1T.T. Liu, BE280A, UCSD Fall 2004Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2004Lecture 6Resolution, Discrete Fourier TransformT.T. Liu, BE280A, UCSD Fall 2004Topics1. Recap of sampling.2. Resolution3. Discrete Fourier Transform2T.T. Liu, BE280A, UCSD Fall 2004Fourier Sampling -- Inverse TransformΧ=*1/ΔkxΔkx=T.T. Liu, BE280A, UCSD Fall 2004Nyquist ConditionFOV (Field of View)1/ΔkxTo avoid overlap, 1/Δkx> FOV, or equivalently, Δkx<1/FOV3T.T. Liu, BE280A, UCSD Fall 2004Intuitive view of Aliasing€ kx= 1/ FOVFOV€ kx= 2 / FOVT.T. Liu, BE280A, UCSD Fall 2004X =*=1/∆k4T.T. Liu, BE280A, UCSD Fall 2004Nyquist ConditionsFOVXFOVY1/∆kX1/∆kY1/∆kY> FOVY1/∆kX> FOVXT.T. Liu, BE280A, UCSD Fall 2004Aliasing5T.T. Liu, BE280A, UCSD Fall 2004ResolutionT.T. Liu, BE280A, UCSD Fall 2004Resolution6T.T. Liu, BE280A, UCSD Fall 2004Windowing€ GWkx,ky( )= G kx,ky( )W kx,ky( )€ gWx, y( )= g x, y( )∗ w(x, y)Windowing the data in Fourier spaceResults in convolution of the object with the inversetransform of the windowT.T. Liu, BE280A, UCSD Fall 2004Resolution7T.T. Liu, BE280A, UCSD Fall 2004Windowing Example€ W kx,ky( )= rectkxWkx rectkyWky € w x, y( )= F−1rectkxWkx rectkyWky = WkxWkysinc Wkxx( )sinc Wkyy( )€ gWx, y( )= g(x, y) ∗∗WkxWkysinc Wkxx( )sinc Wkyy( )T.T. Liu, BE280A, UCSD Fall 2004Windowing Example€ g(x, y) =δ(x) +δ(x −1)[ ]δ(y)€ gWx, y( )=δ(x) +δ(x −1)[ ]δ(y) ∗∗WkxWkysinc Wkxx( )sinc Wkyy( )= WkxWkyδ(x) +δ(x −1)[ ]∗ sinc Wkxx( )( )sinc Wkyy( )= WkxWkysinc Wkxx( )+ sinc Wkx(x −1)( )( )sinc Wkyy( )€ Wkx= 1€ Wkx= 2€ Wkx= 1.58T.T. Liu, BE280A, UCSD Fall 2004Effective Width€ wE=1w(0)w(x)dx−∞∞∫wE€ wE=11sinc(Wkxx)dx−∞∞∫= F sinc(Wkxx)[ ]kx= 0=1WkxrectkxWkx kx= 0=1WkxExample€ 1Wkx€ −1WkxT.T. Liu, BE280A, UCSD Fall 2004Resolution and spatial frequency€ 2Wkx€ With a window of width Wkx the highest spatial frequency is Wkx/2.This corresponds to a spatial period of 2/Wkx.€ 1Wkx= Effective Width9T.T. Liu, BE280A, UCSD Fall 2004Sampling and WindowingX =*=X* =T.T. Liu, BE280A, UCSD Fall 2004Sampling and Windowing€ GSWkx,ky( )= G kx,ky( )1ΔkxΔkycombkxΔkx,kyΔky rectkxWkx,kyWky € gSWx, y( )= WkxWkyg x, y( )∗∗comb(Δkxx,Δkyy) ∗∗sinc(Wkxx)sinc(Wkyy)Sampling and windowing the data in Fourier spaceResults in replication and convolution in object space.10T.T. Liu, BE280A, UCSD Fall 2004Discrete Fourier TransformIdea: If we sample and window in the Fourier domain, weobtain a finite number of discrete Fourier samples. Whenwe reconstruct the object, we should have the samenumber of pixels in our object.Also, the windowing process, has band-limited thesampled Fourier transform, so this allows us to sample thereplicated object at discrete points.T.T. Liu, BE280A, UCSD Fall 20041D Sampling and WindowingΧ Χ =** =11T.T. Liu, BE280A, UCSD Fall 2004Discrete Fourier Transformx==*T.T. Liu, BE280A, UCSD Fall 2004Discrete Fourier Transform€ Nx=WkxΔkx € Nx=FOVX1/Wkx=WkxΔkx€ Wkx € FOV€ Note that FOVXNx=1Wkx=δx, our measure for resolution.12T.T. Liu, BE280A, UCSD Fall 2004DFT€ GDFTkx,ky( )= G kx,ky( )1ΔkxΔkycombkxΔkx,kyΔky rectkxWkx,kyWky ∗∗combkxWkx,kyWky € gDFTx, y( )= WkxWkyg x, y( )∗∗comb(Δkxx,Δkyy) ∗∗sinc(Wkxx)sinc(Wkyy) × WkxWkycomb(Wkxx,Wkxy)Sampling, windowing, and replication in Fourier spaceReplication, convolving, sampling in object spaceT.T. Liu, BE280A, UCSD Fall 2004DFT€ F gD(x)[ ]= gD0FOV∫(x)e− j 2πmΔkxxdx= gDn /Wx( )δ(x − n /Wx)n= −∞∞∑0FOV∫e− j 2πmΔkxxdx= gDn /Wx( )δ(x − n /Wx)e− j 2πmΔkxxdx−∞∞∫n= 0N −1∑= gDn /Wx( )n= 0N −1∑e− j 2πmnΔkx/Wx= gDn[ ]n= 0N −1∑e− j 2πmn / NThis is what MATLAB computes when you use fft13T.T. Liu, BE280A, UCSD Fall 2004DFT Basis Functions€ DFT : G[m] = g n[ ]n= 0N −1∑e− j 2πmn / NBasis Functions are therefore :bm[n] = ej 2πmn / NAre these orthonormal??Inverse DFT : g[n] =1NG m[ ]m= 0N −1∑ej 2πmn / NT.T. Liu, BE280A, UCSD Fall 20042D DFT€ DFT : G[r,s] = g m,n[ ]n= 0N −1∑e− j 2πrm + sn( )/ Nm= 0N −1∑Basis Functions are therefore :br,s[m,n] = ej 2πrm +sn( )/ NAre these orthonormal??Inverse DFT : g[m,n] =1N2G r,s[ ]s= 0N −1∑ej 2πrm +sn( )/ Nr= 0N −1∑In general, the number of points along each dimensionneed not be the same (e.g. N1≠ N2). How does this changethe
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