Exercises

1. ② Derive importance functions and implement the Camera We(), Pdf_We(), Sample_Wi(), and Pdf_Wi() methods for one or more of EnvironmentCamera, OrthographicCamera, or RealisticCamera. Render images using bidirectional path tracing or MMLT and show that given sufficient samples, they converge to the same images as when the scene is rendered with standard path tracing.
2. ② Discuss situations where the current methods for sampling outgoing rays from ProjectionLights and GonioPhotometricLights may be extremely inefficient, choosing many rays in directions where the light source casts no illumination. Use the Distribution2D structure to implement improved sampling techniques for each of them based on sampling from a distribution based on the luminance in their 2D image maps, properly accounting for the transformation from the 2D image map sampling distribution to the distribution of directions on the sphere. Verify that the system still computes the same images (modulo variance) with your new sampling techniques when using an Integrator that calls these methods. Determine how much efficiency is improved by using these sampling methods instead of the default ones.
3. ③ Implement Walter et al.’s lightcuts algorithm in pbrt (Walter et al. 2005, 2006). How do the BSDF interfaces in pbrt need to be generalized to compute the error terms for lightcuts? Do other core system interfaces need to be changed? Compare the efficiency of rendering images with global illumination using your implementation to some of the other bidirectional integrators.
4. ① Experiment with the parameters to the SPPMIntegrator until you get a good feel for how they affect rendering time and the appearance of the final image. At a minimum, experiment with varying the search radius, the number of photons traced, and the number of iterations.
5. ② Another approach to improving the efficiency of photon shooting is to start out by shooting photons from lights in all directions with equal probability but then to dynamically update the probability of sampling directions based on which directions lead to light paths that have high throughput weight and end up being used by visible points. Photons then must be reweighted based on the probability for shooting a photon in a particular direction. (As long as there is always nonzero probability of sending a photon in any direction, this approach doesn’t introduce additional bias into the shooting algorithm.) Derive and implement such an approach. Show that in the limit, your modified SPPMIntegrator computes the same results as the original. How much do these changes improve the rate of convergence?
6. ② The SPPMIntegrator ends up storing all of the BSDFs along camera paths to the first visible point, even though only the last BSDF is needed. First, measure how much memory is used to store unnecessary BSDFs in the current implementation for a variety of scenes. Next, modify the VisiblePoint representation to store the reflectance, BSDF::rho(), at visible points and then compute reflection assuming a Lambertian BSDF. Does this approximation introduce visible error in test scenes? How much memory is saved?
7. ② To find the VisiblePoints around a photon–surface intersection, the SPPMIntegrator uses a uniform grid to store the bounding boxes of visible points expanded by their radii. Investigate other spatial data structures for storing visible points that support efficient photon/nearby visible point queries, and implement an alternate approach. (You may want to specifically consider octrees and kd-trees.) How do performance and memory use compare to the current implementation?
8. ② Implement “final gathering” in the SPPMIntegrator, where camera rays are followed for one more bounce after hitting a diffuse surface. Investigate how many iterations and how many photons per iteration are needed to get good results with this approach for a variety of scenes compared to the current implementation.
9. ② There is actually one case where collisions from the hash() function used by the SPPMIntegrator can cause a problem: if, for example, nearby voxels have a collision, then a VisiblePoint that overlaps both of them will be placed in a linked list twice, and then a photon that is close to them will incorrectly contribute to the pixel’s value twice. Can you prove that this will never happen for the current hash function? If it does happen, can you construct a scene where the error makes a meaningful difference in the final image? How might this problem be addressed?
10. ③ Extend the SPPM integrator to support volumetric scattering. First, read the papers by Knaus and Zwicker (2011) and Jarosz et al. (2011b) and choose one of these approaches. Compare the efficiency and accuracy of images rendered with your implementation to rendering using the bidirectional path tracer or the MMLT integrator.
11. ② One shortcoming of the current SPPMIntegrator implementation is that it’s inefficient for scenes where the camera is only viewing a small part of the overall scene: many photons may need to be traced to find ones that are visible to the camera. Read the paper by Hachisuka and Jensen (2011) on using adaptive Markov chain sampling to generate photon paths and implement their approach. Construct a scene where the current implementation is inefficient and your new one does much better, and render comparison images using equal amounts of computation for each. Are there any scenes where your implementation computes worse results?
12. ③ Extend the BDPT integrator to support subsurface scattering with BSSRDFs. In addition to connecting pairs of vertices by evaluating the geometric term and BSDFs, your modified integrator should also evaluate the BSSRDF when two points are located on an object with the same Material with a non-nullptr-valued BSSRDF. Since two connection techniques lead to paths with a fundamentally different configuration—straight-line transport versus an additional subsurface scattering interaction on the way from on —their area product density should never be compared to each other when computing multiple importance sampling weights.
13. ② Implement Russian roulette to randomly skip tracing visibility rays for low-contribution connections between subpaths in the ConnectBDPT() function. Measure the change in Monte Carlo efficiency compared to the current BDPTIntegrator implementation.
14. ② Modify the BDPT integrator to use the path space regularization technique described by Kaplanyan and Dachsbacher (2013b). (Their method makes it possible for light transport algorithms based on incremental path construction to still handle difficult sampling cases based on chains of specular interactions.) Construct a scene where this approach is useful, and render images to compare results between this approach, SPPM, and an unmodified bidirectional path tracer.
15. ② By adding mutation strategies that don’t necessarily modify all of the sample values , it can be possible to reuse some or all of the paths generated by the previous samples in the MLTIntegrator. For example, if only the PrimarySample values for the light subpath are mutated, then the camera subpath can be reused. (An analogous case holds for the light subpath.) Even if a mutation is proposed for a subpath, if the samples for its first few vertices are left unchanged, then that portion of the path doesn’t need to be retraced. Modify the MLTIntegrator to add one or more of the above sampling strategies, and update the implementation so that it reuses any partial results from the previous sample that remain valid when your new mutation is used. You may want to add both “small step” and “large step” variants of your new mutation. Compare the mean squared error of images rendered by your modified implementation to the MSE of images rendered by the original implementation, comparing to a reference image rendered with a large number of samples. For the same number of samples, your implementation should be faster but will likely have slightly higher error due to additional correlation among samples. Is the Monte Carlo efficiency of your modified version better than the original implementation?
16. ③ In his Ph.D. dissertation, Hoberock proposes a number of alternative scalar contribution functions for Metropolis light transport, including ones that focus on particular types of light transport and ones that adapt the sample density during rendering in order to reduce perceptual error (Hoberock 2008). Read Chapter 6 of his thesis, and implement either the multistage MLT or the noise-aware MLT algorithm that he describes.