**We introduce a novel framework for estimating visual sensitivity using a continuous target-tracking task in concert with a dynamic internal model of human visual performance. Observers used a mouse cursor to track the center of a two-dimensional Gaussian luminance blob as it moved in a random walk in a field of dynamic additive Gaussian luminance noise. To estimate visual sensitivity, we fit a Kalman filter model to the human tracking data under the assumption that humans behave as Bayesian ideal observers. Such observers optimally combine prior information with noisy observations to produce an estimate of target position at each time step. We found that estimates of human sensory noise obtained from the Kalman filter fit were highly correlated with traditional psychophysical measures of human sensitivity ( R^{2} > 97%). Because each frame of the tracking task is effectively a “minitrial,” this technique reduces the amount of time required to assess sensitivity compared with traditional psychophysics. Furthermore, because the task is fast, easy, and fun, it could be used to assess children, certain clinical patients, and other populations that may get impatient with traditional psychophysics. Importantly, the modeling framework provides estimates of decision variable variance that are directly comparable with those obtained from traditional psychophysics. Further, we show that easily computed summary statistics of the tracking data can also accurately predict relative sensitivity (i.e., traditional sensitivity to within a scale factor).**

*Elemente der Psychophysik*was published in 1860 (Fechner, 1860), an enormous amount has been learned about perceptual systems using psychophysics. Much of this knowledge relies on the rich mathematical framework developed to connect stimuli with the type of simple decisions just described (e.g., Green & Swets, 1966). Unfortunately, data collection in psychophysics can be tedious. Forced-choice paradigms are aggravating for novices, and few but authors and paid volunteers are willing to spend hours in the dark answering a single, basic question over and over again. Also, the roughly one bit per second rate of data collection is rather slow compared with other techniques used by those interested in perception and decision making (e.g., EEG).

*R*, to vary as a free parameter. The parameter value (observation noise variance) that maximizes the likelihood of the fit under the model is our estimate of the target position uncertainty that limits the tracking performance of the observer.

*v*,

_{x}*v*) with a one pixel per frame standard deviation. These were summed cumulatively to yield a sequence of

_{y}*x*,

*y*pixel positions. Also visible was a 2 × 2 pixel (2.6 arcmin) square red cursor that the observer controlled with the mouse.

^{2}, respectively.

*y*axis by blob width during the trial. Each row of panels is an individual subject. Because our tracking task has two spatial dimensions, each trial yields a time series for both the horizontal and vertical directions. The first and second columns in the figure show the horizontal and vertical CCGs, respectively, and the black line traces the maximum value of the CCGs across trials. As blob width increases (i.e., lower peak signal-to-noise), the response lag increases, the peak correlation decreases, and the location of the peak correlation becomes more variable. As there were no significant differences between horizontal and vertical tracking in this experiment, the rightmost column of Figure 3 shows the average of the horizontal and vertical responses. Clearly, the tracking gets slower and less precise as the blob width increases (i.e., target visibility decreases).

*R*) as a measure of performance. Figure 5 illustrates the details of the Kalman filter in the context of the tracking task. Our experiment generated two position values at each time step in a trial: (a) the true target position (

*x*) on the screen, and (b) the position of the observer's cursor (

_{t}*x̂*), which was his or her estimate of the target position (plus dynamics due to arm kinematics, motor noise, and noise introduced by spatiotemporal response properties of the input device). The remaining unknowns in the model are the noisy sensory observations, which are internal to the observer and cannot be measured directly. These noisy sensory observations are modulated by a single parameter; the observation noise variance (

_{t}*R*). We fit the observation noise variance (

*R*) of a Kalman filter model (per subject) by maximizing the likelihood of the human data under the model given the true target positions (see Appendix B for details). Note that we have assumed for the purpose of this analysis that the aforementioned contributions of arm kinematics, motor noise, and input device can be described by a temporal filter with fixed properties.

*R*for each combination of observer and blob width. Error distributions on

*R*were computed via bootstrapping (i.e., resampling was performed on observers' data by resampling whole trials).

*Q*and

*R*) are estimated and then the filter can be used to generate estimates (

*x̂*) of the true target positions (

_{t}*x*). In our case, the noisy observations cannot be observed and we estimate the observation noise variance (filter parameter

_{t}*R*) given the true target positions (

*x*), the target position estimates (

_{t}*x̂*), and the target displacement variance (filter parameter

_{t}*Q*). Thus, we essentially use the Kalman filter model in reverse, treating

*x*and

_{t}*x̂*as known instead of

_{t}*y*, in order to accomplish the goal of estimating

_{t}*R*.

*R*(such as that on which our analysis converges), and two others (offset vertically for clarity) using incorrect values. Note that, visually, the standard deviations of the red and green traces are too large and too small, respectively. However, the standard deviation of the black curve is approximately equal to the standard deviation of the blue curve (the human error trace). This point is made clearer by examining the distributions of these residual position values collapsed across time (right column). Note that the black distribution has roughly the same width as the blue distribution, while the others are too big or too small. This is essentially what our fitting accomplishes: finding the Kalman filter parameter,

*R*, that results in a distribution of errors with a standard deviation that is “just right.” (Brett, 1987).

*ϕ*), and the spatial offset of the blob corresponding to

*d′*= 1.0 point (single interval) was interpolated from the fit. The

*d′*for single interval was used because it corresponds directly to the width of the signal + noise (or noise alone) distribution. Because

*P*=

_{R}*ϕ*( $ d \u2032 2 I MathType@MTEF@5@4@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qaceWGKbGbauaapaWaaSbaaSqaa8qacaaIYaGaamysaaWdaeqaaaaa@38EB@ $/2) =

*ϕ*( $ d \u2032 MathType@MTEF@5@4@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qaceWGKbGbauaaaaa@3707@ $/ $ 2 MathType@MTEF@5@4@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaGcaaWdaeaapeGaaGOmaaWcbeaaaaa@3708@ $) where

*P*is the percent rightward choices and

_{R}*d′*

_{2}

*is the 2-interval*

_{I}*d′*, threshold was defined as the change in position necessary to travel from the 50% to the 76% rightward point on the psychometric function.

*d′*of 1.0, thus representing the situation in which the relevant distributions along some decision axis were separated by their common standard deviation. Assuming that the position of the target distribution on the decision axis is roughly a linear transformation of the target's position in space, then this also corresponds to the point at which the targets were separated by roughly one standard deviation of the observer's uncertainty about their position. Thus, the offset thresholds serve as an estimate of the width of the distribution that describes the observer's uncertainty about the target's position. This is exactly what the positional uncertainty estimates represented in the tracking experiment. In fact, it would be reasonable to call the forced-choice thresholds “positional uncertainty estimates” instead. The use of the word threshold is simply a matter of convention in traditional psychophysics.

*y*coordinates) versus those from the traditional psychophysics (

*x*coordinates). The log-log slopes are 0.98 (LKC), 1.12 (JDB), and 1.02 (KLB). The corresponding correlations are 0.985, 0.996, and 0.980, respectively. Obviously, the results are in good agreement; the change in psychophysical thresholds with blob width is accounting for over 96% of the variance in the estimates obtained from the tracking paradigm, the high correlation indicates that the two variables are related by an affine transformation. In our case (see Figure 10), the variables are related by a single scalar multiplier. This suggests to us that the same basic quantity is being measured in both experiments.

*d′*= 1) from the cumulative normal fits, the error bands show ±1 standard error estimated by bootstrapping, and the solid black line show the mean thresholds across subject. Thresholds for all observers decreased with increased stimulus duration at the expected slope of 1/ $ ( n ) MathType@MTEF@5@4@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaGcaaWdaeaadaqadaqaaiaad6gaaiaawIcacaGLPaaaaSWdbeqaaaaa@38C9@ $ (dashed line for reference) until flattening out at roughly 50 to 100 ms, or three to five frames (Watson, 1979; Nachmias, 1981).

*British Journal of Psychology*(Craik, 1947, 1948). Because circuits or, later, computers, are generally much better feedback controllers than humans, there has been less interest in the specifications of human-as-controller with a few exceptions: studies of pilot performance in aviation, motor control, and eye movement research (in some ways a subbranch of motor control, in other ways a subbranch of vision).

*p*(

*right*) = 0.5). The stimuli and templates were rearranged as vectors so that the entire operation could be done as a single dot product as in Ackermann and Landy (2010). The ideal observer was run in exactly the same experiment as the human observers, except that the offsets were a factor of 10 smaller, which was necessary to generate good psychometric functions because of the model's greater sensitivity.

*d′*) is approached at middling blob widths, which is consistent with previous work using grating patches embedded in noise (Simpson, Falkenberg, & Manahilov, 2003).

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*R*, see Figure 5) and therefore also position uncertainty, which is defined as $ R MathType@MTEF@5@4@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaGcaaWdaeaapeGaamOuaaWcbeaaaaa@3724@ $. The two time series produced by the experimental tracking paradigm—target position (

*x*) and subject response (

_{t}*x̂*)—are used in conjunction with the Kalman filter in order to fit observation noise variance by maximizing

_{t}*p*(

*x̂*|

*x*), the probability of the position estimates given the target position under the Kalman filter model.

*x*represents the target position, and

_{t}*y*represents the subjects' noisy sensory observations, which we cannot access directly (see Figure 5).

_{t}*y*

_{1:}

*and the parameters {*

_{t}*Q*,

*R*}, the Kalman filter gives a recursive expression for the mean and variance of

*x*|

_{t}*y*

_{1:}

*, that is, the posterior over*

_{t}*x*at time step

*t*given all the observations

*y*

_{1}, … ,

*y*. The posterior is of course Gaussian, described by mean

_{t}*x̂*and variance

_{t}*P*. The following set of equations perform the dynamic updates of the Kalman filter and result in target position estimates (

_{t}*x̂*).

_{t}*p*(

*x̂*|

*x*). First, we find the asymptotic value of

*P*and then use that to simplify and rewrite the Kalman filter equations in matrix form.

_{t}*P*

_{0}=

*P*

_{∞}; that is, the initial posterior variance will approach some asymptotic posterior variance. A Kalman filter asymptotes in relatively few time steps. In practice, our observers seem to as well, but to be safe we omitted the first second of tracking for each trial to insure that the observers' tracking had reached a steady state. Then the prior variance

*S*, Kalman gain

*K*, and posterior variance

*P*are constant. Thus, the dynamics above can be simplified to: where

*K*depends only on

*R*:

**x̂**and

**x**, to the unknown

*R*. We can use this to write

*p*(

**x̂**|

**x**):

*log*(

*p*(

*x̂*|

*x*)) (below), is used in order to perform the maximum-likelihood estimation of

*R*. where

*n*is the total number of time points (i.e., the length of

**x**and

**x̂**; Note: coefficients

*D*and

*K*are defined in terms of

*Q*and

*R*). The log likelihood for a particular blob width (

*σ*=

*s*) for a given subject is evaluated by taking the sum over all trials with

*σ*=

*s*of

*p*(

**x̂**|

**x**). In our analysis, maximum-likelihood estimation of

*R*is performed for each blob width in order to investigate how the observer's positional uncertainty ( $RMathType@MTEF@5@4@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaGcaaWdaeaapeGaamOuaaWcbeaaaaa@3724@$) changes with increasing blob width (decreasing visibility).

^{1}