An earlier paper [J. Opt. Soc. Am. A 1, 775 (1984)] described a general ideal discriminator (a Stimuli-Defined-Exactly observer) whose sensitivity is limited only by the initial sequence of mechanisms in the visual system-the optics of the eye, the receptor lattice, receptor optics, and photopigment spectral sensitivities. In the present paper, further properties of the model are derived, and a similar model (a Stimuli-Defined-Statistically observer) is developed for conditions with stimulus uncertainty. To test the predictions of the models, two-point intensity discrimination, resolution, and separation discrimination were measured as a function of point-source energy and background luminance. Three important differences between resolution and separation discrimination were found: (1) At moderate to high intensity levels, separation threshold is-much smaller than resolution threshold. (2) At low intensity levels, resolution threshold is smaller than separation threshold. (3) Separation threshold decreases more rapidly than resolution threshold as a function of intensity. All three properties were predicted by the models. However, as expected, the quantitative fits are not accurate. The value of the models is that they allow one to determine, for almost arbitrary visual-discrimination tasks, what aspects of performance are accounted for by preneural factors. Furthermore, the models provide a precise metric of the discrimination information available at the receptors. This permits meaningful comparison of human performance across different tasks. Performance in any given visual-discrimination task may be limited by a large number of physical and physiological factors ranging from the physics of the stimuli up to the receptivefield properties of cortical units and beyond. How does one go about developing and testing reasonable theories of discrimination for a system of such complexity? A rigorous new approach,' based on the theory of ideal observers, 2 -7 forms the basis of the present study. Our approach might best be termed a sequential-mechanisms analysis. The essence of this analysis is to treat the early visual system as a sequence of information-transmission or -processing stages. At the cornea, a visual stimulus contains some finite, total amount of information that could potentially be used to perform some discrimination task. This information is then transmitted through and processed by a long series of stages, beginning with the optical system. The goal of a sequential-mechanisms analysis is to determine quantitatively what information relevant to the discrimination task is transmitted at each stage and, equivalently, what information is lost. In principle, one can thereby determine how much, and in what way, each stage of the visual system contributes to overall discrimination performance. The logic of doing a sequential-mechanisms analysis is not new. Indeed, there are many examples of this kind of research strategy in the vision literature. The classic example is the analysis of the color-discrimination information lost at the receptors because of the existence of a small number of photopigments and the univariance property (e.g., see Brindley 8 ). What is new about our approach is that it uses ideal discriminators together with sound physiological data in order to evaluate rigorously the information transmitted at each stage, beginning at the cornea. The present paper and a previous one' describe our recent attempts to apply a sequential-mechanisms analysis to all the preneural mechanisms of visual processing up through the level of quantum absorption in the photopigments. In particular, we have attempted to determine the contributions of these initial physiological mechanisms to performance in spatial-and intensity-discrimination tasks. The rigorous application of a sequential-mechanisms analysis requires development of quantitative models of the sort illustrated in Fig. 1 . There are three steps in the development of these models. The first is to obtain a complete representation of the stimuli. In order to model the total information available for discrimination, this representation must include a description of the intensity distributions and wavelength compositions of the stimuli over time as well as a description of the quantal noise and other sources of variability in the stimuli. Because of stimulus noise, the same nominal stimulus is not physically the same from trial to trial. Thus the two nominal stimuli in Fig. 1 are represented as sets. The second step is to develop a quantitative representation or model of the physiological mechanisms under consideration. The solid boxes in Fig. 1 represent the initial sequence of mechanisms in the visual pathway up to the photopigments. These are the only mechanisms that will be considered in this paper, but, in principle, what is accomplished for these stages could eventually be extended to additional levels in the visual chain. The third step is to measure all the information that has been transmitted by a mechanism or a sequence of mechanisms. The transmitted information can be measured with a machine, an ideal discriminator, that uses all the information to perform the task optimally. Indeed, the performance of this ideal discriminator is, by definition, a precise measure of all the transmitted information. The design of ideal discriminators is a common problem in statistical-decision theory 9 and is exactly the sort of problem that was solved during development of theories of ideal observers in both vision and audition. 2 -7 As is shown in Fig. 1 , crucial to the analysis is the idea of switching the ideal discriminator to the outputs of the various Fig. 1 . The structure of sequential-mechanisms models. There are three parts: (1) a complete representation of the stimuli, (2) a model of the physiological mechanisms, and (3) an ideal discriminator that can be switched to the output of any mechanism. mechanisms under consideration. By applying the appropriate ideal discriminator both before and after a stage, one can determine exactly how much information relevant to the discrimination was lost at that stage. By comparing the performance of the ideal discriminator with human performance, one can estimate how much information was lost in the stages following the one tapped by the ideal discriminator. In other words, if the model of the physiological mechanisms used in the analysis is sound, then only the differences between ideal and real performance will remain to be explained. A related benefit of carrying out a rigorous sequentialmechanisms analysis is that it provides a better understanding of the stimuli used in discrimination tasks. An ideal discriminator placed at an early level of the visual system measures the maximum amount of information available to the higher levels. This allows one meaningfully to compare the information content of different stimuli (assuming that the physiological model is sound). If altering the stimuli produces little or no change in ideal-discriminator performance, then there is little or no change in the amount of information available to perform the discrimination. Thus we should not be surprised if human performance changes little. On the other hand, if ideal-discriminator performance is greatly affected, we should not be surprised if human performance is greatly affected. In other words, the null expectation is that human performance should follow ideal performance qualitatively. If this does not happen, then we know that the sensory system is ignoring or is insensitive to some dimension(s) of information available in the stimuli. For example, Geislerl showed that the available information at the receptors in hyperacuity tasks is generally much greater than that in resolution tasks. Thus it is not surprising that humans perform much better in the hyperacuity tasks. Note that sequential-mechanisms analyses can be extended to tasks in which the observer is required to classify stimuli into more than two categories (i.e., to identification or classification tasks). In terms of Fig. 1 , we simply increase the number of stimuli, increase the number of response alternatives by an equal amount, and replace the ideal discriminator with an ideal classifier. Geisler' described a sequential-mechanisms model for spatial and intensity discrimination and reported its performance in two-point acuity, hyperacuity, and intensity-discrimination tasks. In the following sections we (1) derive further properties of the original model, (2) develop a model that permits positional and other sorts of stimulus uncertainty, and (3) report the results of psychophysical experiments in which we measured two-point intensity discrimination, resolution, and separation discrimination as a function of stimulus intensity and adapting-background luminance. Although the sequential-mechanisms models described below are applied here only to the various two-point discrimination tasks, the models are quite general and can be applied to almost any spatial-, intensity-, or color-discrimination task. Different sequential-mechanisms models can be distinguished by the range of stimuli to which they apply and by the set of physiological mechanisms that are modeled. In this section we describe two models, one of them a generalization of the other. Both consider the same physiological mechanisms (solid boxes in Fig. 1 ), but they differ in the range of stimuli that are considered. The first model applies to experimental conditions in which the sole source of trial-to-trial variability in a given stimulus is that due to quantal fluctuations. Because the stimuli are precisely defined except for the inherent Poisson randomness of light, we call it a Stimuli-Defined-Exactly, or SDE, model. For most well-controlled psychophysical experiments, it is reasonable to assume that the only source of stimulus noise is quantal fluctuations. Nonetheless, it is important to extend the model to conditions in which there are additional sources of noise or uncertainty. These might include random variations in the position, shape, duration, or wavelength composition of the stimuli. Because of the extra random variation in the stimuli, we call the more general model a Stimuli-Defined-Statistically, or SDS, model. Development of the SDS models is important for several reasons. First, they are the appropriate models for evaluating visual performance in tasks with additional stimulus variation. For example, discrimination or identification in the real world typically involves assigning wide ranges of stimuli to two or more categories. An inspector on an assembly line might have to discriminate between acceptable or deformed parts. Even if the deformations are always of the same simple type, the stimuli to be evaluated are always in different orientations and positions. Second, it is possible that real visual systems inadvertently produce some significant level of internal noise. For example, the first mechanism in Fig. 1  The derivation of our SDE model for foveal stimuli is summarized here; further details are provided elsewhere. 1 The first step is to specify the stimuli. In this presentation, attention is restricted to brief, broad-band stimuli (e.g., white tungsten light) presented monocularly under good fixation. Thus a stimulus is specified by giving its intensity distribution and duration. The Poisson process accurately describes the sole source of stimulus noise-quantal fluctuations. The next step is to develop a quantitative representation of the physiological mechanisms in the solid boxes of Fig. 1 . 0 and Ditchburn" indicate that under good fixation the standard deviation of eye position from trial to trial is between 2 and 5 min of arc. If these random variations in eye position are not accurately monitored by the visual system, they could effectively introduce some position noise into the stimuli across trials. However, one purpose of developing the SDS model was to assess the potential effects of position noise resulting from eye movements. Thus, for the moment, we assume a fixed eye. The optics of the eye at a given eccentricity can be described by a point-spread function, h(x, y). The point-spread function used here was derived from the foveal line-spread function reported by Campbell and Gubisch1 2 for white light and a 3-mm pupil. Their line-spread function is adequately described by the sum of two Gaussian functions; thus the point-spread function is given by where The parameter values are given in the caption to Fig. 6 below. This point-spread function does not include the floor of stray light that is due to scatter in the ocular media. It is unlikely that more than 10% of all incoming light is diffusely scattered. For a wide range of stimulus conditions, including those considered here, we have shown that a 10% level of stray light has minor effects on the performance of our SDE observer. Thus stray light was not included in calculating the present predictions. However, it would have to be included under some stimulus conditions. The ocular media also act as light filters. We have used the estimates of ocular transmittance reported by Wyszecki and Stiles. 13 There is one other potential source of information loss in the optics. Under steady fixation, the accommodative state of the eye fluctuates at around 2 Hz but with an amplitude of less than 0.25 D. 14 By altering the point-spread function above we have shown that this mechanism plays a minor role under most circumstances. Receptor Lattice It is assumed that in the fovea the photoreceptors are placed at the nodes of a triangular array, 15 such that the internodal distance equals the diameter of the receptors (about 0.6 min of arc). The receptors are assumed to be circular with a diameter of 0.6 min of arc at the widest part of the inner segment. It is further assumed that all quanta entering the inner segment pass into the layers of photopigment within that cone. The cones thus perform a small amount of spatial integration (the aperture effect). The Stiles-Crawford effect is not considered because of the 3-mm pupil size and the foveal stimulus presentation. In this paper we consider only chromatically broadband stimuli that stimulate the middle-and long-wavelength cones about equally. In particular, the fraction of effectively absorbed quanta for a stimulus was computed by first finding its 555-nm equivalent. It was then assumed that half of these equivalent quanta are effectively absorbed whether they fall in a middle-or long-wavelength cone. The short-wavelength cones are so few in number, especially at the center of the fovea, that they could be ignored for the spatial-and intensity-discrimination tasks of interest here. Mean Absorbed Quanta All the mechanisms above were combined in order to compute the mean number of quanta absorbed in each receptor. Let 1a(x, y) be the luminance distribution of a stimulus in candelas per square meter; then the mean number of photons effectively absorbed by the ith receptor in the lattice, whose position is denoted by the coordinates (xi, yi), is given by where D is the duration of the stimulus in seconds, S is the pupil area in square millimeters, T 555 is the transmittance of the ocular media at 555 nm, E 555 is the quantum efficiency of the photopigments at 555 nm, h(x, y) is the point-spread function, and r(x, y) is the receptor shape (in this case a cylinder with a diameter of 0.6 min). The symbol * represents the operation of convolution. No visual system is free of intrinsic noise. There are undoubtedly several sources of intrinsic noise in real visual systems. We include in our model an additive, dark-light noise of photonlike events. 1 6 Such noise must be present to some extent in any real visual system, if for no other reason than the occasional thermal breakdown of photopigment molecules. The actual rate of photonlike events in the human photoreceptors is not known at this time. It is included here under the assumption that convincing estimates of its value(s) will become available in the future. Let x 0 be the mean number of noise events per receptor in time period D. Then the mean number of effective quantum absorptions in the ith receptor is given by Poisson Noise In the fixed period of time D, the number of effective quantum absorptions Z(xi, yi)] in the ith photoreceptor of the lattice is described by the Poisson density: where ai is an abbreviation for a (xi, yi). Ideal Discriminator For present purposes, we derive the performance of the ideal discriminator when it is placed at the output of the last of the above-named mechanisms (see Fig. 1 ). On each trial of the experiment the ideal discriminator must decide which of two alternatives (a or A) was presented. It can be proved that the discriminator bases its decision on the likelihood ratio or some monotonic transformation of it. The intuition behind the use of the likelihood ratio is the following. On each trial of the experiment there is some pattern of quantum absorptions across the receptors that can be represented by the vector (Z1, Z 2 ,. . . , Zn ), where Zi [an abbreviation for Z (xi, yi)] is the number of quanta absorbed in the ith receptor. The ideal discriminator first computes the probability of getting this pattern of absorptions assuming that stimulus a was presented. Next, it computes the probability of getting this pattern of absorptions assuming that stimulus / was presented. The ideal discriminator then simply picks the most probable stimulus alternative. (This assumes that the alternatives are equally likely and the discriminator is trying to maximize its proportion of correct responses.) It is not difficult to show' that the likelihood decision rule is equivalent to the following rule. Compute the quantity Z by the formula Z = EI Zi ln(i/ai). (5) If Z exceeds some constant c, respond that stimulus / was present; if Z is less than c, respond that stimulus a was present. Since the decision variable Z is just the weighted sum of the response of each receptor, it can be calculated by a single linear weighting function (receptive field) across the receptor array.1 Sensitivity The sensitivity of the ideal discriminator is given to good approximation by the following formula for d': Threshold in a discrimination task is reached when d' attains some criterion value. In a single-interval forced-choice task, this value is typically 1.36, which corresponds to 75% correct when the two alternatives are presented randomly on each trial with a probability of 0.5. In a two-interval forced-choice task, a d' value of 0.96 corresponds to 75% correct. The SDS model is a direct generalization of the SDE model in which we allow sources of stimulus variability in addition to quantal fluctuations. These additional sources of noise can be described by an uncertainty function. Uncertainty Function Let v = (v1, v 2 , . . , vm ) be the set of parameters that can vary from trial to trial, thus providing the uncertainty. In the case of position variability, v = (x, y)-the spatial position of the stimulus. In general, these parameters can represent any stimulus dimension: size, duration, shape, etc. The uncertainty in a given situation can then be described by an uncertainty function f(v), which gives the probability of obtaining any given parameter values on a trial. Since f (v) is a probability-density function, the volume under it must be 1.0: where the integral is over all the parameters in v. Ideal Discriminator It can be shown (see Appendix A) that the ideal discriminator uses the following decision variable under conditions of stimulus variability: In the case of positional noise, the above likelihood ratio reduces to where C is a constant. If stimulus a is a uniform field (i.e., a simple detection task) further simplification is possible: where ; is the mean number of quanta absorbed per receptor from the uniform field. In all the above cases, if L exceeds a criterion level the ideal discriminator responds that stimulus /3 was present; if L falls below the criterion the discriminator responds that stimulus a was presented. Notice that in Eqs. (8) and (9) the quantities in the exponents are approximately the cross correlations' 7 of the stimulus (the Zi's) with the natural logarithm of the expected stimulus shape. In other words, the likelihood of a stimulus alternative is computed by exponentiating the cross correlation of the signal with the logarithm of the stimulus alternative's shape, multiplying by the uncertainty function, and summing over the uncertainty region. We now turn to the two-point experiments. The performance of the SDE and SDS observers will be described in the discussion section. Thresholds for three types of task: intensity discrimination, separation discrimination, and resolution, all involving twopoint stimuli, were obtained as functions of the intensity of the point sources and of the presence or absence of an adapting background. The two-interval forced-choice paradigm was used for all tasks. Intensity-discrimination, resolution, and separation-discrimination thresholds were obtained at five intensity levels: 3.0, 3.4, 3.8, 4.2, and 4.6 log quanta/flash per point source incident at the cornea. There was also an intensity-discrimination condition in which only the increment was presented (i.e., an absolute threshold condition). Two adapting background levels were used, a dark background and one set to 20.8 Td (7.41 log quanta/sec deg-2 at the cornea). The fixation points remained on continuously at 4.0 log quanta/sec per point at the cornea. One woman and two men, including one of the authors (WG), served as observers. All had normal Snellen acuity and color vision. All three were initially inexperienced in hyperacuity A. DISPLAY CONFIGURATION The two-point stimuli were presented on a Hewlett-Packard 1332A display oscilloscope (P-15 phosphor) under the control of a PDP11 computer that also collected and analyzed the subjects' responses. By the manufacturer's specification, the spot size was 0.3 mm. Observers were seated in a totally dark room at a distance of 504 cm from the display. Thus the spots were 0.2 min of arc in diameter. The adapting background was displayed on an identical scope driven by two oscillators. Light from the adapting background was introduced into the path of the light from the stimulus display by means of a half-silvered mirror. The refresh rate for both screens was set to 100 Hz. Subjects viewed the display monocularly with the right eye through a 3-mm artificial pupil-a diameter picked to minimize the size of the point-spread function.' 2 A bite-barheadrest combination was used to ensure stability of the observer's head. Although all three subjects tested were emmetropes, two of them could (subjectively) maintain somewhat more accurate accommodation over the length of a session with a positive lens inserted just in front of the artificial pupil. The best value of the lens was found by trial and error. For the three subjects they were 0.0, +0.12, and +0.25 D. The stimulus display as seen by the observer is shown in Fig.  2A . It consisted of an almost square background field, four fixation points, and the test stimuli. The test stimuli for the three tasks in the experiment are shown in Figs. 2B-2D . In the separation-and intensity-discrimination tasks the base separation between the two point sources, 0, always remained at 3.0 min of arc. Note that in the intensity-discrimination task the intensity increment was added only to the point source on the right. On each trial, a warning tone was presented 0.5 sec before stimulus presentation. Then the two stimuli were flashed for 100 msec (in random order) separated by a temporal interval of 0.5 sec. This was followed by a 1-sec response interval, at the end of which subjects were given auditory feedback indicating whether their responses were correct or incorrect. Two seconds later, the next warning tone occurred. If the subject failed to respond (a rare event), no feedback was given, and the trial was not included in the data analysis. The total duration of a single trial, including response collection and feedback, was 3.4 sec.
Geisler and Davila REF analyzed the performance limits for an ideal observer case when the signal is defined exactly (SDE) or when the signal is defined statistically (SDS).
Ideal discriminators in spatial vision:two-point stimuli
Journal of the Optical Society of America. A, Optics and image science
Journal of the Optical Society of America. A, Optics and image science
Physics Medicine