# Perceived Brightness and Resolution of Holographic Augmented Reality Retinal Scan Glasses
## 3 Results
### Retinal Scan Luminance and Ambient Contrast Ratio
\[\rm\frac{\rm L_{Lam}\;A_{P}}{\rm f_{e}{}^{2}}=\frac{\rm K_{m}\sum_{\lambda}V( \lambda)\;\Phi_{E_{Scan}}(\lambda)}{4\;\rm f_{e}{}^{2}\tan\left(\frac{\rm g}{ 2}\right)\tan\left(\frac{\rm g}{2}\right)} \tag{11}\]  
After rearranging terms and renaming \(\rm L_{Lam}\) to \(\rm L_{RS}\), we arrive at Equation (12) for the luminance \(\rm L_{RS}\) of an RS display. \[\rm L_{RS}=\frac{\rm K_{m}\sum_{\lambda}V(\lambda)\;\Phi_{E_{Scan}}(\lambda)} {4\;A_{P}\;\tan\left(\frac{\rm g}{2}\right)\tan\left(\frac{\rm g}{2}\right)} \tag{12}\]  
To calculate the pupil area \(\rm A_{P}\), we measure the pupil diameter \(\rm d_{P}\) in mm. \[\mathrm{\Phi_{V_{Scan}}}=\mathrm{K_{m}}\int\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}( \lambda)}\ \mathrm{d}\lambda\approx\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)} \tag{4}\]0  
For an RS display with narrow RGB laser spectral linewidths, we approximate \(\rm\sum_{\lambda}V(\lambda)\;\Phi_{E_{Scan}}(\lambda)\) in Equation (12) with radiant powers \(\rm\Phi_{i}\) and average wavelengths \(\rm\overline{\lambda}_{i}\). \[\mathrm{\Phi_{V_{Scan}}}=\mathrm{K_{m}}\int\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}( \lambda)}\ \mathrm{d}\lambda\approx\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)} \tag{4}\]1  
\(\rm K_{m}\) and \(\rm V(\lambda)\) are constant and tabularized [14]. \(\rm\Phi_{i}\), \(\rm\alpha\), and \(\rm\beta\) must be measured for the RS display. The only remaining variable is the pupil diameter \(\rm d_{Pmm}\). Changes in pupil diameter can be caused by different factors [15]. In most circumstances, however, the main contributor is the ambient luminance \(\rm L_{A}\). We use this fact to express the RS luminance as a function of ambient luminance instead of pupil size. Several mathematical models have been developed for this relationship [16]. We picked the model from [PERSON] and [PERSON] [17] for binocular background adaptation with an adaptive field size a of (60 deg)\({}^{2}\). The model allows us to express pupil diameter \(\rm d_{P}\) in mm as a function of ambient luminance \(\rm L_{A}\) in cd/m\({}^{2}\) for an average observer. \[\mathrm{\Phi_{V_{Scan}}}=\mathrm{K_{m}}\int\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}( \lambda)}\ \mathrm{d}\lambda\approx\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)} \tag{4}\]2  
Plugging Equation (15) into Equation (14) allows us to calculate the RS luminance \(\rm L_{RS}\) as a function of the ambient luminance \(\rm L_{A}\). \[\mathrm{\Phi_{V_{Scan}}}=\mathrm{K_{m}}\int\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}( \lambda)}\ \mathrm{d}\lambda\approx\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)} \tag{4}\]3  
Expressing the RS luminance in terms of the ambient luminance not only eliminates the need for direct pupil size measurements but also simplifies another useful quantity, the \"retinal scan ambient contrast ratio\" \(\text{ACR}_{\text{RS}}\), which is defined by the RS luminance \(\text{L}_{\text{RS}}\), the combiner transparency \(\tau_{C}\), and the ambient luminance \(\text{L}_{\text{A}}\). \[\mathrm{\Phi_{V_{Scan}}}=\mathrm{K_{m}}\int\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}( \lambda)}\ \mathrm{d}\lambda\approx\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)} \tag{4}\]4  
In non-transparent displays, the contrast ratio is determined from the brightest and darkest pixel's luminance. RS displays do not project any light into fully dark pixels. The luminance of the darkest RS display pixel is therefore given by the luminance of the darkest background element within the field of view. This could be measured for a fixed head orientation and homogeneous background in a controlled laboratory setting. For most natural visual scenes, however, such a pixel-precise background luminance would be difficult to determine. For those cases, we use the average ambient luminance \(\text{L}_{\text{A}}\) as an estimate.