**The contrast sensitivity function (CSF) provides a fundamental characterization of spatial vision, important for basic and clinical applications, but its long testing times have prevented easy, widespread assessment. The original quick CSF method was developed using a two-alternative forced choice (2AFC) grating orientation identification task (Lesmes, Lu, Baek, & Albright, 2010), and obtained precise CSF assessments while reducing the testing burden to only 50 trials. In this study, we attempt to further improve the efficiency of the quick CSF method by exploiting the properties of psychometric functions in multiple-alternative forced choice ( m-AFC) tasks. A simulation study evaluated the effect of the number of alternatives m on the efficiency of the sensitivity measurement by the quick CSF method, and a psychophysical study validated the quick CS method in a 10AFC task. We found that increasing the number of alternatives of the forced-choice task greatly improved the efficiency of CSF assessment in both simulation and psychophysical studies. The quick CSF method based on a 10-letter identification task can assess the CSF with an averaged standard deviation of 0.10 decimal log unit in less than 2 minutes.**

**Figure 1**

**Figure 1**

*truncated log parabola*(Lesmes et al., 2010; Watson & Ahumada, 2005; Figure 1d) with four parameters: peak gain

*g*

_{max}, peak spatial frequency

*f*

_{max}, bandwidth at half-height

*β*(in octaves), and low-frequency truncation level

*δ*. Using a Bayesian adaptive algorithm (Cobo-Lewis, 1996; Kim, Pitt, Lu, Steyvers, & Myung, 2014; King-Smith, Grigsby, Vingrys, Benes, & Supowit, 1994; Kontsevich & Tyler, 1999; Kujala & Lukka, 2006; Lesmes, Jeon, Lu, & Dosher, 2006; Watson & Pelli, 1983) to select the optimal test stimulus and update the posterior probabilities of CSF parameters following each trial, the quick CS method directly estimates the entire CSF curve instead of sensitivities at some pre-determined spatial frequencies (See Appendix A for more details). Unlike the conventional methods that select stimuli adaptively in only contrast space, the quick CSF method searches stimuli in both contrast and frequency spaces (Figure 1e and f), making it more efficient. For a 2AFC grating orientation identification task, only 5–10 minutes are needed to obtain a CSF with a 0.10–0.20 decimal log unit standard deviation, comparable to conventional methods using much longer testing times.

*m*-AFC) tasks. Increasing the number of alternatives in

*m*-AFC tasks has two effects on psychometric functions. First, it reduces the guessing rate and therefore makes each trial more informative. Second, it increases the slope of the psychometric function (Figure 2b).

**Figure 2**

**Figure 2**

*m*-AFC tasks. For example, Hall (1983) and Shelton and Scarrow (1984) found that thresholds obtained with a 3AFC auditory task were less variable than those obtained with a 2AFC task. Bi, Lee, and O'Mahony (2010) concluded that the 4AFC task was statistically more powerful than the 2AFC task in food flavor discrimination. Using a contrast sensitivity function assessment test, Jäkel and Wichmann (2006) also reported that a 4AFC task was 3.5 times more efficient than a 2IFC task in contrast detection. In a simulation study, Leek, Hanna, and Marshall (1992) used the up-down staircase procedure with 2AFC, 3AFC, and 4AFC tasks to estimate both the threshold and slope of the psychometric function. They found that the test efficiency for slope estimate, defined as the sweat factor, was highest for the 4AFC task and declined when the number of alternatives decreased.

*m*) in

*m*-AFC tasks would improve the efficiency of the quick CSF method. Indeed, an earlier simulation study found that the average standard deviation of CSFs obtained from the quick CSF method decreased from 0.13 to 0.07 log unit when the slope of the log-Weibull psychometric function increased from 1.55 to 3.5 (Hou et al., 2010).

*m*) in

*m*-AFC tasks on the precision of the quick CSF method. We then report a psychophysical validation experiment of the quick CSF method in a 10AFC task. The CSFs of five normal observers were measured with both the quick CSF and conventional methods. We also compared the efficiency of the quick CSF method based on the 10AFC task with that based on a 2IFC task in a published study (Hou et al., 2010).

**Psychometric functions in**

**m****-AFC**

*d′*= 1.5

^{1}for a simulated observer performing an

*m*-AFC task, with

*m*= 2, 4, 8, 10, and 16. The

*d′*psychometric function of the simulated observer is described by the following equation (Foley & Legge, 1981; Legge, Kersten, & Burgess, 1987; Figure 2a): where

*c*is the contrast of the stimulus,

*τ*(

*f*) is the contrast threshold at

*d′*= 1.5 in the

*f*spatial frequency condition, and

*ζ*is the log-log slope of the

*d′*psychometric function. We set

*ζ =*2.35 in the simulation based on typical values in the literature (Foley & Legge, 1981; Legge et al., 1987; Lu & Dosher, 1999). In this formulation,

*ζ*is independent of the threshold level

*τ*(

*f*); that is, the

*d*′ psychometric functions in different spatial frequency conditions have different thresholds but exactly the same shape and are only shifted horizontally on the low contrast axis.

*m-*AFC task, the probability correct psychometric function of the simulated observer can be derived from the

*d′*psychometric function (Hacker & Ratcliff, 1979): where

*ϕ*() and Φ() are the probability density and cumulative probability density functions of a standard normal distribution,

*d′*(

*c,f*) is the

*d′*value associated with a stimulus with contrast

*c*and spatial frequency

*f*, and

*m*is the number of alternatives. Although the slope of the

*d′*psychometric function

*ζ*is invariant to the number of alternatives in the

*m*-AFC task, the slope of the probability correct psychometric function depends on the number of alternatives,

*m*(Figure 2b).

*λ*that is independent of stimulus level

^{2}(Klein, 2001; Wichmann & Hill, 2001): where

*P*(

*c,f,m*) is the psychometric function without lapse (Equation 2). For the simulated observer and in the quick CSF method,

*λ*was set to 0.04 (Lesmes et al., 2010; Wichmann & Hill, 2001).

*m-*AFC tasks. However, the computational load for integration in Equation 2 is very heavy; Weibull functions are used in our simulation and the quick CSF procedure to approximate these functions and greatly reduce the computational load. In Appendix B, we provide details on Weibull approximations.

*m*-AFC tasks in all possible stimulus conditions can be computed. We then used these probabilities to simulate the response of the observer in the quick CSF procedure, which is used to infer the underlying contrast threshold function,

*τ*(

*f*), of the observer.

*τ*

^{true}(

*f*), which in turn were used to generate the simulated observer's response in each trial by Equation 3.

*ζ*,

*γ*, and

*λ*(Equation 3), is known. Hou et al. (2010) demonstrated that the assumption of the log-invariant psychometric function was largely correct. Only

*τ*(

*f*) was being estimated by the quick CSF method. Again, the detailed quick CSF algorithm is described in Appendix A.

*p*(

_{t}*θ*), and used to construct 1,000 CSF curves. Based on these CSF curve samples, we obtained the empirical distribution of the CSF,

*p*(

_{t}*τ*). Each CSF curve was evaluated at 20 spatial frequencies ranging from 0.5 to 32 cpd, evenly distributed in log space. This resampling procedure automatically takes into account the covariance structure in the posterior distribution of the CSF parameters, and allows us to compute variance of the estimated CSF curve.

*k*th spatial frequency in the

*j*th run,

*τ*is the average of

^{k}*k*= 1, 2, … 20 is the index of spatial frequencies,

*i*= 1, 2, … 1,000 is the index of CSF samples from the posterior distribution of a single quick CSF run, and

*j*= 1, 2, … 500 is the index of the quick CSF runs. The accuracy of the quick CSF method, defined by the bias of the estimated CSF is calculated as the mean differences between the measured and true sensitivities: where $\tau k'=$$\u2211j=1500\tau jk500MathType@MTEF@5@4@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaxacabaaeaaaaaaaaa8qacqaHepaDpaWaaWbaaSqabeaapeGaam4Aaaaaa8aabeqaa8qacaGG0caaaOGaeyypa0ZaaSaaa8aabaWdbmaavadabeWcpaqaa8qacaWGQbGaeyypa0JaaGymaaWdaeaapeGaaGynaiaaicdacaaIWaaan8aabaWdbiabggHiLdaakiabes8a09aadaqhaaWcbaWdbiaadQgaa8aabaWdbiaadUgaaaaak8aabaWdbiaaiwdacaaIWaGaaGimaaaaaaa@499D@$ and

*τ*

^{true,}

*are the estimated and true contrast sensitivity at the*

^{k}*k*th spatial frequency. The standard deviation and bias of sensitivity estimates were averaged across all spatial frequencies and are both in decimal log sensitivity units.

*p*(

_{t}*τ*) from a single quick CSF run. A 68.2% credible interval represents the shortest interval that contains the actual value with 68.2% probability (Clayton & Hills, 1993). Since it is uncommon to repeat the same measurement multiple times in clinical practice, the credible interval of the posterior distribution is a valuable tool to gauge the precision of a test in a single run. Here we report the half width of the 68.2% credible interval (HWCI) for a single quick CSF run. We choose 68.2% credible interval because if the posterior distribution is Gaussian, the 68.2% HWCI is equal to the standard deviation of the distribution. The HWCI is also in unit of decimal log sensitivity.

*m*-AFC tasks with different numbers of alternatives are approximately laminated. The downward shift between curves reflects increasing precision as the number of alternatives increases from 2 to 16.

**Figure 3**

**Figure 3**

*m*-AFC task can substantially improve the efficiency of the quick CSF method. It also suggests that the benefit of large alternative number is even greater with more trials.

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

*m*-AFC task. In this section, we report a psychophysical validation study of the 10AFC quick CSF procedure with a 10-letter identification task, in which observers were asked to identify a randomly chosen letter presented on the screen in each trial.

^{2}, a 1920 × 1080 pixel resolution, and a vertical refresh rate of 60 Hz. A special circuit was used to achieve 14-bit grayscale resolution (Li & Lu, 2012; Li, Lu, Xu, Jin, & Zhou, 2003). Participants viewed the stimuli at a distance of 5 m in a dark room.

*f*denotes radial spatial frequency,

*f*

_{0}= 3 cycles per object (cpo) is the center frequency of the filter, and

*f*

_{cutoff}= 2

*f*

_{0}was chosen such that the full bandwidth at half height is 1 octave. The pixel intensity of each filtered image was normalized by the maximum absolute intensity of the image such that, after normalization, the maximum absolute Weber contrast of the image is 1.0 (Figure 6a). Stimuli with different contrasts were obtained by scaling the intensities of the normalized images with corresponding values. The filtered images were rescaled to 16 different sizes to generate stimuli with 16 evenly spaced (in log space) central spatial frequencies ranging from 1.33 to 32.0 cpd for the quick CSF procedure (Figure 6b). For the conventional method, stimuli at six evenly spaced (in log space) central spatial frequencies ranging from 1.33 to 16.0 cpd were generated.

**Figure 6**

**Figure 6**

^{3}

**Figure 7**

**Figure 7**

*p*< 0.001 for all observers). In addition, all the data points are distributed along the unity line: the slope of the linear regression line is 1.0, 0.987, and 0.99 for CSFs obtained with 10, 20, and 50 quick CSF trials, respectively, which is not significantly different from 1.0. The results show excellent agreement between the quick CSF and conventional methods.

**Figure 8**

**Figure 8**

^{4}In computing bias, CSFs obtained from the conventional method were used as the “true” values.

**Figure 9**

**Figure 9**

*SD*), 0.10 ± 0.04, and 0.06 ± 0.03 log unit after 10, 20, and 50 trials, respectively. The average standard deviation of the CSFs obtained with the 10AFC quick CSF procedure is less than that of the 2IFC quick CSF with 50 trials when the trial number is greater than 8 (

*p*< 0.001). The relative efficiency of the 10AFC quick CSF procedure was 214%, 274%, and 336% at 10, 20, and 50 trials, respectively, relative to the 2IFC quick CSF procedure. For CSFs obtained with the 10AFC procedure, the average HWCI was 0.20 ± 0.06, 0.12 ± 0.04, and 0.06 ± 0.02 log unit after 10, 20, and 50 trials, respectively. For CSFs obtained with the 2IFC procedure, the average HWCI was 0.22 ± 0.06 log unit after 50 trials. With about 10 trials, the HWCI of the CSFs from the 10AFC procedure became narrower than that of the CSFs from the 2IFC procedure with 50 trials (

*p*< 0.001). For comparison, the average HWCI of CSFs from the conventional method with 300 trials is 0.023 ± 0.001 log unit. It took at least 150 trials of the conventional method to reach the same HWCI obtained by the 10AFC quick CSF procedure in 50 trials.

*p*> 0.05 for all trials except trial 38 to 44). No consistent bias was found. The variance of the bias of the CSFs obtained from the 10 AFC quick CSF procedure decreased much faster than that from the 2IFC quick CSF procedure.

*SD*) for repeated measures. The average COR of CSFs obtained from the 2IFC quick CSF procedure were 1.38 ± 0.19, 1.07 ± 0.22, and 0.62 ± 0.17 log10 unit after 10, 20, and 50 trials, respectively. The average COR of CSFs obtained with the 10AFC quick CSF procedure were 0.41 ± 0.18, 0.27 ± 0.11, and 0.16 ± 0.08 log10 unit after 10, 20, and 50 trials, respectively. The CSFs obtained with the 10AFC quick CSF procedure exhibited much lower COR than those obtained with the 2IFC quick CSF procedure, indicating better test–retest reliability.

*m*in

*m*-AFC tasks would improve the efficiency of the quick CSF method. The hypothesis was tested and confirmed in both computer simulations and a human psychophysics experiment.

*m*in an

*m*-AFC task greatly improved the efficiency of the quick CSF procedure. With 50 trials, the relative efficiency of the quick CSF procedure with 4, 8, 10, and 16 AFC was 156%, 211%, 221%, and 255%, respectively, compared to that with a 2AFC task.

*m*-AFC task in estimating AULCSF, we calculated the standard deviation, HWCI and bias of measured AULCSF of our simulated observer in a number of different spatial frequency ranges. Because the most common spatial frequencies used in clinical testing are 1.5, 3, 6, 12, and 18 cpd (American National Standards Institute, 2001; Montes-Mico & Charman, 2001; Pesudovs, Hazel, Doran, & Elliott, 2004), the frequency ranges used in our analysis are low frequencies (1.5–3 cpd), medium frequencies (3–12 cpd), high frequencies (12–18 cpd), and overall frequencies (1.5–18 cpd).

*m*in an

*m*-AFC task improves the precision of the measurement. Similar results are also obtained for AULCSFs in low, medium, and high frequency ranges (see the second, third, and fourth columns of Figure 10). The precision also increases with trial number. With 50 trials, the standard deviation of the estimated CSF is 0.11, 0.07, 0.05, 0.05, and 0.04 log unit for the 2, 4, 8, 10, and 16 AFC tasks, respectively, and the relative efficiencies of the quick CSF procedure for AULCSF with 4, 8, 10, and 16 AFC tasks are 158%, 228%, 237%, and 275%, respectively. The standard deviation and HWCI are essentially the same after about 30 trials for all

*m*-AFC tasks.

**Figure 10**

**Figure 10**

*d*′ psychometric function is related to the internal noise distribution and transducer of the observer (Lu & Dosher, 1998, 2008, 2014; May & Solomon, 2013) and is not easy to manipulate. However, for a single

*d*′ psychometric function, it is possible to reduce the guessing rate and increase the slope of the percent correct psychometric function by increasing the number of alternatives in an

*m*-AFC task, and therefore increase the efficiency of the adaptive procedure. The benefit of a larger number of alternatives in

*m*-AFC tasks may not only apply to the quick CSF procedure, but also to other Bayesian adaptive testing procedures such as QUEST, ZEST, Psi, quick TvC, and quick Partial Report, all of which are based on some underlying parametric psychometric functions (Baek, Lesmes, & Lu, 2014; King-Smith et al., 1994; Kontsevich & Tyler, 1999; Kujala & Lukka, 2006; Lesmes et al., 2006; Lesmes et al., 2010; Watson & Pelli, 1983). It would be worthwhile to perform further studies to test the magnitude of improvements for those methods.

*d′*psychometric function and the task configuration. In a given experimental setting, increasing

*m*in an

*m*-AFC task will lead to reduced guessing rate and increased slope of the percent correct psychometric function. However, across experimental settings, the underlying

*d′*psychometric function may be quite different (Eckstein, Abbey, & Bochud, 2000) and can affect the shape of the probability correct psychometric function (Equation 2). One can't simply look at the

*m*in an

*m*-AFC task to determine if a procedure is more efficient. Figure 11 shows the standard deviation and HWCI curves of CSF obtained by two quick CSF procedures, one is based on a 2AFC task with a slope

*ζ*(the log-log slope of the

*d′*psychometric function, Equation 1) of 2.35, and the other is based on a 4AFC task with a slope

*ζ*= 1.1. The precision of the 4AFC task with a shallower psychometric function is lower than that of the 2AFC task with a steeper psychometric function.

**Figure 11**

**Figure 11**

*m*-AFC tasks (Equation 2). By applying the same filter to all the letters, we have restricted the spatial content of all the stimuli to a two-octave range and reduced the difference in spatial content among letters. This could significantly reduce the probability of confusion. In fact, Gervais, Harvey, and Roberts (1984) found that the differences in spatial frequency content of letters provided the best prediction of the confusion matrix in letter recognition. Furthermore, there is evidence that CSFs measured with narrowband letters was very similar to those measured with sinewave gratings or D6 patterns (Alexander et al., 1994; Hou & Lu, 2014; McAnany & Alexander, 2006).

*m*-AFC tasks is not the only way to increase the efficiency of quick CSF. Kim et al. (2014) have recently developed a hierarchical adaptive design optimization (HADO) procedure that achieves greater accuracy and efficiency in adaptive information gain by exploiting two complementary schemes of inference with past and future data. HADO extends the standalone quick CSF method to a framework that models a higher-level structure across the population, which can be used as an informative prior for each new assessment. In turn, the parameter estimates from each individual enable the update of the higher-level structure. The judicious application of informative priors used by HADO improves the efficiency of the quick CSF method by approximately 30%.

*Psychological Methods*, 9 (2), 250–271.

*Journal of the Optical Society of America A: Optics, Image Science, and Vision*, 11, 2375–2382.

*American National Standard for Ophthalmics–Multifocal Intraocular Lenses*. Fairfax, VA: Optical Laboratories Association.

*Journal of Refractive Surgery*, 14, 397–407.

*Bayesian adaptive estimation of the sensory memory decay function: the quick Partial Report method*. Paper presented at the 2014 Annual meeting of the Vision Science Society, St. Pete Beach, FL.

*Contrast sensitivity of the human eye and its effects on image quality*. Knegsel, The Netherlands: HV Press.

*d*′ and variance of

*d*′ for four-alternative forced choice (4-AFC).

*Journal of Sensory Studies*, 25, 740–750.

*Statistical Methods in Medical Research*, 8, 135–160.

*Visual masking: An integrative approach*. New York: Oxford University Press.

*Vision Research*, 42, 2137–2152.

*Statistical models in epidemiology*. Oxford, UK: Oxford University Press.

*An adaptive method for estimating multiple parameters of a psychometric function*. Paper presented at the 29th Annual Meeting of the Society for Mathematical Psychology, Chapel Hill, NC.

*Journal of the Optical Society of America A: Optics, Image Science, and Vision*, 17, 206–217.

*Vision Research*, 21, 1041–1053.

*Proceedings of the National Academy of Sciences, USA*, 110, 4368–4373.

*Journal of Experimental Psychology: Human Perception and Performance*, 10, 655–666.

*Clinical applications of visual psychophysics*. (pp. 70–106). Cambridge, UK: Cambridge University Press.

*International Ophthalmology Clinics*, 43 (2), 5–15.

*Photographic Science and Engineering*, 16, 221–230.

*d*′ for M-alternative forced choice.

*Perception & Psychophysics*, 26, 168–170.

*The Journal of the Acoustical Society of America*, 73, 663–667.

*Clinical applications of visual psychophysics*. (pp. 11–41). Cambridge, UK: Cambridge University Press.

*Vision Research*, 17, 1049–1055.

*Comparisons of contrast sensitivity functions measured by two different qCSF implements*.

*Investigative Ophthalmology and Visual Science, 55*, E-abstract 770.

*Vision Research*, 47, 22–34.

*Journal of Cataract & Refractive Surgery*, 15, 141–148.

*Proceedings of the National Academy of Sciences, USA*, 111, 2035–2039.

*Perception & Psychophysics*, 14, 313–318.

*Neural Computation*, 26, 2465–2492.

*Vision Research*, 34, 885–912.

*Vision Research*, 37, 1595–1604.

*Perception and Psychophysics*, 63, 1421–1455.

*Perception*, 36, 14.

*Vision Research*, 39, 2729–2737.

*Journal of Mathematical Psychology*, 50, 369–389.

*Emotion*, 14, 978–984.

*Perception and Psychophysics*, 51, 247–256.

*Journal of the Optical Society of America A: Optics, Image Science, and Vision*, 4, 391–404.

*Investigative Ophthalmology and Visual Science*, 54, E-abstract 2762.

*Investigative Ophthalmology and Visual Science*, 53, E-abstract 4358.

*Vision Research*, 46, 3160–3176.

*Journal of Visualized Experiments*, 60, e3312, doi:10.3791/3312.

*Journal of Neuroscience Methods*, 130, 9–18.

*Vision Research*, 38, 1183–1198.

*Journal of the Optical Society of America A: Optics, Image Science, and Vision*, 16, 764–778.

*Psychological Review*, 115 (1), 44–82.

*Visual psychophysics: From laboratory to theory*. Cambridge, MA: The MIT Press.

*Vision Research*, 44, 1333–1350.

*Annals of Ophthalmology*, 13, 1069–1071.

*The British Journal of Ophthalmology*, 70, 553–559.

*Ophthalmology*, 95, 139–143.

*PLoS One*, 8 (10), e74815.

*Vision Research*, 46, 1574–1584.

*Journal of Refractive Surgery*, 17, 646–651.

*Documenta Ophthalmologica*, 103 (3), 175–186.

*Acta Psychologica (Amst)*, 139, 19–37.

*Advances in Ophthalmology*, 41, 103–148.

*International Ophthalmology*, 28, 407–412.

*Ophthalmology*, 113, 1807–1812.

*Brain*, 103, 139–147.

*British Journal of Ophthalmology*, 88, 11–16.

*Investigative Ophthalmology and Visual Science, 56*, E-abstract 2225.

*Spatial vision*(pp. 239–249). Boca Raton, FL: CRC Press.

*Journal of Vision*, 14 (10): 1428, doi:10.1167/14.10.1428. [Abstract]

*Archives of Ophthalmology*, 128, 1576–1582.

*ARVO Investigative Ophthalmology and Visual Science*,

*56*, E-abstract 2224.

*Journal of Ophthalmic & Vision Research*, 5, 175–181.

*Perception & Psychophysics*, 35, 385–392.

*Quarterly Review of Ophthalmology and Otorhinolaryngology*, 8, 4–17.

*The Journal of the Acoustical Society of America*, 49 (2), Suppl 2, 505–508.

*The Journal of the Acoustical Society of America*, 41 (4A), 782–787.

*Vision Research*, 35, 2503–2522.

*Perception & Psychophysics*, 33, 113–120.

*Perception & Psychophysics*, 63, 1293–1313.

*Clinical and Experimental Optometry*, 78, 43–57.

*Eye*, 21, 218–223.

^{1}We normalized Equation 1 by

*d*′ = 1.5 because the true values of the CSF parameters

*d*′ of about 1.5.

^{2}We assume that the observer would make a random guess when she is in lapse. The formulation is slightly different from Klein (2001) and Wichmann and Hill (2001), which did not consider guessing in lapse trials. When

*λ*is low (i.e., 0.04), there is no significant difference between the two definitions.

^{3}The low frequency truncation is not apparent in these CSFs because of the range of spatial frequencies covered in this study (1.33–32 cpd) is relative high compared to the 0.5–16 cpd range used in Hou et al. 2010. The low frequency truncation parameter

*δ*is necessary in CSF tests that include lower spatial frequency conditions.

^{4}The standard deviation reported in Hou et al., (2010) was from a bootstrap procedure because they only had two repeated measures of each CSF.

**1.**

**Define a CSF functional form.**

*τ*(

*f*) is the reciprocal of contrast sensitivity

*S*(

*f*):

*θ*= (

*g*

_{max},

*f*

_{max},

*β*,

*δ*) represents the four CSF parameters: peak gain

*g*

_{max}, peak spatial frequency

*f*

_{max}bandwidth at half-height

*β*(in octaves), and low-frequency truncation level

*δ*.

**2.**

**Define the stimulus and parameter spaces.**The application of Bayesian adaptive inference requires two basic components: (a) a prior probability distribution,

*p*(

*θ*), defined over a four-dimensional space of CSF parameters

*θ*, and (b) a two-dimensional space of possible letter stimuli with contrast

*c*and spatial frequency

*f*.

*c*and 0.5–32 cpd for frequency

*f*. Both parameter and stimuli spaces were sampled evenly in log unit.

**3.**

**Priors.**Before the beginning of the experiment, an initial prior,

*p*

_{t}_{= 0}(

*θ*), which represents the knowledge about the observer's CSF before any data is collected, was defined by a hyperbolic secant function with the best guess of parameters

*θ*and width of

_{i, guess}*θ*

_{i, }_{confidence}for

*i*= 1, 2, 3, and 4 (King-Smith & Rose, 1997; Lesmes et al., 2010). where

*θ*

_{i}=

*g*

_{max},

*f*

_{max},

*β*and

*δ*for

*i*= 1, 2, 3, and 4, respectively,

*θ*

_{i, guess}= 100, 2, 3, and 0.5 for

*i*= 1, 2, 3, and 4, respectively.

*θ*

_{i, confidence}. = 2.48, 3.75, 7.8, and 3.12 for

*i*= 1, 2, 3, and 4, respectively.

**4.**

**Bayesian adaptive inference.**After subject's response is collected in trial

*t*, knowledge about CSF parameters

*p*(

*θ*) is updated, given the evidence provided by the observer's response

*r*= “correct” or “incorrect” to the stimulus

_{x}*x*= (

*c*,

*f*) with contrast

*c*and spatial frequency

*f*in the trial. The outcome of trial

*t*is incorporated into a Bayesian inference step that updates the prior knowledge about CSF parameters

*p*

_{t}_{-1}(

*θ*), where

*p*(

_{t}*θ*|

*r*) is the posterior distribution of parameter vector

_{x}*θ*after obtaining a response

*r*at trial

_{x}*t*;

*p*(

*r*= correct |

_{x}*θ*) = Ψ(

*x;θ*) is the percent correct psychometric function given stimulus

*x*, and

*p*(

*r*= incorrect |

_{x}*θ*) = 1 − Ψ(

*x*,

*θ*);

*p*

_{t}_{−1}(

*θ*) is our prior about

*θ*before trial

*t*, which is also the posterior in trial

*t*−1.

**5.**

**Stimulus search.**To increase the quality of the evidence obtained on each trial, the quick CSF calculates the expected information gain for all possible stimuli

*x*, where

*h*(

*p*) = −

*p*log(

*p*) − (1 −

*p*)log(1 −

*p*) is the information entropy of the distribution

*p*. Before each trial, we find out the candidate stimuli that correspond to the top 10% of the expected information gain over the entire stimulus space. Then we randomly pick one among those candidates as

*x*for presentation. In this way, the quick CSF avoids large regions of the stimulus space that are not likely to provide useful information to the current knowledge about

_{t}*θ*.

**6.**

**Reiteration and stopping rule**. The procedure reiterates steps 4 and 5 until 300 trials are run.

**7.**

**Analysis**. After step 6, we obtain the posterior distribution of CSF parameters

*p*(

_{t}*θ*) (see Figure A1 for the marginal prior and posterior distributions for the four CSF parameters). A resampling procedure is used that samples directly from the posterior distributions of the CSF parameters and generates the CSF estimates based on all the CSF samples. The procedure automatically takes into account the covariance structure of the CSF parameters in the posterior distribution and allows us to compute the credible interval of the estimated CSF functions.

**Figure A1**

**Figure A1**

*d*′ psychometric function

*m*-AFC tasks (Equation 2). The approximation made the simulation about 20 times faster. The terms

*γ*(

*m*) and

*b*(

*m*) are the guessing rate and slope of the Weibull psychometric function in an

*m*-AFC task, and

*τ*

_{w}(

*f*,

*m*) is the Weibull contrast threshold in spatial frequency condition

*f*in an

*m*-AFC task, which can be computed from

*τ*(

*f*). where

*p*

_{1.5}(

*m*) is the fraction of correct responses corresponding to

*d′*= 1.5 in an

*m*-AFC task.

*γ*(

*m*) and

*p*

_{1.5}(

*m*) are listed in Table B1 for a range of

*m*values used in our simulation study.

**Table B1**

*m*= 2, 4, 8, 10, and 16 and

*γ*= 1/

*m*. The Weibull provided an excellent approximation to the psychometric functions in Equation 2 with an average

*r*

^{2}= 0.999. The best fitting

*b*(

*m*)s are listed in Table B1. With predetermined

*τ*(

*f*),

*b*,

*γ*, and

*λ*, the response probabilities of the simulated observer in the

*m*-AFC tasks in all possible stimulus conditions can be computed.