# Measurement of Circadian Effectiveness in Lighting for Office Applications
### Measurement Analysis  
As discussed in the previous section, the measurement setup used in this work permits the spatially resolved estimation of CS as a measure for the circadian effectiveness of the lighting conditions encountered for a sample of typical office workplaces. For this purpose, the \(m\times n\) luminance matrix \(\mathbf{L}\) obtained as output from the imaging photometer must first be converted into a \(m\times n\) illuminance matrix \(\mathbf{E}\), where \(m\times n\) corresponds to the pixel resolution of the measurement device, which, in the present case of the LMK-5 color, is \(1030\times 1380\) effective pixels. For an arbitrary pixel position \((i,j)\), where \(i=1,...,m\) and \(j=1,...,n\), the partial illuminance can be calculated using  
\[E_{ij}=L_{ij}\cdot\Omega_{ij}\cdot\cos\left(\mathcal{E}_{ij}\cdot\frac{\pi}{1 80^{\circ}}\right), \tag{6}\]  
where the matrix \(\mathbf{\Omega}\) summarizes the solid angles associated with each of the different pixel locations, while the matrix \(\mathcal{E}\) gives the angles between an arbitrary pixel position and the imaging photometer's center of view. Note that both matrices are fixed characteristics of the used camera lens system to be determined by calibration and, basically, do not change across measurements. By summing over all pixel values, an estimate of the vertical illuminance at the position of the imaging photometer can thus be determined and is given by  
\[E_{\text{LMK}}=\sum_{i=1}^{m}\sum_{j=1}^{n}E_{ij}. \tag{7}\]  
Figure 4: Image representation and light emission characteristics expressed in terms of a polar luminous intensity graph of the Quadrature® 5 MQ238D2H pendant office luminaires installed in the test offices. Additionally tabulated are the expected luminance values for different viewing angles, Unified Glare Rating (UGR) estimates, as well as the luminaire efficiency \(\eta_{\text{LB}}\) including a breakdown of direct/indirect contributions [100]. The average tristimulus values \(X_{\text{LMK}}\), \(Y_{\text{LMK}}\), and \(Z_{\text{LMK}}\) are obtained in a similar manner by summing over all measured pixel values \(X_{ij}\), \(Y_{ij}\), and \(Z_{ij}\), respectively, and dividing each of these sums by the total number of pixels:  
\[X_{\text{LMK}} =\frac{1}{m\cdot n}\sum_{i=1}^{m}\sum_{j=1}^{n}X_{ij}, \tag{8}\] \[Y_{\text{LMK}} =\frac{1}{m\cdot n}\sum_{i=1}^{m}\sum_{j=1}^{n}Y_{ij},\] \[Z_{\text{LMK}} =\frac{1}{m\cdot n}\sum_{i=1}^{m}\sum_{j=1}^{n}Z_{ij}.\]  
The chromaticity coordinate \(z_{\text{LMK}}\) required for estimating the circadian stimulus \(\text{CS}_{\text{LMK}}\) from the image capture is then obtained from  
\[z_{\text{LMK}}=\frac{Z_{\text{LMK}}}{X_{\text{LMK}}+Y_{\text{LMK}}+Z_{\text{ LMK}}}. \tag{9}\]  
Inserting Equations (7) and (9) into Equation (5) eventually yields  
\[\text{CS}_{\text{LMK}}(z_{\text{LMK}},E_{\text{LMK}})= \begin{cases}0.7-\frac{ 0.7}{1+0.016781\cdot(z_{\text{LMK}}\cdot E_{\text{LMK}}^{0.509265})^{2.268904}}& \text{if $z_{\text{LMK}}>0.195$}\\ 0.7-\frac{0.7}{1+0.011376\cdot(z_{\text{LMK}}\cdot E_{\text{LMK}})^{1.109998}}& \text{if $z_{\text{LMK}}\leq 0.195$}\end{cases}  \tag{10}\]  
as an estimate for the circadian effectiveness of the captured lighting conditions as experienced by an office worker, obtained by applying [PERSON] et al. 's approximation method to the measurement output of the imaging photometer. In addition to the spatially resolved image capture of the luminance photometer, the measurement setup discussed in Section 2.2 also allows for a direct calculation of CS using the original model formalism of Equations (1)-(4) and the spectral irradiance obtained from the CSS-45 measurements.