# Monocular Facial Presentation-Attack-Detection: Classifying Near-Infrared Reflectance Patterns
## 3 Near-Infrared Reflectance Methodology
### Facial Reflectance Modeling  
The facial reflectance can be modeled by a combination of convex and concave Lambertian surfaces. A simple spoof, such as paper printout, is largely flat with slight convexity when bent to the face. This can be modeled as a simple convex sphere. A live face has convex and concave portions. For example, the nose is convex and the eye sockets are concave. This can be modeled as a primary sphere that has secondary convex and concave spheres. When illuminated, these secondary surfaces bias the reflectance distribution, adding scattering effects. These effects are proven to necessarily introduce additional frequency terms using surface reflectance physics. This frequency content is can be detected using texture methods to determine liveliness. A few assumptions are made for simplicity. First, this proof is completed assuming only two-dimensions. Because the models are based off spheres, there is a uniform radius across all points. This means the rate of change in reflectance with respect to the horizontal axis, \(\theta\), is identical to the rate of change with respect to the depth axis, \(\rho\). Demonstrating the surface impacts on circles should intuitively hold true for spheres. Furthermore, not all the components on the face are Lambertian. For example, hair and spoof materials (e.g., fabric) do not perfectly reflect light. This simplification is justified in the experimental results, which demonstrates faces sufficiently generate the expected reflectance distribution. Three Lambertian surface-models are visualized in Figure 3: simple-convex (A), secondary-convex (B), and secondary-concave (C). Note that these models are only relevant for the surface angle range corresponding with positive y values; that is to say, \(\theta\) is bound between the intercept angles \(\theta_{s}\) and \(\pi-\theta_{s}\). Furthermore, the models are valid only for the indicated type of surface. Model A is always valid, and Models B and C are only valid for angles bound within the secondary-surface (elsewhere defaulting back to Model A). Lambertian surfaces reflect diffuse light as a function of the source intensity, angle from the surface normal and albedo [56]. The surface reflection for an active illumination source is calculated in Equation (1). \(SR\) is the surface reflection, \(\frac{\theta_{d}}{\phi}\) is the albedo (reflectivity coefficient), \(I\) is the infrared intensity, and \(\theta_{r}\) is the reflectance angle between incident ray and surface normal. Note that for now the facial albedo is assumed to be a constant dependent associated with the primary material, such as skin, paper, glass, etc. This introduction of sub-components will be accounted for later. \[SR=\frac{\rho_{d}}{\pi}I_{r}\times\cos{(\theta_{r})}. \tag{1}\]  
The surface geometry directly impacts the infrared intensity. By definition, light waves decrease quadratically with the source-distance. This is generalized in Equation (2) by  
Figure 3: Facial surface reflectance models. Model A is the baseline convex surface. Model B introduces a secondary convex surface, increasing reflectivity and phase. Model C introduces a secondary concave surface, decreasing reflectivity and phase. assuming the \(I_{0}\) is the light intensity, and the source-distance is difference between from the further part of the face, \(d_{face}\), and surface height \(y(\theta)\). \[I_{r}=\frac{1}{4\pi}I_{0}\times\frac{1}{(d_{face}-y(\theta))^{2}}. \tag{2}\]  
Calculating the surface height and normal vector can also be modeled as a function of surface angle and radius of curvature. The surface angle \(\theta\) is the angle with respect to the \(x\) axis. The surface radius of curvature \(r_{s}\) is radius of the tangential circle. Note that the circle focal point may be offset on the \(y\) axis. A perfectly circle surface by definition would have focal point at the origin. However, flatter surfaces would have a longer radius of curvature and therefore be an offset on the \(y\) axis.