Bayesian adaptive estimation of the contrast sensitivity function: the quick CSF method

Abstract

The contrast sensitivity function (CSF) predicts functional vision better than acuity, but long testing times prevent its psychophysical assessment in clinical and practical applications. This study presents the quick CSF (qCSF) method, a Bayesian adaptive procedure that applies a strategy developed to estimate multiple parameters of the psychometric function (A. B. Cobo-Lewis, 1996; L. L. Kontsevich & C. W. Tyler, 1999). Before each trial, a one-step-ahead search finds the grating stimulus (defined by frequency and contrast) that maximizes the expected information gain (J. V. Kujala & T. J. Lukka, 2006; L. A. Lesmes et al., 2006), about four CSF parameters. By directly estimating CSF parameters, data collected at one spatial frequency improves sensitivity estimates across all frequencies. A psychophysical study validated that CSFs obtained with 100 qCSF trials ( approximately 10 min) exhibited good precision across spatial frequencies (SD < 2-3 dB) and excellent agreement with CSFs obtained independently (mean RMSE = 0.86 dB). To estimate the broad sensitivity metric provided by the area under the log CSF (AULCSF), only 25 trials were needed to achieve a coefficient of variation of 15-20%. The current study demonstrates the method's value for basic and clinical investigations. Further studies, applying the qCSF to measure wider ranges of normal and abnormal vision, will determine how its efficiency translates to clinical assessment.

Figure 1

CSF parameterization. The spatial contrast sensitivity function, which describes reciprocal contrast threshold as a function of spatial frequency, can be described by four parameters: (1) the peak gain, γ_max; (2) the peak frequency, f_max; (3) the bandwidth (full-width at half-maximum), β; and (4) the truncation (plateau) on the low-frequency side, δ. The qCSF method estimates the spatial CSF by using Bayesian adaptive inference to directly estimate these four parameters.

Figure 2

Simulations. (a) CSF estimates obtained with 25, 50, 100, and 300 trials of the qCSF method. The general shape of the CSF is recovered with as few as 25 trials, but sensitivity estimates are imprecise (shaded regions reflect ±1 SD for individual frequencies). Method convergence with increasing trial numbers (50–300 trials) is supported by (1) the increasing concordance of mean qCSF estimates (red) with the true CSF (blue), and (2) the decreasing area of the error regions. (b) Expected bias of AULCSF estimates as a function of trial number. Evidence for the successful rapid estimation of the AULCSF is provided by (1) the convergence of the mean bias to zero and (2) the decreasing area of the error region (±1 standard deviation) as a function of increasing trial number.

Figure 3

Stimulus sampling. The history of the qCSF’s stimulus sampling pattern is characterized by two-dimensional probability density histograms, aggregated across simulations. Each histogram describes the probability of stimulus presentation, as a function of grating frequency and contrast, for four experimental cutoff points: t = 25, 50, 100, and 300 trials. Even with as few as 25–50 trials, testing is narrowed to a region of the grating stimulus space that correlates with observer sensitivity. For more extensive testing (100–300 trials), stimulus presentation focuses almost exclusively on the true CSF.

Figure 4

Test accuracy. Spatial CSFs obtained with two independent and concurrent adaptive procedures: the qCSF method (blue) and the ψ method (red). CSF estimates obtained from different subjects are presented in different rows; estimates obtained with 25, 50, or 100 trials are presented in different columns. The gray-shaded region reflects the variability of qCSF estimates (8 runs in total), and the red error bars reflect variability of ψ estimates (4 runs in total).

Figure 5

Test precision. (a, b) Test–retest comparisons for the two qCSF runs (of 100 trials) applied in each testing session. (a) Contrast sensitivities measured with the second qCSF run plotted against those obtained in the first run. The Pearson correlation coefficient for these comparisons averaged, r = 96% (SD = 4%), across all testing sessions. (b) A Bland–Altman plot presents the differences between sensitivity estimates obtained from each qCSF run, plotted against their mean. Mean difference <0.01 and the standard error of the difference = 0.175 log units. (c) Coefficient of variability (in percent) of AULCSF estimates obtained from three observers (8 runs each), as a function of trial number. AULCSF estimates converge in agreement with simulations (<15% by the completion of 25 trials).

Figure 6

Clinical application. The qCSF was applied to characterize contrast sensitivity functions in an amblyope. (a) Spatial CSFs were measured in three conditions: (1) one binocular CSF; (2) one monocular CSF measured in the amblyopic eye; and (3) one monocular CSF measured in the fellow eye. Spatial CSFs obtained with only 25 trials demonstrate a severe contrast sensitivity deficit, which is likewise apparent in (b), the AULCSF estimate with as few as 10 trials. AULCSF estimates are approximately stable after 25 trials. One important feature of the qCSF’s stimulus placement strategy is the high rate of evoked performance: this observer completed 75 trials with a comfortable performance level of 84% correct.

Movie 1

The movie demonstrates a simulated 300-trial sequence of the qCSF application in a 2AFC task. In addition to the true CSF (black line), the large leftmost panel presents each trial’s outcome and the subsequently updated qCSF estimate (green line). For each simulated trial, the selected grating stimulus is presented as a dot, whose color represents a correct (green) or incorrect (red) response. The inset presents the results of the qCSF’s stimulus selection algorithm (the pre-trial calculation of expected information gain as a function of grating frequency and contrast), with the updating qCSF estimate (white) overlaid as a reference. The right-hand panels demonstrate the trial-by-trial Bayesian update of the probability density defined over four CSF parameters; in addition to a pair of 2-D marginal densities—defined by peak gain and peak frequency (top), and bandwidth and low-frequency truncation (bottom)—the 1-D marginals for each parameter are presented. The white cross-lines represent the targets of parameter estimation: the observer’s true CSF parameters. The small white dots represent Monte Carlo samples of the probability density, which accelerate the pre-trial calculations (see Appendix A). At the demo’s completion, the main plot’s inset presents the bias of AULCSF estimates (in percent) as a function of trial number for the full-simulated run. Several features of the demo providing evidence for the qCSF’s successful convergence are: (1) the overlap of the CSF estimate with the true CSF; (2) how rapidly the stimulus selection algorithm excludes large regions of the stimulus space and focuses on the region of the stimulus space corresponding to the true CSF; (3) the aggregation of probability mass in the parameter space regions corresponding to the true CSF parameters; and (4) the convergence of the AULCSF error estimate toward 0%.

Figure A1

For one subject’s completed qCSF run, comparison of the prior for trial number 1 (blue) and the posterior following trial number 100 (red) demonstrates that the priors do not dominate the CSF parameter estimates.

Figure B1

See text.