# An Analysis of Visibility Requirements and Reaction Times of Near-Field Projections
## 2 Materials and Methods
### Statistical Methods and Data Analysis  
The evaluation of the study data starts by determining the Weber contrasts of the different greyscales of the content projections from the recorded luminance images to use them for further evaluation (see Figure 4). In further evaluation, threshold values for the safe detection of the projected visual signs are determined from the Weber contrasts. For this purpose, [PERSON]'s [31] small target visibility model calculates the threshold luminance difference between the object and the background. The threshold difference depends on the object size, the background luminance, the presentation time, the observer age, and the contrast polarity. Furthermore, a pre-factor k is chosen based on the desired detection probability. For a detection probability of 90%, the k-factor is determined to be 1.62 [33]. \[\Delta L_{\text{th}}=k\cdot\left(\frac{\sqrt{\phi}}{\alpha}+\sqrt{L}\right)^{2 }\cdot\frac{a(\alpha,L_{u})+t}{t}\cdot F_{CP}\cdot AF \tag{2}\]  
where:  
* \(k\) detection probability factor (\(k=1.62\) for 90 % detection probability);
* \(\sqrt{\phi},\sqrt{L}\) luminous flux, luminance function regarding [PERSON]'s/Weber's law;
* \(\alpha\) target size of displayed content;
* \(a(\alpha,\,L_{U})\) Blondel-Rey constant;
* \(t\) presentation time of displayed content (\(t=350\) ms);
* \(F_{CP}\) contrast polarity factor (\(F_{CP}=1\) for a positive contrast);
* \(AF\) age factor (individual, \(AF=1\) for young observers (age \(<\) 23)). Since the small target visibility model was set up based on laboratory studies and is unable to represent the complexity of a real situation in road traffic, an additional factor is introduced in the form of the visibility level (_VL_) (see (3)), which describes the relationship between the calculated threshold luminance difference \(\Delta L_{th}\) and the actual luminance  
Figure 4: Exemplary luminance image of the projected symbol (battery) taken by a luminance camera LMK 5 Color at mesopic lighting conditions (20 lx) at a distance of 5 m. The black rectangle represents the background area and the blue rectangle the projected content (for the calculation of the Weber contrast the specific shape of the content was used). difference \(\Delta L_{act}\) required in road traffic and is also used as the basis for the design of street lighting installations in the USA [31]. \[VL=\frac{\Delta L_{act}}{\Delta L_{th}} \tag{3}\]  
This study used the constant stimulus method with a forced-choice implementation [34]. The detection probability (proportion of \"yes\" responses) is plotted against stimulus strength (grey level/contrast), resulting in a psychometric function. Since stimuli significantly below the threshold are never detected, and stimuli significantly above the threshold are always detected, a sigmoid function is obtained. This psychometric function can be fitted with logistic regression [35]. The actual luminance difference was determined from the study data. For this purpose, the data were evaluated using logistic regression with the psychometric formula to obtain the detection probability of the different stimuli, according to [PERSON] [36] (see (4)). The advantage of this method is that the contrast and luminance differences for various detection probabilities can be derived from the results obtained. \[P(x_{L})=\gamma_{L}+(1-\gamma_{L})\cdot\left(\frac{1}{1+\left(\frac{x_{L}}{ \alpha_{L}}\right)^{-\beta_{L}}}\right) \tag{4}\]  
where:  
* \(P(x_{L})\) detection probability of stimulus;
* \(x_{L}\) stimulus intensity;
* \(\alpha_{L}\) stimulus with a 50 % probability of a positive response;
* \(\beta_{L}\) steepness of the curve;
* \(\gamma_{L}\) probability of a randomly positive evaluation (here: \(\gamma_{L}\) = 0). Finally, the luminance differences determined for the projection of the symbol and the text in the various scenarios were examined for significant differences. For this purpose, non-parametric tests were applied based on the data available.