Linear composition of BSDF?
- ROBUST MC Chapter 4 > The operator \(A\) that provides a linear mapping in a vector space \(\mathcal F \rightarrow \mathcal F\) is called a linear operator. \(Af\) denotes the application of an operator to a function, resulting another new function. ### Preliminary Light transport operator: \[ \mathbf T = \mathbf K\mathbf G \tag{1} \]
Where \(\mathbf K\) is the local scattering operator, \(L_o = \mathbf K L_i\), both \(L\) defined in the whole ray space \(\mathcal R\):
\[ (\mathbf Kh)(\mathbf{x}, \omega_o) = \int_{S^2} f_s(\mathbf{x}, \omega_i \to \omega_o) \, h(\mathbf{x}, \omega_i) \, d\sigma_{\mathbf{x}}^\perp(\omega_i) \] \(\mathbf G\) is the propagation operator, defining the geometric property of the space, \(L_i = \mathbf G L_o\), which should be symmetric. \[ (\mathbf Gh)(\mathbf{x}, \omega_i) = \begin{cases} h(\mathbf{x}_M(\mathbf{x}, \omega_i), -\omega_i) & \text{if } d_M(\mathbf{x}, \omega_i) < \infty, \\ 0 & \text{otherwise,} \end{cases} \]
Now right the light transport equation in operator form: \[ L = L_e + \mathbf T L \] Inverting the operator equation: \[ \begin{equation} \begin{aligned} (\mathbf I - \mathbf T)L &= L_e \\ L &= (\mathbf I - \mathbf T)^{-1} L_e \end{aligned} \end{equation} \] Defining solution \(\mathbf S = (\mathbf I - \mathbf T)^{-1}\). Given that \(\Vert \mathbf T \Vert \lt 1\), the inversion can be expanded with Neumann series: \[ \mathbf S = (\mathbf I - \mathbf T)^{-1} = \sum^{\infty}_{i = 0} \mathbf T^i = \mathbf I + \mathbf T + \mathbf T^2+ \cdots \] Light transport can then be expanded as well: \[ L = L_e + \mathbf T L_e + \mathbf T^2 L_e + \cdots \tag{2} \]