1T.T. Liu, BE280A, UCSD Fall 2004Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2004Lecture 5SamplingT.T. Liu, BE280A, UCSD Fall 2004Topics1. Overview of Sampling2. 1D Sampling3. 2D Sampling4. Aliasing2T.T. Liu, BE280A, UCSD Fall 2004Analog vs. DigitalThe Analog World:Continuous time/space, continuous valued signals orimages, e.g. vinyl records, photographs, x-ray films.The Digital World:Discrete time/space, discrete-valued signals orimages, e.g. CD-Roms, DVDs, digital photos, digitalx-rays, CT, MRI, ultrasound.T.T. Liu, BE280A, UCSD Fall 2004The Process of Samplingg(x) g[n]=g(n Δx)Δxsample3T.T. Liu, BE280A, UCSD Fall 2004QuestionsHow finely do we need to sample?What happens if we don’t sample finelyenough?Can we reconstruct the original signal orimage from its samples?T.T. Liu, BE280A, UCSD Fall 2004Sampling in the Time Domain4T.T. Liu, BE280A, UCSD Fall 2004Sampling in Image SpaceT.T. Liu, BE280A, UCSD Fall 2004Sampling in k-space5T.T. Liu, BE280A, UCSD Fall 2004Sampling in k-spaceT.T. Liu, BE280A, UCSD Fall 2004Comb Function€ comb(x) =δ(x − n)n= −∞∞∑Other names: Impulse train, bed of nails,shah function.-5 -4 -3 -2 -1 0 1 2 3 4 5x6T.T. Liu, BE280A, UCSD Fall 2004Scaled Comb Function€ combxΔx =δ(xΔx− n)n= −∞∞∑=δ(x − nΔxΔx)n= −∞∞∑= Δxδ(x − nΔx)n= −∞∞∑xΔxT.T. Liu, BE280A, UCSD Fall 20041D spatial sampling€ gS(x) = g(x)1ΔxcombxΔx = g(x)δ(x − nΔx)n= −∞∞∑= g(nΔx)δ(x − nΔx)n= −∞∞∑€ Recall the sifting property g(x)δ(x − a) = g(a)−∞∞∫But we can also write g(a)δ(x − a) = g(a)−∞∞∫δ(x − a)−∞∞∫= g(a)So, g(x )δ(x − a) = g(a)δ(x − a)7T.T. Liu, BE280A, UCSD Fall 20041D spatial samplingg(x)xΔxxcomb(x/Δx)/ ΔxgS(x)T.T. Liu, BE280A, UCSD Fall 2004Fourier Transform of comb(x)€ F comb(x)[ ]= comb(kx)=δ(kx− n)n= −∞∞∑€ F1Δxcomb(xΔx) =1ΔxΔxcomb(kxΔx)=δ(kxΔx − n)n= −∞∞∑=1Δxδ(kx−nΔx)n= −∞∞∑8T.T. Liu, BE280A, UCSD Fall 2004Fourier Transform of comb(x/ Δ x)xΔxcomb(x/ Δx)/ Δxkxcomb(kx Δx)1/Δx1/ΔxFT.T. Liu, BE280A, UCSD Fall 2004Fourier Transform of gS(x)€ F gS(x)[ ]= F g(x)1ΔxcombxΔx = G(kx) ∗ F1ΔxcombxΔx = G(kx) ∗1Δxδkx−nΔx n= −∞∞∑=1ΔxG(kx) ∗δkx−nΔx n= −∞∞∑=1ΔxG kx−nΔx n= −∞∞∑9T.T. Liu, BE280A, UCSD Fall 2004Fourier Transform of gS(x)G(kx)GS(kx)1/Δxkxkx1/ΔxT.T. Liu, BE280A, UCSD Fall 2004Nyquist ConditionG(kx)GS(kx)KS=1/ΔxkxkxB-BTo avoid overlap, we require that 1/Δx>2B or KS > 2B where KS=1/ Δx is the sampling frequencyKS10T.T. Liu, BE280A, UCSD Fall 2004Reconstruction from SamplesKS=1/ΔxGS(kx)Multiply by(1/KS)rect(kx/KS)(1/KS) GS(kx)rect(kx/KS)=G(kx)KST.T. Liu, BE280A, UCSD Fall 2004Reconstruction from Samples€ ˆ G S(kx) =1KSGS(kx)rect(kx/KS) = G(kx)If the Nyquist condition is met, then€ ˆ g S(x) = gS(x) ∗ sinc(Ksx)= g(nΔX)δ(x −n=−∞∞∑nΔX) ∗ sinc(Ksx)= g(nΔX)n=−∞∞∑sinc(Ks(x − nΔx))And the signal can be reconstructed by convolving the samplewith a sinc function11T.T. Liu, BE280A, UCSD Fall 2004Example Cosine Reconstructioncos(2πk0x)k0-k0k0-k0KSk0-k0KSKS>2k0KS=2k0T.T. Liu, BE280A, UCSD Fall 2004Cosine Example with KS=2k012T.T. Liu, BE280A, UCSD Fall 2004Example with Ks=4k0T.T. Liu, BE280A, UCSD Fall 2004Example with Ks=8k013T.T. Liu, BE280A, UCSD Fall 2004Sine Example with Ks=2k0T.T. Liu, BE280A, UCSD Fall 2004Example Sine Reconstruction2jsin(2πk0x)k0-k0k0-k0KSk0-k0KSKS>2k0KS=2k0Aliased!14T.T. Liu, BE280A, UCSD Fall 2004AliasingKSG(kx)kxB-BAliasing occurs when the Nyquist condition is not satisfied.This occurs for KS ≤ 2BT.T. Liu, BE280A, UCSD Fall 2004Aliasing Example15T.T. Liu, BE280A, UCSD Fall 2004Aliasing Examplecos(2πk0x)k0-k0k0-k0KSKS=k0T.T. Liu, BE280A, UCSD Fall 2004Aliasing Examplecos(2πk0x)k0-k0k0-k0KS2k0>KS>k016T.T. Liu, BE280A, UCSD Fall 2004Practical ConsiderationsWhy sample higher than the Nyquist frequency?-- true sinc interpolation is not practical since the sincfunction goes from -infinity to infinity-- the requirements on the low-pass filter are reduced.k0-k0Hardk0-k0EasierKST.T. Liu, BE280A, UCSD Fall 2004Fourier SamplingInstead of sampling thesignal, we sample its FourierTransform???SampleF-1F17T.T. Liu, BE280A, UCSD Fall 2004Fourier Sampling€ GS(kx) = G(kx)1ΔkxcombkxΔkx = G(kx)δ(n= −∞∞∑kx− nΔkx)= G(nΔkx)δ(n= −∞∞∑kx− nΔkx)kx(1 /Δkx) comb(kx /Δkx)ΔkxT.T. Liu, BE280A, UCSD Fall 2004Fourier Sampling -- Inverse TransformΧ=*1/ΔkxΔkx=18T.T. Liu, BE280A, UCSD Fall 2004Fourier Sampling -- Inverse Transform€ gS(x) = F−1GS(kx)[ ]= F−1G(kx)1ΔkxcombkxΔkx = F−1G(kx)[ ]∗ F−11ΔkxcombkxΔkx = g(x) ∗ comb xΔkx( )= g(x) ∗δ(xΔkxn=−∞∞∑− n)= g(x) ∗1Δkxδ(xn=−∞∞∑−nΔkx)=1Δkxg(xn=−∞∞∑−nΔkx)T.T. Liu, BE280A, UCSD Fall 2004Nyquist ConditionFOV (Field of View)1/ΔkxTo avoid overlap, 1/Δkx> FOV, or equivalently, Δkx<1/FOV19T.T. Liu, BE280A, UCSD Fall 2004AliasingFOV (Field of View)1/ΔkxAliasing occurs when 1/Δkx< FOVT.T. Liu, BE280A, UCSD Fall 2004Aliasing ExampleΔkx=11/Δkx=120T.T. Liu, BE280A, UCSD Fall 20042D Comb Function€ comb(x, y) =δ(x − m, y − n)n= −∞∞∑m= −∞∞∑=δ(x − m)δ(y − n)n= −∞∞∑m= −∞∞∑= comb(x)comb(y)T.T. Liu, BE280A, UCSD Fall 2004Scaled 2D Comb Function€ comb(x /Δx, y /Δy) = comb(x /Δx)comb(y /Δy)= ΔxΔyδ(x − mΔx)δ(y − nΔy)n= −∞∞∑m= −∞∞∑ΔxΔy21T.T. Liu, BE280A, UCSD Fall 2004X =*=1/∆kT.T. Liu, BE280A, UCSD Fall 20042D k-space sampling€ GS(kx,ky) = G(kx,ky)1ΔkxΔkycombkxΔkx,kyΔky = G(kx,ky)δ(n=−∞∞∑kx− mΔkx,ky− nΔky)m= −∞∞∑= G (mΔkx,nΔky)δ(n=−∞∞∑kx− mΔkx,ky− nΔky)m= −∞∞∑22T.T. Liu, BE280A, UCSD Fall 20042D k-space sampling€ gS(x, y) = F−1GS(kx,ky)[ ]= F−1G(kx,ky)1ΔkxΔkycombkxΔkx,kyΔky = F−1G(kx,ky)[ ]∗ F−11ΔkxΔkycombkxΔkx,kyΔky = g(x, y) ∗∗comb xΔkx(
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