PPT for my PhD oral defense.

1. Thesis Oral Presentation
Wavelet and its applications
Fan Zhitao
NUS Graduate School
National University of Singapore
March 30, 2015

2. System and frame
A system gives a decomposition from Hilbert space to sequence space.
A frame not only provides this decomposition, but also
ensures the numerial stable reconstruction.
overcomes shortage of orthonormal system, e.g.
Orthonormal Gabor system: no window with good time and
frequency localization (Balian-Low theorem)1
Orthonormal wavelets with compact support: complicated
expression; no symmetry dyadic real wavelet except the trivial
case2
Application of frames
Natural image restoration problems, e.g. denoising3, inpainting4, deblurring5
Biological image processing problems, e.g. 3D molecule reconstruction from
electron microscope images 6
Background 2/46

3. Motivation and contribution
Motivation: Ron and Shen1
developed the dual Gramian
analysis to analyze frame properties for shift-invariant systems
in L2(Rd
).
Goal: Develop the dual Gramian analysis to study frames in a
general Hilbert space.
Benefits
Finding the canonical dual/tight frame by a matrix inverse
Estimating the frame bounds by classical matrix inequalities
Duality principle: transfer the frame property of the system to
Riesz sequence property of its adjoint
1
A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(Rd
), Canadian Journal
of Mathematics, 47:1051-1094, 1995
Background 3/46

4. Contribution on Gabor system
Gabor system
{Ek Ml g = eil·(x−k)g(x − k), k ∈ Zd , l ∈ 2πZd , g ∈ L2(Rd )}
Known works and challenges
Fiber pre-Gramian matrix by Ron and Shen.1
Many classical results, e.g. biorthogonal relationship, Wexler-Raz identity
Design good Gabor windows with good time and frequency localization
Our constribution
The connections of these two definitions of pre-Gramian matrices, by finding
a good orthonormal basis
Fully developed the mixed fiber dual Gramian analysis for two Gabor systems
Classical identities as consequence of duality principle
Constructed Gabor windows, with compact support and arbitrary
smoothness, in particular for multivariate case
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A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in L2(Rd
), Duke Mathematical Journal,
89:237-282, 1997.
Background 4/46

5. Contribution on wavelet system
Wavelet system
{Dk Ej ψ = 2kd/2ψ(2k · −j), k ∈ Zd , j ∈ Zd , ψ ∈ L2(Rd )}
Works and challenges
Orthonormal wavelet with compact support by Daubechies1
UEP/MEP for tight/dual wavelet frame2: complete matrix with polynomial
entries
The multivariate wavelet frame construction remains challenging
Contribution
The dual Gramian analysis is applied to analyze filter banks.
Duality principles lead to a simple way of construction.
Easy construction scheme for multivariate dual/tight wavelet frames with
wavelet such as with small support and symmetric/anti-symmetric.
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I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992
2
A. Ron and Z. Shen, Affine systems in L2(Rd
): the analysis of the analysis operator, Journal ofBackground 5/46

6. Publications and Outline
Paper
Zhitao Fan, Hui Ji, Zuowei Shen, Dual Gramian analysis: duality principle and
unitary extension principle, Mathematics of computation, to appear.
Zhitao Fan, Andreas Heinecke, Zuowei Shen, Duality for frames, Journal of
Fourier Analysis and Applications, to apprear.
Outline
Review of notations and definitions
Dual Gramian analysis
Gabor system
Wavelet system
Background 6/46

7. Notations
H is a separable Hilbert space
A system X is a sequence of H with a certain indexing, e.g.
indexed by Z: {xi }i∈Z = {· · · , x−1, x0, x1, · · · }
indexed by N: {xi }i∈N = {x1, x2, x3, · · · }
A sequence indexed by X
c ∈ 0(X): c(x) ∈ C and (c(x))x∈X with finite support
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

8. Synthesis and analysis operators
The synthesis operator of X
TX : 2(X) → H : c → x∈X c(x)x
The analysis operator of X
T∗
X : H → 2(X) : h → { h, x }x∈X
When either TX or T∗
X is bounded, they are adjoint.
X is a Bessel system.
Background 8/46

9. Riesz and frame properties
A Bessel system X is
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.
2-independent if TX is injective; a Riesz sequence if
A c 2
≤ TX c 2
≤ B c 2
, ∀c ∈ 2(X)
Being an orthonormal sequence if A = B = 1.
A Riesz/orthonormal sequence X is a Riesz/orthonormal
basis if X is fundamental.
Background 9/46

10. Self-adjoint operators
System X is a Bessel system ⇔ either T∗
X TX or TX T∗
X bounded
The frame operator TX T∗
X
1. A Bessel system X is fundamental ⇔ TX T∗
X is injective
2. it is a frame for H ⇔ TX T∗
X has a bounded inverse
3. it is a tight frame ⇔ TX T∗
X = I
The operator T∗
X TX
1. A Bessel system X is 2-independent ⇔ T∗
X TX is injective
2. X is a Riesz sequence ⇔ T∗
X TX has a bounded inverse
3. X is an orthonormal sequence ⇔ T∗
X TX = I
Background 10/46

11. Dual frames
System X is a tight frame, then TX T∗
X = I, i.e.
x∈X
f , x x = f , for f ∈ H.
Two systems
Suppose X and Y = RX are Bessel systems in H
X and Y are dual frames if TY T∗
X = I or TX T∗
Y = I
An example
X is a frame for H
S = TX T∗
X is invertible
S−1X is the canonical dual frame of X
Background 11/46

12. Dual Gramian Analysis
pre-Gramian, Gramian, dual Gramian matrix
Connection to the operator
Results
Duality principle

13. Pre-Gramian matrix
Given 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∗
RX JX =
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 := JRX J∗
X =
x∈X
Rx, e e , x
e∈O,e ∈O
When RX = X, drop “mixed” in the name.
Dual Gramian analysis 13/46

14. The analysis
Let U = TO which is unitary.
UJX c = TX c for all c ∈ 0(X)
T∗
X Ud = J∗
X d for all d ∈ 0(O)
Weak form
TX c, TRX d = d∗GRX,X c for all c, d ∈ 0(X)
T∗
X Uc, T∗
RX Ud = d∗GRX,X c for all c, d ∈ 0(O)
Strong form
If X and RX are Bessel systems
T∗
RX TX c = GRX,X c for all c ∈ 2(X)
U∗TRX T∗
X Uc = GRX,X c for all c ∈ 2(O)
Dual Gramian analysis 14/46

15. The finite example
Let H be Cm
Let {ei }m
i=1 be the canonical orthonormal basis.
Let X = {xk }n
k=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 →
n
k=1
ck xk .
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 → { f , xk }n
k=1.
Dual Gramian analysis 15/46

16. The finite example
Given another system Y = {yk }n
k=1 ⊂ Cm
JY is the associated pre-Gramian matrix
The mixed Gramian matrix and mixed dual Gramian matrix are
GX,Y = J∗
X JY = ( yk , xk )k,k , GX,Y = JX J∗
Y =
n
k=1
xk (j)yk (j )
j,j
which are the matrix representations of the mixed operators T∗
X TY
and TX T∗
Y respectively.
Dual Gramian analysis 16/46

17. Canonical dual and tight frames
The canonical dual frame: S−1
X
Let X be a frame in H with frame bounds A, B and let U be the synthesis
operator of an orthonormal basis of H. Then
system UG−1
X
U∗X is a frame with bounds B−1, A−1 which is the
canonical dual frame of X.
Canonical dual frame
The canonical tight frame: S−1/2
X
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 positive
square root of GX . Then,
system UG
−1/2
X
U∗X forms the canonical tight frame.
Canonical tight frame
Dual Gramian analysis 17/46

18. Estimate upper frame bound
Let I be a countable index set, and let M be a complex valued nonnegative
Hermitian 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 an
orthonormal basis O of H .
(a) Let
B1 : e → e ∈O
| x∈X
e , x x, e |.
Then X is a Bessel system whenever supe∈O B1(e) < ∞ and its Bessel bound is not
larger than (supe∈O B1(e))1/2.
(b) Assume that X is a Bessel system, let
B2 : e → 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.
Dual Gramian analysis 18/46

19. Estimate lower frame bound
The lower frame bound can be obtained when the dual Gramian matrix is diagonally
dominant. For a Hermitian diagonally dominant matrix M,
M−1 ≤ supi∈I |M(i, i)| − j∈Ii
|M(i, j)|
−1
Let X be a system in Hilbert space H satisfying week condition with respect to an
orthonormal basis O of H. Let
˜b1 : e →
x∈X
| e, x |2
−
e =e
|
x∈X
e , x x, e |
−1
.
Then X is a frame whenever supe∈O
˜b1(e) < ∞ and the lower frame bound is not
smaller than (supe∈O
˜b1(e))−1/2.
Dual Gramian analysis 19/46

20. Duality principle
If X = {xk }n
k=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 synthesis
operator of X
The columns is associated with X while the rows is associated with X∗
The analysis properties of X are characterized by the synthesis properties
of X∗.
Duality Principle
Dual Gramian analysis 20/46

21. 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∗
X V
for some unitary operators U and V .
Definition
Up to unitary equivalence,
GRX,X = G(RX)∗,X∗
Dual Gramian analysis 21/46

22. The example by Casazza
Let X = {fk }k∈N be a system in H
Wrt an orthonormal basis {ei }i∈N, it satisfies: i∈N
| fk , ei |2 < ∞ for all
k ∈ N
Suppose {hk }k∈N is another orthonormal basis of H and define
X = {gi := k∈N
fk , ei hk }i∈N
Then X is indeed an adjoint system of X.
The system X satisfies the weak condition:
i∈N
| gi , hk |2
=
i∈N
| fk , ei |2
< ∞ for all k ∈ N.
It is easy to see:
JX = ( gi , hk )k,i = ( fk , ei )k,i = J∗
X
Dual Gramian analysis 22/46

23. Duality results for single systems
GX = GX∗
Let X be a given system in H, and suppose that X∗ is an adjoint system of
X 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 Bessel
and 2-independent.
(c) A system X forms a frame in H ⇔ its adjoint system X∗ forms a
Riesz sequence in H . The frame bounds of X coincide with the
Riesz bounds of X∗.
(d) A system X forms a tight frame in H ⇔ its adjoint system X∗ forms
an orthonormal sequence in H .
Proposition
Dual Gramian analysis 23/46

24. 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
Dual Gramian analysis 24/46

25. Wavelet Systems
Dual Gramian analysis for filter banks
Simple construction scheme for filter banks by duality principle
Multivariate dual/tight wavelet frame construction

26. 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 → (↓N (c ∗ al (−·))(k))(l,k)∈Zr ×Zd
Downsampling: ↓N d(k) = d(Nk) for k ∈ Zd .
The synthesis operator
TX : 2(Zr × Zd
) → 2(Zd
): c → l∈Zr
(↑N c(l, ·)) ∗ al
Upsampling: for fixed l ∈ Zr , ↑N c(l, k) is equal to c(l, N−1k) if N divides all
entries of k ∈ Zd and is equal to 0 otherwise.
Wavelet system 26/46

27. The pre-Gramian matrix of filter bank
The 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 the
NZd -coset of an index.
Two Bessel systems X and Y are dual frames ⇔ the adjoint systems X∗ and
Y ∗ are biorthonormal.
Wavelet system 27/46

28. Construction scheme for filter banks
Let X = X(a, N) and Y = X(b, N), for FIR filters a = {al }r−1
l=0 and
b = {bl }r−1
l=0 in 2(Zd
) and N ∈ N.
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
.
The system X is a tight frame when al = bl for l = 0, . . . , r − 1.
Theorem
Wavelet system 28/46

29. Construction scheme for filter banks
A∗
B = M
M ∈ Cr×r be a diagonal matrix with diagonal c such that
n∈Ωj
c(n) = 1
for every j ∈ Zd /NZd .
Let A = (al (nj ))l∈Zr ,j∈Zr ∈ Cr×r be invertible
Let
B = (bl (nj ))l∈Zr ,j∈Zr = (A∗
)−1
M
Then the filters a = {al }r−1
l=0
and b = {bl }r−1
l=0
defined by A and B generate
dual frames X(a, N) and X(b, N) in 2(Zd ).
Construction
Wavelet system 29/46

30. Multiresolution Analysis (MRA) wavelet
A function φ ∈ L2(Rd
) is called a refinable function if
ˆφ(2·) = ˆa0
ˆφ
The sequence a0 ∈ 2(Zd
) is the refinement mask. ˆa0(0) = 1.
Let V0 ⊂ L2(Rd
) be the closed linear span of E(φ) and Vk := Dk
(V0)
for k ∈ Z. {Vk }k∈Z is called an MRA if
(i)Vk ⊂ Vk+1 (ii) ∪k Vk is dense in L2(Rd
) (iii) ∩k Vk = {0}
e.g. φ ∈ L2(Rd ) is a compactly supported refinable function with ˆφ(0) = 1
The wavelets Ψ = {ψl }r
l=1 ⊂ L2(Rd
)
ˆψl (2·) = ˆal
ˆφ
The sequence al ∈ 2(Zd
) is called the wavelet mask. ˆal (0) = 0.
Wavelet system 30/46

31. Mixed Unitary Extension Principle (MEP)
Let φa, φb be compactly supported refinable functions with
ˆφa(0) = ˆφb(0) = 1 and masks a0, b0.
Let {al }r
l=1, {bl }r
l=1 be the masks of wavelet systems X, Y .
If both X and Y are Bessel systems and
r
l=0
ˆal (ω)ˆbl (ω + ν) = δν,0,
for any ν ∈ {0, π}d
and a.e. ω ∈ Td
, then X and Y are dual
frames.
MEP1
Unitary extension principle(UEP2
): Y = X.
1
A. Ron and Z. Shen, Affine systems in L2(Rd
): dual systems, Journal of Fourier Analysis and
Applications, 3:617-637, 1997.
2
A. 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

32. The connection of UEP/MEP with filter banks
HX (ω) =



ˆa0(ω + ν1) ˆa1(ω + ν1) . . . ˆar (ω + ν1)
ˆa0(ω + ν2) ˆa1(ω + ν2) . . . ˆar (ω + ν2)
...
...
...
...
ˆa0(ω + ν2d ) ˆa1(ω + ν2d ) . . . ˆar (ω + ν2d )



MEP: HX (ω)HY (ω)∗ = I while UEP: HX (ω)HX (ω)∗ = I
The filter bank X = X({2d/2al }r
l=0, 2)
MEP for dual frame filer bank, and UEP for tight frame filter bank
Questions:
Can we start with a refinement mask to construction
wavelets?
When will the filter banks be wavelet masks?
Wavelet system 32/46

33. Dual wavelet frame construction
The construction starts from a real-valued refinement mask a0
satisfying
n∈Ωj
a0(n) = 2−d
,
for all j ∈ Zd
/2Zd
, where Ωj = (2Zd
+ j) ∩ supp(a0).
Examples:
the butterfly subdivision scheme by Dyn (1990)
the interpolatory refinement mask derived from box spline by
Riemenschneider and shen (1997)
Wavelet system 33/46

34. Dual wavelet frame construction
A∗
B = M
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 the
first row of A as its diagonal.
2. (Primary wavelet masks): Complete the matrix A to be an
invertible square matrix, each of whose remaining rows has
entries summing to zero.
3. (Dual wavelet masks): Define ˜A = AM−1
and B = (˜A∗
)−1
.
Wavelet system 34/46

35. Dual wavelet frame construction


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 the
first row of A as its diagonal.
2. (Primary wavelet masks): Complete the matrix A to be an
invertible square matrix, each of whose remaining rows has
entries summing to zero.
3. (Dual wavelet masks): Define ˜A = AM−1
and B = (˜A∗
)−1
.
Wavelet system 34/46

36. Dual wavelet frame construction



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 the
first row of A as its diagonal.
2. (Primary wavelet masks): Complete the matrix A to be an
invertible square matrix, each of whose remaining rows has
entries summing to zero.
3. (Dual wavelet masks): Define ˜A = AM−1
and B = (˜A∗
)−1
.
Wavelet system 34/46

37. A Summary for construction of dual wavelet frames
Suppose the real-valued refinement mask a0 ∈ 2(Zd
) is of finite support
satisfying the mask condition, and the corresponding refinable function
φ ∈ L2(Rd
) is compactly supported with φ(0) = 1. Then
the masks derived by Construction satisfy the MEP condition.
the wavelet systems X and Y generated by those masks are dual
wavelet frames in L2(Rd
).
the number of wavelet is one less than the size of the support of
a0.
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 wavelets is no larger than the support of φ.
Theorem
Wavelet system 35/46

38. Butterfly subdivision scheme
Proposed by Nira Dyn (1990), widely used in computer graphics.
a0 =
1
64







0 0 0 0 −1 −1 0
0 0 −1 0 2 0 −1
0 −1 2 8 8 2 −1
0 0 8 16 8 0 0
−1 2 8 8 2 −1 0
−1 0 2 0 −1 0 0
0 −1 −1 0 0 0 0







−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
0
0.2
0.4
0.6
0.8
1
x
y
z
26 primary and 26 dual wavelets.
Wavelet system 36/46

39. Primary and dual wavelets (part)
Primary wavelets
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−1.5
−1
−0.5
0
0.5
1
1.5
2
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−1.5
−1
−0.5
0
0.5
1
1.5
2
x
y
z
Dual wavelets
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−0.1
−0.05
0
0.05
0.1
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
x
y
zWavelet system 37/46

40. Interpolatory function from box spline
Constructed by Riemenschneider and Shen (1997) from box spline with three
directions of multiplicity two.
a0 =
1
256







0 0 0 −1 −3 −3 −1
0 0 −3 0 6 0 −3
0 −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







−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
0
0.2
0.4
0.6
0.8
1
x
y
z
30 primary wavelets and 30 dual wavelets
Wavelet system 38/46

41. Primary and dual wavelets (part)
Primary wavelets
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−1.5
−1
−0.5
0
0.5
1
1.5
2
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−1.5
−1
−0.5
0
0.5
1
1.5
2
x
y
z
Dual wavelets
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−0.06
−0.04
−0.02
0
0.02
0.04
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−0.1
−0.05
0
0.05
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
x
y
z
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−1
−0.5
0
0.5
x
y
zWavelet system 39/46

42. Construction of tight wavelet frames
A∗
A = M
Suppose the refinement mask a0 ∈ 2(Zd ) with nonnegative entries is of finite support
satisfying the condition, and the corresponding refinable function φ ∈ L2(Rd ) is
supposed to be compactly supported with φ(0) = 1.
The masks derived from the construction satisfy the UEP condition.
The wavelet system X(Ψ) generated by the corresponding masks forms a
tight frame in L2(Rd ).
The number of Ψ is one less than the size of the support of a0.
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 wavelets
is no larger than the support of φ.
Theorem
Wavelet system 40/46

43. 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 support
satisfying the condition, and the corresponding refinable function φ ∈ L2(Rd ) is
supposed to be compactly supported with φ(0) = 1.
The masks derived from the construction satisfy the UEP condition.
The wavelet system X(Ψ) generated by the corresponding masks forms a
tight frame in L2(Rd ).
The number of Ψ is one less than the size of the support of a0.
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 wavelets
is no larger than the support of φ.
Theorem
Wavelet system 40/46

44. Box spline
Piecewise linear box spline
a0 =
1
8



0 1 1
1 2 1
1 1 0



0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
x
y
z
6 wavelets are constructed.
Wavelet system 41/46

45. Tight wavelet frame from piecewise linear box spline
1
8
0 −1 −1
1 2 1
−1 −1 0
, 1
8
0 −1 1
−1 2 −1
1 −1 0
, 1
8
0 1 −1
−1 2 −1
−1 1 0
,
√
3
12
0 −1 −1
−1 0 1
1 1 0
,
√
6
24
0 1 1
−2 0 2
−1 −1 0
,
√
2
8
0 −1 1
0 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
x
y
z
0
0.5
1
1.5
2
0
0.5
1
1.5
2
−0.5
0
0.5
1
x
y
z
0
0.5
1
1.5
2
0
0.5
1
1.5
2
−0.5
0
0.5
1
x
y
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
x
y
z
0
0.5
1
1.5
2
0
0.5
1
1.5
2
−1
−0.5
0
0.5
1
x
y
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
x
y
z
Wavelet system 42/46

46. Box spline in 3D
The box spline in R3
with the refinement mask
a0 = 1
16


0 0 0 0 1 1 0 1 1
1 1 0 1 2 1 0 1 1
1 1 0 1 1 0 0 0 0


14 wavelets are constructed
Wavelet system 43/46

47. Tight wavelet frames in 3D
1
8
0 0 0 0 0 0 0 1 −1
0 0 0 0 0 0 0 0 0
−1 1 0 0 0 0 0 0 0
1
8
0 0 0 0 0 0 0 0 0
−1 1 0 0 0 0 0 1 −1
0 0 0 0 0 0 0 0 0
1
8
0 0 0 0 −1 1 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 1 −1 0 0 0 0
1
8
0 0 0 0 0 0 0 0 0
0 0 0 1 −2 1 0 0 0
0 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 −1
1 1 0 0 0 0 0 0 0
√
2
16
0 0 0 0 1 1 0 0 0
0 0 0 −1 −2 −1 0 0 0
0 0 0 1 1 0 0 0 0
...
Wavelet system 44/46

48. Filter banks to be wavelet masks
Question:
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 there
exists an MRA tight wavelet frame in L2(Rd ) whose
underlying MRA is derived from this low pass filter
the wavelet masks are the rest of the filters in the
filter bank.
Theorem
Wavelet system 45/46

49. Thank you!
Questions?