Our goal in this chapter was not to say, “You can build either a ray tracer or a rasterizer,” but rather that rendering involves sampling of light sources, objects, and rays, and that there are broad algorithmic strategies you can use for accumulating samples and interpolating among them. This provides a stage for all future rendering, where we try to select samples efficiently and with good statistical characteristics.
For sampling the scene along eye rays through pixel centers, we saw that three tests—explicit 3D ray-triangle tests, 2D ray-triangle through incremental barycentric tests, and 2D ray-triangle through incremental edge equation tests—were mathematically equivalent. We also discussed how to implement them so that the mathematical equivalence was preserved even in the context of bounded-precision arithmetic. In each case we computed some value directly related to the barycentric weights and then tested whether the weights corresponded to a point on the interior of the triangle. It is essential that these are mathematically equivalent tests. Were they not, we would not expect all methods to produce the same image! Algorithmically, these approaches led to very different strategies. That is because they allowed amortization in different ways and provoked different memory access patterns.
Sampling is the core of physically based rendering. The kinds of design choices you faced in this chapter echo throughout all aspects of rendering. In fact, they are significant for all high-performance computing, spreading into fields as diverse as biology, finance, and weather simulation. That is because many interesting problems do not admit analytic solutions and must be solved by taking discrete samples. One frequently wants to take many of those samples in parallel to reduce computation latency. So considerations about how to sample over a complex domain, which in our case was the set product of triangles and eye rays, are fundamental to science well beyond image synthesis.
The ray tracer in this chapter is a stripped-down, no-frills ray tracer. But it still works pretty well. Ten years ago you would have had to wait an hour for the teapot to render. It will probably take at most a few seconds on your computer today. This performance increase allows you to more freely experiment with the algorithms in this chapter than people have been able to in the past. It also allows you to exercise clearer software design and to quickly explore more sophisticated algorithms, since you need not spend significant time on low-level optimization to obtain reasonable rendering rates.
Despite the relatively high performance of modern machines, we still considered design choices and compromises related to the tension between abstraction and performance. That is because there are few places where that tension is felt as keenly in computer graphics as at the primary visibility level, and without at least some care our renderers would still have been unacceptably slow. This is largely because primary visibility is driven by large constants—scene complexity and the number of pixels—and because primary visibility is effectively the tail end of the graphics pipeline.
Someday, machines may be fast enough that we don’t have to make as many compromises to achieve acceptable rendering rates as we do today. For example, it would be desirable to operate at a purely algorithmic level without exposing the internal memory layout of our Image class. Whether this day arrives soon depends on both algorithmic and hardware advances. Previous hardware performance increases have in part been due to faster clock speeds and increased duplication of parallel processing and memory units. But today’s semiconductor-based processors are incapable of running at greater clock speeds because they have hit the limits of voltage leakage and inductive capacitance. So future speedups will not come from higher clock rates due to better manufacturing processes on the same substrates. Furthermore, the individual wires within today’s processors are close to one molecule in thickness, so we are near the limits of miniaturization for circuits. Many graphics algorithms are today limited by communication between parallel processing units and between memory and processors. That means that simply increasing the number of ALUs, lanes, or processing cores will not increase performance. In fact, increased parallelism can even decrease performance when runtime is dominated by communication. So we require radically new algorithms or hardware architectures, or much more sophisticated compilers, if we want today’s performance with better abstraction.
There are of course design considerations beyond sample statistics and raw efficiency. For example, we saw that if you’re sampling really small triangles, then micropolygons or tile rasterization seems like a good rendering strategy. However, what if you’re sampling shapes that aren’t triangles and can’t easily be subdivided? Shapes as simple as a sphere fall into this category. In that case, ray casting seems like a very good strategy because you can simply replace ray-triangle intersection with ray-sphere intersection. Any micro-optimization of a rasterizer must be evaluated compared to the question, “What if we could render one nontriangular shape, instead of thousands of small triangles?” At some point, the constants make working with more abstract models like spheres and spline surfaces more preferable than working with many triangles.
When we consider sampling visibility in not just space, but also exposure time and lens position, individual triangles become six-dimensional, nonpolyhedral shapes. While algorithms for rasterizing these have recently been developed, they are certainly more complicated than ray-sampling strategies. We’ve seen that small changes, such as inverting the order of two nested loops, can yield significant algorithmic implications. There are many such changes that one can make to visibility sampling strategies, and many that have been made previously. It is probably best to begin a renderer by considering the desired balance of performance and code manageability, the size of the triangles and target image, and the sampling patterns desired. One can then begin with the simplest visibility algorithm appropriate for those goals, and subsequently experiment with variations.
Many of these variations have already been tried and are discussed in the literature. Only a few of these are cited here. Appel presented the first significant 3D visibility solution of ray casting in 1968. Nearly half a century later, new sampling algorithms appear regularly in top publication venues and the industry is hard at work designing new hardware for visibility sampling. This means that the best strategies may still await discovery, so some of the variations you try should be of your own design!