# Objective Prediction of Human Visual Acuity Using Image Quality Metrics
## 2 Materials and Methods
### The Point Spread Function and the Image of the Eye  
The monochromatic PSF of the subject was computed from the wave aberration function \(W(x,y)\) reconstructed from the Zernike coefficients measured with Visionix VX-120. In this calculation, the first three Zernike coefficients corresponding to the aberration components of piston, tilt X and tilt Y were not considered because they constitute translations  
Figure 2: Characteristics of the studied eyes. (**A**) Type of refractive error (myopia-hyperopia). (**B**) Spherical equivalent associated with age range. (**C**) Number of subjects associated with the subjective decimal VA ranges. and tilts of the reference system, which can be naturally compensated with eye movements. The _PSF_ is obtained as the squared module of the Fourier transform of the generalized pupil function [24]:  
\[PSF(x,y)=\left|FT(P(x,y))\right|^{2} \tag{2}\]  
We considered the Stiles-Crawford effect due to the anatomical structure of photoreceptors [25], which can be modeled by an apodizing filter located at the entrance pupil [26]. Then, the generalized pupil function is given by [27]:  
\[P(x,y)=e^{-0.116~{}R_{p}^{2}(x^{2}+y^{2})}e^{-i\hat{k}W(x,y)}, \tag{3}\]  
where \(R_{p}\) is the pupillary radius, which was provided by the Visionix VX-120 system. In the work of [PERSON] et al. [28], it is shown that under photop conditions the pooled pupil diameter is \(4.07\pm 0.63\) mm so the Zernike expansion coefficients provided by Visionix were transformed into new ones [29; 30] for pupil diameters above 4.07 mm. Finally, we considered that the eye's image of an object could be obtained through the convolution of the function that represents the object \(O(x,y)\) with the \(PSF\) of the optical system. By applying the convolution theorem [24], this image can be obtained as the inverse Fourier transform of the product of the convoluted functions, i.e.,  
\[I(x,y)=\mathcal{F}^{-1}\{\mathcal{F}[O(x,y)].\mathcal{F}[PSF(x,y)]\} \tag{4}\]  
### The Point Spread Function and the Image of the Eye  
Having defined how to obtain the degraded eye's image of an object, the values of the seven studied metrics were determined. We established the metrics result for each eye as the average of the metrics values for the 49 object images. Subjects' subjective VA and the results obtained from calculating the metrics and linear combinations of metrics were fitted with a monotonous logistic function that is commonly used to study image quality [15]. It comes as follows:  
\[VA(Q)=a\left(\frac{1}{2}-\frac{1}{1+\exp[b(Q-c)]}\right)+dQ+e, \tag{5}\]  
where \(Q\) represents any of the used metrics or a linear combination \(Q_{L}\), and \((a,b,c,d,e)\) are the parameters to determine. The first issue to address in the proposal of the linear combinations of full reference metrics is collinearity. Collinearity is a regression analysis problem; if predictors are in a linear combination, the influence of each one on the criterion cannot be distinguished by overlapping them with one another. In that case, the confidence intervals of the estimating coefficients are often wide, which indicates that the obtained estimates are imprecise and probably unstable. The difficulty of assessing the existence of collinearity lies in determining the maximum degree of the permissible relation between independent variables. No consensus on this issue has been reached. [PERSON] et al. [31] considered a series of indicators to analyze the degree of multicollinearity among the regressors of a multivariate linear model. The simplest is the variance inflation factor (_VIF_) between two of the regressive variables, which is defined as:  
\[VIF=\frac{1}{1-R^{2}}, \tag{6}\]  
where \(R^{2}\) is the coefficient of determination between the two variables. According to these authors, if _VIF_ is higher than 10, it can be concluded that the collinearity between the two selected variables is high and will affect multilinear fit by increasing the variance value.