Chapter 5Photon Counting5.1 IntroductionThe purpose of this section is to develop a simple model photon detection.The detector will be a device that responds to the number of photons thatit receives with a response function that is nonlinear. This is a model ofdetectors in whic h an onset threshold and a saturation level. The photonstreamitselfwillbetreatedasaPoissonprocesswithanaveragenumberqof photons received ov er an observation in terval.The use of a photon detector to detect and measure changes in the photonstream will be addressed. If one is to measure the brigh tn ess of a source,a bac k grou nd level, c han ges in source brightness, changes in backgroundlevel and the like, it is necessary to measure changes in the photon stream.This leads to the concepts of detectiv e quan tum efficie ncy (DQE) and noiseequivalent power (NEP), which are terms that are used in the detectionlitera t ure.5.2 S ta tistics of Sp atial Im ag e R ecor dingLet us look first at the basic issues that are in volved in the recording oft wo-dim e nsional image information. Dainty and Sha w have pointed out thatthe basic principles can be understood without reference to the technology8586 CHAPTER 5. PHOTON COUNTINGof an y specific imaging device1.A basic goal for an imaging device is that there should be a one-to-onerelationsh ip bet ween the incident quan ta at eac h location in the imag e planeand some measurable response of the device. We will use a model in whic hthe device is com posed of a spatial distribu tion of pho ton counters.Each counter gathers the photons that fall on its aperture. In the simplestmodel they will be of uniform size and will be distributed to cov er the ima g-ing plane. In a more complicated model the collecting area may vary withlocation, and even be chosen from a random size distribution. The locationsneed not be on an y kind of uniform grid, but can be scattered with somedistribution about the image plane. The response function of the coun terscan also sho w a variation . It is not difficult to see that the model can bequite comp licated in practice. We will use the simplest poss ible model inwhic h the detectors are close-pack ed, have a uniform area a, and hav e iden-tical response functions. This simple model will be adequate to demonstratethe basic princip le s.An exam ple of an arra y of closely packed hexagonal elements with un ifo rmsize is sho w n in Figure 5.1. The array is illuminated by a photon beam witha uniform brigh tness suc h that the average n umber of photons per element isq =4. Note that the n umbers actually range from 0 to 7 because the numberis a random variable. The proportion of cells in a large arra y that receive kphotons is go verned byP [k]=qke−qk!(5.1)Letusassumethateachdetectorhasafinite saturation level L beyondwh ich it will not respond. Then each detector w ill count up to L and no more.The effe ct of saturation can be modeled b y use of the Poisson distribution.In effect, each element has a response function that is linear up to saturatio nand then constant after that.Let X be the number of photons that arriv e at a detector. Then itsresponse is Y = h(X), whereh(x)=½x, x ≤ L − 1L, x ≥ L(5.2)1Dainty, J. C. and R. Shaw, Image Science: principles, analysis and evaluation ofphoto graphic-type imaging processes,AcademicPress,1974. Chapter 1.5.2. STATISTICS OF SPATIAL IMA GE RECORDING 87Figure 5.1: An array of detectors with a number of photon coun ts for eachelement. The number of photons is drawn from a Poisson distribution witha mean value of 4.The a verage response isµY= E[Y ]=∞Xk=0P [X = k]h(k) (5.3)ThenµY=L−1Xk=0kqke−qk!+∞Xk=LLqke−qk!(5.4)We w ou ld like to evaluate this equation in a closed form. If one writes out afew terms it will be seen that the abo ve equation can be written in terms of88 CHAPTER 5. PHOTON COUNTINGL summations.Eachofthesecanbeitselfsummed.µY=∞Xk=1qke−qk!+∞Xk=2qke−qk!+∞Xk=3qke−qk!+ ···+∞Xk=Lqke−qk!(5.5)Each of these terms is equal to the sum from 0 to ∞ less the sum of themissing terms. The sum from 0 to ∞ in each case is just the sum over thePoisson distrib ution, so must equa l 1.µY=(1−eq)+Ã1 −1Xk=0qke−qk!!+Ã1 −2Xk=0qke−qk!!+···+Ã1 −L−1Xk=0qke−qk!!(5.6)or,µY= L¡1 − f1(q, L)e−q¢(5.7)wheref1(L, q)=1LÃ1+1Xk=0qkk!+2Xk=0qkk!+ ···+L−1Xk=0qkk!!(5.8)The response µYvs q is shown in Figure 5.2 for the case L =10. The solidcurv e is the actual response and the dashed curv e is the ideal response asgiven b y equation (5.2). The plot will be similar for other valu e s of L, inwhic h it is nearly linear at the beginning and then bends over as the averagephoton rate approaches the saturation poin t.The saturation threshold ma y not be the only practical limitation of areal photon detector. It ma y also be the case that the detector needs toreceive some minim um number of photons before it begins to respond. Letus consider an array of elements that hav e a lower detec tion threshold of Tand a saturation limit of L+T when the active region is of width L as before.The response function ofh(x)=0,x<Tx − T +1,T≤ x<L+ TL, x ≥ L + T(5.9)The equation for the av er age count now becomesµY=∞Xk=0P [X = k]h(k) (5.10)5.2. STATISTICS OF SPATIAL IMA GE RECORDING 89Figure 5.2: The relationship bet ween the a verage response of the detectorelemen ts of the arra y vs the photon arrival rate for an array with a saturationlevel of L =10. The dashed line has a slope of unity and illustrates the idealresponse.We now substitute (5.1)and(5.9)into(5.10). Following a procedure that issimilar to that above,µY=L+T −2Xk=T(k − T +1)qke−qk!+∞Xk=L+T −1Lqke−qk!(5.11)This equa tion can be rew ritten in the form of equantion (5.7) if w e modifythe functio n f1so thatf1(L, q)=1LÃT −1Xk=0qkk!+TXk=0qkk!+T +1Xk=0qkk!+ ···+L+T −1Xk=0qkk!!(5.12)A plot of the av erage response with this detector is sho wn in Figure 5.3.Note that the response curve now rolls over at both the threshold and thesaturation limit. The response approximates the ideal curv e in the middle oftheactiveregion.The examples above sho w that an array of detectors will have a compositeresponse. This is the response that will be seen b y looking at a region of theimageinamannerthatcausestheeyetointegratetheresponseofalargen umber of cells. The image will begin to look noisy if the ey e can resolv edown to a sma ll nu mber of cells. In effect, the view er forms the averag e,