**Angle perception is an important middle-level visual process, combining line features to generate an integrated shape percept. Previous studies have proposed two theories of angle perception—a combination of lines and a holistic feature following Weber's law. However, both theories failed to explain the dual-peak fluctuations of the just-noticeable difference (JND) across angle sizes. In this study, we found that the human visual system processes the angle feature in two stages: first, by encoding the orientation of the bounding lines and combining them into an angle feature; and second, by estimating the angle in an orthogonal internal reference frame (IRF). The IRF model fits well with the dual-peak fluctuations of the JND that neither the theory of line combinations nor Weber's law can explain. A statistical image analysis of natural images revealed that the IRF was in alignment with the distribution of the angle features in the natural environment, suggesting that the IRF reflects human prior knowledge of angles in the real world. This study provides a new computational framework for angle discrimination, thereby resolving a long-standing debate on angle perception.**

*α*is the reference orientation,

*a*is the variation range of JND, and

*b*is the minimum JND.

*α*.

*k*is the slope of the function and

*a*is the intercept.

*x*-axis respectively;

*i*:

*k*is the number of parameters,

*n*is the number of data points, and

*MSE*is the mean square error of the fitting calculated by the following equation:

^{2}). The resolution of the display was 1280 × 1024, and the pixel size was 0.026° × 0.026°. The refresh rate was 85 Hz. We covered the edge of the monitor by a black cardboard, leaving only a circular aperture of a diameter of 20.6°. This circular aperture prevented observers from using the vertical and horizontal edges of the rectangular screen as cues in the experiment. The stimuli were generated by the Psychophysics Toolbox (Brainard, 1997; Pelli, 1997) based on MATLAB (MathWorks, Natick, MA). A chin rest was used to restrict participants' head movements. Viewing was binocular at a distance of 55 cm. The experiments were conducted in a dark room.

*r*

^{2}of the LCs model was 0.19 (

*p*= 0.033, 95% CI = [0.00, 0.51]) for the data of S. Chen and Levi's study and was 0.08 (

*p*= 0.029, 95% CI = [0.00, 0.35]) in fitting the data of Heeley and Buchanan-Smith's study. Similarly, the

*r*

^{2}of Weber's law was 0.05 (

*p*= 0.292, 95% CI = [0.04, 0.33]) and 0.01 (

*p*= 0.222, 95% CI = [0.01, 0.23]) in fitting the data of S. Chen and Levi's (1996) and Heeley and Buchanan-Smith's (1996) studies, respectively (Figure 1a). Although the two-segmented Weber's law model proposed by S. Chen and Levi seemed to align with their own data (

*r*

^{2}= 0.80,

*p*< 0.001, 95% CI = [0.58, 0.91]), its performance of fitting the data in Heeley and Buchanan-Smith's study was lower (

*r*

^{2}= 0.52,

*p*< 0.001, 95% CI = [0.32, 0.74]; Figure 1a), indicating the model is likely to be overfitting of their own data. The IRF model (Figure 1b) we proposed divided the space into four subregions, accounting for the four sections of the JNDs in angle,

*r*

^{2}= 0.88,

*p*< 0.001, 95% CI = [0.74, 0.95], for the data from S. Chen and Levi's study;

*r*

^{2}= 0.83,

*p*< 0.001, 95% CI = [0.76, 0.93], for the data from Heeley and Buchanan-Smith's study; (Figure 1c). All groups of AIC values passed Anderson-Darling (A-D) test and one-sample Kolmogorov-Smirnov (K-S) test of normality (A-D test:

*p*= 0.484 for AIC values in LCs model,

*p*= 0.258 for AIC values in WL model,

*p*= 0.119 for AIC values in TS model, and

*p*= 0.857 for AIC values in IRF model; K-S test:

*p*= 0.744 for LCs model,

*p*= 0.846 for WL model,

*p*= 0.572 for TS model, and

*p*= 0.956 for IRF model). Given these results, we assumed that AIC values are normally distributed in our cases and conducted a repeated-measures ANOVA to compare AICs across models. Our results demonstrated a significant main effect of AIC values across models,

*F*(3, 15) = 8.1,

*p*= 0.002,

*η*

^{2}= 0.618 (Figure 1d). A posthoc test further verified that the IRF model demonstrated significantly lower AIC values compared with the other three models,

*F*(1, 5) = 33.4,

*p*= 0.002,

*η*

^{2}= 0.87.

*θ*is the angle size. We compared the predicted values with measured JNDs in angle, and found that the LCs model could well predict the JNDs in angle (

*r*

^{2}= 0.82,

*p*< 0.001).

*a*= 1.241 and

*b*= 1.318. The parameters

*a*and

*b*are from the fitting results of our experimental data. If an angle is 45° (Figure 4a), the JNDs of the two bounding lines do not vary according to the same trend as a function of angle orientation, resulting in a very small variation of the quadratic sums of the JNDs across angle orientations. However, when an angle is 90° (Figure 4b), the JNDs of two bounding lines are covariant. The quadratic sums of the JNDs of the two bounding lines fluctuate across a relatively large range. We computed the absolute difference between the highest value and lowest values of the quadratic sum of the JNDs and plotted the JNDs' variation range as a function of angle size (Figure 4c). The simulation results indicated that the JNDs' variation range was the highest at an angle of 90° and the lowest at an angle of 45°. Although S. Chen and Levi's (1996) and Heeley and Buchanan-Smith's (1996) studies measured the JNDs across angle sizes and orientations, they did not analyze the effect of angle orientation at different angle sizes separately. In many cases, the quadratic sums of the JNDs only had a small range of variation across angle orientations. Therefore, the contribution of the orientations of the bounding lines was not very visible in the model fitting of their data.

*a*= 0.251,

*b*= 0.699. The output of the IRF model was calculated by

*ω*is the angular difference between the bounding line and its nearest axis, with

*d*= 0.609,

_{1}*d*= 1.898. The simulated results generated a two-dimensional map of JNDs in angle (Figure 5). The figure supplies an overview of the predicted human angle discrimination performance under all conditions. Comparatively, experimental studies only measured partial conditions on the map and had just a limited view. S. Chen and Levi's and Heeley and Buchanan's studies assessed the conditions varying in the horizontal axis, which highlighted the contributions of the IRF. Our experiment measured the conditions in the vertical direction and made the contributions of orientation sensitivity to the bounding lines more visible.

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