω := A

Examples of IMRA

Example

The sequence of closed subspaces Vj = PW2jB(0,ρ), for any ρ > 0 andj ∈ Z is an IMRA.

Example

Let B(0, r , s) denote the (2-dimensional) annulus centered at the originhaving inner radius r and outer radius s.

Define the sets A =∞⋃

n=1B(0, rn, 2

n−1), with rn = 2n−1 − (1/16)n, and

B = B(0, 1/2).Define Ω := A

Extension Principles

Let φ be a refinable function such that φ is continuous at the origin and,

lim|ξ|→0

φ(ξ) = 1 .

Let H0 be the associated low pass filter, i.e.

D∗Mφ = H0φ.

Assume that there exists a constant B > 0 such that∑l∈Zd |φ(ξ + l)|2 ≤ B a.e. on Rd .

Furthermore, let Hi , i = 1, . . . ,m, be Zd -periodic measurable functionsand define m wavelets, ψi via

D∗Mψi = Hi φ.

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

Unitary Extension Principle

Theorem (Daubechies, Han, Ron and Shen 2003)

Assume Hi ∈ L∞(Td) for all i = 0, . . . ,m, then the following twoconditions are equivalent:

1 The set

D jMTkψi : j ∈ Z, k ∈ Zd , i = 1, . . . ,m

is a Parseval

Frame for L2(Rd).2 For all ξ ∈ σ(V0),

limj→−∞Θ(M∗jξ) = 1.If q ∈ (M∗−1Zd)/Zd \ 0 and ξ + q ∈ σ(V0), then

Θ(M∗ξ)H0(ξ)H0(ξ + q) +m∑

Hi (ξ)Hi (ξ + q) = 0.

Fundamental function

Θ is the so-called fundamental function, defined by

Θ(ξ) =∞∑j=0

m∑i=1

∣∣Hi (M∗jξ)

∣∣2 j−1∏l=0

∣∣∣H0(M∗lξ)

∣∣∣2.

ΘM is the so-called mixed fundamental function, defined by

ΘM(ξ) =∞∑j=0

m∑i=1

Hsi (M

∗jξ)Hai (M∗jξ)

j−1∏l=0

∗lξ)Ha0 (M∗lξ).

Fundamental function

Θ is the so-called fundamental function, defined by

Θ(ξ) =∞∑j=0

m∑i=1

∣∣Hi (M∗jξ)

∣∣2 j−1∏l=0

∣∣∣H0(M∗lξ)

∣∣∣2.ΘM is the so-called mixed fundamental function, defined by

ΘM(ξ) =∞∑j=0

m∑i=1

Hsi (M

∗jξ)Hai (M∗jξ)

j−1∏l=0

∗lξ)Ha0 (M∗lξ).

Mixed Extension Principle

Theorem (Daubechies, Han, Ron and Shen, 2003)

Assume Hai ,H

si ∈ L∞(Td) for all i = 0, . . . ,m. Then the following two

conditions are equivalent,

1 The sets Ψa :=

D jMTkψ

ai : j ∈ Z, k ∈ Zd , i = 1, . . . ,m

Ψs :=

D jMTkψ

si : j ∈ Z, k ∈ Zd , i = 1, . . . ,m

is a pair of dual

frames for L2(Rd).2 For all ξ ∈ σ(V a

0 )⋂

σ(V s0 ),

Ψa and Ψs are Bessel families.limj→−∞ΘM(M∗jξ) = 1.If q ∈ (M∗−1Zd)/Zd \ 0 and ξ + q ∈ σ(V a

0 )⋂

σ(V s0 ), then

ΘM(M∗ξ)Hs0(ξ)H

a0 (ξ + q) +

m∑i=1

Hsi (ξ)H

ai (ξ + q) = 0.

Our versions of Extension Principles

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

Xφψ :=

D jMTkψi : j ∈ N ∪ 0, k ∈ Zd , i = 1, . . . ,m

Tkφ : k ∈ Zd

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

Xφψ :=

D jMTkψi : j ∈ N ∪ 0, k ∈ Zd , i = 1, . . . ,m

Tkφ : k ∈ Zd

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

Xφψ :=

D jMTkψi : j ∈ N ∪ 0, k ∈ Zd , i = 1, . . . ,m

Tkφ : k ∈ Zd

lim|ξ|→0

φ(ξ) = 1 .

D∗Mφ = H0φ.

D∗Mψi = Hi φ.

Xφψ :=

D jMTkψi : j ∈ N ∪ 0, k ∈ Zd , i = 1, . . . ,m

Tkφ : k ∈ Zd

Our version of Unitary Extension Principle

TheoremAssume Hi ∈ L∞(Td) for all i = 1, . . . ,m then Xφψ is a Parseval framefor L2(Rd) if and only if for all q ∈ (M∗−1Zd)/Zd and for a.e.ξ, ξ + q ∈ σ(V0),

m∑i=0

Hi (ξ)Hi (ξ + q) = δq,0.

Our version of Mixed Extension Principle

TheoremLet Ha

i , Hsi ∈ L∞(Td) for i = 1, . . . ,m, then X a

φψ and X sφψ form a pair of

dual frames for L2(Rd) if and only if

1 X aφψ and X s

φψ are Bessel families,

2 For all q ∈ (M∗−1Zd)/Zd and for a.e. ξ, ξ + q ∈ σ(V a0 ) ∩ σ(V s

m∑i=0

Hsi (ξ)H

ai (ξ + q) = δq,0.

An example of dual isotropic wavelet frames

Ha0 = 1 inside the ball of radius b2

Ha0 = 0 on Td \ B(0, b1)

Ha0 |B(0,b1) is radial.

Let ha be a Zd -periodic function defined by,

ha(ξ) =1− Ha

0 (ξ)

|det(M)|1/2.

Using ha we define the analysis high pass filters

Hai (ξ) := e−2πi〈qi−1,ξ〉ha(ξ).

Hs0 = 1 inside the ball of radius b1

Hs0 = 0 on Td \ B(0, b0)

Hs0 |B(0,b0) is radial.

hs = 0 inside the ball of radius b3, for some b3 > 0

hs = 1

|det(M)|1/2 on Td \ B(0, b2)

hs |Td is radial

Using hs we define the synthesis high pass filters:

Hsi (ξ) = e−2πi〈qi−1,ξ〉hs(ξ)

m∑i=0

Hsi (ξ)H

ai (ξ + q) = δq,0.

2D IMRA scaling function and wavelet

(a) Fourier transform of the scalingfunction φ

(b) Fourier transform of the waveletψ1(2.)

Outline

Feature spaces and feature maps

An image is a function f in L2(R3).

To each pixel z ∈ R3, we associate a ‘feature’ vector belonging to afixed finite dimensional space V , called the feature vector space.

The vector space of all V -valued functions h defined on R3 suchthat ‖h(·)‖ is square integrable on R3 is denoted by H. Thus,

‖h‖ =(∫

R3 ‖h(z)‖2dz)1/2

Let A : L2(R3) → H be a bounded linear transformation thatassociates to each f the V -valued function Af defined on R3. Thislinear transformation is called a feature map.

‖h‖ =(∫

R3 ‖h(z)‖2dz)1/2

‖h‖ =(∫

R3 ‖h(z)‖2dz)1/2

‖h‖ =(∫

R3 ‖h(z)‖2dz)1/2

Steerability

DefinitionConsider a bounded linear feature mapping A generating for each image fand each voxel z ∈ R3 a feature vector Af (z) in the Euclidean space V .We shall say that A is a steerable feature mapping if there is a mappingU from the group G of rigid motions into the general linear group GL(V )of V such that for each rigid motion R of R3, the invertibletransformation U(R) verifies,

A[Rf ](z) = U(R) [Af (R(z))] z ∈ R3.

Here R is the following transformation induced by R on L2(R3):

Rf (z) = f (Rz).

An IMRA based steerable feature map

Using the functions φ and ψ, we define the following feature map:

Af (z) = (< f , Tzφ(·/8) >, < f , Tzψ(·/4) >, < f , Tzψ(·/2) >) .

This map is both translation and rotation invariant

Hence, it is steerable.

References

J.R. Romero, S.K. Alexander, S. Baid, S. Jain, and M. Papadakis

The Geometry and the Analytic Properties of Isotropic MultiresolutionAnalysis.

to appear in the special issue on “Mathematical Methods for ImageProcessing” in Advances in Computational Mathematics, Springer, 2009.

S. Jain, Manos Papadakis, and Eric Dussaud

Explicit Schemes in Seismic Migration and Isotropic Multi-scaleRepresentations.

in Radon Transforms, Geometry and Wavelets, Contemporary Mathematics,Volume 464, Pages 177-200, AMS, 2008.

S.K. Alexander, R. Azencott, M. Papadakis

Isotropic Multiresolution Analysis for 3D-Textures and Applications inCardiovascular Imaging.

in Wavelets XII, SPIE Proceedings Vol. 6701, 2007.

Isotropic Multiresolution Analysis: Theory and Applicationssaurabh/Banff-March-2009.pdfTitle...

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