![Page 1: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/1.jpg)
Aproximácia DCT v JPEG
JPEG:•digitálny obraz•bloky 8x8•DCT po blokoch•kompresia + kódovanie•prenos•dekódovanie + dekompresia•inverzná DCT po blokoch
![Page 2: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/2.jpg)
Aproximácia DCT v JPEG
Bázové funkcie DCT 8x8
![Page 3: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/3.jpg)
Aproximácia DCT v JPEG
α α α( ) cos( ) ( ) ( )i i i= − − −2 1 2Ω
generátor jedneho riadku DCT:
pričom: Ω = 2πf fvz/
potom:
αα
( )( ) cos( )01
==
AA Ω
konst.
α ( ) , cos( ), cos( ), cos( ), ...i A A A A= Ω Ω Ω2 3
![Page 4: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/4.jpg)
Aproximácia DCT v JPEG
nahradenie násobenia: cos( )Ω = −2 s
pričom: s = 1 2 3, , , ...
Príklad 1: s =1 cos( ) .Ω = 0 5 Ω = π / 4
N f fvz= =
=/arccos
212
6π
vzoriek na jednu periódu
![Page 5: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/5.jpg)
Aproximácia DCT v JPEG
Ω ≅ 1318116.
N f fvz= =
≅/arccos
.2
14
4 766792π
vzoriek na jednu periódu
s = 2 cos( ) .Ω = 0 25
α α α( ) ( ) ( )i i i= − − −12
1 2
Príklad 2:
![Page 6: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/6.jpg)
Aproximácia DCT v JPEG
α(i)
0 5 10 15 20-1
-0.5
0
0.5
1
0 5 10 15 20-1
-0.5
0
0.5
1
x(n)
i
n
![Page 7: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/7.jpg)
Aproximácia DCT v JPEG
Y TXTt=~ ~X T YTt=
12K
Y TXTt=1
2K~ ~X T YTt=
Y TXTt=1K
~ ~X T YTt=1K
Výpočet dvojrozmernej DCT:• obrazový blok, blok koeficientov• rekonštruovaný blok, blok koeficientov po kvantizácii• K kvantizačná konštanta, T jadro DCT
~X ~YX Y
![Page 8: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/8.jpg)
Aproximácia DCT v JPEG
t i jnn
i jn
( , )( / )( / ) cos
( )/
/= +
12
2 12
1 2
1 2 πi j n= ≤ ≤ −0 0 1,
1 1 0 1≤ ≤ − ≤ ≤ −i n j n,
T =
− − − −− − − −
− − − −− − − −− − − −− − − −− − − −
⌡⌡⌡⌡⌡⌡⌡⌡⌡⌡⌡
h h h h h h h ha b c d d c b ae f f e e f f eb d a c c a d bg g g g g g g gc a d b b d a cf e e f f e e fd c b a a b c d
a = 0 980785.b = 0 831469.c = 0 555570.d = 0 195090.e = 0 923879.f = 0 382683.g = 0 707107.h = 0 707107.
![Page 9: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/9.jpg)
Aproximácia DCT v JPEG
y k l t k i x i j t l jji
( , ) ( , ) ( , ) ( , )===⌠⌠
14 0
7
0
7
~( , ) ( , )~( , ) ( , )x i j t k i y k l t l jlk
===⌠⌠
14 0
7
0
7
DCT:
IDCT:
![Page 10: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/10.jpg)
Aproximácia DCT v JPEG
Dvojkroková aproximácia:
α ( )0 1=α ( ) .1 0 25=α ( ) .2 0 875= −α ( ) .3 0 6875= −α ( ) .4 0 53125=α (5) .= 0 953125α ( ) .6 0 054688= −α ( ) .7 0 980469= −
f = 0 382683.
f ≅ =α ( ) .1 0 25
f ≅ − − =α α( ) ( ) .2 4 0 34375
f ≅ − − + =α α α( ) ( ) ( ) .1 3 6 0 382812
1 vzorkou:
2 vzorkami:
3 vzorkami:
![Page 11: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/11.jpg)
Aproximácia DCT v JPEG
α ( )0 = x ( , , , , , , , )xa xb xc xd xe xf xg xh
α ( )0 1= ( , , , , , , , )a b c d e f g h
xaa xab xac xad xae xaf xag xahxba xbb xbc xbd xbe xbf xbg xbhxca xcb xcc xcd xce xcf xcg xchxda xdb xdc xdd xde xdf xdg xdhxea xeb xec xed xee xef xeg xehxfa xfb xfc xfd xfe xff xfg xfhxga xgb xgc xgd xge xgf xgg xghxha xhb xhc xhd xhe xhf xhg xhh
![Page 12: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/12.jpg)
Aproximácia DCT v JPEG
• akumulácia chýb dvojnásobnou aproximáciou
• samostatný výpočet zrkadlových párov xpq a xqp
Nevýhody dvojkrokovej aproximácie:
Jednokroková aproximácia
![Page 13: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/13.jpg)
Aproximácia DCT v JPEG
aa ab ac ad ae af ag ahba bb bc bd be bf bg bhca cb cc cd ce cf cg chda db dc dd de df dg dhea eb ec ed ee ef eg ehfa fb fc fd fe ff fg fhga gb gc gd ge gf gg ghha hb hc hd he hf hg hh
![Page 14: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/14.jpg)
Aproximácia DCT v JPEGPríklady:
• BABOON• CMAN• LENA• ZEBRA
RMSE x xi j i jji
= −==∑∑
1256
2
0
255
0
255
( ~ ), ,
Spôsob hodnotenia:
![Page 15: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/15.jpg)
Aproximácia DCT v JPEGBABOON
13.5
14
14.5
15
15.5
16
0 50 100
Number of samples
RM
SE
CMAN
8
9
10
11
12
13
0 50 100
Number of samples
RM
SE
LENA
7
8
9
10
11
12
13
0 50 100
Number of samples
RM
SE
ZEBRA
11.5
12
12.5
13
13.5
14
14.5
15
0 50 100
Number of samples
RM
SE
dvojkroková
jednokroková
DCT
Aproximácia1 koeficientomgenerátora
![Page 16: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/16.jpg)
Aproximácia DCT v JPEG
dvojkroková
jednokroková
DCT
Aproximácia2 koeficientamigenerátora
BABOON
14
14.25
14.5
14.75
15
0 10 20 30Number of samples
RM
SE
CMAN
8,5
8,75
9
9,25
9,5
9,75
10
0 10 20 30Number of samples
RM
SE
LENA
7
7.5
8
8.5
9
0 10 20 30Number of samples
RM
SE
ZEBRA
11.5
1212.5
1313.5
1414.5
15
0 10 20 30
Number of samplesR
MSE
![Page 17: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/17.jpg)
Aproximácia DCT v JPEG
dvojkroková
jednokroková
DCT
Aproximácia3 koeficientamigenerátora
BABOON
13,75
14,25
14,75
0 5 10 15 20Number of samples
RM
SE
CMAN
8,25
8,75
9,25
9,75
0 5 10 15 20Number of samples
RM
SE
LENA
7
7,5
8
8,5
9
0 10 20Number of samples
RM
SE
ZEBRA
11,5
12
12,5
13
0 5 10 15 20Number of samples
RM
SE
![Page 18: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/18.jpg)
Aproximácia DCT v JPEG
Prečo nie sú priebehy klesajúce?RMSE vs. súčet chýb
10 vzoriek chyba (~)t t− 15 vzoriek chyba (
~)t t−
a 0.980469 0.000316 0.980469 0.000316b 0.875 -0.043531 0.816895 0.014574c 0.53125 0.02432 0.53125 0.02432d 0.25 -0.05491 0.142761 0.052329e 0.953125 -0.29246 0.922638 0.001241g 0.435547 -0.052864 0.354248 0.028435f 0.6875 0.019606 0.6875 0.019606h 0.6875 0.019606 0.6875 0.019606
Suma -0.116703 Suma 0.160428
Aproximácia1 koeficientomgenerátora
eliminácia chýb
![Page 19: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/19.jpg)
Príklad ZEBRA, typ 1, jednokroková
Originál ZEBRA DCT
ECT 40 vzoriek ECT 5 vzoriekECT 3 vzorky
![Page 20: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/20.jpg)
Príklad LENA, typ 2, dvojkroková
Original LENA 256x256 DCT
ECT 2 vzorky ECT 20 vzoriek
![Page 21: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/21.jpg)
Aproximácia DCT v JPEG
DCT 7.4283
N=2 14.7976N=5 10.2274N=8 7.9551N=10 7.4027N=12 7.4025N=15 7.3825N=20 7.3780
DCT 11.8072
N=2 38.7521N=3 37.2224N=5 13.9402N=10 12.6671N=20 12.6001N=30 12.0134N=40 11.6145N=50 11.8322N=70 11.9923
ZEBRA, typ 1 LENA, typ 2RMSE
![Page 22: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/22.jpg)
Aproximácia ICT v JPEGICT- Integer Cosine Transform
• nahrádza aritmetiku reálnych čísel celočíselnou aritmetikou• znižuje výpočtovú náročnosť• MICT (Modified Integer Cosine Transform) znižuje
počet koeficientov DCT jadra
![Page 23: Aproximácia DCT v JPEG - stuba.sk fileAproximácia DCT v JPEG α 0 = x ( , ,, ,, , , ) xa xb xc xd xe xf xg xh α 01 = (,,, ,, , , ) abcde f gh xaa xab xac xad xae xaf xag xah xba](https://reader031.vdocuments.mx/reader031/viewer/2022021611/5cdec2cb88c993680f8da9a6/html5/thumbnails/23.jpg)
Aproximácia ICT v JPEG
Solution no:
a b c d e f g h K
01 2 0 0 0 1 1 1 1 8 02 28 8 6 4 15 15 15 15 1800 03 164 42 40 4 87 87 87 87 60552 04 24 20 12 6 17 17 17 17 2312 05 56 16 12 8 30 30 30 30 7200 06 72 30 24 8 41 41 41 41 13448 07 84 24 18 12 45 45 45 45 16200 14 140 40 30 20 75 75 75 75 45000 15 180 130 104 20 123 123 123 123 121032 16 88 40 24 22 51 51 51 51 20808 29 176 80 48 44 102 102 102 102 83232 30 192 160 96 48 136 136 136 136 147968 31 216 180 108 54 153 153 153 153 187272 32 240 200 120 60 170 170 170 170 231200