数字图像处理-第08章变换

第8图像1二维离 变换二维连 变二维连续函数f(x,y) f(x,y)xy的函数，且在上绝对可积，则定义积分F(u,v)f(x,y)ej2(uxvy) 为二维连续函数f(x,y) f(x,y)F(u,v)ej2(uxvy)

为F(u,v)的反变换。f(x,y)和F(u,v) 变换对f(x,y)

xX

y【例】求函 XY解：将函数代入到(8.1)XY

其他F(u,v)f(,

j2(uv

dd

ej

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

二维信号的图形表图8.1二维信号f(x（a）信号的频谱 （b）图（a）的灰度图8.2二维离 变尺寸为M×N的离散图像函数的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）幅度谱|F(u,v)|1.3二维离 变换的性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空间位移 f(xx,yy)F(u,v)ej2(ux0vy0)/ 频率位移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 图

F(uN/2)

1NfN

j2x(uN/N

1N (1)xf(x)eNx0

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

x0 (b)中心化前的频谱图 图8.5图像频谱的中心化可分 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

图 二维DFT变换方同样可以通过先求列变换再求行变换得到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

卷积定

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

uv若：f(x,y)F(uv)f(,0F(,0 原图像旋转原图像旋转45，其幅度谱图像也旋转45若：f(x,y)F(uv)g(x,y)G(u,v)则：f(xyg(xyF(uvGuf(x,y)g(x,y)F(u,v)G(u,空 频相

离 变换的矩阵表目的:（1）用矩阵乘法的程序进行FT；（2）理论推导一维DFT的矩阵表F(u)N

f(x)exp

j2

ux x

展开

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

正变换 F fW(忽略二维DFT的矩阵表N1NF(u,v) 1f(x,y)exp j2(N1N x y

N根据可分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 变换在图像滤波中的应 变换在图像压缩中的应变换的低通滤变换的高通滤缩率为变换的缩率为.变换的.压缩率为压缩率为（）f(i,

fg(i,fgg FFT1(F FFg(,)G(,)F(,)2维离散余弦变换 一维离散余弦变 得f(n),0nNfc

f(n1),Nn

0 f图 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

1kN3.2.2二维离散余弦变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

%读入图%显示原图subplot(1,2,2),imshow(log(abs(I)),[0 图8.10离散余弦变换二维DCT的应JPEG,MJPEG和MPEG等标准中都使用了3二维离 -哈达玛变换 （Walsh）变换 哈达玛 哈达玛变换要求图像的大小为N＝2ng(x,u)

n1bi(x)bi(u1(1)iN

其中，bk(z代表z的二进制表示的第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变 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：在正交区间内波形变号次数的1/2称SS i2i i od i eve三种排列的前8个Walsh函数之间的关系4 -列夫变换（K-L变换主分量分析（PCA）二维离散小波变一种窗口大小固定，但形状可改变，因而能满足时频连续小波1a,b(x) 2

xb

a,bR,a

(x)离散小aam bnbam

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

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

f,

【例】应 解 [cA1,c