HOe178-04L7-8

HOe178-04L7-8 - Fourier Transform - review 2D Fourier...

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01/27/2003 2D Fourier Transform 1 2D Fourier Transform Week IV 2-D DFT & Properties 01/27/2003 2D Fourier Transform 2 Fourier Transform - review 1-D: 2-D: Fu fx fxe d x f x Fue du Fuv f xye dxdy fxy Fuve d ud v ju x x x v y x v y af k p kp ≡ℑ = = = = −∞ −∞ −+ + z z zz zz () (,) 2 12 2 2 π 01/27/2003 2D Fourier Transform 3 2D FT: Properties Convolution: f(x,y) g(x,y) = F(u,v) G(u,v) Multiplication: f(x,y) g(x,y) = F(u,v) G(u,v) Separable functions: Suppose f(x,y) = g(x) h(y), Then F(u,v)=G(u)H(v) Shifting: f(x+ x 0 , y+ y 0 ) exp[2 π (x 0 u + y 0 v)] F(u,v) Linearity: a f(x,y) + b g(x,y) a F(u,v) + b G(u,v) 01/27/2003 2D Fourier Transform 4 Separability of the FT dx e dy Fuy e x jv y y = L N M O Q P = −∞ −∞ −∞ z z z 22 2 ππ 01/27/2003 2D Fourier Transform 5 Separability (contd.) f(x,y) F(u,y) F(u,v) Fourier Transform along X. Fourier Transform along Y. We can implement the 2D Fourier transform as a sequence of 1-D Fourier transform operations. 01/27/2003 2D Fourier Transform 6 Eigenfunctions of LSI Systems A function f(x,y) is an Eigenfunction of a system T if T[ f(x,y) ] = α f(x,y) for some constant ( Possibly complex) α. For LSI systems, complex exponentials of the form exp{ j2 π (ux+vy) }, for any (u,v), are the Eigenfunctions.
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01/27/2003 2D Fourier Transform 7 Impulse Response and Eigenfunctions gxy hx sy te dsd t hxye e dxdy Huve ju s v t x v y ju x v y x v y (,) ( , ) () () () =- - = = + -• +- + + z z zz 2 22 2 π ππ Consider a LSI system with impulse response h(x,y). Its output to the complex exponential is 01/27/2003 2D Fourier Transform 8 2-D FT: Example f(x,y) x y Y X A Fuv fxye d xd y Ae d x e d y AXY uX vY e x v y x X jv y Y Xv Y (, ) sin = = = L N M O Q P L N M O Q P -+ -- z z 2 2 0 2 0 01/27/2003 2D Fourier Transform 9 Example (contd.) 01/27/2003 2D Fourier Transform 10 Example2 01/27/2003 2D Fourier Transform 11 Discrete Fourier Transform Consider a sequence {u(n), n=0,1,2,. ...., N-1}. The DFT of u(n) is vk unW k N We un N vkW n N n N N kn N j N k N N kn ( ) ( ) , , ,. ...., , ( ) ( ) , .., == = = = 0 1 2 0 1 01 1 1 1 Where and the inverse is given by 01/27/2003 2D Fourier Transform 12 2-D DFT Often it is convenient to consider a symmetric transform: In 2-D: consider a NXN image N N N kn n N N kn n N = = = - - = - Â Â 1 1 0 1 0 1 and vkl N umn W W umn
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HOe178-04L7-8 - Fourier Transform - review 2D Fourier...

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