What is the purpose in this part of the Monte Carlo path tracing algorithm?

Question

In all of the simple algorithms for path tracing using lots of monte carlo samples the tracing the path part of the algorithm randomly chooses between returning with the emitted value for the current surface and continuing by tracing another ray from that surface's hemisphere (for example in the slides here). Like so:

TracePath(p, d) returns (r,g,b) [and calls itself recursively]:
    Trace ray (p, d) to find nearest intersection p’
    Select with probability (say) 50%:
        Emitted:
            return 2 * (Le_red, Le_green, Le_blue) // 2 = 1/(50%)
        Reflected: 
             generate ray in random direction d’
             return 2 * fr(d ->d’) * (n dot d’) * TracePath(p’, d’)

1. Is this just a way of using russian roulette to terminate a path while remaining unbiased? Surely it would make more sense to count the emissive and reflective properties for all ray paths together and use russian roulette just to decide whether to continue tracing or not.

2. And here's a follow up question: why do some of these algorithms I'm seeing (like in the book 'Physically Based Rendering Techniques') only compute emission once, instead of taking in to account all the emissive properties on an object? The rendering equation is basically

   L_o = L_e + integral of (light exiting other surfaces in to the hemisphere of this surface)

   which seems like it counts the emissive properties in both this L_o and the integral of all the other L_o's, so the algorithms should follow.

Answer 1

In reality, the single emission vs. reflection calculation is a bit too simplistic. To answer the first question, the coin-flip is used to terminate the ray but it leads to much greater biases. The second question is a bit more complex....

In the abstract of Shirley, Wang and Zimmerman TOG 94, the authors briefly summarize the benefits and complexities of Monte Carlo sampling:

In a distribution ray tracer, the crucial part of the direct lighting calculation is the sampling strategy for shadow ray testing. Monte Carlo integration with importance sampling is used to carry out this calculation. Importance sampling involves the design of integrand-specific probability density functions which are used to generate sample points for the numerical quadrature. Probability density functions are presented that aid in the direct lighting calculation from luminaires of various simple shapes. A method for defining a probability density function over a set of luminaires is presented that allows the direct lighting calculation to be carried out with one sample, regardless of the number of luminaires.

If we start dissecting that abstract, here are some of the important points:

1. Lights aren't points: in real life, we're almost never dealing with a point light source (e.g., a single LED).
2. Shadows are usually soft: this is a consequence of the non-point lights. It's very rare to see a truly hard-edged shadow in real life.
3. Noise (especially bright sampling artifacts) are disproportionately distracting: humans have a lot of intuition about how things should look. Look at slide 5 (the glass sphere on a table) in the OP's linked presentation. Note the bright specks in the shadow.

When rendering for more visual realism, both of the sets of reflected visibility rays and lighting calculation rays must be sampled and weighted according to the surface's bidirectional reflectance distribution function.

Note that this is a guided sampling method that's distinctly different from the original question's "generate ray in random direction" method in that it is both:

1. More accurate: the images in the linked PDF suffer a bit from the PDF process. Figure 10 is a reasonable representation of the original - note that lack of bright speckle artifacts that you will sometimes see (as in figure 5 of the original presentation).

2. Significantly faster: as the original presentation notes, unguided Monte Carlo sampling can take quite a while to converge. More sampling rays = much more computation = more time.

Answer 2

After reading the slides (thank you for posting), I'll amend my answer as best I can.

1. Is this just a way of using russian roulette to terminate a path while remaining unbiased? Surely it would make more sense to count the emissive and reflective properties for all ray paths together and use russian roulette just to decide whether to continue tracing or not.

Perhaps the emitted and reflected properties are treated differently because the reflected path depends on the incident path in a way that emitted paths do not (at least for a spectral surface). Does the algorithm take a Bayesian approach and use prior information about the incidence angle as a prior for predicting the reflective angle? Or is this a Feynman integration over all paths to come up with a probability? It's hard to tell without digging deeper into the details of the theory.

My earlier black body comment is quite incorrect. I see that the slides talk about (R, G, B) components; black body emissivities are integrated over all wavelengths.

1. And here's a follow up question: why do some of these algorithms I'm seeing (like in the book 'Physically Based Rendering Techniques') only compute emission once, instead of taking in to account all the emissive properties on an object? The rendering equation is basically

   L_o = L_e + integral of (light exiting other surfaces in to the hemisphere of this surface)

A single emissivity for the surface would assume that there's no functional relationship on wavelength or direction. I don't know how significant it is for rendering photo-realistic images.

The ones that are posted are certainly impressive. I wonder how different they would look if the complexities that you have in mind were included?

Thank you for posting a nice question - I'm voting it up. It's been a long time since I've thought about this kind of problem. I wish I could be more helpful.

Answer 3

1. Yes that is a very basic implementation of Russian Roulette, though normally the probability of terminating would take into account the light intensity (i.e. less light means the value contributes less to the final summation so use a higher probability of terminating).