j i jj ii home page title page - ustcstaff.ustc.edu.cn/~lixustc/course/wavelet/charpter_4.pdf · jj...
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1oÙ õ©E©Û
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õ©E©ÛMallat
Meyeru1986cME5/EÑäk½P~51w¼ê§Ù
? ²£¤ L2(R)IOħâ¦Åý
uÐ"1988c, S. Mallat3EÅÄJÑõ©E©
Û (Multiresolution Analysis) Vg,lmVgþ//`²
Åõ©EÇA5,òdc¤kÅÄEÚ
å5§ÑÅE±9ÅC¯
§=Mallat"
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1 ĵe
½Â Vjj∈Z´L2(R)4fmS. Vjj∈Z¡L2(R)
õ©E©Û (MRA),XJ÷v
(1)i@5: Vj ⊂ Vj+1, j ∈ Z
(2)È5: ∪j∈ZVj = L2(R)
(3)©5: ∩j∈ZVj = 0
(4) 5: f (t) ∈ Vj ⇐⇒ f (2t) ∈ Vj+1, j ∈ Z
(5)²£ØC5: f (t) ∈ V0 ⇒ f (t− k) ∈ V0, k ∈ Z
(6)Ä35:3 φ ∈ V0,¦ φ(t − k), k ∈ Z¤ V0
IOÄ.
Ù¥, φ¡ºÝ¼ê, Vj ¡ºÝm. ÷vþ¡^õ©E
©Û Vjj∈Z¡dºÝ¼ê φ)¤õ©E©Û.
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Vj−1 Vj Vj+1
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½n Vjj∈Z ´dºÝ¼ê φ)¤ L2(R)õ©E©Û.K
é?¿ j ∈ Z,
φj,k(t) = 2j/2φ(2jt− k), k ∈ Z
´ Vj IOÄ.
y²Äk,d φ(t− k) ∈ V0, k ∈ Z,9 5 φj,k ∈ Vj, k ∈ Z.
Ùg,IO5deª
〈φj,k, φj,l〉L2 =
∫R
2jφ(2jt− k)φ(2jt− l)dt
=
∫Rφ(x− k)φ(x− l)dx
= 〈φ0,k, φ0,l〉
= δkl.
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,`² Vj ¥¼êÑd φj,k, k ∈ Z5L«. b f ∈ Vj,
k 5, f (2−jt) ∈ V0. d φ(t − k), k ∈ Z´ V0 IO
Ä,
f (2−jt) =∑k∈Z
〈f (2−j·), φ(· − k)〉φ(t− k)
=∑k∈Z
(∫Rf (2−jx)φ(x− k)dx
)φ(t− k)
=∑k∈Z
(2
j2
∫Rf (y)φ(2jy − k)dy
)2
j2φ(t− k).
u´
f (x) =∑k∈Z
〈f, φjk〉φjk(x).
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2 VºÝ§ÅÈÅì
2.1. VVVºººÝÝݧ§§LLL«««
ºÝ¼ê φ)¤ L2(R)¥õ©E©Û.K φ÷vVºÝ
§
φ(t) =∑k∈Z
hkφ(2t− k),
Ù¥VºÝXê
hk = 2
∫Rφ(t)φ(2t− k)dt.
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VºÝXê5
b Vj; j ∈ Z ´dºÝ¼ê φ )¤ MRA. KVºÝXê
hk, k ∈ Z,äke5
•∑k∈Z
hk−2nhk−2m = 2δnm
•∑k∈Z
|hk|2 = 2
•∑k∈Z
hk = 2
•∑k∈Z
h2k =∑k∈Z
h2k+1 = 1
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y² (1)dVºÝ§
φ(t) =∑k∈Z
hkφ(2t− k),
φ(t− n) =∑k∈Z
hkφ(2t− 2n− k)
=∑k∈Z
hk−2nφ(2t− k)
=1√2
∑k∈Z
hk−2nφ1,k.
Ón,
φ(t−m) =1√2
∑k∈Z
hk−2mφ1,k.
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u´,d φ(t− k), k ∈ Z´IOÄ
δnm = 〈φ(t− n), φ(t−m)〉L2
=1
2
⟨∑k∈Z
hk−2nφ1,k,∑l∈Z
hl−2mφ1,l
⟩L2
=1
2
∑k∈Z
∑l∈Z
hk−2nhl−2m〈φ1,k, φ1,l〉L2
=1
2
∑k∈Z
hk−2nhk−2m.
l ∑k∈Z
hk−2nhk−2m = 2δnm.
(2)AO/,- n = m = 0∑k∈Z
|hk|2 = 2.
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(3)3VºÝ§ü>È©,∫Rφ(t)dt =
∑k∈Z
hk
∫Rφ(2t− k)dt
=1
2
∑k∈Z
hk
∫Rφ(x)dx
=1
2
(∫Rφ(x)dx
)∑k∈Z
hk.
Ï
∫Rφ(t)dt 6= 0 (?),¤±∑
k∈Z
hk = 2.
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(4)d
|∑k∈Z
h2k|2 + |∑k∈Z
h2k+1|2
=∑k∈Z
h2k
∑l∈Z
h2l +∑k∈Z
h2k+1
∑l∈Z
h2l+1
=∑k∈Z
(∑l∈Z
h2l+2k
)h2k +
∑k∈Z
(∑l∈Z
h2l+1+2k)h2k+1
=∑l∈Z
(∑k∈Z
h2k+2lh2k +∑k∈Z
h2k+1+2lh2k+1
)=∑l∈Z
∑k∈Z
hk+2lhk = 2∑l∈Z
δl0 = 2.
9 ∑k∈Z
h2k +∑k∈Z
h2k+1 = 2,
∑k∈Z
h2k =∑k∈Z
h2k+1 = 1.
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2.2. VVVºººÝÝݧ§§ªªªLLL«««
éVºÝ§
φ(t) =∑k∈Z
hkφ(2t− k)
üঠFourierC
φ(ω) =1
2
∑k∈Z
hke−ikω/2φ(ω/2).
- H(ω) =1
2
∑k∈Z
hke−ikω,KkVºÝ§ªL«
φ(ω) = H(ω/2)φ(ω/2).
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H(ω)5
Únb φ ∈ L2(R). K φ(t− k), k ∈ Z´IO8¿^
´ ∑k∈Z
|φ(ω + 2kπ)|2 =1
2π, ω ∈ R.
y² φ(t− k), k ∈ Z´IO8=é?¿ k ∈ Z∫Rφ(t)φ(t− k)dt = δ0k.
d Parsevalª
δ0k =
∫Rφ(t)φ(t− k)dt
=
∫Rφ(ω)φ(ω)eikωdω
=
∫R|φ(ω)|2eikωdω.
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ò R?1¿© Ij = [2πj, 2π(j + 1)], j ∈ Z,Kk
δ0k =∑j∈Z
∫ 2π(j+1)
2πj
|φ(ω)|2eikωdω
=∑j∈Z
∫ 2π
0
|φ(ω + 2πj)|2eikωdω
=
∫ 2π
0
∑j∈Z
|φ(ω + 2πj)|2 eikωdω.
- F (ω) = 2π∑
j∈Z |φ(ω+ 2πj)|2.K φ(t− k), k ∈ Z´IO
8=1
2π
∫ 2π
0
F (ω)eikωdω = δ0k.
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5¿ F ´± 2π±Ï¼ê,ù´Ï
F (ω + 2π) = 2π∑j∈Z
|φ(ω + 2π(j + 1))|2
= 2π∑j′∈Z
|φ(ω + 2πj′)|2
= F (ω).
u´, φ(t− k), k ∈ Z´IO8= F FourierXê
÷v
α−k = δ0k,
d=L²
F (ω) = 1, ω ∈ R.
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H ´± 2π±Ï¼ê,¿÷v
|H(ω)|2 + |H(ω + π)|2 = 1, ω ∈ R.
âVºÝ§ φ(ω) = H(ω/2)φ(ω/2)
1
2π=∑k∈Z
|φ(ω + 2kπ)|2
=∑k∈Z
|H(ω/2 + kπ)|2|φ(ω/2 + kπ)|2
=∑k∈Z
|H(ω/2 + 2kπ)|2|φ(ω/2 + 2kπ)|2+∑k∈Z
|H(ω/2 + (2k + 1)π)|2|φ(ω/2 + (2k + 1)π)|2
= |H(ω/2)|2∑k∈Z
|φ(ω/2 + 2kπ)|2
+|H(ω/2 + π)|2∑k∈Z
|φ(ω/2 + (2k + 1)π)|2
=1
2π(|H(ω/2)|2 + |H(ω/2 + π)|2).
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2.3. ÅÅÅÈÈÈÅÅÅììì
Ú\ gk = (−1)kh1−k, k ∈ Z,9
G(ω) =1
2
∑k∈Z
gke−ikω.
K gk, k ∈ Z÷v
(1)∑k∈Z
gk−2ngk−2m = 2δnm;
(2)∑k∈Z
hk−2ngk−2m = 0;
(3)∑k∈Z
(hn−2khm−2k + gn−2kgm−2k) = 2δnm.
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G(ω)÷v
(1) G(ω) = −e−iωH(ω + π);
(2) |G(ω)|2 + |G(ω + π)|2 = 1;
(3) H(ω)G(ω) +H(ω + π)G(ω + π) = 0.
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3ó§A^¥,~r hk, k ∈ Z gk, k ∈ Z¡$ÏÈÅì
XêpÏÈÅìXê. H(ω) G(ω)K´§ªLy,©O
¡$ÏÈÅìpÏÈÅì. ¢Sþ,
H(0) = G(π) = 1.
d,3MRAµee)ü¼ê H(ω)Ú G(ω),§÷
vÝL«
M(ω)M∗(ω) = I,
Ù¥
M(ω) =
H(ω) H(ω + π)
G(ω) G(ω + π)
.
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3 Åfm L2(R)©)
d MRAüN5±wÑ: Vj ´ Vj+1 ýfm, u´3 Vj 3 Vj+1
¥ÖmWj,
Vj+1 = Vj
⊕Wj.
þãm©)L§48e
u´, Vj+1 = Vl
j⊕k=l
Wk.- j → +∞, l→ −∞, L2(R)©)
L2(R) =−∞⊕
j=+∞Wj.
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ÅmWj, j ∈ Zäk±e5
5: g(t) ∈ Wj ⇐⇒ g(2t) ∈ Wj+1, j ∈ Z.
²£ØC5: g(t) ∈ W0 ⇒ g(t− k) ∈ W0, k ∈ Z.
±þ?ØL², XJUé¼ê ψ, ¦Ùê²£
ψ(t− k), k ∈ Z¤W0IOÄ,K φj,k(t) = 2j/2ψ(2jt−
k), k ∈ Z´Wj IOÄ.l , ψj,k, j, k ∈ Z¤ L2(R)
IOÄ.
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ÅE
½n Vj, j ∈ Z´dºÝ¼ê φ)¤ L2(R)õ©E
©Û, hk, k ∈ Z´VºÝXê. -
ψ(t) =∑k∈Z
gkφ(2t− k),
Ù¥, gk = (−1)kh1−k. K ψ ê²£ ψ(t − k), k ∈ Z
¤ W0 IOÄ, Ù¥ W0 ´ V0 3 V1 ¥Ö. l ,
ψj,k, j, k ∈ Z¤ L2(R)IOÄ.
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y² (1) ψ(t− k), k ∈ Z´IO.
d ψ½Â,
〈ψ0,m, ψ0,n〉L2
= 〈∑k∈Z
(−1)kh1−kφ(2t− 2m− k),∑l∈Z
(−1)lh1−lφ(2t− 2n− l)〉L2
=∑k,l∈Z
(−1)k+lh1−kh1−l〈φ(2t− 2m− k), φ(2t− 2n− l)〉L2
=∑k,l∈Z
(−1)k+l−2m−2nh2m+1−kh2n+1−l〈φ(2t− k), φ(2t− l)〉L2
=1
2
∑k,l∈Z
(−1)k+lh1−k+2mh1−l+2nδkl
=1
2
∑k∈Z
h1−k+2mh1−k+2n.
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Cþ k′ = 1− k + 2m,
〈ψ0,m, ψ0,n〉L2 =1
2
∑k′∈Z
hk′hk′−2m+2n
= δm−n,0
= δmn,
Ïd, ψ0,k, k ∈ Z´IO.
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(2)é?¿ k ∈ Z, ψ(t− k) ∈ W0.
â ψ½Â,´ ψ(t− k) ∈ V1. ÏdI`²
〈φ0,n, ψ0,m〉 = 0, n,m ∈ Z.
〈φ0,n, ψ0,m〉L2 =
⟨∑k∈Z
hkφ(2t− 2n− k),∑l∈Z
(−1)lh1−lφ(2t− 2m− l)
⟩L2
=∑k,l∈Z
(−1)lhkh1−l〈φ(2t− 2n− k), φ(2t− 2m− l)〉L2
=∑k,l∈Z
(−1)l−2mhk−2nh1−l+2m〈φ(2t− k), φ(2t− l)〉L2
=1
2
∑k,l∈Z
(−1)lhk−2nh1−l+2mδkl
=1
2
∑k∈Z
(−1)khk−2nh1−k+2m,
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é?¿ j ≥ 0,?ê¥1 k = n +m− j
(−1)khk−2nh1−k+2m = (−1)n+m−jhm−n−jh1−n+m+j.
1 k = n +m + j + 1
(−1)khk−2nh1−k+2m
= (−1)n+m+j+1hm+j+1−nhm−n−j
= −(−1)n+m−jhm+j+1−nhm−n−j.
ü-,l
〈φ0,n, ψ0,m〉L2 =1
2
∑k∈Z
(−1)khk−2nh1−k+2m = 0.
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(3) W0¥?¿¼ê¤ ψ(t− k)5|Ü.
Äk`²,é?¿ j ∈ Z,k
φ(2t− j) =1
2
∑k∈Z
hj−2kφ(t− k) + (−1)jh1−j+2kψ(t− k).
|^VºÝ§9 ψ½Â,þªdu
φ(2t− j) =1
2
∑k∈Z
hj−2k(∑l∈Z
hlφ(2t− 2k − l))
+(−1)jh1−j+2k(∑l∈Z
(−1)lh1−lφ(2t− 2k − l))
=1
2
∑k,l∈Z
(hlhj−2k + (−1)j+lh1−j+2kh1−l
)φ(2t− 2k − l).
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Ïd,I`² 2k + l = j ,k∑k∈Z
h1−j+2kh1−j+2k + hj−2khj−2k = 2. (∗)
2k + l = j + n, n 6= 0,k∑k∈Z
(−1)nh1−j+2kh1−j+2k−n + hj−2k+nhj−2k = 0. (∗∗)
5¿ (∗)¥?êL«∑γ∈Z
hγhγ = 2.
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n = 2m + 1,31¥- k′= m + j − k,K (∗∗)ªL«
−∑k′∈Z
hj−2k′+nhj−2k′ +∑k∈Z
hj−2k+nhj−2k = 0
n = 2m,31¥- k′= m + j − k,K (∗∗)ªL«∑
k∈Z
h1−j+2kh1−j+2k−n +∑k′∈Z
h−j+2k′h−j+2k′−n.
5¿þª¥?êL«∑γ∈Z
hγ−2mhγ = 0.
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Ùg,é?¿ f ∈ W0 ⊂ V1,k
f (t) =∑j∈Z
cjφ(2t− j)
=∑k∈Z
1
2
∑j∈Z
cjhj−2k
φ(t− k)
+
1
2
∑j∈Z
(−1)jcjh1−j+2k
ψ(t− k).
du f⊥V0,¤±é?¿ k ∈ Z,
1
2
∑j∈Z
cjhj−2k = 0.
l f (t) =∑
k∈Z
(12
∑j∈Z(−1)jcjh1−j+2k
)ψ(t− k).
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ŧªL«
éŧ
ψ(t) =∑k∈Z
gkφ(2t− k)
üঠFourierC
ψ(ω) =1
2
∑k∈Z
gke−ikω/2φ(ω/2)
= G(ω/2)φ(ω/2).
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EÅÄ
(1)E÷võ©E©Û^fmS Vj, j ∈ Z.
(2)ÀJºÝ¼ê φ,¦ φ(t− k), k ∈ Z¤ V0IOÄ.
(3)¦ÑAVºÝXê hk, k ∈ Z.
•|^VºÝ§.
•|^ H(ω) = φ(2ω)
φ(ω),¦Ñ H(ω). 2|^ H(ω) Fourier?êÐm
hk, k ∈ Z.
(4)d hk, k ∈ Z9 φEÅ
ψ(t) =∑k∈Z
(−1)kh1−kφ(2t− k).
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¦ÅªL«
(1)O H(ω).
H(ω) =φ(2ω)
φ(ω)
(2)O G(ω).
G(ω) = −e−iωH(ω + π)
(3)O ψ(ω).
ψ(ω) = G(ω/2)φ(ω/2)
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4 A;.Å
4.1. HaarÅÅÅ
Haarõ©E©Û
Vj ´«m [n/2j, (n + 1)/2j)þu~ê²È¼ê¤
¼êm. äN,
V0 = f (t) : f (t) = ck, k ≤ t < k + 1, k ∈ Z,∑k∈Z
|ck|2 < +∞;
V1 = f (t) : f (t) = dk, k/2 ≤ t < (k+1)/2, k ∈ Z,∑k∈Z
|dk|2 < +∞;
· · · · · ·
HaarºÝ¼ê
φ(t) =
1, 0 ≤ t < 1,
0, Ù§.
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HaarÅE
:dVºÝ§
φ(t) = φ(2t) + φ(2t− 1),
VºÝXê
h0 = 1, h1 = 1.
ddżê
ψ(t) = φ(2t)− φ(2t− 1).
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: O HaarºÝ¼ê FourierC
φ(ω) =1√2π
sin(ω/2)
ω/2e−iω/2,
H(ω) =φ(2ω)
φ(ω)=
1
2(1 + e−iω).
¤±
h0 = 1, h1 = 1.
ddżê
ψ(t) = φ(2t)− φ(2t− 1).
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ψ(t) =
1, 0 ≤ t < 1
2,
−1, 12 ≤ t < 1,
0, Ù§.
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4.2. ShannonÅÅÅ
Shannonõ©E©Û
é?¿ j ∈ Z,-
Vj = f ∈ L2(R) : supp(f ) ⊆ [−2jπ, 2jπ].
ShannonºÝ¼ê
φ(t) :=
1, t = 0,
sin(πt)
πt, t 6= 0.
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φ FourierC
φ(ω) =
1√2π, |ω| ≤ π
0, Ù§.
d Parsevalª,é?¿ k, l ∈ Z
〈φ(t− k), φ(t− l)〉 = 〈 φ(t− k), φ(t− l)〉 =1
2π
∫ π
−πei(l−k)ωdω = δk,l.
Ïd, φ(t− k), k ∈ Z´IO8.
d Shannonæ½n (Ω = π),é?¿ f ∈ V0,k
f (t) =
∞∑k=−∞
f (k)sin(πt− kπ)
πt− kπ=
∞∑k=−∞
f (k)φ(t− k)
Ïd, φ(t− k), k ∈ Z´ V0IOÄ.
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ShannonÅE
¦VºÝXê:
d Shannonæ½n (Ω = 2π)
φ(t) =
∞∑k=−∞
φ(k/2)sin(2πt− kπ)
2πt− kπ
=
∞∑k=−∞
sin(kπ/2)
kπ/2φ(2t− k)
= φ(2t) +∑k∈Z
2(−1)k
(2k + 1)πφ(2t− 2k − 1).
ddVºÝXê h0 = 1,
h2k = 0 (k 6= 0),
h2k+1 =2(−1)k
(2k + 1)π.
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A ShannonÅ
ψ(t) =∑k∈Z
(−1)kh1−kφ(2t− k)
= −φ(2t− 1) +∑k∈Z
2(−1)k
(2k + 1)πφ(2t+ 2k)
=sin π(t− 1/2)− sin 2π(t− 1/2)
π(t− 1/2).
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ShannonÅ´Ãg,Ï Ù1w5' HaarÅ`
õ. ShannonÅvkk|8,¿ |t| → ∞,ªu"
Ý= O( 1|t|),=ÛÜ5é,ò¼ê^ ShannonÅÄÐm
,zÑØUéмêÛÜ5, ShannonÅ^?
Ø.
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4.3. Battle-LemarieÅÅÅ
5^õ©E©Û
é?¿ j ∈ Z,- VjL«ëY3?¿?«m [k/2j, (k+1)/2j]
þ´5²È¼ê¤¼êm.
5^ºÝ¼ê
φ(t) =
t + 1, −1 ≤ t ≤ 0
1− t, 0 < t ≤ 1
0, |t| > 1.
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O φ FourierC
φ(ω) =1√2π
∫Rφ(t)e−itωdt
=1√2π
(∫ 1
0
(1− t)e−itωdt +
∫ 0
−1
(1 + t)e−itωdt
)=
1√2π
(sin(ω/2)
ω/2
)2
.
|^ª ∑k∈Z
1
(ω + 2πk)4=
3− 2 sin2(ω/2)
48 sin4(ω/2), (∗)
∑k∈Z
|φ(ω + 2kπ)|2 =8
π
∑k∈Z
sin4(ω/2)
(ω + 2πk)4
=1
6π(3− 2 sin2(ω/2)).
Ïd φ(t− k), k ∈ ZØ´IO.
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5 HaarºÝ¼ê FourierC
φ(ω) =1√2π
∫ 1
0
e−itωdt =e−iω − 1
−√
2πiω.
d φ(x− k), k ∈ ZIO59
|φ(ω)|2 =1− cosω
πω2=
1
2π
(sin(ω/2)
ω/2
)2
∑k∈Z
sin2(ω2 + πk)
(ω2 + πk)2= 1.
Ïd,k
csc2 ω
2=∑k∈Z
4
(ω + 2πk)2.
? , ∑k∈Z
1
(ω + 2πk)4=
3− 2 sin2(ω/2)
48 sin4(ω/2).
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VºÝ§
d
H(ω) =φ(2ω)
φ(ω)
= cos2(ω/2)
=
(eiω/2 + e−iω/2
2
)2
=1
4(eiω + 2 + e−iω)
h−1 =1
2, h0 = 1, h1 =
1
2.
u´,VºÝ§
φ(t) =1
2φ(2t + 1) + φ(2t) +
1
2φ(2t− 1).
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ºÝ¼ê
½Â
φ∗(ω) =φ(ω)(
2π∑
l |φ(ω + 2lπ)|2)1/2
.
u´,∑k∈Z
|φ∗(ξ + 2kπ)|2 =
∑k |φ(ξ + 2kπ)|2
2π∑
l |φ(ξ + 2lπ)|2=
1
2π.
Ïd, φ∗(t− k), k ∈ Z´IO.
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Battle-LemarieÅ
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5 Å©)-Mallat
5.1. ÄÄÄggg
b f ∈ L2(R)´?n¢S&Ò,ÿ&Ò fj ´ f 3ºÝ
m Vj ¥Cq. é?¿ Vj,k
Vj = Vj−1
⊕Wj−1
= Vj−2
⊕Wj−2
⊕Wj−1
= · · ·
= Vj0⊕
Wj0
⊕Wj0+1
⊕· · ·⊕
Wj−1.
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Ïd,ò fj õ©EL«
fj = fj−1 + wj−1
= fj−2 + wj−2 + wj−1
= · · ·
= fj0 + wj0 + wj0+1 + · · · + wj−1,
Ù¥,
fl =∑k∈Z
cl,kφl,k, l = j0, j0 + 1, · · · , j,
wl =∑k∈Z
dl,kψl,k, l = j0, j0 + 1, · · · , j − 1.
©): fj → fj0, wj0, · · · , wj−1.
: fj0, wj0, · · · , wj−1 → fj.
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5.2. ©©©)))ïïïááá
fj =∑k∈Z
cj,kφj,k ∈ Vj.
fj−1Ú wj−1©O fj 3 Vj−1ÚWj−1¥%C[!,
fj−1 =∑k∈Z
cj−1,kφj−1,k ∈ Vj−1
wj−1 =∑k∈Z
dj−1,kψj−1,k ∈ Wj−1.
=
fj =∑k∈Z
cj,kφj,k =∑k∈Z
cj−1,kφj−1,k +∑k∈Z
dj−1,kψj−1,k.
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©)ïá
cj−1,l =∑k∈Z
cj,k〈φj,k, φj−1,l〉.
dVºÝ§
φ(t) =∑n∈Z
hnφ(2t− n)
φj−1,l(t) = 2(j−1)/2φ(2j−1t− l)
= 2(j−1)/2∑n∈Z
hnφ(2jt− 2l − n)
= 2(j−1)/2∑n∈Z
hn−2lφ(2jt− n)
= 2−1/2∑n∈Z
hn−2lφj,n(t).
u´,
cj−1,l = 2−1/2∑k∈Z
cj,khk−2l.
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dj−1,l =∑k∈Z
cj,k〈φj,k, ψj−1,l〉.
dŧ
ψ(t) =∑n∈Z
gnφ(2t− n)
ψj−1,l(t) = 2(j−1)/2ψ(2j−1t− l)
= 2(j−1)/2∑n∈Z
gnφ(2jt− 2l − n)
= 2(j−1)/2∑n∈Z
gn−2lφ(2jt− n)
= 2−1/2∑n∈Z
gn−2lφj,n(t).
u´,
dj−1,l = 2−1/2∑k∈Z
cj,kgk−2l.
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ïá
cj,l =∑k∈Z
cj−1,k〈φj−1,k, φj,l〉 +∑k∈Z
dj−1,k〈ψj−1,k, φj,l〉.
d
φj−1,k(t) = 2−1/2∑n∈Z
hn−2kφj,n(t)
9
ψj−1,k(t) = 2−1/2∑n∈Z
gn−2kφj,n(t)
cj,l = 2−1/2∑k∈Z
cj−1,khl−2k + 2−1/2∑k∈Z
dj−1,kgl−2k.
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Mallat
©) cj−1,l = 2−1/2
∑k∈Z
cj,khk−2l
dj−1,l = 2−1/2∑k∈Z
cj,kgk−2l
cj,l = 2−1/2∑k∈Z
cj−1,khl−2k + 2−1/2∑k∈Z
dj−1,kgl−2k
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3¢SO¥,kk hk ". Ø h0, h1, · · · , hM−1",
Ù§ hk ". u´d gk = (−1)kh1−k, g2−M , g3−M , · · · , g0, g1
",Ù§ gk ". u´MallatXe/ª
©) cj−1,l = 2−1/2
2l+M−1∑k=2l
cj,khk−2l
dj−1,l = 2−1/22l+1∑
k=2l+2−M
cj,kgk−2l
cj,l = 2−1/2
bl/2c∑k=d(l−M+1)/2e
cj−1,khl−2k + 2−1/2
b(M+l−2)/2c∑k=d(l−1)/2e
dj−1,kgl−2k
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5.3. ©©©)))¢¢¢yyy
©)
1. Щz
Äk,â¢S&Ò f ,(½%Cm Vj. ,À fj ∈ Vj,¦
fj ´ f Vj Z%C,=
fj = Pjf =∑k∈Z
〈f, φj,k〉φj,k.
XÛOºÝXê cj,k = 〈f, φj,k〉?
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½nb Vj, j ∈ Z´dºÝ¼ê φ)¤õ©E©Û,¿ φ
äk;| . XJ f ∈ L2(R)´ëY¼ê,K j ¿©k,
cj,k ≈ mf (k/2j),
Ù¥,
m = 2−j/2∫
Rφ(x)dx.
5¢Sþ,IÀ·%Cm Vj,¦ Vj UZ/A f
«&E.ùI¦æÇ 2j u&Ò NyquistæÇ=
.
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y²Ï φäk;| ,K3M > 0,¦
suppφ ⊂ [−M,M ].
Ïd,
cj,k = 2j/2∫
Rf (t)φ(2jt− k)dt
= 2−j/2∫
Rf (2−jx + 2−jk)φ(x)dx
= 2−j/2∫ M
−Mf (2−jx + 2−jk)φ(x)dx.
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j ¿©,é?¿ x ∈ [−M,M ],k
2−jx + 2−jk ≈ 2−jk.
Ïd,d f ëY5
f (2−jx + 2−jk) ≈ f (2−jk).
l k
cj,k ≈ 2−j/2f (k/2j)
∫ M
−Mφ(x)dx
= 2−j/2f (k/2j)
∫Rφ(x)dx
= mf (k/2j).
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2. S
|^Mallat©)ò fj ©)
fj = fj0 + wj0 + wj0+1 + · · · + wj−1.
3. ª
þãSL§I?1÷v¦%CY² Vj0,ùY²À¢
S¯K ½.
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â?nºÝXê cj0,kÚÅXê dl,k, l = j0, j0+1, · · · , j−1,
|^MallatÅÚ¼ cl,k, l = j0 + 1, · · · , j − 1, j,l
?U&Ò.
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5.4. ÅÅÅ^uuu&&&ÒÒÒ???nnnÄÄÄÚÚÚ½½½
•æ
•Å©)
•Xê?n
•Å
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A^Þ~
1. Å3&ÒÈÅ¥A^
Ä [0, 1]þ&Ò
f (t) = sin(8πt) + sin(12πt) + sin(58πt).
é&Ò f ?1ÈÅ?n,l f ¥ÈK 29Hz¤©,=¦
g(t) = sin(8πt) + sin(12πt).
f g
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À^ DaubechiesÅ (N=2) ,¿Àæm 2−8s,æ:ê
256.
V7 W7
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V6 W6
V5 W5
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Daubechies Haar
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2. Å3&ÒØ ¥A^
Ä [0, 1]þ&Ò
f (t) = sin(2πt) + sin(4πt) + sin(10πt).
©&Ò
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À^ DaubechiesÅ (N=2),¿Àæm 2−8s,æ:ê
256.ò f8©)
f8 = f4 + w4 + w5 + w6 + w7.
ò c4,k, d4,k, d5,k, d6,k, d7,k¥ýéu 0.2êâ
",KØ Ç 17.97%.
Ø &Ò
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5.5. ÅÅÅÈÈÈÅÅÅììì|||
©) cj−1,l = 2−1/2
∑k∈Z
cj,khk−2l
dj−1,l = 2−1/2∑k∈Z
cj,kgk−2l
Ú\eæf: XJ x = (· · · , x−2, x−1, x0, x1, x2, · · · ),K
Dx = (· · · , x−2, x0, x2, · · · )
=
(Dx)l = x2l, l ∈ Z.
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©)òÈLª cj−1 = D(cj ∗ h∗)
dj−1 = D(cj ∗ g∗)
Ù¥,
cj−1 = cj−1,kk∈Z, dj−1 = dj−1,kk∈Z, c
j = cj,kk∈Z,
h∗ = 2−1/2h−kk∈Z, g∗ = 2−1/2g−kk∈Z.
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cj,l = 2−1/2∑k∈Z
cj−1,khl−2k + 2−1/2∑k∈Z
dj−1,kgl−2k
Ú\þæf: XJ x = (· · ·x−2, x−1, x0, x1, x2, · · · ),K
Ux = (· · ·x−2, 0, x−1, x0, 0, x1, 0, x2, 0, · · · )
=
(Ux)k =
0, kÛ,
xk/2, kó.
òÈLª
cj = (Ucj−1) ∗ h + (Udj−1) ∗ g.
Ù¥
h = 2−1/2hkk∈Z, g = 2−1/2gkk∈Z.
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Å©)ÏÈÅì|L«
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6 ºÝ¼êE
b φ ∈ L2(R),é?¿ j ∈ Z,- Vj = spanφ(2jt− k), k ∈ Z.
½nb φ´äk;|8ëY¼ê,¿÷vIO5
^: ∫Rφ(t− k)φ(t− l)dt = δkl, k, l ∈ Z.
Kk ∩j∈ZVj = 0.
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y² d^, φ(t − k), k ∈ Z ´ V0 IOÄ. XJ
f ∈ V0,Kk
f (t) =∑k∈Z
〈f (·), φ(· − k)〉φ(t− k)
=∑k∈Z
(∫Rf (x)φ(x− k)dx
)φ(t− k)
=
∫R
(∑k∈Z
φ(t− k)φ(x− k)
)f (x)dx
=
∫Rk(t, x)f (x)dx,
Ù¥ k(t, x) =∑k∈Z
φ(t− k)φ(x− k).
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|^ Cauchy-Schwartzت
|f (t)| ≤(∫
R|k(t, x)|2dx
)12(∫
R|f (x)|2dx
)12
=
(∫R|k(t, x)|2dx
)12
‖f‖L2.
qdIO5^∫R|k(t, x)|2dx
=
∫R
(∑k∈Z
φ(t− k)φ(x− k)
)(∑l∈Z
φ(t− l)φ(x− l)
)dx
=∑k,l∈Z
φ(t− k)φ(t− l)
(∫Rφ(x− l)φ(x− k)dx
)=∑k∈Z
|φ(t− k)|2.
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u´k
|f (t)| ≤
(∑k∈Z
|φ(t− k)|2)1
2
‖f‖L2.
du φ´äk;|8ëY¼ê,¤±þª¥¦Úkk
,? 3~ê C ¦
maxt∈R
|f (t)| ≤ C‖f‖L2.
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b f ∈ ∩j∈ZVj. Ké?¿ê j,k f ∈ V−j,Ï f (2jt) ∈
V0,¿
|f (2jt)| ≤ C
(∫R|f (2jx)|2dx
)12
= C2−j2
(∫R|f (t)|2dt
)12
, t ∈ R.
l k
maxt∈R
|f (t)| ≤ C2−j2‖f‖L2.
duþªé¤kê j Ѥá,- j → +∞, f = 0. Ïd
k ∩j∈ZVj = 0.
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½nb φ´äk;|8ëY¼ê,¿÷vXe^:
(1). IO5^∫
Rφ(t− k)φ(t− l)dt = δkl, k, l ∈ Z;
(2). IOz^∫
Rφ(t)dt = 1;
(3). VºÝ§ φ(t) =∑k
hkφ(2t− k),kk hk ".
K Vj ¤õ©E©Û.
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y²Iy²È5,= ∪j∈ZVj = L2(R).
- Pj ´ L2(R) Vj ÝKf. y²È5duy²é
?¿ f ∈ L2(R),k
Pjf → f, j → +∞.
qÏ
‖f‖2L2 = ‖f − Pjf‖2
L2 + ‖Pjf‖2L2,
¤±Iy²é?¿ f ∈ L2(R),k
‖Pjf‖L2 → ‖f‖L2, j → +∞.
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1Ú,éA¼ê?1y².
u(t) =
1, a ≤ t ≤ b,
0, Ù¦.
Ù¥ a, b÷v a < b?¿½~ê. d
(Pju)(t) =∑k∈Z
〈u, φj,k〉φj,k(t)
=∑k∈Z
(∫ b
a
φj,k(x)dx
)φj,k(t),
‖Pju‖2L2 =
∑k∈Z
∣∣ ∫ b
a
φj,k(x)dx∣∣2
= 2−j∑k∈Z
∣∣ ∫ 2jb
2ja
φ(t− k)dt∣∣2.
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j ¿©,þªmàÈ©«m [2ja, 2jb]´é,é5`,
φ|8é. ¦Ú¥È©©¤na:
(1) φ(t− k)|8 uÈ©«m,Ï È©".
(2) φ(t− k)|8È©«mà:,ùÜ©ê,
Ñ.
(3) φ(t − k)|8 uÈ©«mS,KdIOz^ÙÈ
©1. ùÜ©ê´ 2j(b− a),k
‖Pju‖2L2 ≈ 2−j2j(b− a) = b− a = ‖u‖L2.
þª¥Ø´d1aÈ©E¤, j C,Ø5.
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1Ú,éF¼ê?1y².
s(t) =∑k
αkuk(t),
Ù¥kk αk ". duk
‖Pjs− s‖L2 = ‖∑k
αk(Pjuk − uk)‖L2 ≤∑k
|αk|‖Pjuk − uk‖L2,
â ‖Pjuk − uk‖L2 → 0,9k αk ", ‖Pjs− s‖L2 → 0.
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1nÚ:é?¿ f ∈ L2(R)?1y². du?¿ f ∈ L2(R)±d
F¼ê%C,¤±é?¿ ε > 0,3F¼ê s,¦
‖f − s‖L2 <ε
3.
d1Ú, j ¿©,k
‖Pjs− s‖ < ε
3,
l k
‖f − Pjf‖ = ‖f − s + s− Pjs + Pjs− Pjf‖
≤ ‖f − s‖ + ‖s− Pjs‖ + ‖Pjs− Pjf‖
≤ ‖f − s‖ + ‖s− Pjs‖ + ‖s− f‖
< ε.
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EºÝ¼êS
½nbP (z) =1
2
∑k
hkzk ´õª,÷ve^:
• P (1) = 1;
• |P (z)|2 + |P (−z)|2 = 1, |z| = 1;
• |P (eit)| > 0, |t| ≤ π/2.
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- φ0´ HaarºÝ¼ê,é?¿ n ∈ N,
φn(t) =∑k
pkφn−1(2t− k).
K¼ê φn3 L2¥Å:Âñ¼ê φ,¿ φ÷v
•IO5^∫
R φ(t− k)φ(t− l)dt = δkl;
•IOz^:∫
Rφ(t)dt = 1;
•VºÝ§: φ(t) =∑k
hkφ(2t− k).
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y²3Sª
φn(t) =∑k
hkφn−1(2t− k)
¥4 n→∞,
φ(t) =∑k
hkφ(2t− k),
d=L² φ÷vVºÝ§.
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y² φ÷vIO5^ÚIOz^,ÄkÄ
φ1(t) =∑k
hkφ0(2t− k).
3þªüà FourierC,
φ1(ω) = P (e−iω/2)φ0(ω/2).
du φ0(0) =1√2π
,P (1) = 1,Ïd
φ1(0) =1√2π,
= φ1÷vIOz^.
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|^ φ1(ω) = P (e−iω/2)φ0(ω/2)∑k∈Z
|φ1(ω + 2πk)|2
=∑k∈Z
|P (e−iω/2+πki)|2|φ0(ω/2 + πk)|2
=∑l∈Z
|P (e−iω/2+2πli)|2|φ0(ω/2 + 2πl)|2
+∑l∈Z
|P (e−iω/2+π(2l+1)i)|2|φ0(ω/2 + π(2l + 1))|2
= |P (e−iω/2)|2∑l∈Z
|φ0(ω/2 + 2πl)|2
+|P (−e−iω/2)|2∑l∈Z
|φ0(ω/2 + π + 2πl)|2.
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d φ0÷vIO5^9éõª P b,∑k∈Z
|φ1(ω + 2πk)|2
=1
2π(|P (e−iω/2)|2 + |P (−e−iω/2)|2)
=1
2π,
= φ1÷vIO5^.
|^þã48y²¤k φn Ñ÷vIO5^ÚIO
z^.l ÏL4 φ÷vùü^.
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~ (Daubechies)
P (z) =1
2
∑k
hkzk,
Ù¥
h0 =1 +
√3
4, h1 =
3 +√
3
4,
h2 =3−
√3
4, h3 =
1−√
3
4.
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