**Purpose**:
The contrast sensitivity function (CSF) is measured traditionally with grating stimuli. Recently, we introduced a new set of digit stimuli to improve the efficiency of CSF tests for people unfamiliar with the Latin alphabet. Given the significant differences between grating and digit stimuli, we conducted this study to evaluate whether the estimated CSFs from the digit test are equivalent to those from the grating test.

**Methods**:
The CSFs of five young (with Psi) and five older (with quick CSF [qCSF]) participants were measured with a two-alternative forced choice (2AFC) grating orientation identification task and a 10-digit identification task. The CSFs obtained from the two tasks were compared.

**Results**:
The estimated CSFs from the two tasks matched well after controlling for stimulus types and performance levels. The root mean square error (RMSE) between the CSFs from the two tasks was 0.093 ± 0.029 (300 trials) and 0.131 ± 0.016 (100 trials) log10 units for young and older observers, respectively. To reach the same standard deviation (0.1 log10 units), the digit CSF test required fewer trials/less time than the classic grating CSF for young (60 vs. 90 trials) and older (15 vs. 21 trials) observers. The complicated behavioral responses of the observer in the 10-AFC digit identification task can be accounted by a model that consists of digit similarity and one single parameter of sensory noise (χ^{2}[99] = 3.42, *P* = 0.999).

**Conclusions**:
The estimated CSFs from the digit test highly matched those obtained from the grating test; however, the digit test is much more efficient.

**Translational Relevance**:
The digit CSF test provides a compatible assessment of the CSF as the traditional grating CSF test with more efficiency.

^{1–5}Since the discovery of multiple spatial frequency channels in vision and the introduction of Fourier analysis into vision science by the pioneering work of Campbell and Robson,

^{6}sinewave gratings have been widely used in vision research

^{7–11}and especially in contrast sensitivity tests, including the various CSF charts, such as the Functional Acuity Contrast Test (FACT),

^{1,12}CSV1000,

^{13}and Vistech Contrast Test,

^{14}and computer-based procedures.

^{15}

^{16–22}The efficiency of CSF tests with grating stimuli is limited by the number of response alternatives. One possible solution is using an

*N*-alternative forced choice orientation discrimination task with

*N*> 2. The problem is that gratings do not have a lot of different and readily named orientations. When more than four orientations are involved, grating orientation identification may introduce some cognitive load for the participants,

^{20}which makes it very difficult to improve the efficiency of the grating test.

^{23}

*N*-alternative forced-choice tasks to improve the efficiency of CSF tests. There is an added advantage that characters have distinct spatial structures and can be readily recognized without much effort. Indeed, a number of recent developments have shown that CSF tests with 10-alterantive forced choice tasks using either Sloan letters

^{24}or digits

^{25}are much more efficient than those with 2AFC grating tests. Because of the presence of multiple spatial channels in the visual system,

^{6}the contrast sensitivity function; that is, contrast sensitivity as a function of spatial frequency, measured with narrow-band stimuli provides a more comprehensive assessment of vision.

^{15,24,26}Many studies have used bandpass filtered letter or digit stimuli to assess the CSF.

^{24,25,27,28}

^{29,30}For the newly developed CSF tests, a very important question is: Are the estimated CSFs from the new tests with bandpass-filtered characters equivalent to those from the grating tests? We attempt to answer the question for the CSF measured with the digit stimuli.

^{31}which is the sum of squared contrast at each spatial location of the stimulus and accounts for effects of luminance distribution and stimulus area.

^{5,32,33}Thus the grating and digit stimuli with the same Michelson contrast value have different total contrast energy and result in drastically different threshold/sensitivity measures. Second, CSF tests in the literature use gratings of either equal size or equal number of cycles across different spatial frequencies.

^{34}Digit stimuli are by definition of equal number of cycles. Third, the contrast thresholds measured in the 10AFC and 2AFC tasks may correspond to different performance levels. Fourth, while most 2AFC grating CSF tests use equivalent and orthogonal stimuli, letter or digit stimuli have different legibilities.

^{35,36}The uneven similarities can make some alternatives more favored than others and, therefore, affect the test precision. Although the bandpass-filtered digit stimuli used in our previous study had a greatly improved similarity structure,

^{25}the effect of nonorthogonal stimuli on the estimated contrast sensitivities has not been systematically investigated to our knowledge.

^{37}and run on an Apple Mac mini computer (Model No. A1347; Apple Inc., Cuppertino, CA). Stimuli were displayed on a 27-inch ASUS PG279Q monitor (Asus Corp. Taipei, Taiwan) with a 2560 × 1440 pixel resolution and a 60 HZ refresh rate. The display was gamma-corrected with a photometer (ST-86A; Photoelectric Instrument Factory of Beijing Normal University, Beijing, China) and had a mean luminance of 91.2 cd/m

^{2}. A bit-stealing algorithm was used to achieve 9-bit gray-scale resolution.

^{38}Observers viewed the display binocularly with their best refractive corrections by wearing glasses in a dark room at a distance of 1.34 m. A chin/forehead rest was used to minimize head movement during the experiment.

^{39}The detailed specifications of the digits have been reported in our parallel study.

^{25}Because the unfiltered digits contains a range of spatial frequencies,

^{26}all digits were filtered with a raised cosine filter

^{24,26}to create bandpass-filtered stimuli (Fig. 1a). The center frequency of the filter is 3 cycles per object (cpo) and the full bandwidth at half height is one octave. The filtered digits, resized to 12°, 6°, 3°, 1.5°, 0.75°, and 0.38°, corresponding to central spatial frequencies of 0.5, 1, 2, 4, 8, and 15.8 cycles per degree (Fig. 1c), were used as signal stimuli in the 10-digit identification task.

*σ*=

*λ*/3) was used to blend the grating into the background. The spatial envelope resulted in a four-cycle full contrast circular window. By rescaling the grating images to different sizes, we generated grating stimuli with center spatial frequencies of 0.5, 1, 2, 4, 8, and 15.8 cycles per degree. The grating stimulus had a random selected phase from 0 to 2π in each trial.

^{40}The Psi method was programmed to estimate the thresholds using Weibull psychometric functions with a fixed slope of 3 in the grating task and a fixed slope of 2.74 in the digit task based on data from pilot studies. Each experimental block lasted approximately 20 minutes and consisted of 300 trials (50 trials × 6 spatial frequencies). Every block was divided into six mini-blocks. The observers could take a break after each mini-block. The spatial frequency of the test stimulus in each trial was selected randomly. Observers completed 10 grating blocks and 10 digit blocks over a total of five days. In each day, the order of the sessions was either “grating, digit, digit, grating” or “digit, grating, grating, digit”, counter-balanced over days for each observer.

^{32,41}When the stimulus area and duration are known, the square root of contrast energy can be simplified to the root mean square (RMS) contrast, which is defined as the standard deviation of the pixel contrast in an image. To equate the contrast energy of grating and digit stimuli, we converted the nominal contrast values of both types of stimulus into RMS contrast. We generated grating images and digit images at 100 different contrast levels and six spatial frequencies and calculated the RMS contrast for every image. By averaging the RMS contrast across orientations/digits and spatial frequencies, we obtained the RMS contrast as a function of nominal contrast for grating and digit stimuli, respectively. Figure 2 shows the relationship between the nominal and RMS contrasts for grating and digit stimuli, which can be described by straight lines with different slopes:

*d*′ performance levels, which represent the distance between the means of the signal and the noise distributions, divided by their common standard deviation.

^{42}The contrast threshold measured by the 2AFC task with the Psi method corresponds to 81.6% correct,

^{43}or a

*d*′ of 1.27. On the other hand, the contrast threshold measured by the 10AFC task corresponds to 66.9% correct, or a

*d*′ of 1.99.

^{24}To equate the

*d*′ performance level of the two tasks, we converted the measured thresholds in the digit task at 66.9% correct (

*d′*= 1.99) into thresholds at 43% correct (

*d′*= 1.27) using a Weibull psychometric function:

*γ*= 0.1, and

*β*= 2.74. (We used a fixed slope of 2.74 for the Psi method in the experiment.) With Equations 1 and 2, we converted the estimated contrast thresholds in both tasks into RMS contrast thresholds at the same

*d*′ performance level of 1.27.

*d*′ performance levels, the CSFs obtained from the digit and grating tasks were very close for all observers.

*F*(1, 4) = 12.99,

*P*= 0.023, with no significant interaction between the stimulus type and spatial frequency,

*F*(5, 20) = 0.358,

*P*= 0.871, which suggests there was a constant difference between the CSFs measured with grating and digit. To quantify the amount of difference between the CSFs obtained from the two tasks, we calculated the RMS error (RMSE) between them. As the CSF using the Psi method was measured with six parallel adaptive procedures in six spatial frequencies, the RMSE was evaluated every six trials, one for each spatial frequency. Figure 4a shows the RMSE as a function of trial number in Experiment 1. The RMSE decreased drastically in the first 30 trials. The average RMSE across all five young observers was 0.131 ± 0.042, 0.112 ± 0.038, 0.095 ± 0.040, and 0.093 ± 0.029 log10 units after 30, 60, 120, and 300 trials, respectively. We also performed a correlation analysis on the estimated contrast sensitivities from the two tasks (Fig. 4b). The Pearson correlation coefficient was 0.991 (

*P*= 5.71 × 10

^{−26}).

^{44–46}to quantify the agreement between the estimated CSFs from the digit and grating tasks (Fig. 4c). The mean difference was −0.072 log10 units. The limits of agreement were ± 0.128 log10 units. Taken together, the results indicated that the estimated CSFs with the grating and digit stimuli were highly matched with a small (approximately 0.1 log10 units) constant difference across spatial frequencies.

*t*-tests have been applied to compare the standard deviations of the CSFs at each trial with 0.1 log10 units for two tasks. The standard deviation of CSFs from the digit task decreased to 0.1 log10 units after 60 trials (

*t*[4] = 1.508,

*P*= 0.206) and became significantly lower than 0.1 log10 units after 96 trials (

*t*[4] = 2.867,

*P*= 0.046). In contrast, the standard deviation of CSFs from the grating task required at least 90 trials to reach 0.1 log10 units (

*t*[4] = 2.556,

*P*= 0.063) and 162 trials to get below 0.1 log10 units (

*t*[4] = 3.007,

*P*= 0.040). The digit test exhibited higher precision.

^{47,48}for measuring agreement among 10 digit assessments. The OCCC is the weighted average of the pairwise concordance correlation coefficient (CCC) between any two assessments, which evaluates the agreement between two tests by computing the weighted average of the pairwise CCC between any two assessments and has desirable characteristics. Concretely, the OCCC is estimated by the following equation

^{47}:

*N*subjects (a random sample from the population of interest) with a continuous scale

*Y*. And

*Y*is the estimate from observer

_{ij}*j*for subject

*i*(

*i*= 1, . . . ,

*N*).

*S*, and

_{j}*S*represent as sample means, variances, and covariances, respectively. In Experiment 1, the mean OCCC of the estimated digit CSF across five young observers was 0.956 ± 0.157.

_{jk}^{25}the RMS contrasts of different digits with the same nominal contrast were different (as evidenced by the SD of RMS contrast function for digit stimuli in Fig. 2). This would undermine the assumption of the single psychometric function. To evaluate if the difference in RMS contrast impaired the fidelity of the estimated CSF, we examined the psychometric function for each digit at each spatial frequency by collecting a large amount of data in Experiment 1.

^{49}

*χ*

^{2}test was used to test the goodness of fit of/between the nested models.

^{49}Both models showed significant good fit to the raw psychometric function for all observers (

*χ*

^{2}test, all

*P*> 0.4, Table 2). Model comparison showed that the two models were equivalent (

*χ*

^{2}[59], all

*P*> 0.10, Table 2), suggesting that the slope of the psychometric functions was the same across all the conditions. The best fitting model for Y1 is showed in Figure 6.

*t*[4] = 0.923,

*P*= 0.408). In Figure 7, we plot the average threshold across all digit conditions against the estimated threshold from the Psi method for all the observers and all the spatial frequency conditions. The two measures matched extremely well, with a RMSE of only 0.038 log10 unit. The diagonal line accounted for 99.4% of the variance.

*i*th column of the confusion matrix represents the frequency of reporting the

*j*th (

*j*= 1, 2, … 10) digit when the

*i*th digit was presented to the observer. The diagonal entries of the confusion matrix indicated the frequency of correct responses. Across all digits, 70.2% responses were correct, with digit 7 being the least confusable digit with a 92% correct response rate. The off-diagonal entries indicated “confusion.” Those with more than 10% response rate were highlighted in red. Digit 8 was the most confusing digit, and was confused with digit 3 and 5 quite often.

- The observer has full knowledge of all filtered digits when they are at the highest contrast;
- The input of an
*i*th digit input is represented in the*j*th digit template with an mean activation that is proportional to the complex wavelet structural similarity indexes (CWSSIM)^{50}of the filtered digits^{25}(Fig. 10b), between the input and the template. - Because we did not use external noise and have collapsed all the spatial frequency conditions in this analysis, we made another simplifying assumption that the gain of the digit templates is 1, and the internal noise in each digit channel is a Gaussian random variable. The internal noise in all channels is independent and identically-distributed with the same standard deviation,
*Na*. - The observer reports the digit associated with the template that has the maximum activation.

*G*(0,1) is a Gaussian random variable with mean of zero and standard deviation of one. After considering the image similarity, the model with one additive noise parameter provided an excellent fit to the confusion matrix (the maximum likelihood procedure, χ

^{2}test,

*χ*

^{2}[99] = 3.42,

*P*= 0.999). The observed confusion matrix is plotted against the model predictions in Figure 8c. The correlation between them is 0.960 (

*P*< 0.001). The results suggested the CSF measured with the 10-AFC digit identification task can faithfully reflect the sensory constraint.

^{25}was used to save the testing time. Older participants who were not familiar with computers were asked to verbally report the identity of the digit stimulus, and their responses were collected by the experimenter via the computer keyboard. The rest of the experiment setup, such as apparatus and stimuli, were exactly the same as those used in Experiment 1.

^{24}and digit stimuli.

^{25}Each participant completed six qCSF runs with grating task and six qCSF runs with digit task over a total of three days. Each qCSF run consisted of 100 trials and lasted approximately 10 minutes. Participants were given five minutes to adapt to the dim test environment before the test in each day. Some participants who usually did not wear spectacles were given an additional 25 minutes to get used to their prescribed optical correction.

*d*′ performance levels, the CSFs obtained from the digit task were close to those from the grating task for all older observers.

*F*[1, 4] = 32.2,

*P*= 0.005), with no significant interaction between stimulus type and spatial frequency (

*F*[5, 20] = 0.231,

*P*= 0.945). We also calculated the RMSE between the CSFs obtained from the two tasks to further quantify the agreement between them. From Figure 10a, we can see that the RMSE decreased rapidly in the first 10 trials. The average RMSE across all older observers was 0.159 ± 0.041, 0.125 ± 0.034, 0.126 ± 0.024, and 0.131 ± 0.016 log10 units after 15, 30, 60, and 100 trials, respectively. We also performed a Bland-Altman analysis

^{44–46}to quantify the agreement between the estimated contrast sensitivities from the digit and grating tasks in Experiment 2 (Fig. 10b). The mean difference was −0.107 log10 units. The limits of agreement were ±0.151 log10 units.

*t*[4]) = 1.867,

*P*= 0.135) and became significantly lower than 0.1 log10 units after 47 trials (

*t*[4] = 3.225,

*P*= 0.032). For the grating task, the standard deviation of the CSFs reached 0.1 log10 units in 21 trials (

*t*[4]= 2.314,

*P*= 0.082) but never got below 0.1 log10 units in the entire run (

*P*s > 0.05 for trials after 21). Again, the digit test was more precise.

^{24}the digit CSF test was more efficient than the grating CSF test. For young observers, with 120 trials, the standard deviation of the estimated contrast sensitivities was 0.078 ± 0.009 and 0.124 ± 0.026 log10 units for the digit and grating tests, respectively. For older observers, after 60 trials the standard deviation of the CSFs was 0.074 ± 0.015 and 0.122 ± 0.061 log10 units for the digit and grating tests, respectively. To reach the same standard deviation, the digit CSF test required fewer trials/less time than the grating CSF. In addition, we found, through a detailed analysis of the psychometric functions for individual digits, that the average thresholds over all digits matched the estimated thresholds in the experiment with a method that assumed a single psychometric function for all the digits and the complicated behavioral responses of the observer in the 10AFC digit identification task can faithfully reflect the sensory constraint of the visual system. The results suggested that the digit CSF test provides a compatible assessment of the CSF as the traditional grating CSF test with more efficiency.

^{28,51}The only difference between the grating and digit CSF tests are the input stimuli and response structure; the system parameters do not change. That is why it is not surprising to find the equivalency of the two tests after we matched the stimulus energy and performance level. Because we have only collected data in the zero external noise condition in this study, we cannot fully determine the system parameters.

^{52–54}It would be interesting to compare these system parameters between grating and digit tasks in future studies with data in a range of external noise conditions.

^{55,56}The digit stimuli used in the current study were developed in a parallel study and had an improved similarity structure compared to those used in the chart reported by Khambhiphanta et al.

^{25}Bandpass filtering

^{26,28}further reduced the standard deviation of the pairwise similarities (CW-SSIM) between digits to 0.126. In this study, we analyzed the confusion matrix by aggregating the data from all observers, and found that it can be well predicted by the CW-SSIM matrix of the stimuli. The result suggested that we can incorporate CW-SSIM into future development of the CSF test to take into account the nonorthogonal nature of the stimuli.

^{35}we found that the slopes of the psychometric functions were the same across the 10 digits and all the spatial frequencies. The finding is consistent with our previous observation that the slope of the psychometric function is constant across different spatial frequency and external noise conditions.

^{28,43,51}The observed slope invariance supports the “homogeneity assumption'' of slope for all pattern-detecting mechanisms.

^{57}It also is an important regularity that we can exploit to model human performance in multiple conditions. For example, the qCSF method makes use of the fixed slope across spatial frequencies to gain testing efficiency.

^{15,24,43}

^{34,58}or Gabors with a fixed number of cycles (and therefore different sizes).

^{58}Pelli and Bex

^{5}suggest that the use of a fixed number of sinewave cycles is better because neurons in the visual cortex are roughly spatial scale-invariant, and fixing stimulus size would introduce additional processes, such as spatial summation as spatial frequency increase.

^{59}The grating stimuli with a fixed number of sinewave cycles was used in the current experiment. In digit or letter CSF tests, the stimuli are scaled to generate tests at different spatial frequencies so that the size of grating stimuli was identical to that of the digit stimuli at same spatial frequency.

**H. Zheng**, None;

**M. Shen**, None;

**X. He**, None;

**R. Cui**, None;

**L.A. Lesmes**, qCSF technology (P), Adaptive Sensory Technology, Inc. (I), AST (E);

**Z.-L. Lu**, qCSF technology (P), Adaptive Sensory Technology, Inc. (I);

**F. Hou**, None

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