Wavelet変換 Wavelet Transformaito/wavelet/slide2.pdfZ-transform Discrete input 8 フーリエ変換 Fourier transform Calculate spectrum from infinite signal – フーリエ変換

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Wavelet変換Wavelet Transform

伊藤　彰則Akinori Ito

[email protected]

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This slide can be downloaded from

● http://www.spcom.ecei.tohoku.ac.jp/~aito/wavelet/slide2.pdf

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参考書 (1)● 中村，山本，吉田：「ウェーブレットによる信号処理と画像処理」，共立出版

● 応用の紹介とプログラムリストが中心，理論的背景はほとんどなし

● 意味不明の比喩を多用「各時代・各国別に美女を探すのが窓フーリエ変換である」

● 応用テーマ：不連続信号検出，相関の検出，ノイズ除去，画像データ圧縮，劣化画像復元

● 芦野，山本：「ウェーブレット解析－誕生・発展・応用」，共立出版

● 理論編と応用編に分かれている．比較的わかりやすい

● Daubechies と Meyer のウェーブレットが中心，その他の紹介はあまりない

● 応用テーマ：不連続信号検出，相関の検出，データ操作

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参考書 (2)● 新井：「ウェーブレット解析の基礎理論」森北出版

● 前半が理論，後半が応用だが理論的背景は薄い

● 理論を飛ばして読む目的には良いが，理論をこれだけで理解するのは難しい

● 対象ウェーブレットを幅広く扱っている

● 応用編はプログラムリストつき

● 応用テーマ：エッジ抽出，データ圧縮，積分方程式の数値解法

● C.K.Chui, 桜井・新井訳：「ウェーブレット入門」，東京電機大学出版局

● 理論のみ，応用事例なし

● 理論を概観する章がある．証明抜きで全体像をつかむには良い

● 扱うのはほとんどスプラインウェーブレット．

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参考書 (3)● 前田，佐野，貴家，原：「ウェーブレット変換とその応用」，朝倉書店

● 前半が理論，後半が応用

● 厳密な理論展開は省略，最小限の証明

● 積分 Wavelet 変換についてちょっと詳しい

● 扱う Wavelet は Haar ， MexicanHat,Poisson など

● 応用テーマ：レーダー，システム同定，雑音抑圧

● B.B.Hubbard, 山田・西野訳：「ウェーブレット入門」，朝倉書店

● インタビューにもとづくウェーブレット発見の物語

● 誰がどういう経緯で何を発見したのかが書いてある

● 著者は数学者ではない

● 数学的記述を全く見ないで読むことも可能

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参考書 (4)● 赤間：「ウェーブレット変換がわかる本」工学社– ウェーブレットの基礎と応用についてわかりやすく解説

– さまざまなウェーブレットを扱う– 理論も比較的詳しいが、わかりやすい感じで書かれている

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Wavelet & Friends

Fourier Transform

WindowFourierTransform

WindowfunctionWindowfunction

Generalized windowTime-Frequency dependency

Generalized windowTime-Frequency dependency

Continuouswavelet transform

functionexpansion

functionexpansion

Fourier Expansion

Functionexpansion

Functionexpansion

Discrete wavelet transform

Multi-resolution analysis

Scaling function

高速 Wavelet 変換Fast wavelet transform(Mallat's algorithm)

Two-scale relation

Gabor Wavelet

DaubechiesWavelet

共役ミラーフィルタQuadrature Mirror Filter

(QMF)

DiscretizationDiscretization

Z-transform

Discrete input

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フーリエ変換 Fourier transform● Calculate spectrum from infinite signal

– フーリエ変換 Fourier transform

– フーリエ逆変換 Inverse Fourier transform

f =∫−∞

∞

f x e−i x dx

f x =∫−∞

∞

f xe i x d

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Analysis of time series● Time-variant spectrum?

Fundamental frequency changes with time● 全体を解析したのでは「だんだん上がる」という分析は不可能

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Window Fourier Transform● FT of windowed signal

– Analyze temporal change of the signal

F t ,=∫−∞

∞

f xw x−t e−i x dx

=∫−∞

∞

f x ,t x dx

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Time-Frequency Analysis● Analyze temporal change of the signal by

shifting the window function

12

Spectrogram● スペクトルの強度を濃淡で表示

t

w

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Window Functions● Functions to extract signal of a specific time

– It is desirable to be zero when |t| is large (compact support)

w x =e−t2

2

w x ={121cos t −1t1

0 otherwise

Gaussian

Han (Hanning)

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Window Functions

w x ={1 −1t10 otherwise

w x ={0.540.46cos t −1t10 otherwise

Hamming

Rectangular

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Window Functions● Condition of w(x) to be a window function

● Center and width

∫ −∞

∞

∣x w x∣dx∞

∣∣w∣∣2=∫−∞

∞∣w x∣2 dx

x∗= 1

∣∣w∣∣2∫−∞

∞x∣w x∣2dx

Δw=√ 1

||w||2∫−∞

∞(x−x∗)2|w( x)|2 dx

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Uncertainty of window FT● Longer window → higher frequency resolution

– Two spectral peaks with similar frequencies can be discriminated

● Shorter window → higher temporal resolution– Quick temporal changes can be captured

● Uncertainty– Temporal width × Frequency width > const

（デモプログラム）

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Analyze the signal by thewindow function

● Analysis using window function y● Gaussian window + exp function

→Gabor transform

F t ,=∫−∞

∞f x w x−t e−i x dx

=∫−∞

∞f x ,t x dx

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Continuous Wavelet transform● Transform using a window function (Wavelet)

– a:周波数の逆数に相当（ dilation）– b: 時間に相当（ shift）

– :Analyzing wavelet

W f b ,a = 1

a∫−∞

∞f x x−b

a dx

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Condition of Wavelet● No bias

● Existence of inverse transform

∫−∞

∞ x dx=0

∫−∞

∞ ∣ x ∣2

∣x∣dx=1

2C∞

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Continuous Inverse Wavelet Transform

● 次の式により逆変換が可能

– ただし f(x) は次の条件を満たす

f x = 2C∫

0

∞ [∫−∞

∞

W f b ,a 1

a x−b

a db] da

a2

f x ∈L R2 namely ∫−∞

∞∣ f x ∣2 dx∞

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Examples of Analyzing Wavelet

● Haar wavelet

● Mexican hat wavelet

x={ 0 x01 0≤ x0.5−1 0.5≤ x10 1x

x=1− x2e− x2

2

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Exercise● cos(kx) を Haar で Wavelet 変換してみよう

● こんな風になるはず– 下の図では a は対数スケールであることに注意

– a は周波数の逆数→ k が大なら a は小

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Continuous Wavelet Transform● Large window on lower frequency

Narrow window on higher frequency

– High frequency resolution at low freq.– High temporal resolution at high freq.– (Constant resolution at log-freq.domain)

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Discrete Wavelet Transform● CWT:1 variable→2 variables

● Express the original function by summing all coefficients→Discrete Wavelet Transform (Wavelet Expansion)

W f b , a = 1

a∫−∞

∞f x x−b

a dx

f x =∑j=−∞

∞

∑k=−∞

∞

W f k2 j ,

12 j 2 j/2 2 j x−k

= ∑j=−∞

∞

∑k=−∞

∞

W f k2 j ,

12 j jk x

CONSTANT

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Wavelet Coefficients● Constant part

● The coefficients are values of CWT at discrete points

c jk=W f k

2 j,

1

2 j f x =∑

j=−∞

∞

∑k=−∞

∞

c jk jk x

LetWaveletcoefficients

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Notes on the Wavelet Coefficients

● a is in proportion to inverse of freq.– Sampling points at a-b domain

– Sampling points at f=1/a-b domain

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Wavelet basis function(1)● Wavelet expansion is a kind of 2-D function

expansion

● Cf.– Taylor-McLaurin expansion

– Fourier expansion

● is the basis function of Wavelet expansion

f x =∑j=−∞

∞

∑k=−∞

∞

c jk jk x

f x = ∑k=−∞

∞

ak x k

f x = ∑k=−∞

∞

ak e−ikx

ψ jk (x)

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Wavelet basis function(2)● Basis functions

– Any function (under a certain conditions) can be composed by a (infinite) weighted sum of the basis functions

– It does not necessarily guarantee uniqueness of coefficients

● Orthonormal basis functions

– Uniquely expands any function

– yjk

based on Haar function is the simplest orthonormal basis

∫−∞

∞ jk x j ' k ' x dx= jj ' kk '

∫−∞

∞2 j /2H 2

j x−k 2 j ' /2H 2j ' x−k dx= jj ' kk '

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Multiresolution analysis● Split the input signal into multiple frequency

bands

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Multiresolution Analysis● Relations between the function classes

● Orthonormal basis of V0 Scaling function

● The simplest scaling function

⊂V−2⊂V −1⊂V 0⊂V 1⊂V 2⊂

f ∈V 0f x = ∑

k=−∞

∞

ck x−k

H x={1 0≤x10 otherwise O

1

1

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Signal Decomposition using the Scaling function

● What functions are expressed by ? 　　　– For any integer k,

　　　 then

A function sampled at the integral points

k≤x<k+1 f (x)= f (k )

ϕ H (x)

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Cautions● We make discussions using the Haar scaling

function for a while– Easy to understand– There are other orthonormal scaling functions

Wavelet function

MRA

Scaling function

Fast wavelet transform=Discrete wavelet transform under a certain conditon

Two-scale relation

Discrete inputs

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Signal Decomposition using the Scaling function

● Decomposition using the scaling function

●Express the original signal as a weighted-sum of the scaling functions

c−6H x6

c−5H x5

c−4H x4

c−3H x3

⋮

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Various Scales

O1

1 O1

1 O1

1

H xH 0.5x H 2x

f x ∈V−1 f x ∈V 0 f x ∈V 1

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Composition of Scaling Function

● Compose the coarse scaling function using the fine scaling functions

● Haar scaling function

x= ∑k=−∞

∞

pk2x−k

H x=H 2x H 2x−1

O

1

1=

O

1

1 O

1

1+

p0=1, p1=1 pk=0 k≠0, k≠1

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Decomposition of scaling function

x∈V 1−V 0

x=∑−∞

∞

qk2x−k

V 1V 0

W 0

Basis function ϕ(2 x−k )

ϕ (x−k )

ψ(x−k )

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Decomposition of scaling functon

● Relations between 　　　　　　　　 ϕ (x ) , ψ(x) , ϕ (2 x )

ψ (x )=∑−∞

∞

qk ϕ (2 x−k )

ϕ ( x)=∑−∞

∞

pk ϕ (2 x−k ) two-scale relation

Haar scaling function

ϕ ( x)=ϕ (2 x )+ϕ (2 x−1)ψ (x )=ϕ (2 x)−ϕ (2 x−1)

x :Haar wavelet function

qk=−1k p1−k

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Decomposition of scaling function

● Express 　　　 using

● Haar Scaling function

ϕ (2 x) ϕ (x ) , ψ(x)

2x−l =12 {∑k=−∞

∞

g 2k−lx−k h2k−l x−k }g−k= pk h−k=qk

H 2x = 12{H x H x}

H 2x−1 = 12{H x −H x}

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Notes

● The function j is not necessarily a mother wavelet even when it is an orthonormal basis of W

0

– Other condition is needed such as

– Composition/decomposition of scaling function is STRONGLY related to the discrete wavelet

–

∫−∞

∞ ∣ x ∣2

∣x∣dx=1

2C∞

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Composition and decomposition of signal(1)

● Signal f N x∈V N

f N x= ∑k=−∞

∞

ckN2N x−k

Decompose fN(x) into

f N−1 x= ∑k=−∞

∞

ckN−12N−1 x−k

g N−1x =∑k=−∞

∞

d kN−12N−1 x−k

f N x= f N−1x g N−1x

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● Relations among – Decomposition

– Composition

c N , cN−1 , d N−1

ckN−1=

12∑

l

c lN g2 k− l=

12∑

l

c lN p l−2 k

d kN−1=

12∑

l

c lN h2 k−l=

12∑

l

clN ql−2 k

ckN=∑

l{cl

N−1 pk−2ld lN−1 qk−2l}

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● In case of the Haar's scaling function– Decomposition

– Composition

ckN−1=1

2c2k

N c2k1N d k

N−1=12c2k

N −c2k1N

c2kN−1=ck

N−1d kN−1 c2k1

N−1=ckN−1−d k

N−1

f N x=∑k

ckN H 2

N x−k

Average of two contiguous points

Half of the difference of two contiguous points

f N−1 x=∑k

ckN−1H 2

N−1 x−k f N−1 x=∑k

d kN−1H 2

N−1 x−k

43

Relation to the discrete wavelet (1)

● Repeat decomposing the signal

f N x=g N−1 x f N−1x=g N−1x g N−2x f N−2x =g N−1x g N−2x g N−3x f N−3 x

...=g N−1x g N−2x ⋯g N−M x f N−M x

If fN does not contain bias,

f N x= ∑j=−∞

N−1

g jx =∑j=−∞

N−1

∑k

d kj 2 j x−k

44

Relation to the discrete wavelet(2)

Decomposition of the signal

Discrete wavelet transform

f N x= ∑j=−∞

N−1

∑k

d kj2 j x−k

f N x= ∑j=−∞

N−1

∑kW f N k

2 j ,12 j 2 j /2 2 j x−k

As the coefficients of orthonormal basis functions are unique,

d kj=W f N k

2 j,

1

2 j 2 j /2

Decompositon using the scaling function = Discrete wavelet transform

45

Fast Wavelet Transform

● Discrete wavelet transform of fN

– When using the Haar's scaling functionfN :piecewise constant function at 2-N interaval

● Sampled signal can be regarded as fN

Fast ⇒ Wavelet Transform　 (Mallat's algorithm)

46

Wavelet in frequency domain● The two-scale relation: time domain● The scaling function in frequency domain

– Higher-order fN: Large bandwidth,

Narrow support

– Lower-order fN: Narrow bandwidth,

Wide support

– yN

compensatesfN

fN

fN+1 y

N

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Preparation● The two-scale relation

● Fourier transform of them

x=∑−∞

∞

qk2x−k x=∑−∞

∞

pk 2x−k

= ∑k=−∞

∞ 12

pk e−i k

2 2 =m02 2

=∑k=−∞

∞ 12

qk e−i k

2 2 =m12 2

m0=∑k=−∞

∞ 12

pk e−i k m1= ∑k=−∞

∞ 12

qk e−ik

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Wavelet in frequency domain(1)

=m02 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

-15 -10 -5 0 5 10 15

∣ H 2 ∣2

=∣1−e−i/2

i/2 ∣2

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=m02 2

0

0.2

0.4

0.6

0.8

1

-15 -10 -5 0 5 10 15

∣m02 ∣2

=∣1e−i/2

2 ∣2

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

-15 -10 -5 0 5 10 15

∣ H ∣2

=m02 2

∣ H 2 ∣2

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=m12 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

-15 -10 -5 0 5 10 15

m12 =1−e−i/2

2

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=m12 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

-15 -10 -5 0 5 10 15

∣ H ∣2

∣ H 2 ∣2

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Wavelet as filters● m

0:Low-pass filter,

m

1:High-pass filter

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

m1m0 QuadratureMirrorFilter(QMF)

54

Wavelet as filters● Combination of high-pass/low-pass filters

and decimation

m1(w)

m0(w)

x

2

2

High frequency

Lowfrequency

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Wavelet other than Haar

● Many kinds of mother wavelets are proposed– Continuous Wavelet Transform:

Gabor, Maxican hat, ...– Discrete Wavelet Trasform:

Spline, Daubechies, Meyer,...

56

Piecewise linear scaling function

● Haar's scaling function generates piecewise constant functions

● What about a scaling function that generates piecewise linear functions?

57


● This scaling function generates piecewise linear functions, but not orthogonal

58


● Two-scale relation

ϕ(x)=ϕ(2 x)+ 2ϕ(2 x−1)+ ϕ(2 x−2)ψ( x)=−ϕ(2 x)+ 2ϕ(2 x−1)−ϕ(2 x−2)

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● The previous scaling function/wavelet function are not orthonormal

Cannot be used as a mother wavelet⇒● Can we generate an orthonormal wavelet

from these scaling/wavelet functions?● Answer: the Battle-Lemarié wavelet

60

Wavelet with Orthonormality and compact support

● Orthonormality

● Compact support– Number of nonzero element in the two-scale

relation is finite–

∫−∞

∞

ψ jk (x)ψ j ' k ' ( x)dx=δ jj ' δkk '

61

Daubechies' Wavelet (2nd order)● Two-scale relation

● f and j cannot be expressed by analytical functions

x =138

2x338

2x−1 3−38

2x−21−38

2x−3

x=1−38

2x−3−38

2x−1338

2x−2−138

2x−3

x x

62

Daubechies' Wavelet in frequency domain

● Daubeches' wavelet as a filter

m0 =138

338

e−i3−38

e−2i1−38

e−3i

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

Haar Daubechies(2)

Sharper than the Haar's wavelet

63

Daubechies' higher-orderwavelet

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3

Daubechies 4 tap wavelet

scaling functionwavelet function

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

D4 wavelet - Fourier amplitudes

scaling functionwavelet function

From Wikipedia, the free encyclopedia

64

Vanishing Moment● The n-th Moment of function f

● The function has A vanishing moment↔

● Wavelet ψ has A vanishing moment=Scaling function φ can express piece-wise xA-1 function

● The 2n-tap Daubechies wavelet function has n vanishing moment

M n( f )=∫−∞

∞

xn f (x)dxM n( f )=∫−∞

∞

xn f (x)dx

M 0( f )=M 1( f )=⋯=M A−1( f )=0

65

Vanishing Moment● Example

– Haar wavelet has 1 vanishing moment

– Thus it can express a piecewise-constant (x0) signal

M 0(ψ H)=∫ψ H (x )dx=∫0

1/2

1dx−∫1/2

1

1dx=0

66

Vanishing Moment

Wavelet Vanishing moment

Explanation

D2 (Haar) 1 Piecewise constant (x0)

D4 2 Piecewise linear (x1)

D6 3 Piecewise quadratic (x2)

D8 4 Piecewise cubic (x3)

https://www.dsprelated.com/showarticle/1006.php

67

Other wavelets

● Simlet ● Coiflet ● Biorthogonal wavelet

–CDF wavelet

68

Symlet● Orthogonal wavelet designed by I. Daubachies● Orthogonal and has compact support

– 2 to 6 tap symlet is identical to Daubecies wavelet– Wavelet functions of 2n-tap Symlet have n vanishing

moment● Function is nearly symmetric

20-tap Symlet 20-tap Daubechies松嶋他「 AIC によるウェーブレット基底関数の選択」，応用統計学 33(2)201-219, 2004

69

Coiflet● Orthogonal wavelet designed by I.

Daubachies– Orthogonal and has compact support– Wavelet and scaling functions of 3n-tap Coiflet

have n vanishing moment– Function is nearly symmetric

6-tap Coiflet

松嶋他「 AIC によるウェーブレット基底関数の選択」，応用統計学 33(2)201-219, 2004

70

Biorthogonal wavelet● Orthogonal wavelet

– It is impossible to design an orthogonal wavelet that has compact support and the function is symmetric.

● Biorthogonal （双直交）wavelet– Use of different filters for analysis and synthesis– Biorthogonality

∫ψ jk (x)ψ j ' k' (x)dx=δ j j 'δ k k '

∫ψ jk(x)~ψ j ' k' (x)dx=δ j j 'δ k k '

71

Biorthogonal Wavelet● Uniqueness of wavelet coefficient

– Orthogonal wavelet

– Biorthogonal wavelet

f ( x)=∑j∑k

c jkψ jk ( x)

∫ψ jk (x) f ( x)dx=∫ψ jk(x)∑j∑k

c jkψ jk( x)dx=c jk

∫~ψ jk (x) f ( x)dx=∫~ψ jk(x)∑j∑k

c jkψ jk( x)dx=c jk

72

CDF wavelet● Cohen-Daubechies-Feauveau wavelet

– Biorthogonal– Compact support, symmetric– LPF and HPF have different tap length– CDF 5/3 wavelet (used in JPEG2000)

http://wavelets.pybytes.com/wavelet/bior2.2/

73

Application of Wavelet(1)● A couple of applications of wavelet

– Detection of discontinuity● Detection of mechanical fracture

– Denoising● Reduce noise without knowing the noise itself● Also useful for signal compression

– Image processing● Split an image into coarse and fine parts● The pyramid algorithm

74

Detection of discontinuity● Where are the discontinuous points?

75

Detection of discontinuity

Haar

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T0W1T0

W2T0W3T

0W4

T0W5

T0W6

T0W7

T0W8

T0

W9T0W10T0V10

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T0W 1

T0W 2

T0W 3T0W 4

T0

W 5

T0

W 6

T0

W 7

T0W 8

T0

W 9

T0W10T0V10

76


Daubechies (4tap)

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T1

W 1T1

W 2T1W 3T

1W 4

T1

W 5

T1

W 6

T1

W 7

T1

W 8

T0V8

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T1W 1T

1W 2

T1

W 3

T1W 4

T1

W 5

T1

W 6

T1W 7

T1

W 8

T0

V8

77


Daubechies (10tap)

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T4

W1T

4W2

T4

W3T4W4

T4W5

T4

W6

T1V6

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T4W 1

T4W 2

T4

W 3

T4W 4

T4W 5

T4W 6

T1

V6

78


Daubechies (20tap)

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T8

W1T8W2

T9W3T9W4

T9

W5

T3

V5

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T8

W1

T8W2

T9

W3

T9W4

T9

W5

T3

V5

79

0 200 400 600 800 1000

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Index

y

Denoising

● Procedure– Calculate the

Wavelet coefficients– Replace the

coefficients smaller than the pre-defined threshold to 0

– Compose the signal again

Discontinuity+NoiseThe discontinuity should be kept

80

Results

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T0

W 1

T0

W 2

T0

W 3T

0W 4

T0

W 5

T0W 6

T0

W 7

T0

W 8

T0

W 9

T0W10T0V10

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T0

W 1

T0

W 2

T0

W 3T

0W 4

T0

W 5

T0W 6

T0

W 7

T0

W 8

T0

W 9

T0W10T0V10

Original Haar

81

Results

0 200 400 600 800 1000

-1.5

-0.5

0.5

1.5

x

X

t

T1

W 1

T1W 2T1W 3

T1

W 4

T1

W 5

T1

W 6

T1

W 7

T1W 8

T0

V8

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T2

W 1

T2

W 2

T2W 3

T2

W 4

T2W 5

T2

W 6

T2

W 7

T1V7

Daubechies 6tapDaubechies 4tap

82

Results

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T8

W 1

T8W 2

T9

W 3

T9

W 4

T9W 5

T3

V5

Daubechies 20tap

0 200 400 600 800 1000

-1.5

-0.5

0.5

x

X

t

T4W 1

T7W 2

T9

W 3

T10W4

T11W5

T14

V5

Coiflet 24tap

83

Denoising of speech signal

ORIGINAL NOISY

84

Denoising of speech signal

Denoised (Haar) Denoised (Daubechies20)

85

Image processing● Example of 2-d wavelet(Haar)

Apply wavelet decomposition for both of X and Y

Half-sizeimage

LL

X-diff.HL

Y-diff.LH

XY-diff.HH

86

Image processing● 2-D analysis, and then?

– Reduce bit length of the high-frequency part ⇒Compression

– Replace the high-frequency part to 0 Smoothing, denoising⇒

– Recursive decomposition

The pyramid algorithm

87

Image compression● Apply wavelet transform to the image● Quantize the wavelet coefficients

– Coarse quantization of the high-frequency part → Smoothed image

● Entropy (reversible) compression

88

Examples● 256x256 image

Haarwavelet

Truncate the coefficients nearly zero

90

しきい値と画質

Th=10 Th=20

91

Wavelet and block noise● Why the block noises

appear?– The image is composed by

the superposition of the basis function

– The shape of the basis function (Haar) appears

● Improvement– Use the basis function with

overlapping (such as Daubachies' wavelet)

92

Results by different wavelets

Haar Daubechies(2)

93

The Wavelet Packet● Wavelet transform: split only the low-

frequency part

Original

L H

LL HLH

LLL HLHLLH

94

The Wavelet Packet● Wavelet packet: Split both the lower and

higher parts

Original

L H

LL LH

LLL LLH

HL HH

LHL LHH HLL HLH HHL HHH

95

Best Basis Selection● Choose "how to decompose the signal" by

the Wavelet Packet– Use the evaluation function of "goodness"

● Example: data compression– Make the small-valued coefficients zero– Evaluation function: Number of coefficients

smaller than a threshold

96

Best Basis Selection

21 29 32 51 16 3 25 43

25 41.5 9.5 34 -4 -9.5 -6.5 -9

33.25 21.75 -8.25 -12.25 -6.75 -7.75 2.75 1.25

27.5 5.75 -10.25 2 -7.25 0.5 2 0.75

L H

97


21 29 32 51 16 3 25 43

25 41.5 9.5 34 -4 -9.5 -6.5 -9

33.25 21.75 -8.25 -12.25 -6.75 -7.75 2.75 1.25

27.5 5.75 -10.25 2 -7.25 0.5 2 0.75

L H

Count the coefficients larger than 5

8

4 3

2 2 2 0

1 1 1 10 0 0 0

98


21 29 32 51 16 3 25 43

25 41.5 9.5 34 -4 -9.5 -6.5 -9

33.25 21.75 -8.25 -12.25 -6.75 -7.75 2.75 1.25

27.5 5.75 -10.25 2 -7.25 0.5 2 0.75

L H

Choose the basis with smaller number of coefficients andsmaller number of split

8

4 3

2 2 2 0

1 1 1 10 0 0 0

99

Note● Wavelet packet is not a simple frequency

analysis

Original signal

12345678High

Low

8765

1234

HH

HL

LH

LL

56

87

12

43

HHH

HHL

HLH

HLL

LHH

LHL

LLH

LLL

6

5

7

8

3

4

2

1

Documents

Wavelet変換 Wavelet Transformaito/wavelet/slide2.pdfZ-transform Discrete input 8 フーリエ変換 Fourier transform Calculate spectrum from infinite signal – フーリエ変換