**Abstract**:

**Abstract**
**A practical model is proposed for predicting the detectability of targets at arbitrary locations in the visual field, in arbitrary gray scale backgrounds, and under photopic viewing conditions. The major factors incorporated into the model include (a) the optical point spread function of the eye, (b) local luminance gain control (Weber's law), (c) the sampling array of retinal ganglion cells, (d) orientation and spatial frequency–dependent contrast masking, (e) broadband contrast masking, and (f) efficient response pooling. The model is tested against previously reported threshold measurements on uniform backgrounds (the ModelFest data set and data from Foley, Varadharajan, Koh, & Farias, 2007) and against new measurements reported here for several ModelFest targets presented on uniform, 1/f noise, and natural backgrounds at retinal eccentricities ranging from 0° to 10°. Although the model has few free parameters, it is able to account quite well for all the threshold measurements.**

*psf*) of the human eye (Navarro, Artal, & Williams, 1993); local luminance gain control (

*G*), which enforces Weber's law for detection on uniform backgrounds (for reviews, see Hood, 1998; Hood & Finkelstein, 1986); the average spatial sampling density of midget retinal ganglion cells in the human retina (Curcio & Allen, 1990; Dacey, 1993; Drasdo, Millican, Katholi, & Curcio, 2007); and the receptive field properties of midget ganglion cells in the nonhuman primate retina (Croner & Kaplan, 1995; Derrington & Lennie 1984). We focus on the midget ganglion cell pathway because of evidence that it is responsible for detection performance under conditions of low to moderate temporal frequency (Merigan, Katz, & Maunsell, 1991; Merigan & Maunsell, 1993). These conditions include the case of interest here: static stimuli presented for the duration of a typical eye fixation (150–400 ms).

_{L}*d*′). The spatial pattern masking is represented by an effective total contrast power (

*P*) that is the sum of three components: a baseline component, a narrowband component, and a broadband component.

_{eff}*d*′ (Burgess & Colborne, 1988; Eckstein, Ahumada, & Watson, 1997a; Lu & Dosher, 1999, 2008). This enforces the psychophysical rule that threshold contrast power increases linearly with background contrast power for white noise backgrounds (Burgess, Wagner, Jennings, & Barlow, 1981; Legge, Kersten, & Burgess, 1987) and for 1/f noise backgrounds (Najemnik & Geisler, 2005). This concludes a brief summary of the model; we now provide more details.

*I*(

_{B}**x**) or the sum of the target and background images

*I*(

_{T}**x**) +

*I*(

_{B}**x**), where we have simplified the notation by letting

**x**= (

*x*,

*y*). Until the final steps of the model, the operations are effectively linear, and hence, the target and background can be processed separately. The retinal images of the target and background are computed by convolving the target and background images with an appropriate optical psf:

*MTF*(

*f*) = 0.78 exp(−0.172

*f*) + 0.22 exp(−0.037

*f*). The optical psf broadens (blur increases) with retinal eccentricity but is relatively constant out to 10° eccentricity, the largest eccentricity measured in the present study. This component of the model could be easily adjusted for greater eccentricities or to take into account individual differences in optics.

*s*

_{0}, is approximately 30 arcsec (0.0083°), and thus, the assumed density of midget ganglion cells in the center of the fovea is 120 cells/°. In reality, there is one on and one off midget ganglion cell for each cone, and hence, the actual density is approximately 240 cells/°. However, with little loss of precision, we represent the pair of on and off cells by a single linear receptive field that produces positive and negative responses. (In the current model, we ignore differences in the density and receptive field sizes of on and off ganglion cells [Dacey & Peterson, 1992].) As can be seen, midget ganglion cell spacing increases approximately linearly with a slope that depends on direction in the visual field.

*ε*,

_{N}*ε*,

_{T}*ε*, and

_{I}*ε*are the eccentricities in the four cardinal directions at which the spacing between ganglion cells reaches twice what it is in the center of the fovea. This spacing is then used to generate the ganglion cell mosaic, a portion of which is shown in Figure 3. The specific algorithm used to generate the mosaic is given in the Appendix. The algorithm produces a mosaic that satisfies the spacing function and does not have any observable artifacts. We represent the mosaic by the function

_{S}*samp*(

**x**).

*g*(

_{a}**y**;

**x**) be a 2-D Gaussian (with a volume of 1.0) centered on retinal location

**x**. Then, the local average luminance at

**x**is

*SD*) of the Gaussian is fixed:

*σ*(

_{L}**x**) =

*σ*. Thus, the effect of light adaptation is represented by a single parameter. The local luminance gain is

_{L}*G*(

_{L}**x**) = 1/

*L*(

**x**). Note that when the background is uniform then luminance gain is the same at all retinal locations (because the Gaussian has a volume of 1.0). To handle low light levels in which Weber's law fails, a constant

*L*

_{0}can be added to the denominator, but for the conditions of interest here, that was not necessary.

*g*(

_{c}**y**;

**x**) and

*g*(

_{s}**y**;

**x**) be 2-D Gaussians representing the center and surround mechanisms of a midget ganglion cell at retinal location

**x**(equations are in the Appendix). The response of ganglion cells to the background alone is given by where

*D*(

**y**;

**x**) is a difference of Gaussians:

*D*(

**y**;

**x**) =

*w*(

_{c}g_{c}**y**;

**x**) − (1 −

*w*)

_{c}*g*(

_{s}**y**;

**x**). For the conditions of interest here, the target contributes little to the local luminance, and hence, the response to the target plus background is simply the sum of the responses to the target and background, and the response of the ganglion cells to the target is given by

*SD*of the center mechanism is given by

*σ*(

_{c}**x**) =

*k*(

_{c}sp**x**) and the surround mechanism by

*σ*(

_{s}**x**) =

*k*(

_{c}sp**x**). We see then that three parameters,

*w*,

_{c}*k*, and

_{c}*k*, describe the receptive field properties of all the ganglion cells.

_{s}*P*that is the weighted sum of three components (see Figure 4): a baseline component

_{eff}*P*

_{0}, a narrowband component

*P*, and a broadband component

_{nb}*P*: where

_{bb}*k*sets the overall strength of pattern masking, and

_{b}*w*sets the relative strength of the narrowband and broadband components.

_{b}*G*(

_{bc}**y**;

**x**) is the continuous (unsampled) ganglion cell center response to the background, and

*f*(

_{T}**y**;

**x**) is the target-specific filter that removes the background power in the ganglion cell responses that do not drive the cortical neurons that encode the target. To determine the target-specific filter, we (a) take the Fourier transform of the ganglion cell center response to the target alone, (b) convert to log polar coordinates (log frequency vs. orientation), (c) convolve (in the frequency domain) with a function that is the product of the amplitude spectrum of a log Gabor (bandwidth 1.5 octaves) and a Gaussian function in orientation (bandwidth 40°), and (d) convert back into standard spatial frequency axes and take the inverse Fourier transform. We convert to log polar coordinates so that the cortical filters at all log frequencies and orientations have the same shape, allowing simple convolution in step (c) (Watson & Solomon, 1997, use a similar trick). In this version of target-dependent filtering, we did not include the effect of the ganglion cell surround because the lowest frequency (DC) is automatically removed by the log-Gabor cortical filtering. However, we have preliminary results that include both center and surround, and the quality of the Model predictions is similar.

*E*(

_{T}**y**;

**x**) is the blurred spatial envelope of the target, and the blurring depends on retinal location (i.e., envelope size increases with eccentricity). The filtered ganglion cell responses are weighted by the blurred envelope of the target under the plausible assumption that only the background power falling within some spatial neighborhood of the target will have a masking effect. The envelope is defined to be the 2-D Gaussian (with arbitrary covariance matrix) that best fits the absolute value of the target (see Appendix). The blurred envelope is obtained by convolving the envelope with a 2-D Gaussian having a

*SD*of the ganglion cell center

*σ*(see Appendix).

_{c}*r*

_{0}is the response of a ganglion cell to a uniform background, which is a constant that depends only on the relative weight of center and surround:

*r*

_{0}= 2

*w*− 1. Subtraction of

_{c}*r*

_{0}guarantees that

*P*is zero for uniform backgrounds. Note that

_{bb}*P*is also zero for uniform backgrounds because the spatial frequency tuning of the cortical neurons is log Gabor (which goes to zero at zero spatial frequency). Generally, the masking power of the background is greatest when the weight on the narrowband component is zero (upper dashed line in Figure 4) and least when the weight on the broadband component is zero (i.e., when the solid curve touches the baseline in Figure 4).

_{nb}*ρ*is a pooling exponent, and

*σ*=

_{eff}*r*can be regarded a signal-to-noise ratio. In this case, if the pooling exponent is 2.0, then Equation 10 is the standard formula for optimal pooling of statistically independent Gaussian signals (“

_{pooled}*d*′ summation”). Following others (Graham, 1977; Quick, 1974; Watson, 1979; Watson & Ahumada, 2005), we allow the pooling exponent to be greater than 2.0 (which is suboptimal) although for the current model the estimated exponent is only slightly larger, 2.4 (see later).

*c*to be the contrast of the target at which the signal-to-noise ratio given by Equation 10 is equal to 1.0, which corresponds to 69% correct. In other words, the predicted contrast threshold is the solution to the equation

_{t}*z*) is the standard normal integral function (this assumes optimal criterion placement).

*σ*as an equivalent noise. However, it could also be regarded as a deterministic gain control, which would make

_{eff}*k*,

_{c}*k*,

_{s}*w*,

_{c}*P*

_{0}, and

*ρ*, determine the predicted contrast thresholds for uniform backgrounds at all retinal locations. The additional three parameters,

*σ*,

_{L}*k*, and

_{b}*w*, determine predicted thresholds for more complex backgrounds. A ninth parameter

_{b}*β*is needed for predicting values of detectability (

*d*′) that do not correspond to the 69% correct threshold.

**x**= (0,0), and then convolve each image separately with a series of Gaussians having

*SDs*that incrementally increase in powers of two. This set of images forms a stack of successively blurred images, each corresponding to a particular discrete

*SD*(resolution). We precompute and save these stacks for each target and background image to be processed. For each target image, we also precompute and save the target-specific spatial frequency filter corresponding to each level of the target stack. Once these stacks are computed and stored, they can be interpolated to rapidly determine the local luminance function, the ganglion cell target response function, and the ganglion cell effective background response function for any target location and fixation location. Specifically, each fixation location and target location specifies a spatial region of the background as well as the spatial coordinates of the samples (ganglion cells) covering that region. The location of a sample specifies a particular continuous

*SD*(resolution). That resolution will fall between two neighboring resolutions in the stack. The value at the sample location is obtained by linearly interpolating between the two values in these neighboring resolution images. This procedure provides a close approximation to the exact calculations. A MatLab implementation of the model is available at http://natural-scenes.cps.utexas.edu/.

*c*be the observed contrast threshold (in dB) for condition

_{i}*i*, and let

*ĉ*(

_{i}**θ**) be the predicted contrast threshold for parameters

**θ**. We minimize the sum of the squared errors

*S*(

**θ**), and thus,

**θ̂**=

*S*(

**θ**). When the background is fixed (e.g., a uniform background), this minimization is straightforward. However, when the background randomly varies from trial to trial (the 1/f noise and natural backgrounds), it is not practical to generate a predicted model response for each trial for each vector of parameter values evaluated during the parameter search.

**θ̂**

_{1}. Once these estimates are obtained, we generate the predicted threshold

*ĉ*(

_{ij}**θ̂**

_{1}) for each specific background patch

*j*in each condition

*i*. Then, for each condition, we rank order the thresholds and select the patch having the median threshold. Let this patch be

*j*. We then estimate the parameters again, where the fixed patch for condition

_{i}*i*is

*j*. These estimates are

_{i}**θ̂**

_{2}. We repeat this process until the estimated parameters converge (usually just a couple of iterations). Simulations show that this procedure is effective in finding the optimal parameters.

^{2}.

*SD*of 8.43 arcmin. The Edge was horizontal and windowed with a Gaussian having a

*SD*of 0.5°. These three targets were taken from the ModelFest stimulus set (ModelFest stimuli #12, #27, and #30, see Watson & Ahumada, 2005).

^{2}, 1920 × 1080). The target contrast is defined to be the difference between the peak and background luminance divided by the background luminance (18 cd/m

^{2}). Depending on the background condition, the 512 × 512 background was either set to mean luminance or randomly selected from either large 1/f noise images (1280 × 1280) or from one of 10 large (4284 × 2844) “Gaussianized” natural images. In all conditions, the pixels on the edge of the 512 × 512 background were set to black; this created a 1-pixel-wide box that cued the location of the background under all conditions. Detection measurements were obtained for uniform backgrounds and for 1/f noise and natural backgrounds of 7.5% and 15% RMS contrast (i.e., five background conditions).

*f*noise image. This was done in the second step by applying the following mapping:

*g*=

_{i}*f*, where

_{j}*g*is the gray level of the natural image pixel having rank order

_{i}*i*out of a total of

*N*pixels,

*f*is the gray level of the 1/f noise pixel having rank order

_{j}*j*out of a total of

*M*pixels, with

*j*= ⌈

*iM*/

*N*⌉. (Note that

*N > M*, and ⌈

*x*⌉ is the “ceiling” function.) This mapping preserved the spatial frequency, orientation, and phase structure of natural images but allowed us to select patches from Gaussianized natural images with similar mean luminance and contrast as patches selected from our large 1/

*f*noise images. Specifically, for each randomly selected 512 × 512 patch of 1/

*f*noise used in the Experiment, we randomly selected a patch of Gaussianized natural image having approximately the same mean luminance (the mean luminance differed by a maximum of 1.45 cd/m

^{2}); the RMS contrasts of the two patches were set to the same value (i.e., 7.5% or 15%).

*f*noise backgrounds were measured in a random order. Then the psychometric functions for the Gaussianized natural backgrounds were measured in a random order.

*c*), steepness parameter (

_{t}*β*), and criterion (

*γ*). Consistent with Equation 14, the probability of a hit is given by and the probability of a false alarm by

*n*is the number of contrast levels of the target, and

*N*(

_{h}*c*),

_{i}*N*(

_{m}*c*),

_{i}*N*(

_{fa}*c*), and

_{i}*N*(

_{cr}*c*) are the numbers of hits, misses, false alarms, and correct rejections, for contrast level

_{i}*c*. We first estimated the parameters by maximizing Equation 17. We found that the values of the steepness parameter were consistent across conditions (see Results) and that there were no systematic variations in the criterion across conditions for a given observer. Thus, the final thresholds for each observer were obtained by setting the steepness parameter to the average across all subjects and conditions, setting the criterion to the average across conditions for that subject, and then finding the maximum likelihood estimate of the thresholds using Equation 17. Importantly, the pattern of thresholds was robust across different versions of this analysis (including ignoring criterion effects and only analyzing percent correct).

_{i}*c*for each of the 60 conditions. Figure 5 plots the square of the estimated contrast thresholds (threshold power) as a function of the square of the background contrast (background power). The open circles represent the average thresholds of three observers for three target stimuli (columns) presented at four retinal eccentricities (colors) in 1/

_{t}*f*noise and Gaussianized natural backgrounds (rows). The colored lines are linear fits to the data (not Model predictions). Note the thresholds measured in uniform backgrounds (background contrast of zero) are the same in both rows of plots and that the vertical scales are different for the different targets. The estimated criterion (bias) for the three observers in units of d′ were 0.362 (JSA), 0.228 (CKB), and 0.277 (SPS).

*f*noise and in natural backgrounds are similar (see Figure 5). However, for the Gabor and Edge targets, masking was somewhat greater in the natural backgrounds. This can be seen clearly in Figure 8, which plots threshold in 1/

*f*noise as a function of threshold in natural backgrounds separately for each target. The points for the Gaussian target fall near or slightly below the diagonal, but the points for the Gabor and Edge target fall above the diagonal. Even though the points do fall off the diagonal, they fall roughly on straight lines, indicating that thresholds in 1/

*f*noise and natural backgrounds differ approximately by a fixed proportionality constant that depends on the target.

*β*in the RV1 model.

*P*

_{0}that scales all thresholds up and down), we estimate the remaining three parameters by fitting the average thresholds from the present Experiment. Finally, in the Discussion section, we keep the parameters fixed (except for

*P*

_{0}) and generate predictions for the results of Foley et al. (2007).

*P*

_{0}to change. The only effect of the baseline noise parameter is to shift the predictions for uniform backgrounds vertically on the dB axis. Although we allowed the baseline noise parameter to vary, the estimated value was well within the range of individual differences for that parameter in the ModelFest data set.

*β*, was determined from the average steepness of the psychometric functions in the Experiment reported here. We find

*β*= 1.685, which we note corresponds to a Weibull slope parameter of 2.13. The remaining five parameters can be estimated from the detection thresholds measured on uniform backgrounds in the fovea. To estimate these parameters we fit the ModelFest data set, which consists of foveal detection thresholds measured for 43 different targets on 16 observers in 10 different laboratories. The fit of the model to the ModelFest data is good, comparable to (slightly worse than) the best nonphysiologically based models (see Watson & Ahumada, 2005).

*SD*of the ganglion cell center mechanism

*σ*is almost exactly equal to the spacing between the (on or off) midget ganglion cells, which in the central visual field is approximately equal to the spacing between the photoreceptors (about a half minute of arc). This is consistent with the anatomical finding that in the central visual field a midget ganglion cell synapses with one midget bipolar cell, which synapses with one cone photoreceptor. The measured width of center mechanisms with single-unit recording is larger than a single cone, but the larger size is expected because of the effect of the optical psf; the measured center mechanism should be the convolution of the physiological center mechanism and the optical psf. Croner and Kaplan (1995) report that in the central 5° the median

_{c}*SD*of the center mechanism is 0.03° and of the surround mechanism is 0.18° (about six times larger than the center). We computed the effective center

*SDs*for our model and find that they range from 0.021° at 0° eccentricity to 0.038° at 5° eccentricity, spanning the value reported by Croner and Kaplan. Similarly, the effective surround

*SD*for the model ranges from 0.077° (3.6 times larger than center) at 0° eccentricity to 0.3° (7.9 times larger) at 5° eccentricity. Finally, Croner and Kaplan report that the relative weight on the center mechanism

*w*is about 0.64 whereas our estimate is 0.53. Thus, we also find greater weight for the center mechanism but not by as large a factor.

_{c}*P*

_{0}to change from 1.4E-3 to 4.5E-4 to account for modest differences in overall sensitivity among different groups of observers.

*SD*of 0.25°. In two other observers, thresholds were measured for a sine-phase Gabor having an envelope

*SD*of 0.18°. The symbols in Figure 12A show the average thresholds. The solid curve shows the prediction of the RV1 model without altering parameters except for the baseline masking power (see figure caption). In their experiment 2, Foley et al. measured thresholds in the fovea for 4 c/° Gabor targets in cosine phase (Figure 12B), sine phase (Figure 12C), and anticosine phase (Figure 12D) for various areas and aspect ratios in two observers. The blue symbols show the thresholds for Gabor targets with a radially symmetric envelope. In this case, the horizontal axis gives the

*SD*of the envelope in all directions. The red symbols show the thresholds for Gabor targets that are elongated parallel to the orientation of the grating. In this case, the horizontal axis gives the

*SD*of the envelope in the parallel direction, with the

*SD*in the perpendicular direction fixed at 0.25°. The green symbols show the thresholds for Gabor targets that are elongated perpendicular to the orientation of the grating. In this case, the horizontal axis gives the

*SD*in the perpendicular direction, with the

*SD*in the parallel direction fixed at 0.25°. The solid curves show the predictions of the RV1 model.

*f*noise backgrounds and in natural backgrounds whose gray scale histogram has been adjusted to match that of 1/

*f*noise. The predictions are good but slightly poorer for the Gabor target than for the Gaussian and Edge targets (see Figure 11). It is interesting to note, however, that the average thresholds reported by Foley et al. (2007) for the Gabor target (Figure 12a) increase slightly faster with eccentricity in better agreement with the RV1 model.

*f*noise backgrounds and that the thresholds for the two kinds of background are similar. The background masking effects in the model are entirely based on the narrowband and broadband power in the ganglion cell responses, not on the specific phase structure, which differs greatly between the natural image and 1/

*f*noise backgrounds. Perhaps the trial-to-trial variation in the backgrounds is hiding the effect of the phase structure. That is, thresholds may be similar in the two types of background only because in some trials the phase structure helps detection and in other trials it hurts detection. However, if this were true, then one might expect shallower psychometric functions for natural backgrounds. In fact, the slope parameter of the psychometric functions is similar for uniform, 1/

*f*noise, and Gaussianized natural backgrounds (see Figure 9). It would appear that for Gaussianized natural backgrounds, the complex phase structure of natural backgrounds has, practically speaking, a relatively minor effect on detection thresholds.

*f*noise) images drops rapidly.

*w*= 0.0, see Equation 6), then the predictions are substantially worse. Conversely, if the parameters are estimated with the weight on the broadband component set to zero, then predictions are also substantially worse. Although the estimated weight is much higher on the narrowband component (

_{b}*w*= 0.962) than the broadband component (1 −

_{b}*w*= 0.038), they both play an important role. In fact, the average total masking power due to the broadband and narrowband components is about equal across the three targets:

_{b}*w*≅ (1 −

_{b}P_{nb}*w*)

_{b}*P*. More specifically, the average ratio of narrowband to broadband masking power is smallest for the Gabor target (0.165), intermediate for the Edge target (0.98), and largest for the Gaussian target (2.27). Although we find that both components are important in the current version of the model, the result may depend on how the target-dependent filter is computed. It is perhaps also worth emphasizing that broadband and narrowband components have no effect on the predictions for uniform backgrounds.

_{bb}*f*noise or natural images.

*f*noise) the target is on average not very similar to the background surrounding the target. This raises the question: How important are crowding effects when looking for specific targets in natural scenes? If one takes an arbitrary target and adds it at a random location in a natural image, then does the target tend to be sufficiently similar to the surrounding background for crowding effects to be strong relative to the more local masking effects? It may be possible to answer this question by analysis of natural image statistics. Of course, in some natural cases, crowding effects are known to be very important (e.g., reading) and in other cases are likely to be very important (e.g., detecting animals, which often mimic the backgrounds in their natural habitat).

*location d′ maps*could be used to determine the best possible search performance, assuming perfect parallel processing of all potential target locations. This is a critical baseline analysis for interpreting the results of visual search and attention experiments (e.g., see Eckstein, 2011; Geisler & Cormack, 2011).

*fixation d′ map*is closely related to the conspicuity area—the spatial region around a target where it can be detected in the background (Bloomfield, 1972; Engel, 1971; Geisler & Chou, 1995; Toet, Kooi, Bijl, & Valeton, 1998). The conspicuity area can be defined as the area of the region where d′ exceeds some fixed criterion. Previous studies (Geisler & Chou, 1995; Toet et al., 1998) have shown that there is a strong negative correlation (on the order of −0.8 to −0.9) between the conspicuity area and the time it takes humans to locate the target even in natural scenes (Toet et al., 1998). This is a powerful result of theoretical importance and of potential practical value. But to be of practical value, one must know the conspicuity area for the particular target at its particular location in the background. The RV1 model might prove useful for estimating conspicuity areas without having to directly measure them in preliminary psychophysical experiments.

*foveal d′ maps*could be used to determine the best possible search accuracy for a given target in a given background given unlimited search time.

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*g*to be the

_{k,n}*k*

^{th}ganglion cell on ring

*n*. The ganglion cell at the fovea will be

*g*

_{1,1}(the only ganglion cell on ring 1). Also, define

*C*(

*g*

_{k}_{,n}) to be the circle centered at

*g*

_{k}_{,n}with a radius (spacing) specified by the equation in Figure 2C (the radius depends on the retinal location of

*g*

_{k}_{,n}). Two rules specify how all ganglion cells on ring

*n*are created given that ring

*n −*1 has been completed. For each rule, there is a special case in which the fovea is the previously created ring.

*g*

_{1,n}, is placed at the intersection furthest from the fovea between

*C*(

*g*

_{k}_{−1,n−1}) and

*C*(

*g*

_{k}_{,n−1}), where k is a randomly chosen positive integer at most as large as the total number of ganglion cells on ring

*n −*1. The special case in which

*n −*1 = 1 (the central ganglion cell) is handled by placing

*g*

_{1,2}at any randomly chosen point on

*C*(

*g*

_{1,1}).

*k*th ganglion cell on ring

*n*, for

*k*> 1, is found by first identifying all intersections between

*C*(

*g*

_{k}_{−1,n}) and the circles of all ganglion cells on ring

*n −*1. In the special case in which

*n −*1 = 1, there will be only two such intersections, one clockwise and the other counterclockwise from

*g*

_{k}_{−1,n}. In this case, choose the intersection that is clockwise from

*g*

_{k}_{−1,n}as the location for

*g*

_{k}_{,n}. In the more general case in which

*n −*1 > 1, we first find the subset of intersections between

*C*(

*g*

_{k}_{−1,n}) and the circles of all ganglion cells on ring

*n −*1 that lie counterclockwise from

*g*

_{k}_{−}

_{1,n}. The location of

*g*

_{k}_{,n}is at the intersection (within this subset) that is furthest from the fovea.

*b*, of 1.5 octaves and the Gaussian has an orientation bandwidth at half height,

_{u}*b*, of 40°. The form of the functions is as follows:

_{θ}*g*(

**y**;

**u**,

**Σ**) that best fits the absolute value of the target: where is the mean vector,

**Σ**is the covariance matrix, and

*g*(

**y**;

**u**,

**Σ**) has a volume of 1.0. To obtain the envelope for a given retinal location, we then blurred this Gaussian with another Gaussian (of volume 1.0) having the size of the ganglion cell center at that retinal location

*σ*(

_{c}**x**). Thus the envelope (which also has a volume of 1.0) is given by