fourier transformation
DESCRIPTION
1. Fourier transformation 1D2. Fourier transformation 2D3. Realization of the FT4. The proprieties of the FT5. The function of convolution 6. The functions of correlation and autocorrelation7. Correlation recognitionTRANSCRIPT
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
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
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]]