## 3 Hemoglobin and Shade Separation from Faces in Images  
Independent component analysis (ICA) is a multivariate analysis technique that estimates the original signal from a mixture of independent signals without any information about the signals themselves or their composition. Here, independence refers to the state in which variables are orthogonal to each other, and the distribution of the variables is invariant. [PERSON] et al. applied ICA to faces in images and separated the pigment components of the skin [30]. In this procedure, the RGB values of a face in an image are used as observation signals in the ICA to estimate the pigment density distribution of the face. Here, we established the relationship between the RGB values of a face in an image and skin pigment density. Figure 1a shows the human skin model proposed by [PERSON] et al. [30]. The human skin is composed of three major layers: the epidermis, dermis and subcutaneous tissue. Skin coloration is determined by various pigments, among which melanin and hemoglobin are the most important. Melanin is found in the epidermis, whereas hemoglobin is found in the dermis. Therefore, under the simplified assumption that the epidermis is composed of melanin and that the dermis is composed of hemoglobin, the spatial distributions of melanin and hemoglobin can be considered independent. Light incident on the skin is divided into surface-reflected light, which is reflected on the skin surface, and internally reflected light, which enters the skin and is repeatedly scattered before being emitted outside the skin. Surface-reflected light is not affected by skin pigmentation and expresses the color of the light source. By contrast, internally  
Figure 1: **(a)** Schematic representation of the skin model adopted in this study. (b) Observed signals and three independent signals [31]. (c) Example of bandpass-filtered pulse-wave signal. reflected light represents the skin color because it is emitted after repeated absorption and scattering by pigments inside the skin. Based on the optical skin model, the relationship between the pixel values and the observed signal (that is, the skin image) was extended to a mathematical model. The pixel values of the skin image can be defined as the sensor response of the camera in terms of the observed signal. The sensor response, \(v_{i}\), of the camera is described by Equation (1),  
\[v_{i}(x,y)=k\int L(x,y,\lambda)s_{i}(\lambda)d\lambda, \tag{1}\]  
where \(i\) is an index identifying the color channel of the sensor; \(\lambda\) is the wavelength; \(L(x,y,\lambda)\) is the spectral irradiance of the reflected light at position \((x,y)\) on the skin surface; \(k\) is a constant indicating the camera gain; and \(s_{i}\) is the spectral sensitivity of the sensor. Applying the modified Lambert-Beer law to the spectral irradiance \(L(x,y,\lambda)\) of the reflected light, Equation (2) can be derived from Equation (1). \[v_{i}(x,y)=k\int e^{-\rho_{m}(x,y)\sigma_{m}(\lambda)l_{e}(\lambda)-\rho_{h}(x, y)\sigma_{h}(\lambda)l_{d}(\lambda)}E(x,y,\lambda)s_{i}(\lambda)d\lambda \tag{2}\]  
where \(E(x,y,\lambda)\) is the spectral irradiance of incident light at position \((x,y)\) on the skin surface; \(\rho_{m}\) and \(\rho_{h}\) are the absorption cross section of melanin and hemoglobin, respectively; and \(l_{e}\) and \(l_{d}\) are the optical path lengths of light passing through the epidermis and dermis, respectively. Since the spectral distribution of the skin is a smooth function along the wavelength axis and the spectral reflectance of the skin is highly correlated with the spectral sensitivity of the camera, the spectral sensitivity can be approximated as a narrow frequency band with \(s_{i}(\lambda)\approx\delta(\lambda_{i})\). Furthermore, assuming a sufficient distance between the illumination and the participant, the directional dependence of the illumination irradiance can be neglected.