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1 JPEG u.ac.jp u.ac.jp/member/yoshi/ (Computer Graphics: CG) (Virtual/Augmented(Mixed) Reality: VR AR MR)

2 (Computer Graphics: CG) (Virtual/Augmented(Mixed) Reality: VR AR MR) 2

3 e 2 x' x 2 x e' x' x' 2 T = x 2 e' 2 e 0 e =, e 2 = x'=tx x=t T x' x = x' - 0 e' =, e' 2 = 2 2-3

4 e k e' k e k e' k k 0 k k k k 0 e k,k 2 e'k,k 2 k 0 k

5 e k,k 2 e' k,k 2 k k 2 0 k e (DCT: Discrete Cosine Transform) π cos i N + k 2 α k π cos i N + 2 ' k, (, 2 ) = k i i α 2 k 2 2 k 2 for k = 0 e' k,k k α = N 2 k 2 k 2 0 for k 0 N N = 8 i 2 i 2 3 i 2 N N 4 e' k (, ) ' (, ) 0, k i i2 e 2 k', k' i i2 = 2 5 i= 0 i2= (for ( k, k2) ( k', k' 2 ) ) 7 i i i 2 5

6 e k,k 2 e k,k 2 e k,k 2 6

7 iux F( u) = f ( x) e dx = f ( x){cos( ux) isin( ux)} dx f(x) e iux = cos(ux) i sin(ux) u F(u) F(u) (Re) cos (Im) sin x F(u) = F*( u) * 7

8 f(x) F(u) f(x) c n x n c n i k x=x i i 2 c=c k k 2 i 2 e 2 x 2 k 2 c x e' c 2 x c e' 2 e e' k,k 2 c = Tx x = T T c x = c 8

9 f(x) F(u) F c ( u) = f ( x)cos( ux) dx 0 [0, ] f(x) f'(x) f'(x) = f'( x) /2 iux Fc ( u) = f ( x)cos( ux) dx = f '( x) e dx 0 2 F c (u) (Re) F c (u) = F( u) 9

10 (DFT) (DCT) i 2 i i i 2 (DFT) (DCT) i (Re) (Im) i 2 i i 2 0

11 (DFT) (DCT) i (Re) (Im) i 2 i i 2 (DFT: Discrete Fourier Transform)

12 (DCT: Discrete Cosine Transform) (DFT: Discrete Fourier Transform) 2

13 (DCT: Discrete Cosine Transform) (DFT) (DCT) DFT DCT 3

14 .. 4

15 (DCT) DCT.. 5

16 6

17 7

18 .. (DCT) 8 8 x DCT x' (x, e' 00 ) DCT k (Re) DFT (Im) e' 00 k 2 e' k,k 2 8

19 (DCT) 8 8 x DCT x' (x, e' 0 ) DCT k (Re) DFT (Im) e' 0 k 2 e' k,k 2 (DCT) 8 8 x DCT x' (x, e' 20 ) DCT k (Re) DFT (Im) e' 20 k 2 e' k,k 2 9

20 (DCT) 8 8 x DCT x' (x, e' 20 ) DCT k (Re) DFT (Im) e' 20 k 2 e' k,k 2 (DCT) 8 8 x DCT x' (x, e' 20 ) DCT k (Re) DFT (Im) e' 20 k 2 e' k,k 2 20

21 (DCT) 8 8 x DCT x' (x, e' 20 ) DCT k (Re) DFT (Im) e' 20 k 2 e' k,k 2.. 2

22 8 8 DCT 8 8 DCT 22

23 k k 2 k 2 k [JPEG Standard Annex K] = 26 = 0 77 DCT [JPEG Standard Annex K] 23

24 DCT DCT DCT DCT DCT DCT 24

25 .. DCT c k,k 2 k k 2 DCT c 00,c 0, c 0,c 02, c,c 20, c 30,c 2, c 2,c 03, c 04,c 3,c 22,c 3,c 40,c 50,.., c 75,c 76, c 67,c 77 25

26 DCT k k 2 DCT 26,0,,0,,-,-,,,0,0,0,......,0, 0, 0, 0 DCT k k 2 DCT 49,8,8,6,4,3,-,0,-,-,,-2,-3,-,,0,0,-,0, 0,0,...,0, 0, 0, 0 26

27 = = 5 26,32,27,. (26),6,-5,.. 27

28 92 50 = = 43 50,-23,-9,-,0,0,,2,2,0,0,,,,0,0,0,0,0,0,0,0,0,0,0,-,0,0,0,0,0,-,0,0,0,0,0,0,0,0,,0,0,0,.0,0,0 27,-2,0,0,-,,2,-2,,0,0,0,-,0,,0,0,-,,0,0,0,0,0,-,0,0,0..0,0,

29 0 0 0 EOB 50, -23,-9,-,0,0,,2,2,0,0,,,,0,0,0,0,0,0,0,0,0,0,0,-, 0,0,0,0, 0,-, 0,0,0,0,0,0,0,0,,0,0,0,.0,0,0 (0,-23) (0,-9) (0,-) (2,) (0,2) (0,2) (2,) (0,) (0,) (,-) (5,-) (8,) EOB (Run, Coef) 0 Run 0 Coef.. 29

30 0/ 0/ A 0.2 B C 0.3 D 0.75 E

31 (DC) 0 size Kingsbury DC Coef Difference Size Typical Huffman codes for Size Additional Bits (in binary) , Size Size 00 0, 3, 2,2, ,0,0, 7,, 4,4,, ,,0,00, 5, 8,8,, ,,0,000,, 023, 52,52,, , 024,024, ,, bit ,, (DC) 0: Size 0 (Code 00), Additional Bits Huffman Code: 00 : Size (00), Additional Bits 0 Huffman Code: : Size 3 (00), Additional Bits 00 Huffman Code : Size 4 (0), Additional Bits 0000 Huffman Code Kingsbury DC Coef Difference Size Typical Huffman codes for Size Additional Bits (in binary) , Size Size 00 0, 3, 2,2, ,0,0, 7,, 4,4,, ,,0,00, 5, 8,8,, ,,0,000,, 023, 52,52,, , 024,024, ,, bit ,, 3

32 (DC) 6.42 bits 6.07 bits Kingsbury (AC) (Run, Coef) (Run, Coef) 0 Run Coef (0, 23) (0, 9) (0, ) (2,) (0,2) (0,2) (2,) (0,) (0,) (, ) (5, ) (8,) EOB (Run, Coef) (Run, Coef Size) Coef Size DC (Run, Coef size) Kingsbury (Run,Size) Code Byte (hex) Code Word (binary) (Run,Size) Code Byte (hex) Code Word (binary) (0,) 0 00 (0,6) (0,2) (Run, 02 Size) 0 (,3) 3 00 (0,3) (5,) 5 00 (EOB) (6,) 6 0 (0,4) 04 0 (0,7) (,) 00 (2,2) (0,5) (7,) 7 00 (,2) 2 0 (,4) 4 00 (2,) 2 00 (3,) 3 00 (ZRL) F0 00 (4,)

33 (AC) (Run, Coef)= (0, 7) : (Run Size)=(0,3) (Run, Coef)= (,3): (Run, Size) = (,2) + 0 Coef size (Run, Coef size) (Run, Size) (Run, Coef size) bit AC Coef Size Additional Bits (in binary) 0 0, 0, 3, 2,2,3 2 00,0,0, 7,, 4,4,, ,,0,00, (Run,Size) Code Word (binary) (0,) 00 (0,2) 0 (0,3) 00 (EOB) 00 (0,4) 0 (,) 00 (0,5) 00 (,2) 0 (2,) 00 (Run Size)=(0,) (Run Size)=(0,2) (Run Size)=(,) Kingsbury 33

34 8 8 DCT (DCT) DCT DCT (0,-23) (0,-9) (0,-) (2,) (0,2) (0,2) (2,) (0,) (0,) (,-) (5,-) (8,) EOB u.ac.jp/~yizawa/infsys/basic/index.htm u.ac.jp/~yizawa/infsys/advanced/index.htm u.ac.jp/~yizawa/infsys/ref u.ac.jp/ yizawa/infsys/ref_contents/index.htm u.ac.jp/~asano/kougi/0a/tokuron/ Prof. Bernd Girod Image Communication I JPEG Entropy Coding and 2D DCT by Nick Kingsbury Wikipedia JPEG, 34

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