1TT Liu, BE280A, UCSD Fall 2006Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2006CT/Fourier Lecture 4TT Liu, BE280A, UCSD Fall 2006Topics• Aliasing• Sampling Requirements in CT• Fanbeam and Spiral2TT Liu, BE280A, UCSD Fall 2006Aliasing ExampleTT Liu, BE280A, UCSD Fall 2006AliasingKSG(kx)kxB-BAliasing occurs when the Nyquist condition is not satisfied.This occurs for KS ≤ 2B3TT Liu, BE280A, UCSD Fall 2006Aliasing Examplecos(2πk0x)k0-k0k0-k0KSKS=k0TT Liu, BE280A, UCSD Fall 2006Aliasing Examplecos(2πk0x)k0-k0k0-k0KS2k0>KS>k04TT Liu, BE280A, UCSD Fall 2006Anti-aliasing FilterTo minimize aliasing, filter the signal before sampling.Typically, choice of highest sampling rate is determined bytechnical considerations and cost.Once the sampling rate has been determined, a low-passfilter is used to set the bandwidth of the signal.Example: CD sampling rate = 44.1 KHz; But real-worldmusic content has frequencies above 22 KHz. Apply ananti-aliasing filter that starts rolling off around 20 KHz.http://www.indiana.edu/~emusic/etext/digital_audio/chapter5_rate.shtmlTT Liu, BE280A, UCSD Fall 2006OversamplingWhy sample higher than the Nyquist frequency?-- the requirements on the analog low-pass filter arereduced.-- final filtering and downsampling can be done in thedigital domain where it is easier to get nearly ideal filters.5TT Liu, BE280A, UCSD Fall 2006Oversamplingk0-k0Hardk0-k0EasierKSTT Liu, BE280A, UCSD Fall 2006Suetens 2002Example1. C onsider the function ! g(x) = cos22"k0x( ). Sketch this function. You sample this signal in the spatial domain with a sampling rate ! KS=1/#x (e.g. samples spaced at intervals of ! #x). What is the minimum sampling rate that you can use without aliasing? Give an intuitive explanation for your answer.6TT Liu, BE280A, UCSD Fall 2006ArtifactsSuetens 2002ObjectEffect of NoiseAliasing due toinsufficientnumber ofdetectorsAliasing due toinsufficientnumber of viewsTT Liu, BE280A, UCSD Fall 2006Detector Sampling RequirementsSuetens 2002Samplinginterval ΔrBeamwidth Δs7TT Liu, BE280A, UCSD Fall 2006Smoothing of ProjectionSuetens 2002ProjectionBeam WidthSmoothedProjection2/(Δs)TT Liu, BE280A, UCSD Fall 2006Smoothing of ProjectionSuetens 2002! gs(l,") = rect(l /#s) $ g l,"( )Gs(kx,") = #ssinc(kx#s)G(kx,")8TT Liu, BE280A, UCSD Fall 2006Sampling RequirementsSuetens 2002SmoothedProjectionDetectorsΔr≤ Δs/2SampledSmooth ProjectionTT Liu, BE280A, UCSD Fall 2006View AliasingKak and Slaney9TT Liu, BE280A, UCSD Fall 2006View Sampling RequirementsSuetens 2002View Sampling -- how many views?Basic idea is that to make the maximum angularsampling the same as the projection sampling.! "FOVNviews= #rNviews,360="FOV#r="Nproj (for 360 degrees)Nviews,180="Nproj2 (for 180 degrees)TT Liu, BE280A, UCSD Fall 2006ExampleSuetens 2002! beamwidth "s = 1 mmField of View (FOV) = 50 cm"r = "s/2 = 0.5 mm500 mm/ 0.5 mm = N = 1000 detector samples#* N = 3146 views per 360 degrees $ 1500 views per 180 degreesCT "Rule of Thumb"Nview= Ndet ectors= Npixels10TT Liu, BE280A, UCSD Fall 2006Fan BeamSuetens 2002! "=#+$! r = R sin"rTT Liu, BE280A, UCSD Fall 2006Fan BeamSuetens 2002! "=#+$! r = R sin"r0r! "! "! "= 0! "max! "max11TT Liu, BE280A, UCSD Fall 2006Spiral CTSuetens 2002From http://www.sprawls.org/resources/CTIMG/classroom.htmNearest NeighborInterpolationLinearInterpolationTT Liu, BE280A, UCSD Fall 2006Longitudinal Aliasing in Spiral CTSuetens 2002From http://www.sprawls.org/resources/CTIMG/classroom.htm12TT Liu, BE280A, UCSD Fall 2006Multislice CTSuetens
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