The preceding sections have shown that, for the detection of a signal in a noise background, the ideal observer will use a strategy of cross-correlating the stimulus with a template of the signal to be detected and make a decision about the presence or absence of the signal using a likelihood or Bayesian decision making strategy. The following subsections will review the literature investigating whether human observers can operate in the same way as ideal observers.
1.4.2.1 Can human observers cross-correlate?
As discussed in section 1.3, the optimum strategy for the detection of a SKE in Gaussian white noise is to cross-correlate the received image with a template of the signal (Green & Swets, 1966, p. 165) and use likelihood or Bayesian reasoning to select the option with the highest a posteriori probability and this is, therefore, the strategy that the ideal observer would use for this task. But can the human observer operate in the same way as an ideal observer?
Burgess & Ghandeharian (1984a) proposed that the Bayesian ideal observer would make use of a priori information and in the signal known exactly scenario this will include information about the size, shape and location of the signal. Thus, the Bayesian ideal observer will match a template
of the signal to the received stimulus and make a decision based on the most probable hypothesis; the strategy of cross-correlation or template matching. Burgess & Ghandeharian (1984a), in the rst of a series of four papers, asked the question of whether human observers could also use a cross- correlation strategy and investigated this by comparing the performance of human observers against the predicted performance of the ideal cross-correlating observer and an alternative strategy, the ideal auto-correlating observer. Whereas the cross-correlating observer cross-correlates the received stimulus with a known template of the signal, the auto-correlating observer is simply an energy detector and, hence, cannot use all the known properties of the signal in the same way as a cross-correlating observer, as illustrated in Equations 1.45 and 1.46.
Cross correlator : say yes if Xr · s > criterion (1.45)
Energy detector : say yes if Xr2> criterion (1.46)
Burgess & Ghandeharian (1984a) calculated the performance of the ideal cross-correlating observer and derived the performance of the auto-correlating observer, which we will refer to as an energy detector, using Monte Carlo simulations. They used a 2AFC protocol with static noise in one eld and static noise plus the signal in the other eld and compared the performance of human observers against the two ideal observers in two conditions; with phase information about the signal and without this information. An energy detector is unable to use properties of the signal, such as phase, and would predict the same performance in both conditions. This was not the case and, as shown in Figure 1.11, observers given phase information performed better than without phase information and, indeed, performed better than the ideal energy detector.
This supported the hypothesis of Burgess & Ghandeharian (1984a) that human observers could use a template with information, in this case phase, about the signal and supported the theory that humans can perform cross-correlation detection when given enough information about the signal that they can form a good template of it.
1.4.2.2 Can human observers make use of a priori and a posteriori information? Whilst Burgess & Ghandeharian (1984a) demonstrated the ability of the human observer to cross- correlate, the format of the task didn't provide evidence that they could also use a Bayesian strategy in the decision making process, however, this was tested in Burgess & Ghandeharian (1984b) and Burgess (1985). The second paper of the series, Burgess & Ghandeharian (1984b) introduced uncertainty about the signal by carrying out the experiment with multiple possible
Figure 1.11: Figure reproduced from Burgess & Ghandeharian (1984a) showing that the perfor- mance of a human observer, for the detection of a two cycle sine wave, when given signal phase information ( lled data points) is better than a human observer not given phase information (open data points). The performance of the human observer given phase information also exceeds that of the performance of a theoretical auto-correlating observer (energy detector) (dotted line) sup- porting the hypothesis that the human observers can cross-correlate.
locations ranging from 2 to 1800. The optimal, ideal observer, strategy was proposed as cross- correlating the signal locations with a known template of the signal and weighting the cross- correlation with the probability of the signal being in that location, a strategy referred to as the maximum a posteriori (MAP) decision strategy. The ideal observer performance was then further weighted by 50% to re ect the likely human observer e ciency. For the ideal observer, signal location uncertainty reduces performance as the number of locations increases. Once again, human performance was compared against the weighted ideal observer performance and the results for the human observers showed a good t with that of the weighted ideal observer, showing similar decrements in performance as uncertainty increased. The observation that ine ciency remained at 50% regardless of the number of locations supported the hypothesis that human observers can also carry out a MAP strategy as employed by the Bayesian ideal observer.
These conclusions from the rst two papers in the series (Burgess & Ghandeharian, 1984a,b) were further supported by the third paper in the series (Burgess, 1985), where signal uncertainty was introduced by increasing the the number of possible signal types to ten, with the signals selected from a Hadamard function set (Pratt, 1978). The study used an alternative forced choice paradigm with a signal selected from the set in a known location but conducted using three methods; detection with signal known, detection with signal unknown and an identi cation task with signal unknown.
Figure 1.12: Example gure, reproduced from Burgess (1985), showing percent correct against the signal to noise ratio for the 2AFC and 10AFC identi cation tasks. The theoretical comparison for the ideal observer is shown by the solid line and the comparison for two human observers (AB and RA) are shown by the open and lled symbols.
Burgess (1985) predicted the ideal observer performance by calculation for the signal known exactly condition and using a Monte Carlo simulation for the signal unknown condition and com- pared this to the human observer performance. As would be expected, the eect of increased signal uncertainty was to reduce ideal observer performance. The same performance decrements were seen for the human observers, who were found to operate with, on average, 33% e ciency when compared to the ideal observer. Figure 1.12, reproduced from Burgess (1985), illustrates this, showing the close agreement between the ideal observer (weighted for human e ciency) and the two human observers for two alternative forced choice and the 10 alternative forced choice conditions.
As in the second paper (Burgess & Ghandeharian, 1984b), the human response varied linearly with that of the ideal observer showing that signal uncertainty has the same eect on the human observer as on the ideal observer and provides support for the hypothesis that human observers can utilise prior signal knowledge to cross-correlate with the received stimuli and use a Bayesian MAP strategy for decision making.