Thesis oral defense

Thesis Oral PresentationWavelet and its applications

Fan Zhitao

NUS Graduate SchoolNational University of Singapore

March 30, 2015

System and frameI A system gives a decomposition from Hilbert space to sequence space.I A frame not only provides this decomposition, but also

I ensures the numerial stable reconstruction.I overcomes shortage of orthonormal system, e.g.

I Orthonormal Gabor system: no window with good time andfrequency localization (Balian-Low theorem)1

I Orthonormal wavelets with compact support: complicatedexpression; no symmetry dyadic real wavelet except the trivialcase2

Application of framesI Natural image restoration problems, e.g. denoising3, inpainting4, deblurring5I Biological image processing problems, e.g. 3D molecule reconstruction from

electron microscope images 6

1I. Daubechies, Ten Lectures on Wavelets, SIAM, 19922I. Daubechies, Ten Lectures on Wavelets, SIAM, 19923Z. Gong, Z. Shen and K.C. Toh, Image restoration with mixed or unknown noises, Multiscale Modeling

and Simulation: A SIAM Interdisciplinary Journal, 12(2):458-487, 20144J. Cai, R. Chan and Z. Shen, A framelet-based image inpainting algorithm, Applied and Computational

Harmonic Analysis, 24:131-149, 20085J. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring from a single image using sparse

approximation, CVPR, 20096M. Li, Z. Fan, H. Ji and Z. Shen, Wavelet frame based algorithm for 3D reconstruction in electron

microscopy, SIAM Journal on Scientific Computing, 36(1):45-69, 2014.

Background 2/46

Motivation and contribution

I Motivation: Ron and Shen1 developed the dual Gramiananalysis to analyze frame properties for shift-invariant systemsin L2(Rd).

I Goal: Develop the dual Gramian analysis to study frames in ageneral Hilbert space.

BenefitsI Finding the canonical dual/tight frame by a matrix inverseI Estimating the frame bounds by classical matrix inequalitiesI Duality principle: transfer the frame property of the system to

Riesz sequence property of its adjoint

1A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(Rd ), Canadian Journalof Mathematics, 47:1051-1094, 1995

Background 3/46

Contribution on Gabor systemGabor system

EkM lg = eil·(x−k)g(x − k), k ∈ Zd , l ∈ 2πZd , g ∈ L2(Rd )

Known works and challengesI Fiber pre-Gramian matrix by Ron and Shen.1I Many classical results, e.g. biorthogonal relationship, Wexler-Raz identityI Design good Gabor windows with good time and frequency localization

Our constributionI The connections of these two definitions of pre-Gramian matrices, by finding

a good orthonormal basisI Fully developed the mixed fiber dual Gramian analysis for two Gabor systemsI Classical identities as consequence of duality principleI Constructed Gabor windows, with compact support and arbitrary

smoothness, in particular for multivariate case

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1A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in L2(Rd ), Duke Mathematical Journal,89:237-282, 1997.

Background 4/46

Contribution on wavelet systemWavelet system

DkE jψ = 2kd/2ψ(2k · −j), k ∈ Zd , j ∈ Zd , ψ ∈ L2(Rd )

Works and challengesI Orthonormal wavelet with compact support by Daubechies1I UEP/MEP for tight/dual wavelet frame2: complete matrix with polynomial

entriesI The multivariate wavelet frame construction remains challenging

ContributionI The dual Gramian analysis is applied to analyze filter banks.I Duality principles lead to a simple way of construction.I Easy construction scheme for multivariate dual/tight wavelet frames with

wavelet such as with small support and symmetric/anti-symmetric.

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1I. Daubechies, Ten Lectures on Wavelets, SIAM, 19922A. Ron and Z. Shen, Affine systems in L2(Rd ): the analysis of the analysis operator, Journal of

Functional Analysis, 148:408-447, 1997.Background 5/46

Publications and Outline

PaperI Zhitao Fan, Hui Ji, Zuowei Shen, Dual Gramian analysis: duality principle and

unitary extension principle, Mathematics of computation, to appear.

I Zhitao Fan, Andreas Heinecke, Zuowei Shen, Duality for frames, Journal ofFourier Analysis and Applications, to apprear.

OutlineI Review of notations and definitionsI Dual Gramian analysisI Gabor systemI Wavelet system

Background 6/46

Notations

H is a separable Hilbert space

A system X is a sequence of H with a certain indexing, e.g.I indexed by Z: xii∈Z = · · · , x−1, x0, x1, · · · I indexed by N: xii∈N = x1, x2, x3, · · ·

A sequence indexed by XI c ∈ `0(X): c(x) ∈ C and (c(x))x∈X with finite supportI c ∈ `2(X): a sequence that is square summable, i.e.

∑x∈X |c(x)|2 <∞

RX : another system in H indexed by X , i.e. Rx ∈ H for x ∈ X .

A matrix

(a(i , j))i∈M,j∈N =

j ∈ N

...i ∈ M · · · a(i , j) · · ·

...

Background 7/46

Synthesis and analysis operators

The synthesis operator of X

TX : `2(X )→ H : c 7→∑

x∈X c(x)x

The analysis operator of X

T ∗X : H → `2(X ) : h 7→ 〈h, x〉x∈X

I When either TX or T ∗X is bounded, they are adjoint.I X is a Bessel system.

Background 8/46

Riesz and frame properties

A Bessel system X isI fundamental if T ∗X is injective; a frame if

A‖f ‖2 ≤ ‖T ∗X f ‖2 ≤ B‖f ‖2, ∀f ∈ H

Being a tight frame if A = B = 1.I `2-independent if TX is injective; a Riesz sequence if

A‖c‖2 ≤ ‖TXc‖2 ≤ B‖c‖2, ∀c ∈ `2(X )

Being an orthonormal sequence if A = B = 1.I A Riesz/orthonormal sequence X is a Riesz/orthonormalbasis if X is fundamental.

Background 9/46

Self-adjoint operators

System X is a Bessel system ⇔ either T ∗XTX or TXT ∗X bounded

The frame operator TXT ∗X1. A Bessel system X is fundamental ⇔ TXT ∗X is injective

2. it is a frame for H ⇔ TXT ∗X has a bounded inverse

3. it is a tight frame ⇔ TXT ∗X = I

The operator T ∗XTX

1. A Bessel system X is `2-independent ⇔ T ∗XTX is injective

2. X is a Riesz sequence ⇔ T ∗XTX has a bounded inverse

3. X is an orthonormal sequence ⇔ T ∗XTX = I

Background 10/46

Dual frames

System X is a tight frame, then TXT ∗X = I, i.e.∑x∈X〈f , x〉 x = f , for f ∈ H.

Two systemsI Suppose X and Y = RX are Bessel systems in HI X and Y are dual frames if TY T ∗X = I or TXT ∗Y = I

An exampleI X is a frame for HI S = TXT ∗X is invertibleI S−1X is the canonical dual frame of X

Background 11/46

Dual Gramian AnalysisI pre-Gramian, Gramian, dual Gramian matrixI Connection to the operatorI ResultsI Duality principle

Pre-Gramian matrixGiven a system X and an orthonormal basis O of H, the pre-Gramian (wrt O)

JX = (〈x , e〉)e∈O,x∈X

Given two systems X and RX in H, the mixed Gramian matrix is

GRX ,X := J∗RXJX =

(∑e∈O

⟨x ′, e⟩〈e,Rx〉

)x∈X ,x′∈X

=(⟨

x ′,RX⟩)

x∈X ,x′∈X

If both X and RX satisfy the weak assumption∑x∈X | 〈x , e〉 |

2 <∞ for all e ∈ O

The mixed dual Gramian matrix (wrt O) is

GRX ,X := JRXJ∗X =

(∑x∈X

〈Rx , e〉⟨e′, x⟩)

e∈O,e′∈O

When RX = X , drop “mixed” in the name.Dual Gramian analysis 13/46

The analysis

Let U = TO which is unitary.

UJXc = TXc for all c ∈ `0(X )

T ∗XUd = J∗Xd for all d ∈ `0(O)

Weak form

〈TXc,TRXd〉 = d∗GRX ,Xc for all c, d ∈ `0(X )〈T ∗XUc,T ∗RXUd〉 = d∗GRX ,Xc for all c, d ∈ `0(O)

Strong formIf X and RX are Bessel systems

T ∗RXTXc = GRX ,Xc for all c ∈ `2(X )U∗TRXT ∗XUc = GRX ,Xc for all c ∈ `2(O)

Dual Gramian analysis 14/46

The finite exampleI Let H be Cm

I Let eimi=1 be the canonical orthonormal basis.I Let X = xknk=1 ⊂ Cm.

Then, the pre-Gramian matrix of X is

JX =

x1(1) · · · xn(1)...

. . ....

x1(m) · · · xn(m)

which is the matrix representation of the synthesis operator

TX : `2(X)→ Cm : c 7→∑n

k=1 ckxk .

The adjoint matrix J∗X

J∗X =

x1(1) · · · x1(m)...

. . ....

xn(1) · · · xn(m)

is the matrix representation of the analysis operator

T∗X : Cm → `2(X) : f 7→ 〈f , xk〉nk=1.


The finite example

I Given another system Y = yknk=1 ⊂ Cm

I JY is the associated pre-Gramian matrix

The mixed Gramian matrix and mixed dual Gramian matrix are

GX ,Y = J∗XJY = (〈yk′ , xk〉)k,k′ , GX ,Y = JXJ∗Y =(∑n

k=1 xk(j)yk(j′))j,j′

which are the matrix representations of the mixed operators T ∗XTYand TXT ∗Y respectively.


Canonical dual and tight frames

I The canonical dual frame: S−1X

Let X be a frame in H with frame bounds A,B and let U be the synthesisoperator of an orthonormal basis of H. Then

I system UG−1X U∗X is a frame with bounds B−1,A−1 which is thecanonical dual frame of X .

Canonical dual frame

I The canonical tight frame: S−1/2X

Let X be a frame in H and let U be the synthesis operator of an or-thonormal basis of H. Let G−1/2

X denote the inverse of the positivesquare root of GX . Then,

I system UG−1/2X U∗X forms the canonical tight frame.

Canonical tight frame


Estimate upper frame boundLet I be a countable index set, and let M be a complex valued nonnegativeHermitian matrix with its rows and columns indexed by I.

supi∈I(∑

j∈I|M(i,j)|2

)1/2≤‖M‖≤supi∈I

∑j∈I|M(i,j)|

Let X be a system in a Hilbert space H satisfying the weak condition with respect to anorthonormal basis O of H .(a) Let

B1 : e 7→∑

e′∈O |∑

x∈X 〈e′, x〉〈x , e〉 |.

Then X is a Bessel system whenever supe∈O B1(e) <∞ and its Bessel bound is notlarger than (supe∈O B1(e))1/2.

(b) Assume that X is a Bessel system, let

B2 : e 7→(∑

e′∈O |∑

x∈X 〈e′, x〉〈x , e〉 |2

)1/2.

Then K = (supe∈O B2(e))1/2 <∞ and the Bessel bound is not smaller than K .


Estimate lower frame boundThe lower frame bound can be obtained when the dual Gramian matrix is diagonallydominant. For a Hermitian diagonally dominant matrix M,

‖M−1‖ ≤ supi∈I(|M(i , i)| −

∑j∈I\i |M(i , j)|

)−1

Let X be a system in Hilbert space H satisfying week condition with respect to anorthonormal basis O of H. Let

b1 : e 7→

(∑x∈X

| 〈e, x〉 |2 −∑e′ 6=e

|∑x∈X

⟨e′, x⟩〈x , e〉 |

)−1.

Then X is a frame whenever supe∈O b1(e) < ∞ and the lower frame bound is notsmaller than (supe∈O b1(e))−1/2.


Duality principleIf X = xknk=1 ⊂ Cm, wrt the standard orthonormal bases,

JX =

x1(1) · · · xn(1)...

. . ....

x1(m) · · · xn(m)

A possible adjoint system of X is

X∗ = (xk(i))k=1,...,n : i = 1, . . . ,m ⊂ Cn

Systems X and X∗ are adjoint if for some matrix representation of the synthesisoperator of X

I The columns is associated with X while the rows is associated with X∗

I The analysis properties of X are characterized by the synthesis propertiesof X∗.

Duality Principle


Adjoint system

A system X∗ is called an adjoint system of X , if(a) Exists orthonormal basis O′, such that X∗ and O′ satisfy the

weak condition∑x ′∈X ′

| 〈x ′, e′〉 |2 <∞ for all e′ ∈ O′.

(b) The pre-Gramian JX∗ of X∗ with respect to O′ satisfies

JX∗ = UJ∗XV

for some unitary operators U and V .

Definition

I Up to unitary equivalence,

GRX ,X = G(RX)∗,X∗


The example by Casazza

I Let X = fkk∈N be a system in HI Wrt an orthonormal basis eii∈N, it satisfies:

∑i∈N | 〈fk , ei 〉 |

2 <∞ for allk ∈ N

Suppose hkk∈N is another orthonormal basis of H and define

X ′ = gi :=∑

k∈N 〈fk , ei 〉 hki∈N

Then X ′ is indeed an adjoint system of X .I The system X ′ satisfies the weak condition:∑

i∈N

| 〈gi , hk〉 |2 =∑i∈N

| 〈fk , ei 〉 |2 <∞ for all k ∈ N.

I It is easy to see:

JX ′ = (〈gi , hk〉)k,i = (〈fk , ei 〉)k,i = J∗X


Duality results for single systems

GX = GX∗

Let X be a given system in H, and suppose that X∗ is an adjoint system ofX in H′. Then(a) A system X is Bessel in H ⇔ its adjoint system X∗ is Bessel in H′

with the same Bessel bound.

(b) A Bessel system X is fundamental ⇔ its adjoint system X∗ is Besseland `2-independent.

(c) A system X forms a frame in H ⇔ its adjoint system X∗ forms aRiesz sequence in H′. The frame bounds of X coincide with theRiesz bounds of X∗.

(d) A system X forms a tight frame in H ⇔ its adjoint system X∗ formsan orthonormal sequence in H′.

Proposition


Duality results for dual frames

I = GX ,RX = GX∗,(RX)∗

Suppose X and RX are Bessel systems in H.

X and RX are dual frames ⇔ X∗ is biorthonormal to (RX )∗

Theorem


Wavelet SystemsI Dual Gramian analysis for filter banksI Simple construction scheme for filter banks by duality principleI Multivariate dual/tight wavelet frame construction

Filter banks

Filter banks in `2(Zd):

X = X (a,N) := (al(n − Nk))n∈Zd : l ∈ Zr , k ∈ Zd

The analysis operator

T ∗X : `2(Zd)→ `2(Zr × Zd) : c 7→ (↓N (c ∗ al(−·))(k))(l,k)∈Zr×Zd

I Downsampling: ↓N d(k) = d(Nk) for k ∈ Zd .

The synthesis operator

TX : `2(Zr × Zd)→ `2(Zd) : c 7→∑

l∈Zr(↑N c(l , ·)) ∗ al

I Upsampling: for fixed l ∈ Zr , ↑N c(l , k) is equal to c(l ,N−1k) if N divides allentries of k ∈ Zd and is equal to 0 otherwise.

Wavelet system 26/46

The pre-Gramian matrix of filter bankThe pre-Gramian matrix of X (wrt the canonical orthonormal basis) is

JX = (al (n − Nk))n∈Zd ,(l,k)∈Zr×Zd

Suppose al ’s are FIR filters. JX is formed by shifts of a small block matrix

A =

a0(n1) a0(n2) · · · a0(nm)a1(n1) a1(n2) · · · a1(nm)

......

. . ....

ar−1(n1) ar−1(n2) · · · ar−1(nm)

An adjoint system

X∗ = (al (n))(l,n)∈Zr×Ωj : j ∈ Zd

with Ωj := j + NZd . I.e. concatenation of the columns of A indexed by theNZd -coset of an index.

Two Bessel systems X and Y are dual frames⇔ the adjoint systems X∗ andY ∗ are biorthonormal.


Construction scheme for filter banks

Let X = X (a,N) and Y = X (b,N), for FIR filters a = alr−1l=0 andb = blr−1l=0 in `2(Zd) and N ∈ N.

I Then X and Y are dual frames in `2(Zd), if

A∗B = M

where M is a diagonal matrix with diagnal c satisfying∑n∈Ωj

c(n) = 1

for all j ∈ Zd/NZd .I The system X is a tight frame when al = bl for l = 0, . . . , r − 1.

Theorem


Construction scheme for filter banks

A∗B = M

I M ∈ Cr×r be a diagonal matrix with diagonal c such that∑n∈Ωj

c(n) = 1

for every j ∈ Zd/NZd .I Let A = (al (nj ))l∈Zr ,j∈Zr ∈ Cr×r be invertibleI Let

B = (bl (nj ))l∈Zr ,j∈Zr = (A∗)−1M

Then the filters a = alr−1l=0 and b = blr−1l=0 defined by A and B generatedual frames X(a,N) and X(b,N) in `2(Zd ).

Construction


Multiresolution Analysis (MRA) wavelet

A function φ ∈ L2(Rd) is called a refinable function if

φ(2·) = a0φ

The sequence a0 ∈ `2(Zd) is the refinement mask. a0(0) = 1.

Let V0 ⊂ L2(Rd) be the closed linear span of E (φ) and Vk := Dk(V0)for k ∈ Z. Vkk∈Z is called an MRA if

(i)Vk ⊂ Vk+1 (ii) ∪kVk is dense in L2(Rd) (iii) ∩kVk = 0

e.g. φ ∈ L2(Rd ) is a compactly supported refinable function with φ(0) = 1

The wavelets Ψ = ψlrl=1 ⊂ L2(Rd)

ψl(2·) = al φ

The sequence al ∈ `2(Zd) is called the wavelet mask. al(0) = 0.


Mixed Unitary Extension Principle (MEP)

I Let φa, φb be compactly supported refinable functions withφa(0) = φb(0) = 1 and masks a0, b0.

I Let alrl=1, blrl=1 be the masks of wavelet systems X ,Y .

If both X and Y are Bessel systems andr∑

l=0al(ω)bl(ω + ν) = δν,0,

for any ν ∈ 0, πd and a.e. ω ∈ Td , then X and Y are dualframes.

MEP1

I Unitary extension principle(UEP2): Y = X .1A. Ron and Z. Shen, Affine systems in L2(Rd ): dual systems, Journal of Fourier Analysis and

Applications, 3:617-637, 1997.2A. Ron and Z. Shen, Affine systems in L2(Rd ): the analysis of the analysis operator, Journal of

Functional Analysis, 148:408-447, 1997.Wavelet system 31/46

The connection of UEP/MEP with filter banks

HX (ω) =

a0(ω + ν1) a1(ω + ν1) . . . ar (ω + ν1)a0(ω + ν2) a1(ω + ν2) . . . ar (ω + ν2)

......

. . ....

a0(ω + ν2d ) a1(ω + ν2d ) . . . ar (ω + ν2d )

I MEP: HX (ω)HY (ω)∗ = I while UEP: HX (ω)HX (ω)∗ = I

I The filter bank X = X(2d/2alrl=0, 2)

I MEP for dual frame filer bank, and UEP for tight frame filter bank

Questions:I Can we start with a refinement mask to constructionwavelets?

I When will the filter banks be wavelet masks?


Dual wavelet frame construction

The construction starts from a real-valued refinement mask a0satisfying ∑

n∈Ωja0(n) = 2−d ,

for all j ∈ Zd/2Zd , where Ωj = (2Zd + j) ∩ supp(a0).

Examples:I the butterfly subdivision scheme by Dyn (1990)I the interpolatory refinement mask derived from box spline by

Riemenschneider and shen (1997)



A∗B = M

1. (Initialization): Define the first row of a matrix A by collectingthe non-zero entries of a0. Let M be the diagonal matrix with thefirst row of A as its diagonal.

2. (Primary wavelet masks): Complete the matrix A to be aninvertible square matrix, each of whose remaining rows hasentries summing to zero.

3. (Dual wavelet masks): Define A = AM−1 and B = (A∗)−1.



a0(n1) · · · a0(nm)∗ B =

a0(n1). . .

a0(nm)

1. (Initialization): Define the first row of a matrix A by collecting

the non-zero entries of a0. Let M be the diagonal matrix with thefirst row of A as its diagonal.





a0(n1) · · · a0(nm)...

...am(n1) · · · am(nm)

∗

B =

a0(n1). . .

a0(nm)

1. (Initialization): Define the first row of a matrix A by collecting

the non-zero entries of a0. Let M be the diagonal matrix with thefirst row of A as its diagonal.




A Summary for construction of dual wavelet frames

Suppose the real-valued refinement mask a0 ∈ `2(Zd) is of finite supportsatisfying the mask condition, and the corresponding refinable functionφ ∈ L2(Rd) is compactly supported with φ(0) = 1. Then

I the masks derived by Construction satisfy the MEP condition.I the wavelet systems X and Y generated by those masks are dual

wavelet frames in L2(Rd).I the number of wavelet is one less than the size of the support of

a0.I the support of the derived masks is no larger than the support of

a0 and if the support of φ is convex, then the support of theprimary and dual wavelets is no larger than the support of φ.

Theorem


Butterfly subdivision scheme

Proposed by Nira Dyn (1990), widely used in computer graphics.

a0 =164

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Interpolatory function from box splineConstructed by Riemenschneider and Shen (1997) from box spline with threedirections of multiplicity two.

a0 =1256

0 0 0 −1 −3 −3 −10 0 −3 0 6 0 −30 −3 6 33 33 6 −3−1 0 33 64 33 0 −1−3 6 33 33 6 −3 0−3 0 6 0 −3 0 0−1 −3 −3 −1 0 0 0

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Primary and dual wavelets (part)

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1

2

3−1.5

−1

−0.5

0

0.5

1

1.5

2

xy

z

−3−2

−10

12

3

−3

−2

−1

0

1

2

3−1.5

−1

−0.5

0

0.5

1

1.5

2

xy

z

Dual wavelets

−3−2

−10

12

3

−3

−2

−1

0

1

2

3

−0.06

−0.04

−0.02

0

0.02

0.04

xy

z

−3−2

−10

12

3

−3

−2

−1

0

1

2

3

−0.1

−0.05

0

0.05

xy

z

−3−2

−10

12

3

−3

−2

−1

0

1

2

3

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

xy

z

−3−2

−10

12

3

−3

−2

−1

0

1

2

3

−1

−0.5

0

0.5

xy

zWavelet system 39/46

Construction of tight wavelet frames

A∗A = M

Suppose the refinement mask a0 ∈ `2(Zd ) with nonnegative entries is of finite supportsatisfying the condition, and the corresponding refinable function φ ∈ L2(Rd ) issupposed to be compactly supported with φ(0) = 1.

I The masks derived from the construction satisfy the UEP condition.I The wavelet system X(Ψ) generated by the corresponding masks forms a

tight frame in L2(Rd ).I The number of Ψ is one less than the size of the support of a0.I The support of the derived masks is no larger than the support of a0 and if

the support of φ is convex, then the support of the primary and dual waveletsis no larger than the support of φ.

Theorem


Construction of tight wavelet frames

(AM−1/2)∗AM−1/2 = I

Suppose the refinement mask a0 ∈ `2(Zd ) with nonnegative entries is of finite supportsatisfying the condition, and the corresponding refinable function φ ∈ L2(Rd ) issupposed to be compactly supported with φ(0) = 1.

I The masks derived from the construction satisfy the UEP condition.I The wavelet system X(Ψ) generated by the corresponding masks forms a

tight frame in L2(Rd ).I The number of Ψ is one less than the size of the support of a0.I The support of the derived masks is no larger than the support of a0 and if

the support of φ is convex, then the support of the primary and dual waveletsis no larger than the support of φ.

Theorem


Box spline

Piecewise linear box spline

a0 = 18

0 1 11 2 11 1 0

0

0.5

1

1.5

2

0

0.5

1

1.5

20

0.2

0.4

0.6

0.8

1

xy

z

6 wavelets are constructed.


Tight wavelet frame from piecewise linear box spline

18

( 0 −1 −11 2 1−1 −1 0

), 1

8

( 0 −1 1−1 2 −11 −1 0

), 1

8

( 0 1 −1−1 2 −1−1 1 0

),

√3

12

( 0 −1 −1−1 0 11 1 0

),

√6

24

( 0 1 1−2 0 2−1 −1 0

),√

28

( 0 −1 10 0 0−1 1 0

).

0

0.5

1

1.5

2

0

0.5

1

1.5

2−0.5

0

0.5

1

xy

z

0

0.5

1

1.5

2

0

0.5

1

1.5

2−0.5

0

0.5

1

xy

z

0

0.5

1

1.5

2

0

0.5

1

1.5

2−0.5

0

0.5

1

xy

z

0

0.5

1

1.5

2

0

0.5

1

1.5

2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

xy

z

0

0.5

1

1.5

2

0

0.5

1

1.5

2−1

−0.5

0

0.5

1

xy

z

0

0.5

1

1.5

2

0

0.5

1

1.5

2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

xy

z


Box spline in 3D

The box spline in R3 with the refinement mask

a0 = 116

0 0 0 0 1 1 0 1 11 1 0 1 2 1 0 1 11 1 0 1 1 0 0 0 0

14 wavelets are constructed


Tight wavelet frames in 3D

18

(0 0 0 0 0 0 0 1 −10 0 0 0 0 0 0 0 0−1 1 0 0 0 0 0 0 0

)18

(0 0 0 0 0 0 0 0 0−1 1 0 0 0 0 0 1 −10 0 0 0 0 0 0 0 0

)18

(0 0 0 0 −1 1 0 0 00 0 0 0 0 0 0 0 00 0 0 1 −1 0 0 0 0

)18

(0 0 0 0 0 0 0 0 00 0 0 1 −2 1 0 0 00 0 0 0 0 0 0 0 0

)√2

16

(0 0 0 0 0 0 0 1 1−1 −1 0 0 0 0 0 −1 −11 1 0 0 0 0 0 0 0

)√2

16

(0 0 0 0 1 1 0 0 00 0 0 −1 −2 −1 0 0 00 0 0 1 1 0 0 0 0

)

...


Filter banks to be wavelet masks

Question:I When will the filter banks be wavelet masks?

Suppose a given FIR filter bank satisfies the UEP condi-tion. If one of the filters is a low pass filter, then thereexists an MRA tight wavelet frame in L2(Rd) whose

I underlying MRA is derived from this low pass filterI the wavelet masks are the rest of the filters in the

filter bank.

Theorem


Thank you!

Questions?

Thesis oral defense

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