1!TT Liu, BE280A, UCSD Fall 2010!Bioengineering 280A"Principles of Biomedical Imaging"Fall Quarter 2010"CT/Fourier Lecture 5"TT Liu, BE280A, UCSD Fall 2010!Topics!• Sampling Requirements in CT!• Sampling Theory!• Aliasing!TT Liu, BE280A, UCSD Fall 2010!CT Sampling Requirements!Suetens 2002!What should the size of the detectors be?!How many detectors do we need?!How many views do we need?!TT Liu, BE280A, UCSD Fall 2010!View Aliasing!Kak and Slaney!2!TT Liu, BE280A, UCSD Fall 2010!Aliasing!Kak and Slaney!TT Liu, BE280A, UCSD Fall 2010!Artifacts!Suetens 2002!Object!Effect of Noise!Aliasing due to insufficient number of detectors!Aliasing due to insufficient number of views!TT Liu, BE280A, UCSD Fall 2010!Analog vs. Digital!The Analog World:!Continuous time/space, continuous valued signals or images, e.g. vinyl records, photographs, x-ray films.!The Digital World:!Discrete time/space, discrete-valued signals or images, e.g. CD-Roms, DVDs, digital photos, digital x-rays, CT, MRI, ultrasound. !TT Liu, BE280A, UCSD Fall 2010!The Process of Sampling!g(x)! g[n]=g(n Δx)!Δx!sample!3!TT Liu, BE280A, UCSD Fall 2010!Questions!How finely do we need to sample?!What happens if we don’t sample finely enough? !Can we reconstruct the original signal or image from its samples? !TT Liu, BE280A, UCSD Fall 2010!Sampling in the Time Domain!TT Liu, BE280A, UCSD Fall 2010!Sampling in Image Space!TT Liu, BE280A, UCSD Fall 2010!Sampling in k-space!4!TT Liu, BE280A, UCSD Fall 2010!Sampling in k-space!TT Liu, BE280A, UCSD Fall 2010!Comb 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 5!x!TT Liu, BE280A, UCSD Fall 2010!Scaled Comb Function!€ combxΔx⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =δ(xΔx− n)n=−∞∞∑=δ(x − nΔxΔx)n=−∞∞∑= Δxδ(x − nΔx)n=−∞∞∑x!Δx!TT Liu, BE280A, UCSD Fall 2010!1D 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)5!TT Liu, BE280A, UCSD Fall 2010!1D spatial sampling!g(x)!x!Δx!x!comb(x/Δx)/ Δx!gS(x)!TT Liu, BE280A, UCSD Fall 2010!Fourier 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=−∞∞∑TT Liu, BE280A, UCSD Fall 2010!Fourier Transform of comb(x/ Δx)!x!Δx!comb(x/ Δx)/ Δx!kx!comb(kx Δx)!1/Δx!1/Δx!F!TT Liu, BE280A, UCSD Fall 2010!Fourier 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=−∞∞∑6!TT Liu, BE280A, UCSD Fall 2010!Fourier Transform of gS(x)!G(kx)!GS(kx)!1/Δx!kx!kx!1/Δx!TT Liu, BE280A, UCSD Fall 2010!Nyquist Condition!G(kx)!GS(kx)!KS=1/Δx!kx!kx!B!-B!To avoid overlap, we require that 1/Δx>2B or !KS > 2B where KS=1/ Δx is the sampling frequency!KS!TT Liu, BE280A, UCSD Fall 2010!Example!Assume that the highest spatial frequency in an object! is B = 2 cm-1.!Thus, smallest spatial period is 0.5 cm. .!Nyquist theorem says we need to sample with !Δx < 1/2B = 0.25 cm!This corresponds to 2 samples per spatial period. !TT Liu, BE280A, UCSD Fall 2010!Reconstruction from Samples!KS=1/Δx!GS(kx)!Multiply by (1/KS)rect(kx/KS)!(1/KS) GS(kx)rect(kx/KS)!=G(kx)!KS!7!TT Liu, BE280A, UCSD Fall 2010!Example Cosine Reconstruction!cos(2πk0x)!k0!-k0!k0!-k0!KS!k0!-k0!KS!KS>2k0!KS=2k0!KS!TT Liu, BE280A, UCSD Fall 2010!Reconstruction 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 sample with a sinc function !TT Liu, BE280A, UCSD Fall 2010!Reconstruction from Samples!g(x)!Sample at! Δx!x!gS(x)!€ sinc(Ksx) = sinc( x / Δx)€ ˆ g S( x)TT Liu, BE280A, UCSD Fall 2010!Cosine Example with KS=2k0!8!TT Liu, BE280A, UCSD Fall 2010!Example with Ks=4k0!TT Liu, BE280A, UCSD Fall 2010!Example with Ks=8k0!TT Liu, BE280A, UCSD Fall 2010!Aliasing!KS!G(kx)!kx!B!-B!Aliasing occurs when the Nyquist condition is not satisfied. This occurs for KS ≤ 2B !TT Liu, BE280A, UCSD Fall 2010!Aliasing Example!9!TT Liu, BE280A, UCSD Fall 2010!Aliasing Example!cos(2πk0x)!k0!-k0!k0!-k0!KS!KS=k0!TT Liu, BE280A, UCSD Fall 2010!Aliasing Example!cos(2πk0x)!k0!-k0!k0!-k0!KS!2k0>KS>k0!TT Liu, BE280A, UCSD Fall 2010!Suetens 2002!Example!1. Consider the function € g(x) = cos22πk0x( ). Sketch this function. You sample this signal in the spatial domain with a sampling rate € KS=1/Δx (e.g. samples spaced at intervals of € Δx). What is the minimum sampling rate that you can use without aliasing? Give an intuitive explanation for your answer. TT Liu, BE280A, UCSD Fall 2010!Detector Sampling Requirements!Suetens 2002!Sampling interval Δr!Beamwidth Δs !10!TT Liu, BE280A, UCSD Fall 2010!Smoothing of Projection!Suetens 2002!Projection!Beam !Width!Smoothed!Projection!2/(Δs)!TT Liu, BE280A, UCSD Fall 2010!Smoothing of Projection!Suetens 2002!€ gs(l,θ) = rect(l /Δs) ∗ g l,θ( )Gs(kx,θ) = Δssinc(kxΔs)G(kx,θ)TT Liu, BE280A, UCSD Fall 2010!Sampling Requirements!Suetens 2002!Smoothed!Projection!Detectors!Δr≤ Δs/2!Sampled!Smooth !Projection!TT Liu, BE280A, UCSD Fall 2010!View Aliasing!Kak and Slaney!11!TT Liu, BE280A, UCSD Fall 2010!View Sampling Requirements!Suetens 2002!View Sampling -- how many views?!Basic idea is that to make the maximum angular sampling the same as the projection sampling. !€ πFOVNviews= ΔrNviews,360=πFOVΔr=πNproj (for 360 degrees)Nviews,180=πNproj2 (for 180 degrees)TT Liu, BE280A, UCSD Fall 2010!Example!Suetens 2002!€ beamwidth Δs = 1 mmField of View (FOV) = 50 cmΔr = Δs/2 = 0.5
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