1 TT Liu, BE280A, UCSD Fall 2015 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2015 CT/Fourier Lecture 4 TT Liu, BE280A, UCSD Fall 2015 Convolution/Modulation Theorem Fg( x) ∗ h( x ) { } = g(u) ∗ h( x − u) du −∞ ∞ ∫ [ ] e − j 2πk x x −∞ ∞ ∫ dx = g(u) h( x − u) −∞ ∞ ∫ e − j 2πk x x −∞ ∞ ∫ dxdu = g(u)H(k x )e − j 2πk x u −∞ ∞ ∫ du = G(k x ) H( k x ) Convolution in the spatial domain transforms into multiplication in the frequency domain. Dual is modulation Fg( x) h( x) { } = Gk x ( ) ∗ H( k x ) TT Liu, BE280A, UCSD Fall 2015 2D Convolution/Multiplication Convolution Fg( x, y ) ∗∗h( x, y) [ ] = G( k x , k y ) H( k x , k y ) Multiplication Fg( x, y )h( x, y) [ ] = G( k x , k y ) ∗∗H( k x , k y ) TT Liu, BE280A, UCSD Fall 2015 Application of Convolution Thm. Λ( x) = 1 − x x < 1 0 otherwise $ % & F ( Λ( x)) = 1 − x ( ) −1 1 ∫ e − j 2πk x x dx = ?? -1 1
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TT Liu, BE280A, UCSD Fall 2015
Bioengineering 280A ���Principles of Biomedical Imaging���
���Fall Quarter 2015���
CT/Fourier Lecture 4
TT Liu, BE280A, UCSD Fall 2015
Convolution/Modulation Theorem
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F g(x)∗ h(x){ } = g(u)∗ h(x − u)du−∞
∞
∫[ ]e− j 2πkxx−∞
∞
∫ dx
= g(u) h(x − u)−∞
∞
∫ e− j2πkxx−∞
∞
∫ dxdu
= g(u)H(kx )e− j2πkxu
−∞
∞
∫ du
=G(kx )H(kx )
Convolution in the spatial domain transforms into multiplication in the frequency domain. Dual is modulation
F g(x − a, y− b){ }= F g(x, y)∗δ x − a, y− b( ){ }=G(kx ,ky )e
− j2π (kxa+kyb)
€
Shifting the function doesn't change its spectral content, sothe magnitude of the transform is unchanged.Each frequency component is shifted by a. This correspondsto a relative phase shift of - 2πa /(spatial period) = - 2πakxFor example, consider exp( j2πkxx). Shifting this by a yieldsexp( j2πkx (x − a)) = exp( j2πkxx)exp(− j2πakx )