# Monocular Facial Presentation-Attack-Detection: Classifying Near-Infrared Reflectance Patterns
## 3 Near-Infrared Reflectance Methodology
### Facial Reflectance Modeling
Assume that the surface angle which intersects the \(x\) axis can be defined as \(\theta_{s}\). This means that the tangent circle focal point is then offset by \(r_{s}\sin\theta_{s}\). Recall the three Lambertian-surface models from Figure 3. Model A is simplest, where the height is the \(x\) axis projection and the surface-normal is the surface angle. Introducing secondary surfaces, however, now it requires projecting a translation as a function of to the secondary surface radius and angle to the secondary-arc focal point. Model B presents a constructive interaction where the secondary convex arc has focal point at \(r_{v},\theta_{v}\). The surface height is increased by the projected difference between the original arc and sub-arc focal point, and the surface-normal is increased by \(\theta_{v}\) in phase. Model C makes a similar assumption that the concave-arc focal point is the location \(r_{c},\theta_{c}\). This time the translation is destructive; the secondary-surface height is decreased by the projected difference between the original arc and sub-arc focal point, and surface-normal is decreased in phase by \(\theta_{c}\). \[d(\theta)=\left\{ \begin{array}{ll}d_{face}-r_{s}\cdot(\sin(\theta)-\sin( \theta_{s})),&\text{Model A}\\ \\ d_{face}-r_{s}\cdot(\sin(\theta)-\sin(\theta_{s}))\\ \\ -|r_{s}-r_{c}|\cdot\sin(\theta-\theta_{c}),&\text{Model C}\end{array} \right\}. \tag{3}\]  
Surface reflectance is a function of distance as applied in Equation (3). More specifically, this distance is a function the furthest face distance \(d_{face}\), surface radius \(r_{s}\), surface angle \(\theta_{s}\), and surface bounding angle \(\theta_{s}\) (noting secondary surfaces also have radii \(r_{v},r_{c}\) with offset angles \(\theta_{v},\theta_{c}\)). Intuitively, one can see the radius term, \(r_{s}\), becomes dominant for flatter surfaces. This will inherently reduce the valid surface angles to be closer to the \(y\)-axis (i.e., \(\frac{\pi}{2}\)), fundamentally acting as a low-pass filter. This is particularly relevant for simple spoofs, such as paper mask and display-replay, that are generally flat. That is to say the simple spoofs should only have low frequency content. \[I(\theta)=SR(\theta)+I_{amb}. \tag{4}\]  
For completeness, the surface reflectance also needs to factor the ambient light. This is simply a summation of the infrared surface reflectance and light present at the surface \(I_{amb}\), as applied in Equation (4). Ambient here is simplified to be diffuse (i.e., it is approximately uniform across the image). These equations can be combined for an explicit total reflectance equation. Adding secondary surfaces necessarily generates new frequency content in the reflectance profile as applied in Equation (5) (see next-page). This, in theory, should generate more variance in the distribution, a behavior which the texture classifier should easily identify. Note here the albedo, source intensity and ambient are constants and should not impact the light variance. Furthermore, the illumination source angle, \(\theta_{I}\), is visualized as 90 degrees in the surface models figure but only acts as a phase shift. Position should theoretically matter less than the geometry of the object. \[ \begin{array}{c}I(\theta)=\frac{\theta_{A}}{4\pi^{2}}I_{0}\times\\ \left\{\begin{array}{cc}\frac{1}{(d_{face}-r_{s}(\sin(\theta)-\sin(\theta_{c}) ))^{2}}&\text{Model A}\\ \times\cos{(\theta_{I}-\theta)}+I_{amb},&\\ \frac{1}{(d_{face}-r_{s}(\sin(\theta)-\sin(\theta_{c}))-|r_{s}-r_{c}|\sin( \theta+\theta_{c})))^{2}}&\text{Model B}\\ \times\cos{(\theta_{I}-(\theta+\theta_{c}))}+I_{amb},&\\ \frac{1}{(d_{face}-r_{s}(\sin(\theta)-\sin(\theta_{c}))+|r_{s}-r_{c}|\sin( \theta-\theta_{c})))^{2}}&\text{Model C}\\ \times\cos{(\theta_{I}-(\theta-\theta_{c}))}+I_{amb},&\end{array}\right\}.\end{array}  \tag{5}\]  
To verify the geometry is more important than position or light intensity, the surface reflectance derivative can be taken.