Adaptive detection mechanisms in globally statistically nonstationary-oriented noise

Transcription

1 Zhang et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A 1549 Adaptive detection mechanisms in globally statistically nonstationary-oriented noise Yani Zhang, Craig K. Abbey, and Miguel P. Eckstein Vision and Image Understanding Laboratory, Department of Psychology, University of California, Santa Barbara, California Received November 4, 2005; revised January 18, 2006; accepted January 22, 2006; posted February 3, 2006 (Doc. ID 65804) Studies have shown that human observers can adapt their detection strategies on the basis of the statistical properties of noisy backgrounds. One common property of such studies is that the backgrounds studied are (or are assumed to be) statistically stationary. Less is known about how humans detect signals in the more complex setting of nonstationary backgrounds. We investigated detection performance in the presence of a globally nonstationary oriented noise background. We controlled for noise-correlation effects by considering a stationary background with a power spectrum matched to the average spectrum of the nonstationary process. Performance of a nonadaptive linear filter that was unable to make use of differences in local statistics yielded constant performance in both the stationary and the nonstationary backgrounds. In contrast, performance of an ideal observer that uses local noise statistics yielded substantially higher (140%) detectability with the nonstationary backgrounds than the stationary ones. Human observers showed significantly higher (33%) detection performance in the nonstationary backgrounds, suggesting that they can adapt their detection mechanisms to the local orientation properties Optical Society of America OCIS codes: , , , INTRODUCTION There is a large body of work studying how human observers detect or search for signals in noisy backgrounds. 1 4 Several studies have shown that observers can adapt their strategy on the basis of statistical properties of the noise when integrating image information to make perceptual decisions about the presence or location of a target. 5 7 One commonality of most of these studies is that the backgrounds are statistically stationary. Thus, the pixel variance is assumed to be constant across the image, and the correlation between two pixels (over an ensemble of images) depends only on their position relative to each other. This guarantees (under Gaussian assumptions) that the optimal strategy to perform the task reduces to the dot product of a prewhitened matched filter and the image data at all possible signal locations. In natural image backgrounds where stationarity cannot be guaranteed, most studies have assumed stationarity or have pooled the statistical properties from different image locations Thus, we know less about how humans detect signals in nonstationary backgrounds where the statistical properties vary across image locations. A stationary spatial random process is one that is invariant in some sense to spatial translations. Strict stationarity is defined as translation invariance of the full process distribution, whereas the weaker condition of wide-sense stationarity requires only invariance of the first and second moments of the distribution. 13,14 Both of these definitions require a process of infinite spatial extent, a condition impossible to achieve in any practical setting. Therefore it is common to specify stationarity within a bounded region such as the confines of an image displayed on a monitor. In this setting, the term global stationarity refers to stationarity within the domain of the image. It is also possible for stationarity to exist in a smaller subregion of an image, which is referred to as local stationarity. An alternative approach to defining stationarity in a square or rectangular region is to assume a periodic extension of the region as is done for Fourierseries expansions. This is often referred to as cyclic stationarity because it leads to wrap-around effects at edges of the region. Many natural and medical images present statistically nonstationary properties that must be accommodated to most effectively use these images to perform visual tasks. One form of nonstationarity that occurs regularly is nonstationary oriented textures. Figure 1 15 shows both natural [Figs. 1(a) and 1(b)] and medical [Figs. 1(c) and 1(d)] image examples with nonstationary oriented textures. Little is known about how human observers visually detect signals in nonstationary noise. We investigate the ability of human observers to optimize their detection strategy of a Gaussian signal in the presence of globally stationary (GS) versus globally nonstationary (GNS) textured noise fields in four-alternative forced-choice (4AFC) detection tasks. In one condition, we use GNS textured noise with a correlation structure that changes in orientation across the four locations. Each of the four locations is locally stationary but GNS, since the correlation structure has a different orientation at each location. For comparison, and to control for the global effects of textured noise, we also consider a GS noise texture that has been matched to the average local power spectrum of the nonstationary noise across the four locations. The two image sets used in this study are created from a general filtering algorithm that can be used to generate GS or GNS noise textures. The filtering operation is de /06/ /$ Optical Society of America

2 1550 J. Opt. Soc. Am. A/ Vol. 23, No. 7/ July 2006 Zhang et al. Fig. 1. Examples of oriented textures. 15 (a) Cracks in a tree trunk, (b) ridges in a finger print, (c) ribs in a chest x ray, (d) glandular tissue and ducts in a mammogram. signed so that linear models that derive a single template assuming global stationarity across the four locations will yield approximately the same level of performance in both conditions. We refer to these models as nonadaptive models because they do not use different strategies (templates) based on the varying local noise statistics in the nonstationary images. In contrast, adaptive models that use the varying local noise statistics in the nonstationary noise to derive individual templates for each location will perform better with the nonstationary noise backgrounds. Comparison of human signal detection performance across the two experimental conditions allows us to determine whether human observers have the ability to adapt their strategy to the local statistical properties of the GNS noise. 2. METHODS A. Generation of Stationary and Nonstationary Backgrounds The images we use to evaluate detection performance in stationary and nonstationary noise are created using a general filtering approach. In this approach, a stationary or nonstationary background is generated from a sample of white noise by applying a corresponding stationary or nonstationary filter. Let n be a sample of Gaussian white noise, with n i,j the intensity of pixel i,j. Each pixel value is sampled independently from a Gaussian distribution with a mean of zero and a variance of 2 0. The filtering operation can be written as b i,j = i,j n i,j f i,j i,j. 1 At its most general, this equation allows for a different filter at each position in the background. A stationary filter constrains the form of the filters to be shifted versions of a common profile, f i,j i,j = f i i,j j. This work relies heavily on elongated Gaussian filters in which the base filter function f is defined to be f i, j = exp 2 1 i 2 x + j y 2, 3 2 Fig. 2. Local filtering of images. The local filtering approach is used to generate (a) GNS and (b) GS textures. Each panel shows a sample background with four possible signal locations indicated (center of white outlined boxes) and the four corresponding local filters. For the stationary background, all local filters are identical. where i=i i and j=j j. This definition assumes that distances are measured in terms of pixel size. The parameters of the Gaussian, x and y, characterize the width of the Gaussian in the x and y directions. Throughout this work, these parameters are set to x =7 (0.30 deg of visual angle at 40 cm viewing distance) and y = deg, which yield an elongation factor of 7 in the x direction. We can create a nonstationary-oriented filtering operation from this base filter by rotating it with a locationdependent rotation angle, i,j. The resulting filter function is given by f i,j i, j = exp 2 1 i cos i,j j sin i,j x + i sin i,j + j cos i,j y

3 Zhang et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A 1551 Note that the dependence on location is encompassed entirely by the rotation angle i,j. We use these filters to create the GNS test images. In Fig. 2(a), we show an example of a generated background texture. The different oriented filters from four possible target locations (local stationary areas) are also shown. As a control for the effects of noise correlations, we also consider a stationary filter matched to the average power spectrum of the four filters at the possible target locations. The resulting filter is given by f stat i, j = FFT FFT f loc FFT f loc FFT f loc FFT f loc 4 2 1/2, 5 where FFT stands for the fast Fourier transform and f locm, m=1...4, are the oriented filters [local filters, as shown in Fig. 2(a)] at each of the four possible signal locations. In Fig. 2(b), a sample background texture is shown with four copies of the stationary filter defined in Eq. (5). We refer to this type of image as the GS image. The stationary filter f stat was obtained from the combination of the four oriented filters; i.e., the performance of model observers that assumed statistical stationarity (making no use of differences in local statistics) would be the same for both test image sets. B. Generation of Image Sets 1. Background Generation The background-generation procedures described above were used to create sets of images for detectability studies. The GNS test images were created by filtering a white-noise background using filters with varying orientations ranging from 1 to 180 deg. To generate a GNS image with local stationary regions, we divided the whole image into four classes of areas (as shown on the left side of Fig. 3): nontransition area, x-transition area, y-transition area, and x-and-y-transition area. The nontransition area has constant values for the filter s orientation. The x-transition area has filter orientation values changing linearly horizontally across the image. The y-transition area has values changing linearly vertically across the image. The x-and-y-transition area has predetermined edge values and a center point value for the filter orientation. The values within the x-and-y-transition area are determined according to the distance from the location of the center to the edge points. Nontransition areas (locally stationary areas, in size) served as the possible signal locations. The orientations for the four oriented filters in the nontransition areas are 37, 74, 111, and 149 deg. The middle image in Fig. 3 shows the filter orientation map in gray values (pixel values represent the orientation of the filters ranging from 1 to 180 deg), and the right image in Fig. 3 shows the filtered white-noise backgrounds with the varying orientation filters. We preserved local stationarity in the GNS images to allow for easier calculation of the model observer performance. The GS test images were created by filtering a white noise by using a stationary filter f stat. 2. Test Image Generation The nonstationary and stationary procedures for generating images are depicted in Fig. 4. After a background is generated, low-contrast white noise is added, with a pixel standard deviation of n =7.5 corresponding to a rootmean-square (rms) contrast of To this combination a signal is added in one of four possible locations. Images used in psychophysical studies also had four pairs of location cues to help focus the observer s attention on the possible locations. The final image, g, can be written as g i, j = b i, j + n i, j + s m i, j, where b is the nonstationary or stationary background, n is the additional white noise, and s m is the signal in location m. In each trial, m is a random integer variable from 1 to 4. For each test condition (GNS and GS) 10,000 background images with the low-contrast white noise added were created and stored. The signal was added at run time to ensure that the model observer and human observer used the same data set since human observer experiment requires cues around each of the possible signal locations. C. Detection Task The task is to localize the signal in one of four possible locations [four-alternative forced-choice (4 AFC) task]. The signal is a Gaussian with a standard deviation of six pixels 0.26 deg. The peak contrast of the signal is set to 6 Fig. 3. Orientation transition map generation for the GNS test image. Left image: layout of transition areas. Middle image: orientation map; gray values represent the orientation (from 1 to 180 deg). Right image: GNS test image sample.

4 1552 J. Opt. Soc. Am. A/ Vol. 23, No. 7/ July 2006 Zhang et al. Fig. 4. Image generation procedure. The procedure for generating (a) nonstationary and (b) stationary images. For each sample image, an initial sample of white noise is generated and then filtered with the local filtering algorithm. A second low-contrast white-noise field is then added to the result. The final step consists of adding the signal profile to one location at random and location cues. The four location cues were superimposed only on the images used in the psychophysical studies. be The signal is randomly added to one of four possible locations in the background images. These locations are indicated by fiduciary marks for the human observers and are spaced 112 pixels 4.78 deg apart vertically and horizontally (Fig. 4). The signal is known to the human observer and does not vary from trial to trial [signal known exactly 2 (SKE)]. D. Model Observers Model observers are mathematical algorithms that process an input image (or images) and render a decision about the presence or location of a signal. There are many classes of models that generate explicit predictions about the human visual detectability of a signal embedded in a noisy background. 2,5,6,16,17 In this paper, we used the following two commonly used linear models: a nonprewhitening matched filter with an eye filter 18 (NPWE) and a prewhitening matched filter (PW). 6,19 Linear model observers have the following decisionmaking procedure for the SKE forced-choice task. On each trial the template, w i, is applied to each of the possible signal locations, g i, i=1,...,4. The response, i,isexpressed as N 2 w i,ng i,n = w i g i, i =1,...,4. 7 The superscript t on w refers to transpose, and the subscript n represents the element of the template w and the data vector g. We classified model observers into two types, adaptive and nonadaptive, according to their ability to explicitly use the local statistics of nonstationary backgrounds. 1. Nonadaptive Models in Nonstationary Backgrounds Nonprewhitening matched filter with an eye filter. The NPWE model uses information about the signal along with a model of human visual sensitivity to different spatial frequencies to formulate a detection template. Since it does not include information about the background, it is unable to accommodate any sort of noise statistical properties, including nonstationarity. Frequency dependence of the human visual system is modeled as a filtering operation with the visual contrast sensitivity function, 18,20 also known as an eye filter. The eye filter acts on both the stimulus and the mean signal, resulting in a detection template that can be obtained by filtering the signal profile with the square of the eye filter. This can be written as w NPWE = Ẽ 2 s, where Ẽ is the eye filter in the Fourier domain and s is the signal in the Fourier domain. The spatial domain template w can be obtained using an inverse Fourier transform of w. The eye filter used for the NPWE model is given by 8 i = n=1 E = exp c, 9 where = u 2 +v 2 is the radial spatial frequency in cycles per deg. In this paper we select an eye filter with parameters close to that used by Burgess. 10,18 The parameters we used are c=0.013, =1.4, and =1.6.

5 Zhang et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A 1553 Prewhitening matched filter with stationarity assumption over regions of interest. The PW model observer derives a template that takes into account knowledge about not only the signal profile but also the background noise statistics. The template is calculated from the inverse of the image covariance matrix. We restricted the image covariance matrix into the locally stationary areas only. For the PW model observer with stationarity assumption across four possible signal locations, i.e., nonadaptive PW, we calculated only one covariance matrix K, and the PW observer single template is given by w PW = K 1 s, 10 where K is the image covariance matrix and s is the known signal. In this study, instead of using samples, 21 we calculate K (64 64 in size) analytically 22 using the following formula: K i,j = 2 f stat,i t f stat,j, 0 i, j 64 64, 11 where f stat is obtained using Eq. (5). f stat, i represents the filter f stat centered at the ith location, and f stat, j represents the filter f stat centered at the jth location. 2. Adaptive Models in Nonstationary Backgrounds Ideal observer (adaptive prewhitening matched filter to individual regions of interest). The ideal observer (IO) derives individual templates for each of the locally stationary areas. In this study, we approximate the IO by restricting the image covariance matrix to each locally stationary area (64 64 pixel in size). In the absence of internal noise and assuming an invertible image covariance matrix, the IO template, w IO loc, is given by w IO loc = K 1 loc s, loc = , 12 where K loc is the corresponding image covariance matrix for each location. Throughout this study we adopt each individual filter, f loc [as shown in Fig. 2(a)] to get the corresponding covariance matrix K loc analytically by the following: K loc,i,j = 2 f loc,i t f loc, j, 0 i, j 64 64, 13 where f loc,i represents the filter centered at the ith location within the area, f loc, j represents the filter centered at the jth location within the area, and is the standard deviation of the white-noise background. 3. Model Observer Templates We study two classes of background image sets, GNS and GS. NPWE model observer templates are the same for these two different test image sets, since the NPWE model does not use the information about the background statistics. For GNS backgrounds, the IO generates four different templates corresponding to the statistics of each possible signal location. For GS backgrounds, the IO renders one template, since the statistics of four possible signal locations are the same in this case. The nonadaptive PW model creates one template for both test image sets. In Fig. 5, the top figure shows the signal as an image (left) and a surface (right). The bottom figure shows the model observer templates for the GNS (column two) and GS (col- Fig. 5. Top: signal as an image (left) and a surface (right). Bottom: model observer templates as images and surfaces for the GNS (column 2) and GS (column 3) test images. Row 1, NPWE templates; row 2, PW templates; row 3, IO templates.

6 1554 J. Opt. Soc. Am. A/ Vol. 23, No. 7/ July 2006 Zhang et al. Fig. 6. Model observer performance (NPWE, PW, IO). Standard errors ranged from to and are too small to display. Left: comparison within model observers Right: comparison across model observers. umn three) test images. The IO s four templates for the GNS noise are rotated versions of one another. E. Model Observer Performance 1. Index of Detectability d from Decision Variable We assume that each possible signal location is within an independent background sample. Another common assumption is that the template responses are Gaussian distributed. Under these assumptions, model observer response to the signal location and to the noise-only locations can be represented as being sampled from two univariate Gaussian distributions. The distribution of model responses to the signal location often has a larger mean than that to the noise location. Performance can then be defined in terms of the distance in standarddeviation units between the signal-plus-noise and the noise-only distribution. This metric, d, is known as the index of detectability 23 : d = s n, 14 where s is the mean model response to the signal-plusnoise background location, n is the mean model response to the noise background location only, and is the standard deviation of the model responses assumed to be the same for the signal-present (signal-plus-noise) and noise-only locations. 2. Index of Detectability d mafc from Proportion of Correct We refer to the index of detectability transformed from proportion correct (Pc) as d mafc to differentiate it from the index of detectability d calculated from the decision variable [Eq. (14)]. Pc is measured by counting the proportion of trials in which the model observer correctly localized the signal. The relationship between d mafc and Pc is expressed as + Pc d mafc,m = x d mafc x M 1 dx, 15 where x =1/ 2 e x 2 /2, x is the cumulative Gaussian distribution function, x = x y dy, and M=4 is the number of possible signal locations in the experiment. When the decision variables are indeed Gaussian distributed, of equal variance, and independent, the index of detectability calculated from the decision variable, d, will be equal to that calculated from transformations of Pc, d mafc. In Section 3, we will use d mafc for human observer performance and most of the model observer performance with one exception (due to a ceiling effect with Pc). The translation from Pc to d mafc was accomplished by using a look-up table that was precomputed using Eq. (15) through numerical integration. F. Human Psychophysical Study The experiments were conducted in a darkened room with a viewing distance of 40 cm. The images were displayed on an Image System M17LMAX monochrome monitor with a resolution of 0.3 mm/ pixel (Image Systems, Minnetonka, Minn.). Each pixel represents a visual angle of deg. The relationship between digital gray level and luminance was linearized using a Dome Imaging System board and a luminance calibration system. The luminance range was from 0.00 to cd/m 2. There were two experimental conditions: (1) GS and (2) GNS. On each trial an image was randomly (without replacement) sampled from the test image database and displayed, and a copy of the signal was shown on top of the test image. No time limitation was imposed on the observer to make a decision. Three naïve observers participated in the study. They had no knowledge of the goals of the experiment. The subjects had normal corrected vision. Observers were given feedback about their decisions (correct/wrong). An experimental session consisted of 100 trials and lasted approximately 6 8 min. Observers participated in ten training sessions with 1000 trials total. After training, each observer participated in 100 experimental sessions of 100 trials per condition. Two observers finished 10,000 trials per condition. The third observer finished only 5000 trials per condition. Human performance Pc was measured by calculating the proportion of trials in which the observer correctly localized the signal. An index of detectability d mafc was then calculated from Pc through Eq. (15).

7 Zhang et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A RESULTS A. Model Observer Performance We define performance improvement (PI) from the GS to the GNS test image set as PI = fd GNS /d GS Model observer performance was calculated from the 10,000 test images for each condition. Model observer performances (index of detectability) are shown in Fig. 6. Performance of both the NPWE and the PW models (left graph) does not vary much across the two test conditions (GNS versus GS). The PI is 0.7% for the NPWE and 2% for the PW. In contrast, the detectability of the IO is substantially better for the detection task with GNS images than with GS images with a PI of 142%. Figure 6 (right graph) compares NPWE, PW, and IO performance. For both test image types, the performance of the NPWE is lower than that of the PW and IO. 24 B. Human Observer Performance The first two graphs in the top row of Fig. 7 show human observers performance (detectability) across groups of 10 sessions (1000 trials). The third graph in the top row of Fig. 7 shows observer 3 s performance across groups of five sessions (500 trials). Error bars for each session group were calculated from the standard error across sessions. For all groups of sessions, human observer performance was significantly higher p 0.05 for the GNS test images than that for the GS test images. The bottom left graph in Fig. 7 shows overall performance for three human observers. The PI from the stationary noise condition to the nonstationary noise condition averaged across human observers was approximately 30%. Efficiency is a measure that compares the performance of a human observer with that of an IO. Human observer efficiency with respect to the IO performance (defined as d mafc human 2 /d mafc IO 2 ) was 0.30 for the GS test image set and 0.01 for the GNS test image set. C. Comparison of Human and Model with Addition of Internal Noise The addition of internal noise has been used to degrade model observer performance to a level that is comparable with human observer performance. 9,25 28 Here, internal noise in the SKE task was implemented by adding a random variable to the scalar decision variables [ in Eq. (7)]. For each location and trial, was sampled independently from a normal distribution with zero mean and a standard deviation proportional to the decision variable s standard deviation. With the internal noise the models decision variable is given by in = e +, where in is the decision variable after injection of internal noise, e is the decision variable prior to the inclusion of internal noise [calculated by Eq. (7)], and is sampled from N 0; e as used in previous work. 9,25 The bottom right graph of Fig. 7 compares the human performance averaged across all three observers with the IO performance with added internal noise. The internal noise constant was set so that the IO performance matched the average human performance for the GS test image set. The same constant Fig. 7. Human model observer performance. Top row: human observer performance by session groups for three individual observers (1000 trials per group for observers 1 and 2; 500 trials per group for observer 3). Bottom left: performance across all trials per condition for three different human observers (10,000 trials for observers 1 and 2; 5000 trials for observer 3). Standard errors ranged from to and are too small to display. Bottom right: comparison of human performance averaged across the three observers and the IO performance with internal noise.

8 1556 J. Opt. Soc. Am. A/ Vol. 23, No. 7/ July 2006 Zhang et al. was then used to add internal noise to the IO for the GNS test image set. After internal noise is added, the IO still resulted in higher absolute performance for the GNS test image set than the human observer. Performance improvement for the IO with internal noise is 125%, which is similar to that before the internal noise injection. 4. DISCUSSION Results show that human performance is greater when the signal is detected in the nonstationary images than when detected in the stationary images. Unlike that of human observers, performance of two models that do not adapt to the local statistics of the noise (NPWE and nonadaptive PW) does not differ across the two test image conditions. Our results suggest that human observers have the ability to adapt their strategy to the local statistics of the noise in the nonstationary backgrounds. This process might involve estimation of the main orientation 29 in the local noise to generate an individualized template for that local region. However, human observers ability to adapt their template to the local noise properties is very suboptimal when compared with an IO that uses specific templates optimally individualized for the local statistics in the nonstationary background. Human performance d mafc improved by 30% from the stationary to the nonstationary noise, while the IO performance improved by 142%. Reasons for the human suboptimality might involve preprocessing of image data by a channel mechanism (spatial frequency and orientation tuned) that reduces the available visual information to the observer. In this context, the adaptation to different local orientations might involve use of distinct orientation channels (off-orientation looking) in the same way that has been reported with spatial frequency (off-frequency looking). 33 A second possible source of suboptimality might be human inability to optimally estimate the local noise statistics. The findings suggest that when searching for targets in natural nonstationary images (see Fig. 1 for examples), human observers do adapt their strategies for different regions of the image on the basis of local noise statistics. This ability to adapt to the local noise in human observers may explain in part previous findings showing that phase randomization in the Fourier domain of some medical image backgrounds (prior to addition of a signal) has a detrimental effect on human signal detection performance. Bochud et al. 34 showed that performance detecting a simulated mass degraded when mammographic backgrounds were randomized in phase. Similar results have been recently reported with chest x-ray backgrounds. 35 If the phase of a nonstationary background is randomized, the resulting image will be statistically stationary. 36 After phase randomization there will be no difference in the local noise statistics across image regions, and there will be no benefit for an adapting observer (model or human) to derive individualized templates for each region. Thus it appears that human observers are adapting their strategy to different local statistics of images to improve their performance. The phase randomization makes the images statistically stationary, not allowing human observers to benefit from the adapting process and thus degrading detection performance. Our experimental results suggest that when strong nonstationary effects are present in the images, model observers that derive a single template assuming global stationarity might be inadequate to model human performance. Instead, models that adapt to the local statistics are needed. Clearly, in this study our simulated images presented strong statistically nonstationarities by design. Finding the circumstances and types of images that require these adaptive models to capture human detection performance in nonstationary noise should be a worthwhile topic of future investigation. Finally, in the present experiment, the statistical nonstationarity was spatial and did not change from trial to trial. Future research should investigate whether observers are able to adapt to changing statistical properties in orientation across trials and time. 5. SUMMARY AND CONCLUSIONS In this paper we studied human signal detectability in nonstationary backgrounds where the statistical properties vary across image locations, and it is therefore advantageous to adapt visual detection strategies to the local statistics of the image. Human performance was significantly higher in the nonstationary backgrounds than in the stationary backgrounds, with an observed increase of 33% in detectability. The results suggest that human observers do adapt their detection strategy to local proper- Table 1. Comparison of d mafc and d for the Ideal Observer under Nine Signal Contrasts Signal Contrast d mafc d Relative Error (%) Fig. 8. Comparison between d and d mafc for the IO with GNS test images.

9 Zhang et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A 1557 ties of image backgrounds. We conclude that humans can adapt their strategy to the local statistical properties of nonstationary backgrounds, albeit suboptimally, and model observers that derive their templates on the basis of global stationary assumptions may be inadequate as a general predictor of human performance in nonstationary backgrounds. APPENDIX A: COMPARISON BETWEEN d AND d mafc FOR THE IDEAL OBSERVER FOR GLOBALLY NONSTATIONARY TEST IMAGES Calculation of the index of detectability from Pc [Eq. (15)] for the IO was not possible because Pc from the Monte Carlo simulations reached 100%. We thus calculated the detectability index directly from the decision variable as expressed by Eq. (14). One limitation is that, for the four possible locations, the mean template response to the noise n varies to a small extent. We therefore calculated an average index of detectability d across the signal location and each of the three remaining noise locations. For example, when the signal was in the first location, then there were three d s: d 1 = s,1 n,2, d 2 = s,1 n,3, d 3 = s,1 n,4. This calculation was repeated for each of the four possible signal locations and resulted in a total of 12 d s. The index of detectability, d, in Fig. 6 for the IO was calculated from the average across all 12 d s. To verify that this method of estimation of d did not lead to large discrepancies between the index of detectability calculated from the decision variable and that calculated from Pc using Eq. (15), we computed the index of detectability using both methods for the IO for lower signal contrasts. The following table shows the comparison between d mafc and d at nine different signal contrasts. For most of the cases, the differences between them are less than 3% (as shown in the fourth column of Table 1). 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