CSE 2340 Lectures

1
? 8 ? ??????
2
???????

• ????
• ??????
• ????
• ????????

3
????
4
?????
(A1 A2) (B1 B2) (C1 C2) (D1 D2) (E1 E2)
(o1 o2 a) (A1 0) (B1 0) (C1 0) (D1 0) (E1
0)
5
???????

• ????????(??)??????,???????????(?????????????)?????
  ????,?????,?????????????????????????,??????????
• ? ?????X, ?????T,
• ? Y T X, Y????????.
• ?????????????,????DCT?K-L??

6
????

• ? ?????X, ?????T, ????.
• ? Y T X, Y????????.
• ? ????T??
• T T I, ? T T-1,
• ? ??T??????.
• ?Y???????X????
• X T Y

7
??????
8
???????
9
??????(DCT)
(??, ?u0?, C(u)v2 / 2, ??, C(u) 1)
( 1 1 1 1 1 1 1 1 ) (f (0) f (1) f (7))
F(0)
(
)
(f (0) f (1) f (7) )
F(1)
(
)
(f (0) f (1) f (7))
F(2)

10
DCT Basis Functions(1)
11
DCT Basis Functions(2)
12
?????????

• DCT???????????,??F(0)?????,???????.
  F(0)??????????????
• DCT?????????
• DCT???????,???

13
DCT??1
14
DCT??2
15
???????(IDCT)
(??, ?u0?, C(u)v2 / 2, ??, C(u) 1)
16
2D??????(DCT)
f0,0 f0,7 f7,0 f7,7
? f(i, j) ?64????????,
?DCT???
IDCT???
(??, ?u0,v0 ?, C(u)C(v)v2 / 2, ??, C(u)
C(v)1)
17
(No Transcript)
18
????F(u,v)???

• Fu,v??DCT??,??F0,0?????,???????.
  F0,0??????????????
• (uv)???????????,(uv)???????????.
• ?????????????,?????0.
• 8x8????????.

19
DCT????
????? F(u,v)
20
DCT?DFT???
DFT
DCT
21
??
22
DCT???

• ???????????????????,??????????????,???????????
• ?????????????DCT?????,?? ???????,?????????????????
  ???,???????
• ??????????,??????????????????????????,??????????,?
  ????????????

23
????
24
?????

• ???????????????????????????????,?????????????????
  ???????

25
???

• 1807 Joseph Fourier????????????????????,????????
  ?
• 1910 Alfred Haar??Haar??
• 1945 Gabor??STFT(short time Fourier transform)
• 1980 Morlet???CWT (continuous wavelet transform)

26
???(?)

• 1986?,?????Y.Meyer???????????????????,????(dilatio
  ns)???(translations)?? 2?j????????????????L2(R)???
  ???,??????????
• 1988?,S.Mallat??????????????????(multiresolution
  analysis)???, ???????????????????????????????????
  ??,??Mallat?????????????????????????,?????????????
  ??????????????

27
?????
28
???????

• ?????
• ????

29
Example of 1D Haar DWT

• xn,i10,13,25,26,29,21,7,15

xn-1,i, dn-1,i11.5, 25.5, 25, 11, -1.5, -0.5,
4, -4
xn-2,i , dn-2,i, dn-1,i18.5,18,-7,7,-1.5,-0.5,
4,-4
xn-3,i , dn-3,i, dn-2,i, dn-1,i18.25,0.25,-7,7
,-1.5,-0.5,4,-4
30
1D DWT

• ??1?DWT ???????????????,(????????)??????21??????

31
??(5,3)???????
????? h0(n)(-1 2 6 2 -1)/8 h0(n)(1 2 1)/2
????? h1(n)(-1 2 -1)/2 h1(n)(-1 -2 6 2 -1)/8
n Low-pass, h0 (n) h0 (n)
n High-pass, h1 (n) n h1 (n)
0 3/4 1
-1 1 1 3/4 1
1/4 1/2 -2, 0
-1/2 0, 2 -1/4 2
-1/8
-1, 3 -1/8
32
(5,3) ????????(????)
h0(n)(-1 2 6 2 -1)/8
??????? 0 0 -1/8 ¼ ¾ ¼ -¼ ¼ ¾ ¼ -1/8
0 0 0
h1(n)(-1 2 -1)/2
??????? 0 0 0 0 -½ 1 -½ 0
-½ 1 -½ 0 0 0
33
1D DWT?????

• ????(????)
• ??????????(g0, g1) ???????????
• ?????????

• ????????????????
• (h0, h1) ??????????????????????????,?????????? (
  perfect reconstruction,PR)???

34
(5,3) ????????(????)
h0(n)(1 2 1)/2
?? 0 0 0 ¼ 0 ¼ 0
¼ 0 ¼ 0 0 0 0
??????? 0 0 1/8 ¼ ¼ ¼ ¼ ¼ ¼ ¼ 1/8
0 0 0
h1(n)(-1 -2 6 -2 -1)/8
?? 0 0 0 0 0
1 0 0 0 1 0 0 0 0
??????? 0 0 -1/8 -¼ 3/4 -¼ -¼ -¼ 3/4
-¼ -1/8 0 0 0
35
??Daubechies(9,7)???????
n Low-pass, h0 (n)
Low-pass, h0 (n) 0
0.602949018236360
1.115087052457000 1 0.266864118442875
0.591271763114250 2 -0.078223266528990
-0.057543526228500 3 -0.016864118442875
-0.091271763114250 4 0.0267487574108
10 n High-pass, h1 (n)
n High-pass, h1 (n)
-1 1.115087052457000 1 0.60294901823
6360 -2, 0 -0.591271763114250 0,
2 -0.266864118442875 -3, 1
-0.057543526228500 -1, 3 -0.078223266528990
-4, 2 0.091271763114250 -2,
4 0.016864118442875 -3,
5 0.026748757410810
36
??1D????
37
???????2D DWT
38
?????????????????

• ??????HL?LH????????,HL????????????????????????,?LH
  ??????????????????????

39
?????DWT

• ?????????1-D DWT, ???
• ???????????1-D DWT,???

??4???(??)
40
Lena ?????????
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
?????????
45
??DWT ???

• ??????(multi-resolution)???????.
• ????????????,??????????????,?????????????.
• ??DWT??????,?????????????????????
• ???????????????,???????????,???????????????????,??
  ??????

46
????????
47
??

• 1993?Shapiro??
• Embedded Zerotree Wavelet(EZW)
• 1996,Said and Pearlman, SPIHT
• 2000,Taubman,EBCOT,?Jpeg2000??
• ??? MZW
• ?????
• ???????????
• ??????,????????
• ???????????

48
????

• ????????
• ????
• ????????4?
• ???t
• ???z
• ????p
• ????n

49
??????
50
??????(SAQ)

• ?????????
• ???????????,????????????????????????
• ??????????,??????????????
• ???????????
• ????,?????Ti,????????????,??????,????????,????????
  ?
• ????,?????????????,?????Ti,3Ti/2,??0??,???1??

51
EZW??
52
EZW??