## 2 Photometric and Melanopic Relationships among Different Standards  
### Theoretical Considerations  
In this section, basic expressions required to develop illumination projects, considering both the photopic and melanopic pathways, are clearly and briefly introduced. The method that we propose is calculated from the traditional relationship between radiometric and photometric magnitudes and from the current standards as follows:  
\[\text{Photropic irradiance}\left(\frac{\text{W}}{\text{m}^{2}}\right)\text{E}_{ \text{e,photopic}}=\int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{V}( \lambda)\text{d}\lambda, \tag{1}\]  
\[\text{Photropic illuminance (lux) with K}_{\text{m}}=\frac{683\text{m}}{\text{W}}\ \text{E}_{\text{photopic}}=683\times\int_{\lambda=380}^{780}\text{SPD}(\lambda) \times\text{V}(\lambda)\text{d}\lambda, \tag{2}\]  
\[\text{Melanopic irradiance}\left(\frac{\text{W}}{\text{m}^{2}}\right)\text{E}_{ \text{e, melanopic}}=\int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{S}_{ \text{mel}}(\lambda)\text{d}\lambda, \tag{3}\]  
\[\text{Melanopic action factor (MAF)}:\ \text{MAF}=\ \frac{\text{E}_{\text{e, melanopic}}}{\text{E}_{\text{e,photopic}}}=\frac{\int_{\lambda=380}^{780} \text{SPD}(\lambda)\times\text{S}_{\text{mel}}(\lambda)\text{d}\lambda}{ \int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{V}(\lambda)\text{d} \lambda}. \tag{4}\]  
To establish a full analogy between photopic and melanopic components, K\({}_{\text{melanopic}}\) must be defined. The K\({}_{\text{melanopic}}\) factor is well defined by the normalizations imposed by the different standards. Equivalent melanopic lux (EML) is defined by the WELL standard [19] with reference to the equi-energy illuminant (E) and can be calculated as follows:  
\[\text{EML}=\text{R}\times\text{E}_{\text{photopic}}, \tag{5}\]  
where \(\text{R}=\frac{\int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{S}_{ \text{mel}}(\lambda)\text{d}\lambda}{\int_{\lambda=380}^{780}\text{SPD}( \lambda)\times\text{V}(\lambda)\text{d}\lambda}\times\text{C}\
i\text{C}= \frac{\int_{\lambda=380}^{780}\text{SPD}_{\text{E}}(\lambda)\times\text{V}( \lambda)\text{d}\lambda}{\int_{\lambda=380}^{780}\text{SPD}_{\text{E}}(\lambda) \times\text{S}_{\text{mel}}(\lambda)\text{d}\lambda}\)\(\text{yields}\)\(\text{R}_{\text{E}}=1\) to the equal-energy illuminant. For any source, in terms of melanopic parameters, the EML is defined as follows:  
\[\text{EML}=\text{R}\times\text{E}_{\text{photopic}}=\frac{\int_{ \lambda=380}^{780}\text{SPD}(\lambda)\times\text{S}_{\text{mel}}(\lambda) \text{d}\lambda}{\int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{V}( \lambda)\text{d}\lambda}\times\frac{\int_{\lambda=380}^{780}\text{SPD}_{\text{E}} (\lambda)\times\text{V}(\lambda)\text{d}\lambda}{\int_{\lambda=380}^{780}\text{ SPD}_{\text{E}}(\lambda)\times\text{S}_{\text{mel}}(\lambda)\text{d}\lambda}\times 683\times \tag{6}\] \[\int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{V}(\lambda) \text{d}\lambda=1.218\times 683\times\int_{\lambda=380}^{780}\text{SPD}(\lambda) \times\text{S}_{\text{mel}}(\lambda)\text{d}\lambda=831.8\times\] \[\int_{\lambda=380}^{780}\text{SPD}(\lambda)\times\text{S}_{\text {mel}}(\lambda)\text{d}\lambda.\]  
With K\({}_{\text{melanopic,E}}=831.8\) Im/W\({}_{\text{melanopic,E}}\). In addition, EML's relation to photometric illuminance is as follows:  
\[\text{EML}=1.218\times\text{MAF}\times\text{E}_{\text{photopic}}. \tag{7}\]  
In accordance with EDI as defined by the CIE standard [16], normalization is proposed with the melanopic illuminance provided by the standard illuminant D\({}_{65}\) (daylight CCT = 6500 K). A light-source type D\({}_{65}\) furnishing photopic illuminance E\({}_{\rm photopic,D65}\) to provide the same melanopic illuminance E\({}_{\rm melanopic,D65}\) can be calculated.