Fourier transformationFourier transformation

1. Fourier transformation 1D1. Fourier transformation 1D2. Fourier transformation 2D2. Fourier transformation 2D3. Realization of the FT3. Realization of the FT4. The proprieties of the FT4. The proprieties of the FT5. The function of convolution 5. The function of convolution 6. The functions of correlation6. The functions of correlation and autocorrelationand autocorrelation7. Correlation recognition7. Correlation recognition

Fourier transformation 1DFourier transformation 1D

FFxx FFwwxxexpexp[-[-jjwwxxxx)])]dwdwxx

FFwwxx FFxxexpexp[[jjwwxxxx]]ddxx

The signal u(t) and The signal u(t) and his approximationhis approximation

Fourier transformation 1DFourier transformation 1D

FFwwxx FFxxexpexp[[jj((wwxxxx)])]dxdx

eexpxp[[jj((wwxxx)]x)] cos coswwxxxx + jsin + jsinwwxxxx

FFwwxx FFxxexpexp[[jj((wwxxxx)])]dx dx

==FFxx[cos[coswwxxxx + jsin + jsinwwxxxx]dx ]dx

FFxxcoscoswwxxxxdx +dx + jjFFxxsinsinwwxxxxdxdx==

==RRwwxx + jX + jXwwxx

Fourier transformation 2DFourier transformation 2D

FFwwxx,w,wyy FFxx,y,yexpexp[[jj((wwxxxx+w+wyyyy]]dxdxdydy

Fourier transformation 2DFourier transformation 2D

FFwwxx,w,wyy FFxx,y,yexpexp[[jj((wwxxxx+w+wyyyy]]dxdxdydy

Fwx,wy Rwx,wy+jXwx,wy =

Fwx,wyexp[jQwx,wy] =

FFxx,y,ycoscoswwxxxx++wyydxdxdydy++ jjFFxx,y,ysinsinwwxxxx++wyydxdxdydy

Fwx,wy [Rwx,wy]2 + [Xwx, wy]21∕2

Qwx,wy arctg[Xwx,wy∕Rwx,wy]

Realization of the FTRealization of the FT

1. 1. FFwwxx,y,y FFxx,y,yexpexp[[jj((wwxxxx]]dxdx

2. 2. FFwwxx,w,wyy FFwwxx,y,yexpexp[[jj(w(wyyyy]dy]dy

INVERSE 2D FTINVERSE 2D FT

FFxx,y,y FFwwxx,w,wyyexpexp[[jj((wwxxxx+w+wyyyy]]ddwwxxdwdwyy

The proprieties of the FTThe proprieties of the FT

1. The functional properties1. The functional properties

For a symmetric function F(x,y),For a symmetric function F(x,y),F(x,y)=F(-x,-y)F(x,y)=F(-x,-y)

FTFT{F(x,y)}= {F(x,y)}= F(F(wwxx,,wwyy) = ) =

==F(F(wwxx,,wwyy)exp[jQ()exp[jQ(wwxx,,wwyy)])]

==FTFT{F(-x,-y)} = {F(-x,-y)} = F(-F(-wwxx,, --wwyy))

For a non symmetric function F(x,y),For a non symmetric function F(x,y),

F(x,y)≠F(-x, -y)F(x,y)≠F(-x, -y)

FTFT{F(-x,-y)}= {F(-x,-y)}= FF*(w*(wxx,,wwyy) = ) =

==F(F(wwxx,,wwyy)exp[)exp[jQ(jQ(wwxx,,wwyy)])]

2. The linearity2. The linearity

FTFT{a{a11FF11(x,y)+...+a(x,y)+...+ann F Fnn(x,y)} = (x,y)} =

aa11FTFT{F{F11(x,y)}+...+a(x,y)}+...+annFTFT{F{Fnn(x, y)}(x, y)}

3. Scale change3. Scale change

FTFT{F({F(aax,x,bby)} = [F(y)} = [F(wwxx//aa, , wwyy//bb)]/()]/(aa..bb))

4. Translation4. Translation

FTFT{F(x-a,y-b)}=F({F(x-a,y-b)}=F(wwxx,,wwyy)exp(-)exp(-jj((wwxxa + a + wwyyb)}b)}

5. Fourier transformation of convolution5. Fourier transformation of convolution

FTFT{F(x,y){F(x,y)**H(x, y)}=F(H(x, y)}=F(wwxx,,wwyy)H()H(wwxx,,wwyy))

6. Fourier transformation of product6. Fourier transformation of product

FTFT{F(x,y)H(x,y)}=F({F(x,y)H(x,y)}=F(wwxx,,wwyy))**H(H(wwxx,,wwyy))

FUNCTION OF CORRELATIONFUNCTION OF CORRELATION

C(x,y)=F(x,y)#H(x,y)=C(x,y)=F(x,y)#H(x,y)== FF-1-1{{FF[F(x,y)][[F(x,y)][FF*[H(x,y)]}= *[H(x,y)]}=

= = FF--11{{FF[F(x,y)][[F(x,y)][FF[[H*H*(x,y)]}= (x,y)]}=

= = FF--11{F(u,v){F(u,v)H*H*(u,v)}=(u,v)}=

= = FF--11{{F(u,v)F(u,v)exp[exp[jQjQFF]]HH(u,v)(u,v)exp[jQexp[jQHH]}]} = =

= = FF--11[[F(u,v)F(u,v)HH(u,v)(u,v)exp[exp[jj{{QQFF--QQHH}}]]]] = =

= = FFxx,y,yHH**(x-(x-ξξ,y-,y-ηη))dxdxdydy

FUNCTION OF FUNCTION OF AUTOCORRELATIONAUTOCORRELATION

CCAA(x,y)(x,y) == F(x,y)#F(x,y)F(x,y)#F(x,y) ==

= = FF--11{{FF[F(x,y)][[F(x,y)][FF[F[F**(x,y)]}= (x,y)]}=

FF--11{{FF[F(x,y)][[F(x,y)][FF*[F(x,y)]}= *[F(x,y)]}=

= = FF--11{F(u,v)F{F(u,v)F**(u,v)}=(u,v)}=

= = FF--11{{F(u,v)F(u,v)exp[exp[jQjQFF]]F(u,v)F(u,v)exp[jQexp[jQFF]}]} = =

FF--11[[F(u,v)F(u,v)22]]

CORRELATION RECOGNITIONCORRELATION RECOGNITION

CCiiMMx,yx,y F Fx,yx,y#H#Hiix,yx,y

Max[CMax[CiMiMx,yx,y], i], i1,N1,N