In Metropolis Light Transport, we need to approximate the integral of the measurement contribution function. I've read that this can be done using a traditional approach like path tracing. However, how exactly do we need to implement this approximation?
I've read the explanation in PBR3, but I still don't understand how this approximation is done exactly.
More precisely, let's make the following definitions (ignoring rigor): Let $M\subseteq\mathbb R^3$ denote the scene surfaces, $\sigma_M$ denote the surface measure on $M$, $$\Omega_n:=\left\{x\in M^{\mathbb N_0}:x_i\ne x_j\text{ for all }0\le i<j\le n\text{ and }x_N=x_n\text{ for all }N\ge n\right\}$$ $$\iota_n:\Omega_n\to M^{\left\{0,\:\ldots\:,\:n\right\}}\;,\;\;\;x\mapsto(x_0,\ldots,x_n)$$ $$\mu_n(A):=\sigma_M^{\otimes\left\{0,\:\ldots\:,\:n\right\}}(\iota_n(A))\;\;\;\text{for }A\subseteq\Omega_n$$ $$\Omega:=\biguplus_{n\in\mathbb N}\Omega_n$$ $$\mu(A):=\sum_{n=1}^\infty\mu_n(A\cap\Omega_n)\;\;\;\text{for }A\subseteq\Omega$$ Moreover, with the usual notation, let $$T(x_0,\ldots,x_n):=\prod_{i=1}^{n-1}f_r(x_{i+1}\to x_i\to x_{i-1})G(x_i\leftrightarrow x_{i+1})$$ $$f(x):=L_e(x_n\to x_{n-1})T(x_0,\ldots,x_n)\;,\;\;\;\text{if }x\in\Omega_n$$ $$h_j(x):=G(x_0\leftrightarrow x_1)W_e^{(j)}(x_0\to x_1)$$ Then, the measurement $I_j$ of the $j$th pixel is $$I_j=\int fh_j\:{\rm d}\mu.$$
Now, the question is: How exactly is $$b:=\int f\:{\rm d}\mu$$ approximated using (unidirectional) path tracing?
