1TT Liu, BE280A, UCSD Fall 2006Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2006X-Rays Lecture 2TT Liu, BE280A, UCSD Fall 2006Topics• Review topics from last lecture• Attenuation• Contrast• Noise• Image Equation2TT Liu, BE280A, UCSD Fall 2006TT Liu, BE280A, UCSD Fall 2006X-Ray ProductionPrince and Links 2005Collisional transfersRadiative transfers3TT Liu, BE280A, UCSD Fall 2006X-Ray SpectrumPrince and Links 2005Lowerenergyphotons areabsorbed byanode, tube,and otherfiltersTT Liu, BE280A, UCSD Fall 2006Interaction with MatterPhotoelectric effectdominates at low x-rayenergies and high atomicnumbers.Typical energy range for diagnostic x-rays is below 200keV.The two most important types of interaction are photoeletricabsorption and Compton scattering.Compton scatteringdominates at high x-rayenergies and low atomicnumbers, not much contrasthttp://www.eee.ntu.ac.uk/research/vision/asobania ! Ee"= h#" EB4TT Liu, BE280A, UCSD Fall 2006X-Ray Imaging ChainSuetens 2002Reduces effects of Compton scatteringTT Liu, BE280A, UCSD Fall 2006Attenuation510 50 100 15010.1AttenuationCoefficient500BoneMuscleFatAdapted from www.cis.rit.edu/class/simg215/xrays.ppt Photon Energy (keV)Photoelectric effectdominatesCompton ScatteringdominatesMore AttenuationLess Attenuation5TT Liu, BE280A, UCSD Fall 2006Intensity! I = E"EnergyPhoton flux rate! "=NA#tUnit TimeUnit AreaNumber of photonsTT Liu, BE280A, UCSD Fall 2006Intensity! "= S(# E 0$%)d# E X-ray spectrum! I = S(" E 0#$)" E d" E6TT Liu, BE280A, UCSD Fall 2006Attenuation! n =µN"x photons lost per unit lengthµ=n /N"x fraction of photons lost per unit length! "N = #n! dNdx= "µN! N(x) = N0e"µx! I("x) = I0e#µ"xFor mono-energetic case, intensity isTT Liu, BE280A, UCSD Fall 2006Attenuation! dNdx= "µ(x)N! N(x) = N0exp "µ# x ( )0x$d# x ( )Inhomogeneous Slab! I(x) = I0exp "µ# x ( )0x$d# x ( )Attenuation depends on energy, so also need to integrateover energies! I(x) = S0" E ( )" E 0#$exp %µ" x ;" E ( )0x$d" x ( )d" E7TT Liu, BE280A, UCSD Fall 2006ContrastBushberg et al 2001TT Liu, BE280A, UCSD Fall 2006Contrast8TT Liu, BE280A, UCSD Fall 2006! A = N0exp("µx)B = N0exp("µ(x + z))CS=B " AA=N0exp("µ(x + z)) " N0exp("µx)N0exp("µx)= exp("µz) "1Subject/LocalContrastBushberg et al 2001Background intensityObject intensityTT Liu, BE280A, UCSD Fall 2006Noise and Image QualityPrince and Links 2005Bushberg et al 20019TT Liu, BE280A, UCSD Fall 2006What is Noise?Fluctuations in either the imaging system or the objectbeing imaged.Quantization Noise: Due to conversion from analogwaveform to digital number.Quantum Noise: Random fluctuation in the number ofphotons emitted and recorded.Thermal Noise: Random fluctuations present in allelectronic systems. Also, sample noise in MRIOther types: flicker, burst, avalanche - observed insemiconductor devices.Structured Noise: physiological sources, interferenceTT Liu, BE280A, UCSD Fall 2006Histograms and Distributions3rd grade heights 6th grade heightsBushberg et al 200110TT Liu, BE280A, UCSD Fall 2006Gaussian Distribution1, 2, and 3 standard deviation intervals correspond to 68%,95%, and 99% of the observationsBushberg et al 2001TT Liu, BE280A, UCSD Fall 2006Poisson ProcessEvents occur at random instants of time at an average rateof λ events per second.Examples: arrival of customers to an ATM, emission ofphotons from an x-ray source, lightning strikes in athunderstorm.Assumptions:1) Probability of more than 1 event in an small timeinterval is small.2) Probability of event occurring in a given small timeinterval is independent of another event occuring inother small time intervals.11TT Liu, BE280A, UCSD Fall 2006Poisson Process! P N (t) = k[ ]="t( )kk!exp(#"t)"= Average rate of events per second"t = Average number of events at time t"t = Variance in number of eventsProbability of interarrival timesP T > t[ ]= e#"tTT Liu, BE280A, UCSD Fall 2006Example! A service center receives an average of 15 inquiriesper minute. Find the probability that 3 inquiries arrivein the first 10 seconds. "=15 /60 = 0.25"t = 0.25(10) = 2.5P[N (t =10) = 3) =(2.5)33!exp(#2.5) = .213812TT Liu, BE280A, UCSD Fall 2006Quantum NoiseFluctuation in the number of photons emitted by the x-raysource and recorded by the detector.! Pk=N0kexp("N0)k!Pk: Probability of emitting k photons in a given time interval.N0: Average number of photons emitted in that time interval = #tTT Liu, BE280A, UCSD Fall 2006Transmitted Photons! Qk=tN0( )kexp("tN0)k!Qk: Probability of k photons making it through object N0: Average number of photons emitted in that time interval = #tt = exp("µdz) = fraction of photons transmitted$13TT Liu, BE280A, UCSD Fall 2006Mean and Variance! For a Poisson process, the mean = variance, i.e. X ="2Therefore, the standard deviation is given by "= X For X - ray systems, if the mean number of counts is N, then thestandard deviation in the number of counts is "= N.TT Liu, BE280A, UCSD Fall 200614TT Liu, BE280A, UCSD Fall 2006! Poisson Distribution describes x - ray counting statistics.Gaussian distribution is good approximation to Poisson when "= X Bushberg et al 2001TT Liu, BE280A, UCSD Fall 200611015TT Liu, BE280A, UCSD Fall 2006TT Liu, BE280A, UCSD Fall 2006Contrast and SNR for X-Rays! Contrast = C =It" IbIb SNR =It" Ib#b! Ib=Nb" EA#t$ var Ib( )= var(Nb)EA#t% & ' ( ) * 2= NbEA#t% & ' ( ) * 2! "b= std Ib( )= NbEA#t$ % & ' ( ) ! SNR =CIb"b= C Nb= C #ARt$Photons/Roentegen/cm2Area Exposure inRoentgensDetectorefficiencyFractiontransmitted16TT Liu, BE280A, UCSD Fall 2006Example! " = 637 #106 photons R-1cm$2R = 50 mRt = 0.05%= 0.25A = 1mm2C = 0.1 (10% contrast)SNR = 0.1 6.37 #108& .05 & .25 & .01 = 6.320log106.3( )= 16 dBTT Liu, BE280A, UCSD Fall 2006! C =It" IbIb=N0exp "µ1(L " W ) +µ2W( )( )" exp("µ1L)( )N0exp("µ1L)SNR = C N0A exp("µ1L)µ1µ2LWArea A17TT Liu, BE280A, UCSD Fall 2006Magnification of Object! M(z) =dz=Source to Image Distance (SID)Source to Object Distance (SOD)Bushberg et al 2001zdTT Liu, BE280A, UCSD Fall 2006Magnification of ObjectBushberg et al 2001M = 1: I(x,y) = t(x,y)M = 2: I(x,y) = t(x/2,y/2)In general, I(x,y) = t(x/M(z),y/M(z))t(x,y) I(x,y)I(x,y)18TT Liu, BE280A, UCSD Fall 2006 Prince and Link 2005TT Liu, BE280A, UCSD Fall 2006Source magnification! m(z) = "d " zz= "BA= 1" M(z)Bushberg et al 2001d=z19TT Liu, BE280A, UCSD Fall 2006Image of a point object! Id(x, y) = limm "0ks(x / m, y /m)=#(x, y)s(x,y)s(x,y)! Id(x, y) = ks(x, y)m=1! Id(x, y) = ks(x / m,
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