Two of the factors limiting progress in understanding the mechanisms of visual search are the difficulty of controlling and manipulating the retinal stimulus when the eyes are free to move and the lack of an ideal observer theory for fixation selection during search. Recently, we developed a method to precisely control retinal stimulation with gaze-contingent displays (J. S. Perry & W. S. Geisler, 2002), and we derived a theory of optimal eye movements in visual search (J. Najemnik & W. S. Geisler, 2005). Here, we report a parametric study of visual search for sine-wave targets added to spatial noise backgrounds that have spectral characteristics similar to natural images (the amplitude spectrum of the noise falls inversely with spatial frequency). Search time, search accuracy, and eye fixations were measured as a function of target spatial frequency, 1/*f* noise contrast, and the resolution falloff of the display from the point of fixation. The results are systematic and similar for the two observers. We find that many aspects of search performance and eye movement pattern are similar to those of an ideal searcher that has the same falloff in resolution with retinal eccentricity as the human visual system.

*f*noise). We varied the spatial frequency of the target, the contrast of the noise background, and the rate of falloff in display resolution from the point of gaze. Our aims were to obtain a broad picture of search performance under naturalistic conditions and to obtain an estimate of how much information can be removed from the periphery without affecting eye movement patterns or search time. Najemnik and Geisler (2005) showed that modest differences in peripheral detection sensitivity across stimulus conditions can substantially affect search time; thus, our expectation was that mild reductions in peripheral resolution would significantly increase search time.

*f*noise. First, the Fourier amplitude spectra of natural images fall off approximately as 1/

*f*(Burton & Moorehead, 1987; Field, 1987), and thus, searching for targets in 1/

*f*noise should be representative, in at least some ways, of search in the natural environment. Second, there is substantial literature concerning the detection and identification of targets in broadband noise; this literature provides a solid foundation for understanding search performance in broadband noise (Burgess, Wagner, Jennings, & Barlow, 1981; Lu & Dosher, 1999; Pelli & Farell, 1999). Third, it is possible to derive an ideal observer theory of visual search for targets in broadband noise (Najemnik & Geisler, 2005); this ideal searcher provides the appropriate benchmark against which to compare real performance and a useful starting point for proposing realistic (suboptimal) models of visual search.

*f*noise background, as a function of target spatial frequency, noise contrast, and the rate of falloff in display resolution from the fixation location. There were two observers with normal vision; one was an author while the other was naive to the aims of the study.

*f*noise); the remaining display pixels were set to the mean luminance (20 cd/m

^{2}). The 1/

*f*noise was created by filtering white noise, truncating the waveform at ±2

*SD,*scaling to obtain the desired rms amplitude and then adding a constant to obtain the mean luminance. Four noise contrast levels were tested: 0.25, 0.125, 0.0625, and 0.03125 rms.

*f*noise was displayed on each search trial.

*j*th level of the Gaussian pyramid is given by

*f*is spatial frequency in cycles per degree,

*σ*

_{0}= 0.248

*w*

_{pix}/

*w*

_{deg},

*w*

_{pix}is the width of the display in pixels, and

*w*

_{deg}is the width of the display in degrees. The half-height resolution of the

*j*th level of the Gaussian pyramid is

*j*= 0 in Equation 1. We use this transfer function for interpolation between the original image and the first level of the pyramid. In effect, we are assuming that the original image is the first level in a Gaussian pyramid for an image with twice the resolution of the original image. This trick keeps the interpolation procedure between the original image (Level 0) and Level 1 of the pyramid consistent with the interpolation procedure between other neighboring levels of the pyramid (e.g., between Levels 1 and 2).

*r,*is given by

*r*desired at that pixel location.

*ɛ*

_{2}is the eccentricity (in degrees) at which the display resolution drops to one half of its value at the fixation point. The value of

*ɛ*

_{2}controls the rate of falloff in display resolution; the smaller the value of

*ɛ*

_{2}, the faster the rate of falloff. Six values of

*ɛ*

_{2}were tested: 2, 4, 6, 8, 12, and 16 deg.

*e*

_{2}) is in the range of 2.0–2.5 deg. Thus, the most rapid falloff of display resolution in our experiment is only slightly more rapid than the falloff in resolution of the visual system. However, it is important to keep in mind that the display resolution combines multiplicatively with visual resolution. For example, if

*ɛ*

_{2}is set equal to

*e*

_{2}, then at an eccentricity of 2–2.5 deg, the total effective resolution is reduced by a factor of 4 rather than a factor of 2.

*ɛ*

_{2}= 4 deg, background contrast = 0.25 rms, target frequency = 6 cpd) when fixation is to the right and below the center of the display. Figure 2B illustrates the appearance of the same display when the fixation is on the target. The insets show enlargements of the region containing the target.

*ɛ*

_{2}) required greater image processing, and thus, the update rate was only 45 times per second. However, these differences had no noticeable effect on the display because the display frame rate always remained at 60 Hz noninterlaced. For more details, see Perry and Geisler (2002). This subjective impression is consistent with a recent gaze-contingent, blur-detection experiment (Loschky & McConkie, 2005).

*ɛ*

_{2}values). One block was run for each of the 96 stimulus conditions, then, after all 96 conditions were completed, the second block was run for each of the 96 conditions, but the order of conditions was reversed. The observer knew the target spatial frequency, background noise contrast, and value of

*ɛ*

_{2}in each block.

*n*possible nonoverlapping target locations and that the searcher's goal is to find the target as quickly as possible, with the constraint that the average target localization accuracy exceeds some particular criterion value.

*d*′). Najemnik & Geisler (2005) measured the visibility maps of two observers, for a 6-cpd sine-wave target, as a function of target contrast, eccentricity, and 1/

*f*noise contrast. They also found that modest changes in the maps had substantial effects on ideal search performance. Unfortunately, it was not practical to directly measure the visibility maps for the large number of conditions in this study.

*ɛ*

_{2}correspond to higher rates of falloff in display resolution; see Figure 1). Figure 5 plots the error rates, which were generally low except for the hardest search conditions (high target spatial frequency, high background contrast, and small display half-resolution).

*r*= .996 for M.E.W.;

*r*= .998 for W.S.G.).

*r*= .91). Thus, search time varies across the various stimulus conditions both because of the number of fixations and because of the duration of the fixations. However, the dominant factor tends to be the number of fixations. This can be seen in Figure 8A, which plots average fixation duration as a function of average number of fixations to find the target, for all 96 stimulus conditions, for both observers. The fixation durations vary by a factor of approximately 1.5 (from about 200 ms to a little more than 300 ms), whereas the number of fixations varies by a factor of approximately 10.

*f*noise (which has the amplitude spectrum typical of natural images). The second aim was to examine the role of the peripheral visual field in visual search by systematically manipulating peripheral spatial information using gaze-contingent display technology.

*f*noise (i.e., they are approximately parallel on a logarithmic axis; see Figure 6). This result also holds for number of fixations, which is not surprising given the very high correlation (.997) between search time and number of fixations in this study. A second property is that search times (and number of fixations) tend to jump up rather more sharply when the contrast increases from 0.125 to 0.25 than for the other steps in contrast (see Figures 4 and 7). A third property is that for the 1- and 2-cpd targets, increasing the background contrast causes an increase in the number of fixations, yet varying the rate of falloff in display resolution does not (i.e., search time is flat as a function of

*ɛ*

_{2}, yet increases with background noise contrast).

*f*noise as a function of target and noise contrast, and they found that human performance approaches that of the ideal searcher. As can be seen in Figure 9, the absolute performance level of the human observers in the present experiment also approaches optimal for the conditions with the 6-cpd target. Thus, this study extends the previous finding of a close match between human and ideal search performance to a wider range of conditions.

*ɛ*

_{2}= 6 deg) was sufficient to cause a reliable increase in search time. This level of display foveation was subjectively invisible for medium and low noise contrasts. This is not surprising given that the gaze-contingent displays were free of artifacts and that the half-resolution eccentricity for the human visual system is approximately 2.3 deg. The fact that accentuating the human falloff in resolution by a small amount causes a significant drop in performance confirms the conclusion from the ideal-observer analysis that peripheral information is being used efficiently in guiding eye movements. This is quite different from reading tasks (McConkie & Rayner, 1975), where peripheral information plays little role, but is qualitatively consistent with other analyses of peripheral information use in search (Eckstein et al., 2001; Rajashekar et al., 2002).

*ɛ*

_{2}of 6 deg often produced an undetectable level of blur is consistent with the recent blur-detection experiments of Loschky, McConkie, Yang, & Miller (2005), who report that blur is undetectable when

*ɛ*

_{2}= 6 deg and only detectable 5% of the time when

*ɛ*

_{2}= 3 deg. However, their study used a divided attention task, which may have underestimated sensitivity to blur. Nonetheless, their results in conjunction with ours suggest that even when blur goes unnoticed in the periphery, it can affect sensitivity for detecting peripheral targets and, hence, affect search performance.

*d*′) of an imaginary target location to zero (or near zero) and giving this location a prior probability corresponding to the probability of a target absent trial. (Note that the posterior probability of the “target absent” location climbs during the search as other locations are ruled out.) It remains to be seen how efficient humans are in this task; however, because of the greater memory demands in target absent trials, humans may be less efficient, especially under conditions where the target it relatively difficult to detect.

*P*[1], …,

*P*[

*N*], and the output is a list of fixation points,

*F*. Initially, list

*F*and a working list

*T*are set to be empty, and an index

*i*is set to 0. Also, we note that

*P*′ is a temporary list that accumulates all the eye positions corresponding to a given single fixation. The algorithm proceeds as follows:

- If
*i*exceeds*N*, the algorithm ends. Otherwise, compute the centroid*C*of eye positions*P*[*i*],*P*[*i*+ 1], …,*P*[*i*+*k*], such that*P*[*i*+*k*] is the last eye position that occurs less than 75 ms after*P*[*i*]. If there is not 75 ms worth of eye positions following*P*[*i*], the algorithm ends. - If the standard deviation of the distances of
*P*[*i*],*P*[*i*+ 1], …,*P*[*i*+*k*] from*C*is greater than*a*degrees,*P*[*i*] is not the beginning of a fixation; thus, ignore this eye position by setting*i*=*i*+ 1; continue at 1. - If the standard deviation of the distances of
*P*[*i*],*P*[*i*+ 1], …,*P*[*i*+*k*] from*C*is less than*a*degrees,*P*[*i*] is the beginning of a fixation; hence, save these*k*+ 1 eye positions in a list*P*′, and set*i*=*i*+*k*+ 1. - If the distance from
*P*[*i*] to*C*is less than*b*degrees, add*P*[*i*] to*P*′. Set*i*=*i*+ 1. If*i*does not exceed*N*, continue at 4; otherwise, compute the mean of the eye positions in*P*′ and add it to a list of fixations*F*; the algorithm ends. - If the distance from
*P*[*i*] to*C*is greater than*c*degrees, compute the mean of the eye positions in*P*′ and add it to a list of fixations*F*; continue at 1. - If the distance from
*P*[*i*] to*C*is less than*c*degrees and greater than*b*degrees, clear the list*T*of potential fixation eye positions. - If
*i*exceeds*N*, compute the mean of the eye positions in*P*′ and add it to a list of fixations*F*; the algorithm ends. Otherwise, add*P*[*i*] to the list*T*. Set*i*=*i*+ 1. If the time difference between the first and last eye positions in*T*is less than 50 ms, continue at 7. - Compute the mean of the eye positions in
*T*. If this mean is less than*b*degrees from*C*, add this mean eye position to*P*′ and clear the list*T*; continue at 4. Otherwise, compute the mean of the eye positions in*P*′ and add it to the list of fixations*F*; continue at 1.

*a*= 0.1 deg,

*b*= 0.2 deg, and

*c*= 0.3 deg.

*d*′):

*c*is the rms contrast of the target (i.e.,

*c*

^{2}is the contrast power),

*e*

_{ n}is the stimulus noise contrast power, and

*ɛ*is the eccentricity in degrees (see Supplement to Najemnik & Geisler, 2005). (We note that

*d*′ is monotonically related to the proportion of correct responses in the detection task.)

*α*is a function of the narrow band of background noise that affects the responses of the target-matched template, and it can be estimated by measuring the mean and the variance of the template responses to a large number of samples of the target embedded in the actual background noise used in the experiments. The value of

*α*is approximately 0.022.

*T*(

*f, ɛ*) is given by substituting Equation 3 into Equation 2.

*c*

_{T}is the contrast threshold for the detection. Furthermore, we found that the slopes and intercepts of these linear functions vary systematically with eccentricity:

*c*

_{s}(0) is the contrast sensitivity in the fovea,

*k*is a constant whose value is typically in the neighborhood of 0.1,

*f*is spatial frequency in cpd, and

*e*

_{2}is the half-resolution eccentricity, which is typically around 2.0–2.5 deg (see, e.g., Geisler & Perry, 1998). Writing Equation B7 in terms of the threshold contrast power, we have

*b*

_{0}is the threshold contrast in the fovea at zero spatial frequency. Therefore, we predict from Equation B8 that ln[

*b*(

*ɛ*)] should be a linear function of eccentricity:

*e*

_{2}to a representative value from the literature (

*e*

_{2}= 2.3 deg), we can estimate

*k*

_{b}and

*b*

_{0}by setting Equation B9 equal to Equation B5:

*k*

_{b}= 0.168,

*b*

_{0}= 0.0295. To generate estimates of the slopes of the masking functions for other spatial frequency targets, we assume that an equation similar to Equation B9 holds for slopes:

*k*

_{a}= 0.088 and

*a*

_{0}= 0.44.

*d*′ for each point in the display relative to the current point of fixation. We substitute these

*d*′ values into Equation B1 to estimate the equivalent internal noise power

*β*(

*c,e*

_{ n}

*,ɛ*) in the unfoveated display. The equivalent internal noise power in the foveated display is obtained by dividing

*β*(

*c,e*

_{ n}

*,ɛ*) by the square of the foveated transfer function (see Equations 2, 3, and B2). The equivalent external noise power is obtained by multiplying the contrast noise power by the estimated constant

*α*. In simulating ideal search performance, the external noise was taken to be static noise and the internal noise was taken to be dynamic noise that was statistically independent in space and time. For more details, see Najemnik and Geisler (2005).