數(shù)字圖像處理-第08章變換

第8圖像1二維離 變換二維連 變二維連續(xù)函數(shù)f(x,y) f(x,y)xy的函數(shù)，且在上絕對(duì)可積，則定義積分F(u,v)f(x,y)ej2(uxvy) 為二維連續(xù)函數(shù)f(x,y) f(x,y)F(u,v)ej2(uxvy)

為F(u,v)的反變換。f(x,y)和F(u,v) 變換對(duì)f(x,y)

xX

y【例】求函 XY解：將函數(shù)代入到(8.1)XY

其他F(u,v)f(,

j2(uv

dd

ej

ej2vyAXYsin(uX)ejuxsin(vY)e F(u,v)

二維信號(hào)的圖形表圖8.1二維信號(hào)f(x（a）信號(hào)的頻譜 （b）圖（a）的灰度圖8.2二維離 變尺寸為M×N的離散圖像函數(shù)的F(u,v) Mf(x,y)ej2(ux/Mvy/N) MNx0M1Nf(x,y) F(u,v)ej2(ux/Mvy/Nu0

F(u,v)R(u,v)jI(u,v)(u,v)arctan[I(u,v)R(u,v)

F(u,v

(u,v)

12(u,

(u,v

arctan[I(u,v)R(u,v)DFT（2）幅度譜|F(u,v)|1.3二維離 變換的性a1f1(x,y)a2f2(x,y)11(u,v)a2F2(u,f(ax,by)

Fx,y F(u,v)Ff(x,y)FFf(x f(x,y)F1F(u,v)F1F(u, 9空間位移 f(xx,yy)F(u,v)ej2(ux0vy0)/ 頻率位移f(x,y)ej2(u0xv0y)/

F(uu0,vv0f(x,y)(1)xyF(uN,vN F(u,v)=F(u+aN,v+bN),

f*(x,y)F*(u,v)

f(r,0)F(,0N1NF(0,0) f(x,y)f(x,N x0f(x,y)*h(x,y)<=>f(x,y)h(x,y)<=> F(u)

0xXf(x)0 f(x)ej2uxdxAXej2uxdx0

sinuXe

F

sin 圖

F(uN/2)

1NfN

j2x(uN/N

1N (1)xf(x)eNx0

j2 （ DFT[f(x,y)(1)xy]F(uM/2,vN/ DFT的原點(diǎn)，即F(0,0)被設(shè)置在u=M/2和v=N/21NF(0,0) f(x,y)

x0 (b)中心化前的頻譜圖 圖8.5圖像頻譜的中心化可分 M NF(u,v)

ej2ux/ 1f(x,y)ej2vy/MM

N

M

F(x,v)ej2ux/MF(x,v)

N f(x,y)ej2vy/N

圖 二維DFT變換方同樣可以通過(guò)先求列變換再求行變換得到2DDFT

0 0 f(x,y) 00 00x方向

F(u,v)

0 0

22 221 1 F(u,y)

y方向

22

2

1 1

22 2 f(x,y)*g(x,y) f(m,n)g(xm,ym0

卷積定

M和N的取DFT[f(x,y)*g(x,y)]F(u,v)G(u, 圖 變xyf(x,y)f(,F(u,v)F(,

uv若：f(x,y)F(uv)f(,0F(,0 原圖像旋轉(zhuǎn)原圖像旋轉(zhuǎn)45，其幅度譜圖像也旋轉(zhuǎn)45若：f(x,y)F(uv)g(x,y)G(u,v)則：f(xyg(xyF(uvGuf(x,y)g(x,y)F(u,v)G(u,空 頻相

離 變換的矩陣表目的:（1）用矩陣乘法的程序進(jìn)行FT；（2）理論推導(dǎo)一維DFT的矩陣表F(u)N

f(x)exp

j2

ux x

展開(kāi)

令jjw

N F(u) f(x)wuxxF(0)f(x)w0 f(0)w0 f(1)w0 f(N1)wNF(1)NF(1)f(x)wxxf(0)w0 f(1)w1 f(N1)wN1NF(N1)f(x)w(N1)xf(0)w0f(1)wN1f(N1)w(N1)(N 令

f f F

f f(NF(N W

wN w w0

wN

正變換 F fW(忽略二維DFT的矩陣表N1NF(u,v) 1f(x,y)exp j2(N1N x y

N根據(jù)可分F(,y)f(x,y)exp j2N

N2 F(u,v)

y

F(u,y)exp vy

jjw

N

W w(N1)(N1)

F

F(N

F(N

F(N1,N f f

ff

f

f(N

f(N

FT：F IFT：fW1FW（忽略 變換在圖像濾波中的應(yīng) 變換在圖像壓縮中的應(yīng)變換的低通濾變換的高通濾縮率為變換的縮率為.變換的.壓縮率為壓縮率為（）f(i,

fg(i,fgg FFT1(F FFg(,)G(,)F(,)2維離散余弦變換 一維離散余弦變 得f(n),0nNfc

f(n1),Nn

0 f圖 延拓示意f(n)＝f(n),0nN f(2Nn1),Nn2Nfc(2Nn1)＝fc nknkFc

2NN

fc(n)W2

22N22＝2

f(n)W

＋m

f(2Nm1)WN 2Fc(2

＝

f(n)W ＋2iN2N

f(i)W(2N＝＝22

f(n)cos(2n2N(2n2N

f(n)

21 k2

1kN3.2.2二維離散余弦變MN1NF(u,v)C(u)C(v) 2 Mf(x,y)cosu(x1)cosv(yMN1Nx0y 2

21N2MC(u)C(v)F(u,v)cosu(x1)cosv(y11Nf(x,y)

MN

2 2

%讀入圖%顯示原圖subplot(1,2,2),imshow(log(abs(I)),[0 圖8.10離散余弦變換二維DCT的應(yīng)JPEG,MJPEG和MPEG等標(biāo)準(zhǔn)中都使用了3二維離 -哈達(dá)瑪變換 （Walsh）變換 哈達(dá)瑪 哈達(dá)瑪變換要求圖像的大小為N＝2ng(x,u)

n1bi(x)bi(u1(1)iN

其中，bk(z代表z的二進(jìn)制表示的第k

nH(u)

bi(x)bi(uf(x)(1)i

N

nbi(x)bi(uf(x)H(u)(1)i

H(u,v)

1N1NNx0

n[bi(x)bi(u)bi(y)bif(x,y)(1)i

1N1N

n[bi(x)bi(u)bi(y)bif(x,y)

H(u,v)(1)iNu0

HN HN

N NN

HN21111110H8

11111173 11422111111611

11 211 5變 1N

nbi(x)bn1i(ug(x,u)

(1)iN

n1N1N [bi(x)bn1i(u)bi(y)bn1iW(u,v) f(x,y)(1)iNx0

n1N1N [bi(x)bn1i(u)bi(y)bn1if(x,y) W(u,v)(1)iNu0

1 1 H

2

Walw(0,t1 Walw(1,t)1 -Walw(2,t1 -Walw(3,t1 -Walw(4,t1 -Walw(4,t)1 -Walw(5,t)1 -1Walw(6,t) -Walw(7,t)1 -列率：在正交區(qū)間內(nèi)波形變號(hào)次數(shù)的1/2稱(chēng)SS i2i i od i eve三種排列的前8個(gè)Walsh函數(shù)之間的關(guān)系4 -列夫變換（K-L變換主分量分析（PCA）二維離散小波變一種窗口大小固定，但形狀可改變，因而能滿(mǎn)足時(shí)頻連續(xù)小波1a,b(x) 2

xb

a,bR,a

(x)離散小aam bnbam

a 10R,n m,n(x) 2 xnb0

f(x)m,n(x)dxf(x),m,n(x)f(k)

f,

【例】應(yīng) 解 [cA1,c