: ,

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4


:

.-.


2004

Ņ....3

1.   ۅ...5

2.   -Ņ....9

3.   ۅ..12

4.   ȅ.13

4.1. 充...13

4.2. ...16

4.3. , ..16

18


- (wavelet transform), 90- , , . . - ( ) .

, , , . , , , , , . , - ( ) .

200- : , , , -. - , . , "" , . .

- ? - "" "" . (, ) . , .

, . ("") . - - 1946-47 Jean Ville , , Dennis Gabor. 1950-70- - .

70- - (Jean Morlet) , . , , . , . - . (wavelets) - . (Yves Meyer), (Ingrid Daubechies), (Ronald Coifman), (Stephane Mallat) .

, , .. , .. , .. .

1.

1. (multiresolutional analysis) L2(Rd), d³1,

, (1.1)

:

1. , 蠠 L2(Rd),

2. fÎ L2(Rd), jÎ Z, f(x)ÎVj ,

f(2x) ÎVj-1,

3. fÎ L2(Rd), kÎ Zd, f(x)ÎV0 , f(x-k)ÎV0,

4. (scaling) jÎV0, {j(x-k)}kÎZd 򠠠

V0.

4 :

4. jÎV0, {j(x-k)}kÎZd 򠠠 V0.

Wj Vj Vj-1,

, (1.2)

L2(Rd)

(1.3)

n, (1.1) :

(1.4)

(1.5)

, , , j=0

, V0Î L2(Rd) (1.6)

(1.4). V0 .

j - (-) . y - - , {y(x-k)}kÎZ W0.

, m=0..M-1. (1.7)

4 , , -, j V-1 . {jj,k(x)=2-j/2j(2-jx-k)}kÎZ Vj,

. (1.8)

, (1.8) . (1.8)

, (1.9)

, (1.10)

2p- m0 :

. (1.11)

-, {j(x-k)}kÎZ ,

(1.12)

(1.13)

. (1.14)

(1.9),

(1.15)

, (1.15) ,

. (1.16)

2p- m0 (1.14), x/2 x,

(1.17)

hk (1.11). , 

(1.18)

y :

, (1.19)

, k=0,,L-1 , (1.20)

y

, (1.21)

, (1.22)

,  蠠 젠 젠 堠 jÎZ

{yj,k(x)=2-j/2y(2-jx-k)}kÎZ Wj.

(1.17) (quadrature mirror filters, QMF) H G, . QMF H G . L (1.11) (1.22) , .

j y , , . , j y . , H G, , j y.


4.


, , .

K(x,y) :

(4.1)

(4.2)

(4.3)

4.1 d/dx

. d/dx. , , , , , i, l, jÎ Z d/dx

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

, (1.8) (1.19),

(4.12)

(4.13)

(4.14)

d/dx , , d/dx V0.

4.1. 1. (4.11), , lÎ Z (5.8) :

(4.15)

(4.16)

(4.17)

2. , (4.15)-(4.16) , .

. =1, (4.15)-(4.16) , (4.11) . () , .

2. , (4.12) (4.13) () (5.10) (5.11) , . : .

4.1 [2].

(4.15)-(4.16) . , , (4.15) .

4.2 dn/dxn

d/dx, dn/dxn V0, ..

, lÎ Z, (4.18)

.

4.2. 1. (4.18) , , lÎ Z

(4.19)

(4.20)

(4.17).

2. M ≥ (n+1)/2, . (4.18) , (4.19)-(4.20) , . n

(4.21)

(4.22)

(4.23)

n

(4.24)

(4.25)

3. M ≥ (n+1)/2, 2 , (4.18) .

      

 

,

, f(x) , . :

, {φ1, φ1,} ; :

Kij

,

f g

, ,

fi gi :

, , i=1,2,

, i=1,2,

, R:

, , ,

K. n n :

, i=1,2,,n


1


function [a,r]=dif_r(wname)

[LO_D,HI_D,LO_R,HI_R] = wfilters(wname);

% a2k-1

len=length(LO_D);

a=zeros(len-1,1);

for k=1:len-1;

for i=0:len-2*k;

a(2*k-1)=a(2*k-1)+2*LO_D(i+1)*LO_D(i+2*k);

end;

end;

% rl

f=zeros(len-2,1);

f(1)=-1/2;

R=zeros(len-2);

for l=len-2:-1:2;

R(l,l)=-1;

if (2*l<=len-2)

R(l,2*l)=2;

end;

for n=1:2:len-1;

if (abs(2*l-n)<len-2);

if ((2*l-n)<0);

R(l,abs(2*l-n))=-a(n)+R(l,abs(2*l-n));

else

R(l,2*l-n)=a(n)+R(l,2*l-n);

end;

end;

if (abs(2*l+n)<len-2);

if ((2*l+n)<0);

R(l,abs(2*l+n))=-a(n)+R(l,abs(2*l+n));

else

R(l,2*l+n)=a(n)+R(1,2*l+n);

end;

end;

end;

end;

for j=1:len-2;

R(1,j)=j;

end;

r=inv(R)*f;


2


function [al, bet, gam]=difcoef(wname,N)

% rl

[LO_D,HI_D,LO_R,HI_R] = wfilters(wname);

[a,r]=dif_r(wname);

L=length(LO_D);

% αl, βl, γl

J=length(r):-1:1;

R=[-r(J);0; r];

K=L+1;

al=zeros(2*L+1,1);

bet=al;

gam=al;

for i=-L+1:L+1;

for k=L+1:2*L;

for k1=L+1:2*L;

if(((2*i+k-k1+L)<length(R)+1)&&((2*i+k-k1+L)>0))

al(i+L)=HI_D(k-L)*HI_D(k1-L)*R(2*i+k-k1+L)+al(i+L);

bet(i+L)=HI_D(k-L)*LO_D(k1-L)*R(2*i+k-k1+L)+bet(i+L);

gam(i+L)=LO_D(k-L)*HI_D(k1-L)*R(2*i+k-k1+L)+gam(i+L);

end;

end;

end;

end;


3


1.    M=2.
a1=1.1250 a3=-0.1250
r1=-0.6667 r2=0.0833

2.    M=3.
a1=1.1719 a3=-0.1953 a5=0.0234

r1=-0.7452 r2=0.1452 r3=-0.0146 r4=-0.0003

3.    M=4.
a1=1.19628906249870 a3=-0.23925781249914
a5=0.04785156250041 a7=-0.00488281249997

r1=-0.79300950497055 r2=0.19199897079726 r3=-0.03358020705113

r4= 0.00222404967066 r5=0.00017220619000 r6=-0.00000084085054

4.    M=5.
a1=1.21124267578280 a3=-0.26916503906311 a5=0.06921386718738

a7=-0.01235961914130 a9=0.00106811523422

r1=-0.82590601185686 r2=0.22882018706986 r3=-0.05335257193327

r4=0.00746139636621 r5=-0.00023923581985 r6=-0.00005404730164

r7=-0.00000025241171 r8=-0.00000000026960

5.    M=6.

a1=1.22133636474683 a3=-0.29079437255810 a5=0.08723831176674

a7=-0.02077102661228 a9=0.00323104858448 a11=-0.00024032592766

r1=-0.85013666156022 r2=0.25855294414318 r3=-0.07244058999853

r4=0.01454551104340 r5=-0.00158856154379 r6=0.00000429689148

r7=0.00001202657519 r8=0.00000042069120 r9=-0.00000000289967

r10=0.00000000000070

6.    M=2.
a1=1.20035616471068 a3=-0.24753371156550 a5=0.05401594511476

a7=-0.00724698442340 a9=0.00043220193586 a11=-0.00002361577240

r1=-0.80177838961957 r2=0.20214744976459 r3=-0.03943577686925

r4=0.00404789045961 r5=-0.00008445623632 r6=0.00000255044096

r7=0.00000088836508 r8=0.00000000237860 r9=-0.00000000002099

r10=0.00000000000000

7.    M=2.

a1=1.12499999999971 a3=-0.12499999999971

r1=-0.66666666666616 r2=0.08333333333308

8.    M=3.

a1=1.17187500000666 a3=-0.19531250000432 a5=0.02343749999766
r1=-0.74520547946903 r2=0.14520547945865 r3=-0.01461187214494
r4=-0.00034246575336

9.    M=4.

a1=1.19628906249990 a3=-0.23925781249985 a5=0.04785156249993

a7=-0.00488281249998

r1=-0.79300950497424 r2=0.19199897079876 r3=-0.03358020705098

r4=0.00222404967071 r5=0.00017220619000 r6=-0.00000084085054

4


1.    M=2.

α-3=-0.00520833333331

β-3 =-0.00139556871057

γ-3=0.01943776462271

α-2=0.04687500000004

β-2=0.02222890204378

γ-2=-0.04027109795592

α-1=0.71874999999873

β-1=-0.03887552924536

γ-1=0.00279113742108

α1=-0.71874999999873

β1=-0.00279113742108

γ1=0.03887552924536

α2=-0.04687500000004

β2=0.04027109795592

γ2=-0.02222890204378

α3=0.00520833333331

β3=-0.01943776462271

γ3=0.00139556871057

2.    M=3.

α-5= -0.00000401327055

β-5 =0.00000042496289

γ-5=-0.00003790058109

α-4=0.00173507063342

β-4=-0.00018594182937

γ-4= 0.01618803080395

α-3= -0.01438088613327

β-3= 0.00249383057321

γ-3= -0.05023776816965

α-2= 0.09779091752885

β-2=-0.02225975249164

γ-2=0.03807446337594

α-1=0.84450449488848

β-1=0.05176823864378

γ-1=0.02782997442973

α1= -0.84450449488848

β1= -0.02782997442973

γ1=-0.05176823864378

α2=-0.09779091752885

β2= -0.03807446337594

γ2= 0.02225975249164

α3= 0.01438088613327

β3= 0.05023776816965

γ3= -0.00249383057321

α4= -0.00173507063342

β4=-0.01618803080395

γ4=0.00018594182937

α5=0.00000401327055

β5=0.00003790058109

γ5=-0.00000042496289

M=4.

α-7=0.00000000205286

β-7 =0.00000000009443

γ-7=-0.00000004462725

α-6=-0.00000544992677

β-6 =-0.00000025123058

γ-6=0.00011822433115

α-5=-0.00041543477135

β-5 =-0.00001769213018

γ-5=0.00969983443149

α-4=0.00432716179594

β-4=0.00030224225713

γ-4= -0.04151919818136

α-3=-0.02134228538239

β-3=-0.00242879427312

γ-3= 0.05677199535135

α-2=0.14516544960962

β-2=0.01699891329704

γ-2=-0.00862627283270

α-1=0.93050197130889

β-1=-0.04758076037403

γ-1=-0.04917088083201

α1=-0.93050197130889

β1= 0.04917088083201

γ1=0.04758076037403

a2=-0.14516544960962

β2= 0.00862627283270

γ2=-0.01699891329704

a3=0.02134228538239

β3= -0.05677199535135

γ3=0.00242879427312

α4=-0.00432716179594

β4=0.04151919818136

γ4=-0.00030224225713

a5=0.00041543477135

β5=-0.00969983443149

γ5=0.00001769213018

a6=0.00000544992677

β6=-0.00011822433115

γ6=0.00000025123058

α7=-0.00000000205286

β7= 0.00000004462725

γ7=-0.00000000009443

3.    M=2.

α-3=-0.00520833333331

β-3 =-0.00139556871057

γ-3=0.01943776462271

α-2=0.04687500000004

β-2=0.02222890204378

γ-2=-0.04027109795592

α-1=0.71874999999873

β-1=-0.03887552924536

γ-1=0.00279113742108

α1=-0.71874999999873

β1=-0.00279113742108

γ1=0.03887552924536

α2=-0.04687500000004

β2=0.04027109795592

γ2=-0.02222890204378

α3=0.00520833333331

β3=-0.01943776462271

γ3=0.00139556871057



1.    Beylkin G. Wavelets and Fast Numerical Algorithms

2.    Beylkin G. Wavelets, Multiresolution Analysis and Fast Numerical Algorithms

3.    Beylkin G. In The Representation.of Operators in Bases of Compactly Supported Wavelets

4.    Bradley K. Alpert A Class of Bases in L2 for the Sparse Representation of Integral Operators

5.    .., .., .. // 2001, 5. .465-500


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