Introduction to Multiresolution Analysis (MRA)bigdft.org/images/1/1b/Day-2-RS-FK-MRA.pdf · Outline...

Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation References

Introduction to Multiresolution Analysis (MRA)

R. SchneiderF. Kruger

TUB - Technical University of Berlin

November 22, 2007

R. Schneider F. Kruger Introduction to Multiresolution Analysis (MRA) November 22, 2007 1 / 33

Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesOutline

Outline

Introduction and Example

Multiresolution Analysis

Discrete Wavelet Transform (DWT)

Finite Calculation


Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesIntroduction and Example




Finite Calculation



goal approximation of functions (e.g. signals, images,orbitals)

idea coarse approximation (trend) + fine improvement(detail) with detail << trend

imagination building a house. start with big pieces and fill in withmiddle sized and at the end with little pieces



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15detail 1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15detail 2



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25detail 3



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2detail 4



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9step 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1detail 5



Details

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3 detail 10



In the example a very simple set of basis functions is used:

ϕ(x) =1[0,1)(x) =

{1, x ∈ [0, 1)0, else

−1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5Haar function



ϕjk(x) =2j/21[2−jk,2−j(k+1))(x) =

{2j/2, x ∈ [2−jk, 2−j(k + 1))0, else

=2j/2ϕ(2jx− k).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5

0

0.5

1

1.5

2

2.5Haar function, j = 2, k = 3



The details are given by the functions

ψjk(x) =2j/2ψ(2jx− k)

with

ψ(x) =

1, x ∈ [0, 0.5)−1, x ∈ [0.5, 1)0, else.

−1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5Haar wavelet

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5Haar wavelet, j = 2, k = 3



Refinement Equation

ϕ is called Haar function and ψ is called the Haar wavelet. Theysatisfy the so-called refinement equations :

ϕjk(x) =2−1/2(ϕj+1

2k (x) + ϕj+12k+1(x))

ψjk(x) =2−1/2(ϕj+1

2k (x)− ϕj+12k+1(x))

−1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5Refinement for the Haar function

φ2−1/2 φ0

1

2−1/2 φ11

−1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5Refinement for Haar wavelet

ψ2−1/2 φ0

1

2−1/2 φ11


Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesMultiresolution Analysis




Finite Calculation




Given a function

ϕ ∈ L2(R) = {f | ‖f‖2 =( ∫

R|f(x)|2dx

)1/2<∞}.

We consider the shifts and dilatations of ϕ :

ϕjk(x) = 2j/2ϕ(2jx− k), j, k ∈ Z.

We write

Vj = span{ϕjk | k ∈ Z}.



If every f ∈ L2(R) can be arbitrarily accurately approximated byϕj

k’s, i.e.

∞⋃j≥j0

Vj = L2(R)

holds and ϕ fulfills a refinement equation

ϕ =∑k∈Z

hkϕ1k

then we say that ϕ or the V ′j s, respectively, build a multi resolutionanalysis (MRA).



Orthonormal Wavelets

Because of the refinement equation it holds Vj ⊂ Vj+1 for everyj ∈ Z. Therefore there exists the orthogonal space Wj of Vj inVj+1. Therefore

Vj ⊥Wj , Vj ⊕Wj = Vj+1.

Wj is called the detail space or the wavelet space for Vj . Afunction ψ that satisfies

1.∫

R ψ(x)dx = 02. {ψ(· − k) | k ∈ Z} is an orthonormal basis of W0

is called (orthonormal) wavelet or mother wavelet for the functionϕ. ϕ is also called scaling function or generator function or fatherwavelet. The Haar wavelet is a wavelet for the Haar function, forexample.



If the translations of ψ are not orthonormal we need biorthogonalwavelets. But we do not go into details for this case.



Multi Levels

Let J be the level at which we want to approximate, i.e. weproject into the space VJ . Then we have

VJ =VJ−1 ⊕WJ−1

=VJ−2 ⊕WJ−2 ⊕WJ−1

...

=V0 ⊕J−1⊕j=0

Wj .



If we go to infinity we have

L2(R) = V0 ⊕∞⊕

j=0

Wj .

We can imagine that we start at the very coarse level 0 andimprove the result by successively adding the finer becomingdetails.



Filters

Because of the fact that Vj ⊂ Vj+1 and Wj ⊂ Vj+1 in a MRA wehave the refinement equations

ϕjk =

∑l

hlϕj+12k+l

ψjk =

∑l

glϕj+12k+l.

(hl)l∈Z and (gl)l∈Z are called filters. If ϕ has compact support hhas finite length. If additionally ψ has compact support g has alsofinite length.



Reconstruction

If {ϕjk | k ∈ Z} and {ψj

k | k ∈ Z} are orthonormal bases, i.e.

〈ϕjk, ϕ

jl 〉 =

∫Rϕj

k(x)ϕjl (x)dx =δk,l

〈ψjk, ψ

jl 〉 =δk,l

we have the reconstruction

ϕj+1k =

∑l

hk−2lϕjl +

∑l

gk−2lψjl .

There is also a very simple correlation between g and h:

gk = (−1)1−kh1−k


Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesDiscrete Wavelet Transform (DWT)




Finite Calculation



Orthogonal Projection

Assume that {ϕk | k ∈ Z} and {ψk | k ∈ Z} are orthonormal.Given f ∈ L2(R). We consider the orthogonal projections PJ ontoVJ and QJ onto WJ . Let

λjk =〈ϕj

k, f〉 =∫

Rϕj

k(x)f(x)dx and

µjk =〈ψj

k, f〉.

Then it holds

fJ = PJf =∑

k

λJkϕ

Jk ∈ VJ

QJf = (PJ+1 − PJ)f =∑

k

µJkψ

Jk ∈WJ .



We can easily get the coefficients in the coarser levels and thedetail spaces by using successively the synthese equation.

fJ =∑

k

λJkϕ

Jk

=∑

k

λJk (

∑l

hk−2lϕJ−1l +

∑l

gk−2lψJ−1l )

=∑

l

λJ−1l ϕJ−1

l +∑

l

µJ−1l ψJ−1

l

...

=∑

l

λ0l ϕ

0l +

J−1∑j=0

∑l

µjlψ

jl



The occurring coefficients are given by

λjl =

∑k

λj+1k hk−2l and (1)

µjl =

∑k

λj+1k gk−2l, j = 0, . . . , J − 1. (2)

(1) and (2) can be written as(λj

µj

)= Tλj+1.



Discrete Inverse Wavelet Tansform (DIWT)

Given the coefficients λjk and µj

k we get the cofficients λj+1k back

by

λj+1k =

∑l

hk−2lλjl +

∑l

gk−2lµjl .

This describes again a linear transformation

λj+1 = T−1

(λj

µj

)= T T

(λj

µj

).



Note: Because of the D(I)WT it is not really necessary to knowexplicitly the functions ϕ and ψ. It is sufficient to know the filtersh and g.



Schemes

DWT DIWT

λJ λ0 µ0

↓ ↘ ↓ ↙λJ−1 µJ−1 λ1 µ1

↓ ↘ ↓ ↙λJ−2 µJ−2 λ2 µ2

↓ ↘ ↓ ↙...

......

...↓ ↘ ↓ ↙λ1 µ1 λJ−1 µJ−1

↓ ↘ ↓ ↙λ0 µ0 λJ


Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesFinite Calculation




Finite Calculation


Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesFinite Calculation

In the previous chapter we still had infinite many coefficientsλj

k, µjk. For practical calculations we have to make their size finite.

There are mainly two ways to do this.

1. Zeropadding: You consider only a finite region and assumethat all coefficients out of this region are zero.

2. Periodizing: You assume that your data is periodic and youcalculate only on one period.

Then the D(I)WT is a finite transform if the filters are finite. Thenumber of arithmetic operations for one transformation is O(N) ifN is the size of the input data. To transform on J levels you haveO(J ·N) arithmetic operations.


Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation ReferencesReferences

References

S. Mallat, A Wavelet Tour of Signal Processing, 2nd. ed., AcademicPress, 1999


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