Chapter 1IntroductionThis is an in troductory course in probabilit y and random processes for imag-ing scientists. The purpose is to develop an understanding and abilit y inmodeling noise and random processes within the con text of imaging systems.The focus will be on stationary random processes in both one dim ension(time) and two dimensions (spatial). Power spectrum estimation will be de-v eloped and applied to signal characte rization in the frequency domain. Theeffect of linear Þltering will be modeled and applied to signal detection andmaximization of SNR. The matc hed Þlter and the Wiener Þlter will be devel-oped. Signal detect ion and ampliÞcation will be modeled, using noise Þgureand SNR as measures of system quality. A t completion of the course, thestudent should hav e the ability to model signals and noise within imagingsystems.The particular que stion of detection and measurement of quan tum ßuxwill be addressed through the mechanism of P oisson models. The conceptsbehind the performance of quan tum -coun t ing detectors will be developedand modeled. The nature of thresholds will be used to inv estigate saturationeffects. Detective Quan tum Efficiency (DQE) will be used to characterizethe performance of detectors.Repeated Bernouilli trials w ill be used to introduce the concept of randomprocesses. This naturally extends to the Gaussian noise model and also linksto the Poisson model. This combination of related concepts is very importantin modeling the performance of imaging systems.Random processes co me in many ßa v ors. We will concentrate on wide-sense stationary random processes because they are the most widely useful.Restrictingourfocustothisclassenablesustocoverabroadterritoryinthe78 CHAPTER 1. INTR ODUCTIONsm all amoun t of time that is available in this introductory course. There aremany ways to model wss processes, with two major categories being paramet-ric source models and spectral models. The parametric source models tak ethe viewpoin t that if one can describe a machine that could ha ve generatedthe process, then one will know a lot about its output. The spectral m odeldescribes the observed power vs frequency distribution without reference tothe source description. These approaches are, of course, closely related. Weshall sho w the relationship with a variet y of Þltered wh ite noise systems.The detection of phe nom ena, namely signals, in noise is of great interest.An extension is the estim ation of the parameters of signals in noise. We shallintroduce the classical system s for the detection of signals in noise and theestimation o f para mete r values.Most of the discussion is done within the framework of one-dimensionalprocesses. This might seem limiting i n an im agin g systems context . H ow ever,many images are transmitted, analyzed and even displayed by transformingin to a one-dimensional form. Consider television as one prominent example.Howe ver, two-dimensional structures do ha v e differen t properties. We willdiscuss some me thods for the modeling of random Þelds, which are partic-ularly useful in the consideration of geomet ric objects, textures and otherpatterns that are naturally two-dim ensional.These notes are undergoing con tinuous development, with additions andchanges each time this course is taught. They are not considered to beÞni shed by any means. However, they are intended to gather the lecturerecord in some sense and be a point of departure for studen ts in the course.Chapter 2P r ob a bility Modeling2.1 In troduc tionProbability models are used exten sively to describe the behavior and pe r-formance of systems. We will be particularly inte rested in applications toimaging systems, but the tec hniques are widely useful in science and engi-neering.Consider the problem of designing a photon-counting detector. A simplemodel of reality postulates a photon ßux of Φ photons per square meter persecond. Then one w ould expect to collect q = ΦAT photon s with a detec torplaced behind an aperture of size A that is kept open for an observ ationof time T. Im agine doing the photon counting experiment for a sequence ofvalues for the parame ter AT. The results ma y produce a plot like that sho wnin Figure 2.1. We note that as AT is increased the observation (normalizedby plotting q/AT ) seem s to settle around a value close to 4. There is arandom nature to the observations, but we ma y still be able to draw usefulconclusions from them. We may eve n be able to mak e useful predictionsabout the performance of an imaging system. This is but one example ofa situation in which w e would like the ability to analyze and understandrandom behav ior. The tools for this are based on the concepts of probabilitymodels.Probability modeling is an abstract tool that can be applied to man ydifferent tasks by interpretation of its sy mbolism in the domain of in terest.There are sev eral possible viewpoints of the abstract model. In this coursewe will adopt th e viewpoint of m odel ing an experiment.910 CHAPTER 2. PR OBABILITY MODELINGFigure 2.1: Observed values of photon coun ts divided by AT.2.2 Structure of a Pro babilit y ModelAn experim ent is a process that can be done repeatedly and which will pro-duce observable results. We will call each repetition a trial.Theresultofeach trial m ay be di fferent, even when the condition s are carefully controlled .Consider, for example, tossing a fair die and observ ing the top face . This isan experiment in which there are six possible results, corresponding to whichof the faces is up. Note, how ev er, that an outcome does not need to be anumerical q uan tity. Votin g can be modeled as an experiment tha t selectsbetween”yes”and”no,”forexample.We will use the term outcome for each of the possible results. Eachexperiment has a set of possible outcomes, which we will denote by U.Forthe die-tossing experiment the space of outcomes is U = {ei,i =1,...,6}corresponding to the six possible faces. On eac h trial exactly one element ofU must occur.It is common to refer to the set U as the sample space of the expe r im ent.A trial is then equivalent to a sel ection from the sample space.Suppose that a large number N trials of an experiment is to be conducted.2.2. STRUCTURE OF A PR OBABILITY MODEL 11Is there any prediction that can be made about the observations? On thebasis of experience or other information, w e may at least hav e an expectat ionof the frac tion of the results that correspond to