Global Illumination (3) Path Tracing

Overview Light Transport Notation Path Tracing Photon Mapping

Ray Tracing L(D)?S*E Rays cast from eye into scene –Why? –Because most rays cast from light wouldn’t reach eye objectslights

Ray Tracing L(D)?S*E Shadow rays cast at each intersection –Why? –Because most rays wouldn’t reach the light source objectslights

Ray Tracing L(D)?S*E Workload badly distributed –Why? –Because # of rays grows exponentially, and their result becomes less influential objectslights

Tough Cases Caustics –Light focuses through a specular surface onto a diffuse surface –LSDE –Which direction should secondary rays be cast to detect caustic?

Tough Cases Bleeding –Color of diffuse surface reflected in another diffuse surface –LDDE –Which direction should secondary rays be cast to detect bleeding?

Path Tracing Kajiya, SIGGRAPH 86 Diffuse reflection spawns infinite rays Pick one ray at random Cuts a path through the dense ray tree Still cast an extra shadow ray toward light source at each step in path Trace 40 paths per pixel objectslights

Path Tracing pure Monte Carlo reverse ray-tracing –rays are randomly diffused multiple samples taken at each pixel (distributed ray-tracing) can produce all global illumination effects, but is slower than Photon Mapping –more noise at same number of iterations

Monte Carlo (MC) Methods Stochastic techniques –meaning they are based on the use of random numbers and probability statistics to investigate problems. Used in everything from economics to nuclear physics to regulating the flow of traffic. To call something a "Monte Carlo" experiment, all you need to do is use random numbers to examine some problem.

Monte Carlo (MC) Methods MC methods to model physical problems allows us to examine more complex systems than we otherwise can. Solving equations which describe the interactions between two atoms is fairly simple; solving the same equations for hundreds or thousands of atoms is impossible. With MC methods, a large system can be sampled in a number of random configurations, and that data can be used to describe the system as a whole. "Hit and miss" integration is the simplest type of MC method to understand

Advantages of MC Integration Few restrictions on the integrand – Doesn’t need to be continuous, smooth,... – Only need to be able to evaluate at a point Extends to high-dimensional problems –Same convergence Conceptually straightforward Efficient for solving at just a few points

Disadvantages of MC Integration Noisy Slow convergence Good implementation is hard – Debugging code – Debugging maths – Choosing appropriate techniques

MC in Graphics: what can we integrate? Pixel: antialiasing Light sources: soft shadows Lens: depth of field Time: motion blur BRDF: glossy reflection Hemisphere: indirect lighting

Path Tracing An extension to ray tracing Simulates global illumination –Can handle all possible light bounces L(S|D)*E Stochastically samples all light paths Handles area light sources, diffuse reflections Approximates the integral of illumination

Send a ray through a pixel Trace the ray to its first intersected object From the intersected object, send out –One ray to each light source –One additional ray a diffusely reflected ray, a specularly reflected ray, or a transmissive ray Trace the ray and recursively follow it, as above Produces a ray “path” – not a ray tree Path Tracing

Trace only one secondary ray per recursion But send many primary rays per pixel (performs anti-aliasing as well) Monte Carlo Path Tracing

Light ray Eye Image plane Light

Illuminance ray Eye Image plane Light Specular? Diffuse? Transmission?

Path Tracing This provides a Monte Carlo approach to global illumination Stochastic samples are taken that should represent the actual surface properties

Path Tracing The lighting distribution is sampled by tracing rays stochastically along all possible light paths Averaging a large number of sample rays gives an estimate of the integral of all light paths through the pixel

Selecting the ray to trace How do we select which ray to trace? Each material has a k d, k s, and k t –Let k tot = k d + k s + k t –Select a random number R in the range (0, k tot ) –if (R < k d ) then send diffuse ray –else if (R < k d + k s ) then send specular ray –else send transmission ray

Computing the Diffuse reflection We can compute a random direction as follows: –Given two random numbers  1 in [0,1] and  2 in [0,1], the randomly reflected direction  d is given by  d = (  ) = (cos -1 (sqrt(  1 )), 2  2 )

Problems Need to trace a lot of rays to get an accurate image Typically trace 100 – 1000 rays per pixel

Problems Need to trace a lot of rays to get an accurate image Typically trace 100 – 1000 rays per pixel

Radiosity Diffuse surfaces Subdivide scene Radiosity assumed constant over a patch Form-factor between patches Geometry and visibility Big Matrix system

Radiosity vs. Monte Carlo We have an integral equation on an infinite space Finite elements (Radiosity) –Project onto finite basis of functions –Linear system Monte Carlo –Probabilistic sampling

Radiosity vs. Monte Carlo We have an integral equation on an infinite space Finite elements (Radiosity) –Project onto finite basis of functions –Linear system –View-independent (no angular information) Monte Carlo –Probabilistic sampling –View-dependent (but angular information)