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?
Are we selecting random points on the view plane, shoot rays from the eye through these points and calculate the contribution of the resulting path to a light source? This would yield the contribution of a single path. Maybe we create N paths in this way and average their contributions.
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?