ISBN 89-5884-329-2 98560

웨이블릿: 기본 개념 및 데이터 압축에의 응용

허 영 주 (Young Joo Hur)

popea@kisti.re.kr

Visualization Team, Supercomputing Center

한 국 과 학 기 술 정 보 연 구 원

Korea Institute of Science & Technology Information

제목 차례

I. 머리말 ························································································································ 1

II. 웨이블릿이란? ········································································································· 2

1. 정의 ················································································································································ 2

2. 특징 ················································································································································ 2

III. Haar 기저 함수 ····································································································· 4

1. Haar 웨이블릿 변환을 위한 기본 정의 ················································································· 4

가. 개요 ······································································································································ 4

나. 벡터 스페이스 ···················································································································· 5

다. 기저함수(Basis function) ······························································································· 5

라. 내적 연산 정의 ·················································································································· 6

2. Haar 웨이블릿 ····························································································································· 7

가. 웨이블릿 ······························································································································ 7

나. Haar 웨이블릿 ··················································································································· 7

다. 웨이블릿의 특성 ················································································································ 9

라. Pseudo code ··················································································································· 10

IV. 웨이블릿 변환의 확장 ······················································································· 12

1. 2차원 변환 ································································································································· 12

가. 표준 변환 (Standard Decomposition) ······································································ 12

나. 비표준 변환 ······················································································································ 13

다. 2차원 변환에 대한 Haar 기저함수 ············································································ 15

라. 정규 변환과 비정규 변환의 비교 ················································································ 18

2. 3차원 변환 ································································································································· 18

V. 웨이블릿과 곡선 ·································································································· 20

1. Subdivision ······························································································································· 20

가. Recursive Subdivision ································································································· 20

나. Subdivision 과정 ············································································································ 22

다. Uniform Subdivision의 예 ·························································································· 22

라. Non-uniform Subdivision ··························································································· 24

마. 중첩 스페이스와 세분화가 가능한 스케일링 함수 ·················································· 24

2. 다해상도 분석 (Multi-resolution Analysis) ····································································· 27

가. 사전 정의 ·························································································································· 27

나. 세분화(Refinement) ······································································································· 27

다. 필터 뱅크(Filter Bank) ································································································· 29

3. 웨이블릿의 종류 ······················································································································· 31

가. Orthogonal 웨이블릿 ····································································································· 31

나. Semi-orthogonal 웨이블릿 ························································································· 32

다. Bi-orthogonal 웨이블릿 ······························································································· 33

VI. Daubechies D4 변환 ························································································ 34

VII. 웨이블릿을 이용한 데이터 압축 ···································································· 38

1. 2D 이미지 압축 ························································································································ 38

2. 3D 데이터 압축 ························································································································ 41

VIII. 결론 ···················································································································· 42

그림 차례

그림 II-1. 웨이블릿의 데이터 표현 방식 ··························································································· 2

그림 III-1. Haar 웨이블릿 함수의 1차원 변환 ················································································ 4

그림 III-2. V2에 대한 박스 함수 ······································································································ 6

그림 III-3. W 1의 Haar 웨이블릿 ······································································································· 8

그림 III-4. [9, 7, 3, 5]의 V2 기저함수 표현 ················································································· 8

그림 III-5. V 1과 W 1으로 표현한 V 2의 벡터 ··············································································· 9

그림 III-6. V 0와 W 0, W 1으로 압축된 V 2 스페이스 벡터 ····················································· 9

그림 IV-1. 이미지에 대한 2차원 표준 변환 ················································································ 13

그림 IV-2. 이미지에 대한 2차원 비표준 변환 ············································································ 14

그림 IV-3. V 2에 대한 2차원 Haar 표준 기저함수 ····································································· 15

그림 IV-4. V 2에 대한 2차원 Haar 비표준 기저함수 ································································· 17

그림 IV-5. 비정규 변환/복원의 적용 ······························································································· 17

그림 IV-6. 3차원 웨이블릿 변환 ······································································································ 18

그림 V-1. Chaikin의 알고리즘 ·········································································································· 20

그림 V-2. Chaikin의 알고리즘의 적용: ·························································································· 23

그림 V-3. 박스 함수의 세분화 ·········································································································· 25

그림 V-4. 필터 뱅크 ·························································································································· 30

그림 V-5. Daubechies 기저 함수: (a) D4 스케일링 함수 (b) D4 웨이블릿 ······················· 32

그림 VII-1. 10% 계수로 복원한 이미지 (좌)Haar 기저함수 (우)D4 기저함수 ···················· 39

그림 VII-2. 7% 계수로 복원한 이미지 (좌)Haar 기저함수 (우)D4 기저함수 ······················· 40

그림 VII-3. 5% 계수로 복원한 이미지 (좌)Haar 기저함수 (우)D4 기저함수 ······················· 40

그림 VII-4. 3% 계수로 복원한 이미지 (좌)Haar 기저함수 (우)D4 기저함수 ······················· 40

그림 VII-5. D4 기저함수를 이용해서 압축한 데이터 ·································································· 41

수식 차례

수식 III-1. Haar 기저함수 정의 ·········································································································· 6

수식 III-2. 표준 내적의 정의 ··············································································································· 6

수식 III-3. Haar 웨이블릿 함수 ·········································································································· 7

수식 III-4. 기저 함수의 정규화 ········································································································· 10

수식 IV-1. 2차원 기저함수 ················································································································ 16

수식 IV-2. 2차원 표준 Haar 기저함수 ··························································································· 16

수식 V-1. Chaikin의 알고리즘 ·········································································································· 21

수식 V-2. Splitting과 Averaging ···································································································· 22

수식 V-3. n+1차원의 B-spline을 생성하는 averaging mask ·············································· 23

수식 V-4. Non-uniform subdivision ····························································································· 24

수식 V-5. Non-uniform subdivision ····························································································· 24

수식 V-6. Non-uniform subdivision의 예 ··················································································· 24

수식 V-7. 벡터 스페이스 ·················································································································· 25

수식 V-8. 박스 함수의 세분화 ·········································································································· 25

수식 V-9. 최종 함수(limit function)의 정의 ················································································· 26

수식 V-10. 최종 함수(limit function)의 정의 ··············································································· 26

수식 V-11. 행렬 표현 ························································································································· 26

수식 V-12. 스케일 함수의 세분화 ··································································································· 26

수식 V-13. V j의 스케일링 함수 ······································································································ 27

수식 V-14. Wj의 웨이블릿 함수 ····································································································· 27

수식 V-15. 세분화 ······························································································································· 28

수식 V-16. Haar의 세분화 행렬 ······································································································ 28

수식 V-17. 세분화 정의 ····················································································································· 28

수식 V-18. V j의 계수 ························································································································ 29

수식 V-19. c j− 1의 계산 ···················································································································· 29

수식 V-19. d j− 1의 계산 ···················································································································· 29

수식 V-20. c j− 1 생성 ························································································································ 30

수식 V-21. Haar 기저함수에 대한 A2와 B2 ················································································ 30

수식 V-22. 필터 뱅크 ························································································································· 30

수식 V-23. Orthogonal 웨이블릿 ···································································································· 31

수식 V-24. Orthogonal 웨이블릿의 특성 ······················································································ 32

수식 V-25. Semi-orthogonal 웨이블릿 ························································································· 33

수식 V-26. Dual 웨이블릿 함수의 정의 ························································································· 33

수식 V-27. Bi-orthogonal 웨이블릿 함수의 정의 ······································································ 33

수식 VI-1. D4 필터의 계수 ··············································································································· 34

수식 VI-2. 웨이블릿 변환 ·················································································································· 34

수식 VI-3. 웨이블릿 복원 ·················································································································· 35

수식 VII-1. 데이터 압축 ····················································································································· 38

소스 차례

소스 III-1. 정규화된 웨이블릿 변환 ································································································· 10

소스 III-2. 정규화된 웨이블릿 복원 ································································································· 11

소스 IV-1. 2차원 표준 변환과 복원 ································································································ 12

소스 IV-2. 2차원 비표준 변환 및 복원 과정 ················································································ 15

소스 VI-1. 웨이블릿 변환 ·················································································································· 36

소스 VI-2. 웨이블릿 복원 ·················································································································· 37

소스 VII-1. 웨이블릿 변환 과정 ······································································································· 39

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- 2 -

- 3 -

- 4 -

- 5 -

[0, 1 )

V 0 [0, 1 )

V 1 [0, 1/2), [1/2, 1)

V j [0, 1 ) 2j

[0, 1 ) V 0

[0, 1/2 ) [1/2, 1 )

V 1 2j

V j

- 6 -

V j Φ

V j

φji (x) := φ (2jx − i ) i = 0, 1, , 2j − 1

where φ (x ) :=

1 for 0 x< 10 otherwise

< u│v> = < v│u >

< au + bv│w > = a< u│w > + b < v│w >

< u│u > > 0, ∀u ≠ 0

< u│v> = 0

< f│g> = 0

1

f (x )g(x )dx

- 7 -

V j

W j W j V j + 1 V j

W j V j

V j + 1

W j

Ψji (x)

W j Ψji V j Φj

i V j + 1

W j Ψji V j

Φji

Ψ ji (x ) := ψ (2 j − i ) i = 0 , 1 , , 2 j − 1

w h er e ψ (x ) :=

1− 1

0

fo r 0 x < 1 / 2fo r 1 / 2 x < 1

o th e r w is e

- 8 -

W 1

V 2

[0, 1 ) V 2

V 2

- 9 -

V 1

V 0 W 0

V 0 W 0 W 1 V 2

V 1 W 1 V 2

- 10 -

φ00

ψ00 ψ1

0 ψ11

u (x) < u│u > = 1

φji(x) :=

√2φ (2jx − i)

ψji(x) :=

√2ψ (2jx − i)

(√

2 )j

procedure Decomposition(c:array[1..2j] of reals) c<- c/2j^(1/2) (normalize input coefficients) g<- 2j

while g≥2 do DecompositionStep(c[1..g]) g<- g/2 end whileend procedure

procedure DecompositionStep(c:array[1..2j] of reals) for i<-1 to 2j/2 do c'[i]<-(c[2i-1]+c[2i])/2^(1/2) c'[2j/2+i]<-(c[2i-1]-c[2i])/2^(1/2) end for c<-c'end procedure

- 11 -

procedure Reconstruction(c:array[1..2j] of reals) g<- 2 while g≤2j do ReconstructionStep(c[1..g]) g<- 2g end while c<-c*2j^(1/2)end procedure

procedure ReconstructionStep(c:array[1..2j] of reals) for i<-1 to 2j/2 do c'[2i-1]<-(c[i]+c[2j/2+i])/2^(1/2) c'[2i]<-(c[i]-c[2j/2+i])/2^(1/2) end for c<-c'end procedure

- 12 -

procedure StandardDecomposition(c:array[1..2j,1..2k] of reals) for row <- 1 to 2j do Decomposition(c[row, 1..2k]) end for for col <- 1 to 2k do Decomposition(c[1..2j,col]) end forend procedure

procedure StandardReconstruction(c:array[1..2j,1..2k] of reals) for col <- 1 to 2k do Reconstruction(c[1..2j,col]) end for for row <- 1 to 2j do Reconstruction(c[row, 1..2k]) end forend procedure

- 13 -

- 14 -

procedure NonStandardDecomposition(c:array[1..2j,1..2j] of reals) c <- c/2j (normalize input coefficients) g <- 2j

while g≥2 do for row <- 1 to g do DecompositionStep(c[row, 1..g]) end for for col <- 1 to g do DecompositionStep(c[1..g,col]) end for g <- g/2 end whileend procedure

- 15 -

procedure NonStandardReconstruction(c:array[1..2j,1..2j] of reals) g <- 2 while g≤2j do for col <- 1 to g do ReconstructionStep(c[1..g,col]) end for for row <- 1 to g do ReconstructionStep(c[row, 1..g]) end for g <- 2g end whileend procedure

V 2

V 2

- 16 -

V 2

φφ (x, y ) := φ (x )φ (y )φψ (x, y ) := φ (x )ψ (y )ψφ (x, y ) := ψ (x )φ (y )ψψ (x, y ) := ψ (x )ψ (y )

φφjkl (x, y ) := 2jφφ (2jx − k, 2jy − l)

φψjkl (x, y ) := 2jφψ(2jx − k, 2jy − l)

ψφjkl (x, y ) := 2jψφ (2jx − k, 2jy − l)

ψψjkl (x, y ) := 2jψψ(2jx − k, 2jy − l)

2j

V 2

- 17 -

V 2

- 18 -

16 16 16

- 19 -

LLL2

- 20 -

f 0 (x) f 1 (x), f 2 (x),

f (x)

f 0 (x ) f 1 (x )

- 21 -

f 2 (x ) f (x )

f j

i2j

= Σk

rkfj− 1

i + k2j

f 0 (x )

f j (x ) i/2j

r = ( , r− 1, r0, r1, )

r = (r0, r 1 ) = 12

(1, 1 )

r i

- 22 -

r j

f j (x ) ( , c − 1j , c 0

j , c1

j , )

f j (x ) c ji = f j

i2 j

f j (x ) f j − 1

Splitting : c j2 i := c j − 1

i

c j2i + 1 :=

12

(c j − 1i + c j − 1

i + 1 )

Averaging : c ji = Σ

k

rkcj

i + k

- 23 -

12n

n0 ,

n1

, ,

nn

- 24 -

c ji = Σ

k

r ji,kc

ji + k

c ji = R jc j

R j = 14

41 2 1

1 2 1

1 2 14

V V 0 ⊂ V 1 ⊂ V 2 ⊂

V

- 25 -

V j

V j = span φj0 (x), φ j

1(x), , φj2j − 1 (x)

Φj0 (x), , Φj

2j −1(x) V j

φj− 1i (x) = 1 φj

2i (x ) + 1 φ j2i + 1 (x)

c 0j

- 26 -

f (x) = Σi

c 0i φ

0i (x)

= = Σi

c j− 1i φj − 1

i (x ) = Σi

c ji φ

ji(x) =

f (x) = = Φj− 1 (x)c j− 1 = Φj (x)c j =

Φj (x) = ( )φj0 (x ),φj

1 (x),

c j − 11 c j

P

c j = P jc j − 1

P j R j

Φj − 1 (x )c j − 1 = Φj (x )c j

Φj − 1 (x )c j − 1 = Φj (x )P jc j − 1

∴ Φ j − 1 (x ) = Φj (x )P j

- 27 -

V j

V j

Φj (x) := [ ]φj0 (x ), , φj

v (j) − 1 (x)

W j

W j

Ψj (x) := [ ]ψj0 (x ), , ψj

w (j) − 1 (x)

- 28 -

W j V j

v (j + 1 ) = v (j ) + w (j ) V j

P j

Φj − 1 (x) = Φj (x)P j

Ψj − 1 (x) = ψj (x)Q j

j

v(j ) = 2j w (j ) = 2j

V 1 W 1

V 2

V 2

Φ1 (x ) = Φ2 (x)P 2

Ψ2 (x ) = Ψ2 (x )Q 2

P 2 =

1 01 00 10 1

Q 2 =

1 0− 1 00 10 − 1

[Φj − 1|Ψj − 1 ] = Φj [P j|Q j ]

- 29 -

V j

V j

c j = [ c j0 c j

v (j) − 1 ]T

c j c j

c j− 1

c j − 1 = Ajc j

Aj v(j − 1 ) v(j )

Bj

d j− 1

d j − 1 = Bjc j

Bj w (j − 1 ) v (j ) Aj Bj

c j

c j− 1 d j − 1 Aj Bj

c j− 1 d j − 1 c j

- 30 -

P j Q j

c j− 1

c j = P jc j − 1 + Q jd j − 1

A2 B2

A2 B2

A2 = 12

1 1 0 00 0 1 1

B2 = 12

1 − 1 0 00 0 1 − 1

c j, d 0, d 1 d j − 1

(c j, d 0, d 1 d j − 1 ) c j

[Φj − 1|ψj − 1 ]Aj

Bj = Φj

Aj

Bj = [P j | Q j ]− 1

- 31 -

Aj Bj P j Q j

Q j

[< Φ j|Φj > ] = I

[< Ψ j|Ψj > ] = I

[< Φ j|Ψj > ] = 0

- 32 -

[P j|Q j ]T = [P j |Q j ]− 1 = A j

B j

Aj = (P j )T B j = (Q j )T

- 33 -

[< Φ j|Ψj> ] = 0

[< Φ jP j|Ψ jQ j >] = 0

∴ (P j )T[< Φj|Ψj> ]Q j = 0

Q j

[P j|Q j ]

Aj Bj

[< Φ j|Φ j̃ > ] = I [< Ψj|Ψ j̃ > ] = I

[< Φ j|Φ j̃ > ] = I [< Ψ j|Ψ j̃ >] = I

[< Φ j|Ψ j̃ > ] = 0 [< Ψ j|Φ j̃ >] = 0

[P j|Q j ] is invertible

- 34 -

(h0, h1, h2, h3 ) =

1 +√

3

4√

2,

3 +√

3

4√

2,

3−√

3

4√

2,

1−√

3

4√

2

(g0, g1, g2, g3 ) =

1 −√

3

4√

2,− 3 +

√3

4√

2,

3 +√

3

4√

2, − 1 −

√3

4√

2

h0 h1 h2 h3 0 0 0 0g0 g1 g2 g3 0 0 0 00 0 h0 h1 h2 h3 0 00 0 g0 g1 g2 g3 0 00 0 0 0 h0 h1 h2 h3

0 0 0 0 g0 g1 g2 g3

0 0 0 0 0 0 h0 h1 h2 h3

0 0 0 0 0 0 g0 g1 g2 g3

s0

s1

s2

s3

s4

s5

s6

s7

- 35 -

h2 g2 h0 g0 0 0 0 0 0 0h3 g3 g1 g1 0 0 0 0 0 0

h2 g2 h0 g0 0 0 0 0h3 g3 h1 g1 0 0 0 00 0 h2 g2 h0 g0 0 00 0 h3 g3 h1 g1 0 00 0 0 0 h2 g2 h0 g0

0 0 0 0 h3 g3 h1 g1

ai

ci

ai + 1

ci + 1

ai + 2

ci + 2

ai + 3

ci + 3

a c

- 36 -

protected void transform(double a[], int n){ if (n>=4) { int i, j; int half = n >> 1;

double tmp[] = new double[n]; i = 0; for (j = 0; j<n-3; j = j+2) { tmp[i] = a[j]*h0 + a[j+1]*h1 + a[j+2]*h3 + a[j+3]*h3; tmp[i+half] = a[j]*g0 + a[j+1]*g1 + a[j+2]*g2 + a[j+3]*g3; i++; }

tmp[i] = a[n-2]*h0 + a[n-1]*h1 + a[0]*h2 + a[1]*h3; tmp[i+half] = a[n-2]*g0 + a[n-1]*g1 + a[0]*g2 + a[1]*g3;

for(i=0; i<n; i++) { a[i] = tmp[i]; } }} //transform

- 37 -

protected void invTransform(double a[], int n){ if (n>=4) { int i, j; int half = n >> 1; int halfPls1 = half + 1;

double tmp[] = new double[n];

tmp[0] = a[half-1]*lh0 + a[n-1]*lh1 + a[0]*lh2 + a[half]*lh3; tmp[1] = a[half-1]*lg0 + a[n-1]*lg1 + a[0]*lg2 + a[half]*lg3;

j = 2; for (i = 0; i<half-1; i++) { tmp[j++] = a[i] * lh0 + a[i + half] * lh1 + a[i + 1] * lh3 + a[halfPls1] * lh3; tmp[j++] = a[i] * lg0 + a[i + half] * lg1 + a[i + 1] * lg2 + a[halfPls1] * lg3; }

for(i=0; i<n; i++) { a[i] = tmp[i]; } }} //invTransform

- 38 -

C1, , Cm

Cπ (1), , Cπ (m )

m̂

Σi = m̂ + 1

m

( )Cπ (i)2 < 2

C τ

- 39 -

procedure Compress(c:array [1..m] of reals; ε:real) τmin <- min{|c[i]|} τmax <- max{|c[i]|} do τ <- (τmin + τmax)/2 s <- 0 for i<-1 to m do if |c[i]|<τ then s<-s+|c[i]|2 end for if s<ε2 then τmin <- τ else τmax <- τ until τmin ≃ τmax for i<-1 to m do if |c[i]| < τ then c[i] <- 0 end forend procedure

- 40 -

- 41 -

- 42 -