I want to do volume rendering using path tracing with a absorption only model. And my data is in a 3D grid, which is turned into a voxel medium in my renderer.
However, I can't seem to be correct implementing the algorithm.
According to this note (however absorption-only heterogeneous case is not mentioned here), the radiance at point $x$, direction $\omega$, with emission-absorption model in a homogeneous medium is:
$L(x,\omega) = \int_{x}^{y} \tau(x,x') \epsilon(x')dx + \tau(x,y)L_e(x,\omega)$
where $\tau$ is the attenuation, y is a point on surface, and $\epsilon$ is emission at point $x'$.
So I come up with the heterogenous, absorption-only equation as:
$L(x, \omega) = \tau(x,y)L_e(x,\omega)$
since I think the emission term should be omitted. However this equation seems not correct intuitively and do not give correct image.
According to this equation, the radiance at every point $x'$ along the ray within the volume is not counted, which is where I think the problem lies.
So what is the correct heterogeneous volume path tracing equation? How to derive the correct equation?
I give more details of my implementation here. The scene is simply a cornell box where only a voxel medium is inside the box. I trace a ray from the eye(camera), and the ray go through the medium and intersect a surface(the wall). To simplify the procedure, I set max_bounce to 1 which means the path is
eye -> medium -> surface -> light.
So the value of this sample is simply the throughput of the medium times the throughput of the surface times direct light at surface intersection.
However I got the result image
