Topics. View Aliasing. CT Sampling Requirements. Sampling Requirements in CT Sampling Theory Aliasing

Transcription

1 Topics Bioengineering 280A Principles of Biomedical Imaging Sampling Requirements in CT Sampling Theory Aliasing Fall Quarter 2007 CT/Fourier Lecture 4 CT Sampling Requirements View Aliasing What should the size of the detectors be? How many detectors do we need? How many views do we need? Kak and Slaney 1

2 Aliasing Artifacts Object Effect of Noise Aliasing due to insufficient number of detectors Aliasing due to insufficient number of views Kak and Slaney Analog vs. Digital The Process of Sampling The Analog World: Continuous time/space continuous valued signals or images e.g. vinyl records photographs -ray films. g() The Digital World: Discrete time/space discrete-valued signals or images e.g. CD-Roms DVDs digital photos digital -rays CT MRI ultrasound. Δ sample g[n]=g(n Δ) 2

3 Questions Sampling in the Time Domain 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? Sampling in Image Space Sampling in k-space 3

4 Sampling in k-space Comb Function comb() = "( # n) % n=# Other names: Impulse train bed of nails shah function. Scaled Comb Function comb # & % ( = )( " ' " * n) n=* = )( * n" ) " n=* = " )( * n") n=* Δ 1D spatial sampling g S () = g() 1 " comb # & % ( " ' Recall the sifting property = g() )( * n") n=* = g(n"))( * n") n=* % # g()"( # a) = g(a) But we can also write % g(a)"( # a) = g(a) % "( # a) = g(a) So g()"( # a) = g(a)"( # a) # # 4

5 1D spatial sampling g() comb(/δ)/ Δ Fourier Transform of comb() F[ comb() ] = comb(k ) % = "(k # n) n=# Δ g S () # F 1 " comb( % " ) & ' ( = 1 " "comb(k ") = )(k " * n) n=* = 1 " n=* )(k * n " ) Fourier Transform of comb(/ Δ) comb(/ Δ)/ Δ Δ F 1/Δ comb(k Δ) 1/Δ k Fourier Transform of g S () [ ] = F g() 1 " comb % F g S () ) * # & (. " '- ) = G(k ) / F 1 " comb # & % (. * " '- = G(k ) / 1 " = 1 " = 1 " 2 3 n= n= n=12 # 0 k 1 n & % ( " ' # G(k ) /0 k 1 n & % ( " ' # G k 1 n & % ( " ' 5

6 Fourier Transform of g S () Nyquist Condition G(k ) G(k ) k -B B k G S (k ) 1/Δ G S (k ) k k =1/Δ 1/Δ To avoid overlap we require that 1/Δ>2B or > 2B where =1/ Δ is the sampling frequency Eample Reconstruction from Samples 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 Δ < 1/2B = 0.25 cm =1/Δ G S (k ) Multiply by (1/ )rect(k / ) This corresponds to 2 samples per spatial period. (1/ ) G S (k )rect(k / ) =G(k ) 6

7 Eample Cosine Reconstruction -k 0 k 0 cos(2πk 0 ) >2k 0 Reconstruction from Samples If the Nyquist condition is met then G ˆ S (k ) = 1 G S (k )rect(k / ) = G(k ) And the signal can be reconstructed by convolving the sample with a sinc function -k 0 k 0 -k 0 k 0 =2k 0 g ˆ S () = g S ()" sinc(k s ) ( & = * ' g(n#x)( % n#x) -" sinc(k s ) ) n=%& & ' = g(n#x) sinc(k s ( % n#)) n=%& g() Reconstruction from Samples g S () Cosine Eample with =2k 0 Sample at Δ ˆ g S () sinc(k s ) = sinc( / ") 7

8 Eample with K s =4k 0 Eample with K s =8k 0 Aliasing Aliasing Eample -B B G(k ) k Aliasing occurs when the Nyquist condition is not satisfied. This occurs for 2B 8

9 Aliasing Eample Aliasing Eample cos(2πk 0 ) cos(2πk 0 ) -k 0 k 0 -k 0 k 0 =k 0 2k 0 > >k 0 -k 0 k 0 -k 0 k 0 Eample Detector Sampling Requirements 1. C onsider the function g() = cos 2 ( 2"k 0 ). Sketch this function. You sample this signal in the spatial domain with a sampling rate =1/# (e.g. samples spaced at intervals of # ). What is the minimum sampling rate that you can use without aliasing? Give an intuitive eplanation for your answer. Sampling interval Δr Beamwidth Δs 9

10 Smoothing of Projection Smoothing of Projection Projection g s (l") = rect(l /#s) g( l" ) G s (k ") = #ssinc(k #s)g(k ") Beam Width 2/(Δs) Smoothed Projection Sampling Requirements View Aliasing Smoothed Projection Detectors Δr Δs/2 Sampled Smooth Projection Kak and Slaney 10

11 View Sampling Requirements View Sampling -- how many views? Basic idea is that to make the maimum angular sampling the same as the projection sampling. "FOV N views = #r N views360 = "FOV #r N views180 = "N proj 2 = "N proj (for 360 degrees) (for 180 degrees) Eample beamwidth "s = 1 mm Field of View (FOV) = 50 cm "r = "s/2 = 0.5 mm 500 mm/ 0.5 mm = N = 1000 detector samples # * N = 3146 views per 360 degrees 1500 views per 180 degrees CT "Rule of Thumb" N view = N det ectors = N piels 11