Aperture Optimization in Emission Imaging Using Optimal LROC Observers P. Khurd, A. Rangarajan and G. Gindi I. I NTRODUCTION We address the problem of optimizing emission imaging systems based on task performance figures of merit (FOM). For example, one might optimize SPECT collimator properties to maximize detection of a lesion (signal) embedded in a complex patient background (though we shall address a simpler problem). In such system optimization research involving detection tasks, there is an escalatory path toward making studies more realistic. One seeks to (1) model the imaging process and data noise accurately (2) make the ensemble of possible backgrounds and signals as realistic as possible (3) make the task as realistic as possible - involving in our case localization as well as detection. To optimize the system, we need an observer and a scalar FOM to gauge detection performance as we vary system parameters. For a host of reasons, the mathematical ideal observer has been used. It is optimal in several senses and can, in principle, incorporate the ever more sophisticated models needed to escalate verisimilitude as per points (1)-(3) above. The ideal observer lends itself to computer simulation and forms an upper limit on possible human observer performance. Here, we consider a new form of ideal observer that we recently proposed [1] . It extends the detection-only task to a more realistic one of detection plus localization. We revisit an old problem in system optimization: For a planar pinhole imaging system, optimize the pinhole size to attain the best task performance. This problem was addressed by [2] and others for detection only, and here we consider what happens when a localization task is added. II. OPTIMAL LROC OBSERVER We first revisit the well-known case of an ideal observer for a 2-class (signal present? or absent?) detection problem. Let b be the possibly stochastic background object and s an additive signal (lesion). The vector g is the observed image. Then p(g|b) is the signal-absent likelihood and p(g|b + s) the signal-present likelihood. These likelihoods can be quite complex, incorporating many of the sources of variability discussed in points (1) -(3). The likelihood ratio t(g)= p(g|b+s) p(g|b) is an ideal observer that delivers a scalar test statistic t(g) which is compared to a threshold τ to decide signal-present if t(g) >τ and signal absent if t(g) <τ . At each τ , one can calculate a true positive fraction (TPF) that is the probability of deciding signal present when it is indeed present, and a false positive fraction (FPF) that is the probability of deciding that a signal is present when it is absent. By sweeping τ , one generates the well-known ROC curve TPF vs FPF. The area under this curve, AROC, is a FOM for this 2-class detection problem. The likelihood ratio is ideal in the sense that it maximizes AROC amongst all possible observers, and is also optimal in several other senses. G. Gindi is with the Department of Radiology, SUNY Stony Brook, NY, USA (telephone: 631-444-2539, email:gindi@clio.mil.sunysb.edu). P. Khurd is with the Department of Radiology, Univ. Pennsylvania , and A. Rangarajan is with the Department of C.I.S.E., University of Florida If we now demand that the observer both detect and localize the signal, the ideal observer changes [1]. Let f be the object, so that f = b for the signal-absent case. Under the signal-present hypothesis, f = b + s l , with the signal s l centered at location l, l ∈{1, ··· ,L}. We shall model signal location uncertainty with a pmf p R (l) on the location l. Our observer or decision strategy will detect the presence or absence of the signal in the data g and, if a signal is detected, it will report a location l(g) where it deems the signal to be present. If the true location of the signal falls within a tolerance region T (l(g)) about the reported location l(g), we will say that the observer has correctly localized the signal. Note that the inclusion of a tolerance region (typically a radius about the reported location) is an inherent part of a detection-localization task, and any FOM is dependent on the size of the tolerance region. The test statistic is given by t(g)= max l∈{1,··· ,L} X j∈T (l) p(g|b + s j )p R (j ) p(g|b) (1) with l(g) being the location at which t(g) is maximum. Note that Eq.(1) takes into account search tolerance. Also, this equation applies to a case of a discrete object and image, but can be generalized [1] to the realistic case of a continuous object and discrete image. One compares t(g) to τ and decides signal present at location l(g) if t(g) >τ and signal absent if t(g) <τ . Object Aperture Image Fig. 1. Pinhole imaging system. The object is one sample from an ensemble with correlated backgound noise and variable signal location. The arrows indicate the signal location in object and image spaces. Given the test statistics and reported locations, one can calculate the correct localization fraction CLF, which is the probability that a signal is correctly reported and localized given that it is present, as well as the FPF. By sweeping τ , one generates the LROC curve, a plot of CLF vs FPF. The area under the LROC curve, ALROC, is considered a FOM for the localization-detection problem. The decision strategy above is ideal in that no strategy can yield a greater ALROC and is also optimal in several other senses [1]. We refer to the decision strategy above as the “optimal LROC observer”, an ideal observer for the detection-localization task. To actually calculate ALROC, we use a Monte-Carlo (MC) procedure: For a given tolerance and aperture size, and with the signal absent or at one of its many locations, we generate many realizations of g and calculate t(g) for each g. Histograms of