When a target stimulus could potentially occur in one of several spatial locations, directing the observer’s attention to the target region prior to target onset generally leads to improved performance in accuracy (Bashinski & Bacharach, 1980; Carrasco, Penpeci-Talgar, & Eckstein, 2000; Cheal, Lyon, & Gottlob, 1994; Dosher & Lu, 2000b; Downing, 1988; Eckstein, Shimozaki, & Abbey, 2002; Enns & Di Lollo, 1997; Henderson, 1991; Lu & Dosher, 1998; Shiu & Pashler, 1994) and/or response time (Egly & Homa, 1991; Eriksen & Hoffman, 1972; Henderson & Macquistan, 1993; Posner, Nissen, & Ogden, 1978). How does spatial attention improve human performance? The full answer to this question lies first in a clear behavioral analysis of the impact of attention, and second in an integrated mechanistic account at multiple levels of the nervous system (Corbetta & Shulman, 2001).

At the behavioral level, we recently concluded that the primary role of spatial attention is to exclude external noise in the target region (Dosher & Lu, 2000a; Dosher & Lu, 2000b; Lu & Dosher, 2000; Lu, Lesmes, & Dosher, 2002), although attention can also increase the gain on the target stimulus (Carrasco et al., 2000; Lu & Dosher, 1998; Lu & Dosher, 2000; Lu, Liu, & Dosher, 2000; Morrone, Denti, & Spinelli, 2002), especially in peripheral cuing conditions. In this study, we are concerned with the functional mechanism by which spatial attention excludes unwanted information.

That spatial attention excludes unwanted information has also been consistently demonstrated at the neuronal level in monkey single cell recording from V2 (Luck, Chelazzi, Hillyard, & Desimone, 1997; Reynolds, Chelazzi, & Desimone, 1999), V4 (Haenny, Maunsell, & Schiller, 1988; Luck et al., 1997; Moran & Desimone, 1985; Reynolds et al., 1999; Spitzer, Desimone, & Moran, 1988), IT (Moran & Desimone, 1985), and MT and MST (Treue & Andersen, 1996) (for a review, see Desimone & Duncan, 1995), and at the neural population level by functional imaging (Kastner, De Weerd, Desimone, & Ungerleider, 1998; Kastner & Ungerleider, 2000). Animal single-unit recording studies suggest that spatial attention excludes unwanted information in two different ways: sharpening selectivity of the cellular signal (e.g., in orientation/spatial frequency) (Haenny et al., 1988; Spitzer et al., 1988), and/or weighing the input from the attended region/object more heavily in a competitive neural network without changing cellular tuning characteristics (Desimone & Duncan, 1995; Luck et al., 1997; Moran & Desimone, 1985; Reynolds et al., 1999; Treue & Maunsell, 1996).

In this study, we ask, at the overall observer level, does spatial attention exclude unwanted information (external noise) via sharpening of spatial frequency characteristics of the perceptual template, analogous to cellular retuning? The effect of spatial attention was investigated using a temporal cuing paradigm in combination with systematic manipulations of the spatial frequency characteristics of external noise added to the stimulus. Four T-like pseudo-character stimuli, one of them the target, occurred simultaneously on the screen, all embedded in filtered external noise. Each pseudo-character was in one of four possible, randomly chosen orientations. The observer was asked to identify the orientation of the target stimulus, whose location was cued either before (attended) or after (unattended) the target presentation. The pass-band of the filtered external noise was systematically manipulated to estimate the spatial frequency characteristics of the perceptual template (Henning, Hertz, & Hinton, 1981; Lu & Dosher, 2001; Pantle & Sekuler, 1968; Parish & Sperling, 1991; Solomon & Pelli, 1994; Stromeyer & Julesz, 1972; Wilson, McFarlane, & Phillips, 1983). Contrast threshold – signal contrast required to support a particular performance criterion level (e.g., 62.5% correct) – was measured in each condition. An observer model (Lu & Dosher, 2001) was used to quantitatively estimate the spatial-frequency selectivity of the perceptual template in both attended and unattended conditions from the threshold measurements. We found that spatial attention excluded unwanted information (external noise) without changing the spatial frequency selectivity of the perceptual template. Spatial attention also increased the gain to the contrast of the target stimulus.

All stimuli were presented on a Nanao Technology FlexScan-6600 monitor with a P4 phosphor, a refresh rate of 120 frames/s and a luminance dynamic range from 1 to 53 cd/m² (background = 27 cd/m²). Observers viewed the displays binocularly with natural pupil at a viewing distance of approximately 72 cm.

In each experimental trial, four T-like stimuli (Figure 1a) were simultaneously presented on the computer screen for 33 ms at 5.85-deg eccentricities in four spatial regions (Figure 2a). Each T was made of three 0.14 × 0.57 deg line segments and placed at one of four independently chosen random orientations. The observer was required to report the orientation of only one T, indicated by the arrow cue in the center of the display. In the attended condition, the cue occurred 167 ms before the target onset (Figure 2b); in the unattended condition, the cue occurred 75 ms after (Figure 2c). Across trials, the cue pointed to each of the four locations with equal probability. Two independent external noise frames, each lasting 33 ms, were shown, one immediately before and one immediately after the presentation of the Ts in each spatial region. Twelve filters, six low-pass and six high-pass, were used to generate filtered external noise (Figure 1c and 1d). The method of constant stimuli (Woodworth & Schlosberg, 1954) with seven signal contrast levels was used to estimate threshold contrasts in all the external noise and attention conditions.

The 12 digital filters were all ideal, with cutoff frequencies, , , , , , and c/deg (Figure 1c and 1d). The gains of the i^th low- and the i^th high-pass filters are

In each trial, one digital filter was chosen and used to generate eight filtered external noise images used in the four spatial regions: (1) A matrix was filled with real numbers, each a sample of a Gaussian random variable with mean 0 and standard deviation . The contrast range of the display system was from –1.0 to +1.0. Pixel contrast distribution with a standard deviation of 0.33 conformed reasonably to Gaussian. (2) The Fourier transformation of the noise image was computed. (3) One of the 12 digital filters (Equation 1) was applied to the output form (2). (4) An inverse Fourier transformation was performed on the filtered image to produce an external noise image in real space. The filtered external noise values were then sampled at 256 equally spaced linear contrast levels from –100% to +100%.

Three observers with informed consent and normal or corrected-to-normal vision participated in 5 practice and then 10 experimental sessions, each consisting of 672 trials with equal sampling of all the conditions (2 cues × 12 filters × 7 contrasts).

Four templates are required to identify the orientation of the Ts. For a noisy template matching process, we can derive the four optimal templates with maximum sensitivity in identification.

Denote the four templates as with normalized total energy

where is assumed to be Gaussian distributed due to internal and/or external noise (for discussion, see The Perceptual Template Model).

The i^th stimulus can be correctly identified by the i^th template if

For each , the maximum sensitivity for identifying the i^th stimulus is obtained when the expected total difference between its output to the matched signal and that to the nonmatched signals

To derive the template that maximizes Equation 6 and simultaneously satisfies the normalization constraint (Equation 3), we use the method of Lagrange multipliers (Riley, Hobson, & Bence, 2002). We define

where is a Lagrange multiplier. For an optimal template, the first-order derivative of Equation 7 is zero¹:

We can solve Equation 8 to derive the optimal templates:

Each optimal template is therefore made of a T in a given orientation (Figure 3a) minus the average of the Ts in all four orientations (Figure 3b). Like the Ts, the four optimal templates are made of the same shape in four orientations (Figure 3c).

A very important property of the optimal templates is that is the same for and all . In other words, the expected “responses” of a template to all the nonmatching Ts are identical. We calculated the Fourier magnitude spectra of the optimal templates as a function of spatial frequency. The resulting functions were identical for the four optimal templates and are plotted as a single function in Figure 3d.

Performance of human observers is generally suboptimal due to various processing limitations. In a Perceptual Template Model (PTM), observer inefficiencies are modeled as a perceptual template (suited to the signal stimulus), a nonlinear transducer function, and internal additive and multiplicative noises (Figure 4a). Originally developed to account for human performance in detecting or identifying stimuli embedded in white Gaussian external noise (Lu & Dosher, 1998; Lu & Dosher, 1999), the PTM has recently been extended and used to derive the spatial-frequency characteristics of perceptual templates from measurements of human performance in detecting or identifying stimuli embedded in filtered external noise (Lu & Dosher, 2001). Here, we briefly describe the PTM in the context of filtered external noise. The detailed formal development and validity tests of the PTM can be found in our previous publications (Dosher & Lu, 1999; Dosher & Lu, 2000a; Lu & Dosher, 1998; Lu & Dosher, 1999; Lu & Dosher, 2001).

A PTM (Figure 4a) consists of five components: (1) a perceptual template characterized by its gain in each spatial frequency range (j = 0, ..., 5), (2) a nonlinear transducer function () whose output is a power function of its input, (3) a multiplicative internal noise whose amplitude is proportional to the total energy in the stimulus (), (4) an additive internal noise with mean amplitude 0 and standard deviation , and (5) a task-dependent decision process based on the noisy output. Although four templates are required to identify the orientations of the Ts, the spatial frequency characteristics of the four templates were assumed to be the same (see Optimal templates). The functional form of this model is briefed in the equations below, and is illustrated in the pattern of predictions in Figure 4b.

The input stimulus in each trial consisted of a signal with a normalized mean Fourier magnitude spectrum of in each spatial frequency range (Figure 1b) and contrast , and filtered external noise with expected Fourier power spectrum . The signal in the stimulus is processed through the perceptual template and the nonlinear transducer to yield a signal output, :

where is the relative efficiency of the signal stimulus. The value of alpha scales the degree of match between the template and the signal in other domains that are not explicitly measured in the experiment (e.g., time) (Lu & Dosher, 2001). can be regarded as a parameter of the PTM model.

The filtered noise in the stimulus passes through the perceptual template and the nonlinear transducer with output variance:

The total variance of all the noise sources at the decision stage is the sum of the variances of all the noise sources: the external noise , the internal multiplicative noise and the internal additive noise :

Finally, the noisy “signal” with variance (across trials) is submitted to the decision process with signal discriminability, d’, determined by the signal-to-noise ratio²:

Threshold contrast – signal contrast required for the observer to reach a particular performance criterion level d’– can be expressed as a function of the pass-band and the cutoff spatial-frequency by inverting Equation 13:

Figure 4b shows the contrast threshold () predictions of an example PTM model with a known template. Thresholds are shown for three criterion performance levels as functions of cutoff frequencies of the low-pass and high-pass filtered external noise – the so-called “TvF” (threshold versus frequency) functions. To derive the spatial frequency characteristics of the perceptual template for an observer, TvF at one or several performance criterion levels are measured and then fit using Equation 14 with , , , , and as parameters (Lu & Dosher, 2001)Performance signatures of attention mechanisms

Within the PTM framework, attention may improve performance via four different mechanisms: (1) stimulus enhancement, which is mathematically equivalent to internal additive noise reduction in the PTM, modeled as multiplying by, (2) uniform external noise exclusion across all spatial-frequencies (multiplying by ), (3) changing the spatial-frequency characteristics of the perceptual template (replacing with in the attended condition), and (4) multiplicative internal noise reduction (multiplying by ). The effect of the four mechanisms of attention can be expressed in a single equation after modifying the corresponding terms in Equation 14:

The mechanism of stimulus enhancement has also been termed “increased contrast gain” and observed at the cellular level (Colby, Duhamel, & Goldberg, 1996; Luck et al., 1997; Maunsell, G., Nealey, & DePriest, 1991,; McAdams & Maunsell, 1999; Motter, 1993; Reynolds, Pasternak, & Desimone, 2000). The mechanism of spatial-frequency nonspecific external noise exclusion is consistent with increased weighting of input from the attended region/object without changing cellular tuning characteristics observed at the single unit level (Desimone & Duncan, 1995; Luck et al., 1997; Moran & Desimone, 1985; Reynolds et al., 1999; Treue & Maunsell, 1996). A spatial-frequency specific external noise exclusion mechanism parallels findings of attentional re-tuning of neuronal signal selectivity (Haenny et al., 1988; Spitzer et al., 1988). The mechanism of internal multiplicative noise reduction corresponds to a change of response gain control (Reynolds et al., 2000).

The performance signatures of the four attention mechanisms are shown in Figure 5 by invoking each of the four attention mechanisms for a PTM with known parameters. Stimulus enhancement increases the gain on the input stimulus, including both the signal and the external noise; it only affects thresholds when there is not much external noise added to the signal stimuli – at low cutoff values for low-pass filtered external noise and at high cutoff values for high-pass filtered external noise (Figure 5a). Both spatial-frequency specific and nonspecific external noise exclusion are only effective in the presence of significant amount of external noise. Therefore, external noise exclusion only affects thresholds at high cutoff values for low-pass filtered external noise and at low cutoff values for high-pass filtered noise (Figure 5b). Changing the spatial frequency characteristics of the perceptual template only affects thresholds in intermediate spatial frequencies (Figure 5c). Because internal multiplicative noise is proportional to the total amount of signal and external noise energy in the input stimulus, internal multiplicative noise reduction affects thresholds across all cutoff frequencies (Figure 5d).

Figure 5. Signature performance patterns of the four mechanisms of attention. Red curves: TvF functions in the attended condition. Blue curves: TvF functions in the unattended condition.

For stimulus enhancement and the two mechanisms of external noise exclusion, the magnitude of attention effects is invariant to the criterion performance level at which threshold is defined (Figure 5: left and right columns). For internal multiplicative noise reduction, the magnitude of attention effects depends critically on the criterion performance level. Therefore, measuring TvF functions at multiple threshold performance levels would allow us to distinguish mixtures of attention mechanisms (Dosher & Lu, 1999; Lu & Dosher, 1999).

PTM models with all possible combinations of the four mechanisms of attention were fitted to the measured TvF functions. The results were compared statistically using nested model tests based on F-statistics (Hays, 1988). The best-fitting model, statistically equivalent to the fullest yet with minimum number of parameters, identified the mechanism(s) of attention.

Data from each observer yielded 24 psychometric functions (percentage correct orientation identification versus signal contrast), corresponding to the 24 experimental conditions (2 cue 12 types of external noise). Threshold contrasts and error bars at three criterion performance levels, 50%, 62.5%, and 75% correct, corresponding to d's of 0.84, 1.24, and 1.68 in four-alternative forced-identification, were estimated from the psychometric functions (Wichmann & Hill, 2001a; Wichmann & Hill, 2001b). Threshold contrasts at all three performance levels are displayed in Figure 6a as functions of the cutoff spatial frequency of the filtered external noise. The four TvF functions in each panel of Figure 6a correspond to the low-pass and high-pass external noise conditions in both attended and unattended conditions.

The shape of the TvF functions reflects the spatial frequency selectivity of the perceptual template used by the human observer to perform the task in this experiment. The observed TvF functions appeared highly regular. Each TvF function in the low-pass series had three segments: At very low cutoff frequencies, the thresholds were constant and low (low-contrast signals supported performance) because the perceptual template did not pass external noise at very low spatial frequencies; performance is limited mostly by internal noise. At high cutoff frequencies, the thresholds were constant and high (high signal contrasts were required to support performance) because the perceptual template only passed external noise up to a certain spatial frequency; external noise with even higher spatial frequencies did not pass through the perceptual template. In mid cutoff frequencies, threshold increased with cutoff frequency. Similar properties held for the TvF functions in the high-pass conditions as cutoff frequency decreased.

Significant precuing advantages were observed in all external noise conditions; precuing reduced threshold by 27.3%, 20.4%, and 18.5% for the three subjects on average across all the noise conditions. The magnitude of threshold reduction did not depend on the performance level (Figure 6a). In Figure 6b, log contrast thresholds in the attended condition are plotted against those in the unattended condition at three performance levels for each of the eight highest external noise conditions: the four highest cutoff-frequencies in low-pass and the four lowest cutoff-frequencies in high-pass. All the points in the log scatter-plot fall on a straight line with slope 1.0, suggesting a constant threshold ratio between the attended and the unattended conditions across different performance levels and external noise conditions. The ratio constancy across different performance levels implies that attention did not alter multiplicative noise; rather, it improved performance via a mixture of stimulus enhancement and external noise exclusion (Dosher & Lu, 1999; Lu & Dosher, 1999). The ratio constancy across external noise conditions is further illustrated in Figure 6c, in which we plot the ratio of threshold attended versus threshold unattended as a function of cutoff frequency. In these high external noise conditions, observer performance was affected heavily by the amount of external noise passed through the perceptual template. However, all data points in Figure 6c fall on a straight line. This ratio constancy suggests that attention reduced the amount of external noise in all high noise conditions by the same proportion without changing the spatial frequency characteristics of the perceptual template.

The PTM (Equation 15) was used to fully test and quantify the qualitative conclusion and to rule out alternative interpretations. The TvF functions of all observers were best fit by a mixture of stimulus enhancement and uniform external noise exclusion mechanism of attention (r²=0.9798, 0.9802, and 0.9811). For observers KY and QL, this model is superior to all its subset models (p < .001), and the model that assumes all four mechanisms does not improve the fit to the data (p >.15). For observer SM, this model is superior to all its subsets (p <.001), and the model that assumes all four mechanisms provides only a marginally better fit to the data (p >.07).

The smooth curves in Figure 6a represent the predictions of the best-fitting PTM model, whose parameters are listed in Table 1. In this model, spatial attention reduced internal additive noise to 55.6%, 61.0%, and 64.7% of its magnitude in the unattended condition, and excluded external noise to 85.3%, 89.4%, and 90.7% of its unattended level for observers KY, QL, and SM, respectively. More importantly, spatial attention uniformly excluded external noise across all the spatial frequencies – it did NOT significantly change the spatial frequency characteristics of the perceptual template. This last point is best illustrated in Figure 7, where perceptual templates separately estimated for the attended (circles) and unattended (triangles) conditions, and jointly estimated for both conditions (heavy line), all fall on top of each other, confirming the qualitative observation (Figure 6b and 6c) that spatial attention did not change the perceptual template. In addition, the Fourier magnitude spectra of the best-fitting perceptual templates are quite similar to that of the optimal templates (Figure 3d; re-plotted as shaded areas in Figure 7), rather than that of the input signal stimuli (Figure 1b). Relatively little gain at low spatial frequencies (the so-called “DC suppression”) in letter identification has also been observed by Solomon and Pelli (1994) and others. Our analysis suggests that this might be resulting from the use of perceptual templates that are similar to the optimal templates by human observers.

One technical point is worth noting in interpreting these results: The “true” impact of external noise exclusion is best gauged by raising to its th power because both the external noise and pass through the nonlinear transducer function. For KY, QL, and SM, 0.7505, 0.7913, 0.8253, reflecting the magnitude of threshold reduction in the attended condition in high external noise.

In this study, we found that precuing of spatial location reduced contrast threshold by about 23% on average across all the observers and external noise conditions. Even though precuing was highly effective in excluding external noise in the attended condition, it did not significantly change the spatial frequency selectivity of the perceptual template of the observer. On the other hand, the magnitude of the observed precuing effect is relatively large compared to reported cases in which attentional improvements on perceptual sensitivity are reported (e.g., Bashinski & Bacharach, 1980; Dosher & Lu, 2000b; Downing, 1988; Henderson, 1996; Lu & Dosher, 1998; Lu & Dosher, 2000; Lyon, 1990; Shaw, 1984). For example, Bashinski and Bacharach (1980) reported an attention effect equivalent to 17% in two-alternative forced-choice (2AFC) when they required observers to detect a briefly displayed “O”; Henderson (1996) reported about 5% improvements in 2AFC discrimination of a briefly presented and masked X or O when he compared the valid and invalid cued conditions. In a pseudo-character identification task very similar to the one reported in the current study, we (Lu & Dosher, 2000) found that precuing reduced contrast threshold by about 16.6% in the presence of high external noise. Recently, larger magnitude attention effects have been reported in paradigms that require observers to perform two perceptual tasks with different judgment frames (Han, Dosher, & Lu, 2003; Morrone et al., 2002). Attention processes other than spatial attention (e.g., dual task central deficits) might be involved in producing these attention effects. Whether attention changes spatial frequency selectivity of the perceptual template in those paradigms is under investigation.

Studies of spatial attention using the temporal cuing paradigm have typically used simultaneous cuing as the baseline condition to measure effects of spatial attention (Cheal et al., 1994; Lu & Dosher, 2000). In this study, the baseline condition used a 75-ms postcue. Given that observers typically report 3 to 4 items in whole report and the time constant of partial report superiority effect is more than 300 ms (Sperling, 1960), we estimated that memory contributed at most 5% of the observed cuing effects in this study. In a pilot experiment, we investigated effects of temporal cuing in zero external noise over a large range of cue-target SOAs (from 175 ms to –105 ms) using the same stimulus parameters as those used in the main experiment. We found that contrast thresholds were essentially identical in the whole range of SOAs. We found that contrast thresholds were essentially identical in the whole range of SOAs. The significant cuing effects in the zero external noise condition observed in the main experiment have not been found in other recent studies. We suspect that this may reflect differences in the proportion of trials in low external noise in these experiments.

The finding that spatial attention does not change the spatial frequency selectivity of the perceptual template suggests that attention might have changed the selectivity of the perceptual template in other feature dimensions, such as orientation (but see Baldassi & Verghese, 2003). The result is, however, consistent with results from several single-unit recording studies that document attentionally increased neuronal sensitivity without changing neuronal selectivity (Colby et al., 1996; Luck et al., 1997; Maunsell et al., 1991; McAdams & Maunsell, 1999; Motter, 1993; Reynolds et al., 2000). On the other hand, one could not have concluded that spatial attention does not change spatial frequency selectivity of the perceptual template at the observer level by simply generalizing the results from single-unit studies to the overall observer level. Finding similar principles at multiple levels of the nervous system provides more insight into the brain mechanisms of attention.

Of particular interest is a recent neuronal model of attention based on competitive neural interaction in macaque monkey area V2 and V4 (Reynolds et al., 1999). The model accounts for both increased contrast gain in the attended region when the distractor is outside of the cell’s receptive field (Reynolds et al., 2000) and the apparent change of the spatial extent of the receptive field (Moran & Desimone, 1985) when both the target and the distractor fall in the cell’s receptive field (Reynolds et al., 1999). The presence of external noise in the target region may be analogous to the situation in which both target and distractors are in a cell’s receptive field (but see (Pelli, Palomares, & Majaj, in press). Competitive mechanisms similar to those proposed by Reynolds et al. (1999) could operate to reduce the effective gain of the perceptual template to the external noise in the attended condition in (nonsignal) feature dimensions.

Two recent psychophysical studies also concluded that spatial attention does not change the spatial frequency selectivity of the perceptual template. In one study, Eckstein et al. (2002) found that the spatial profiles of the perceptual template were the same in the valid and invalid cued locations in a simple two-location cueing “Posner” paradigm (Posner, 1980). In another study, Talgar, Pelli, and Carrasco (2004) concluded that, compared to neutral cuing, peripheral cuing did not change the spatial frequency tuning of the letter channels. In the first study, the cuing effects were attributed to changes in decision criteria or information weighting, not in sensitivity or information coding (Eckstein et al., 2002; Sperling & Dosher, 1986). In the second study, structural location uncertainty was not controlled. The observed cuing effect might substantially reflect reduction of location uncertainty in the decision process. It is therefore possible that changes of the tuning characteristics might occur when attention produces true sensitivity improvements. In contrast to these two studies, the primary focus of our research on spatial attention in this study and several previous publications has been on elucidating mechanisms of attention using paradigms that eliminate “structural decision uncertainty.” In these paradigms, all the potential target locations are consistently marked prior to each trial, and the observers are explicitly cued of the target locations in all the conditions before response. The procedure eliminates structural uncertainty for an ideal observer with no functional capacity limitations (Palmer, Ames, & Lindsey, 1993). This allows us to attribute the observed effects of spatial cuing to some form of capacity limitations rather than reduction of decision uncertainty. In a previous study using the same cuing paradigm as the current study, it was concluded that simultaneous cuing successfully eliminated uncertainty about the target location – it excluded from decision both the external noise and signal in the nontarget locations. And the advantages of precuing, therefore, reflected additional benefits in the target region, a limited capacity attentive process that only occurs in the target region (Lu et al., 2002). Consistent with the previous conclusion, we suggest that the effects of attention observed in this study reflect stimulus enhancement and external noise exclusion in the target region.

In summary, we found that spatial attention improved human performance via stimulus enhancement and external noise exclusion at the target location, without modifying the spatial frequency selectivity of the perceptual template. Similar principles describe attention mechanisms at multiple levels of the nervous systems. As a final note, we observed that most of the single-unit results on attention have been obtained when the animal was in a single steady-attention state (e.g., attending a particular spatial region throughout large blocks of trials), yet most behavioral studies on human observers have used paradigms that require alterations of the observer’s attention state (e.g., switching of spatial attention). It would be extremely important to conduct studies using identical or similar attention manipulations in different studies.

This research was supported by U.S. Air Force Office of Scientific Research, Life Science Directorate, Visual Information Processing Program. This work was first reported in the Psychonomics Society Meetings, 1999.

¹The second-order derivative of Equation 7 is also required to be negative for the optimal template.

²That the expected “responses” of an optimal template to all the nonmatching stimuli are identical led us to simplify the formulation of the PTM, calculating a single d’ for the identification task.