Path tracing methods

Monte Carlo methods have a long history in computer graphics, dating back to the work of Kajiya [101] and Cook [40]. In contrast to other techniques available

3.2 ● Monte Carlo light transport simulation 39 at the time, these algorithms enabled researchers and practitioners to simulate effects that had been difficult to generate previously, such as: depth-of-field, motion blur, glossy reflections and global illumination.

In terms of the path integral formulation introduced in section2.5, the goal of a rendering algorithm is to compute the pixel values by estimating the corresponding integrals over path space, introduced in equation (2.20),

I_j= ∫_PW_j(x)¯ f(x)dμ(¯ x).¯ (3.22) Specifically, the algorithms are designed to sample light paths ¯X_iaccording to some probability function p(x)¯ defined on the path space, and then evaluating the resulting Monte Carlo estimator: Although not all Monte Carlo rendering algorithms were originally derived in this form, which is a relatively recent construction, it is often informative to study and compare them using this formulation [78, 213]. In the path integral formulation, a rendering algorithm can be viewed as a sampling mechanism with the goal of constructing a proper probability distribution p(x)¯ that produces a Monte Carlo estimator with high efficiency. To ensure that the resulting estimator has low variance, p(x)¯ should be roughly proportional to the integrand (see section3.1.2). However, as the measurement contribution function, f(x), is a complicated function consisting of many terms, and the¯ domain of the integrand is generally infinite (arbitrary long light paths may exist), it is generally not an easy task to design a suitable p(x).¯

Path tracing

One of the most fundamental Monte Carlo rendering algorithms,Path tracing, constructsp(x)¯ by sequentially sampling path vertices starting from the sensor.

First, a starting location on the image plane is sampled. Then the algorithm proceeds by generating a random walk into the scene. To sample the next vertex along the path, a conditional distribution function is used to sample a outgoing direction from the current vertex. If the resulting ray intersects some participating media, a media vertex is sampled at a random distance proportional to the transmittance of the media, otherwise the next surface intersection is found along the ray. The conditional distribution function used to sample an outgoing direction from the current vertex is set to be proportional to the local BRDF if the current vertex is at a surface or the local phase function if the current vertex is in media. The random walk procedure proceeds by

Sensor

Light

(a) Path tracing

Sensor

Light

(b) Bidirectional path tracing Figure 3.1: Illustration of path tracing and bidirectional path tracing.a) Path tracing constructs a random walk into the scene starting at the sensor (green line). At each intersection next event estimation tries to find a valid (visible) connection to a light source (dashed purple lines).b) Bidirectional path tracing, construct two random walks. One starting at the sensor (green line) and one starting at a light source in the scene (blue line). All possible connections between vertices on the two paths are then considered. Valid connection paths (dashed purple lines), not intersecting any scene geometry, are considered in the final estimator.

adding vertices in this fashion until a random termination criteria is met, also known asRussian roulette[102]. The random termination criteria is designed so that it ensures that shorter paths (say paths of 3−5 vertices) are sampled more often than longer paths, which are often less significant.

In practice, the basic path tracing algorithm is often improved by using a technique known asnext event estimation. Using this method, the path tracer also creates additional connections to light sources in the scene at every sampled surface or volume intersection of the path. Each original path from the camera can thus produce many valid light path samples with varying length. As the explicit connections are only valid if there are no occluding objects between the vertex and the light source, path tracing methods generally only work well in scenes where the light sources are directly visible from sampled vertices of the random walk starting at the camera. This can be a major problem in many scenes which are indirectly illuminated, for example, an interior room lit indirectly through occluded light sources. The presence of specular or glossy materials can also be a problem, as they make it difficult to find light paths that ends up at light sources. For example, rendering a scene where the light source is located inside a glass enclosure often results in very noisy results using these methods.

3.2 ● Monte Carlo light transport simulation 41 Bidirectional path tracing

An alternative to traditional Path tracing is to use Bidirectional path trac-ing [125,214]. A bidirectional path tracer constructs p(x)¯ by first generating two independent random walks. One starting from the camera and one starting from light sources in the scene. A large number of valid light paths are then generated by explicitly connecting each sampled interaction vertex along the two subpaths, creating a number of different lengths for the generated paths.

This sampling strategy can lead to estimators with significantly less variance than for unidirectional path tracing, as it enables a larger variety of light paths to be sampled. Furthermore, by combining the sampled paths using multiple importance sampling, robust estimates can be obtained for scenes where one of the sampled connection lengths is much better than the others. However, bidi-rectional path tracing can still be inefficient when rendering scenes containing very glossy transfer (light paths reflected through glossy materials) or scenes that contain prominentspecular-diffuse-specularreflection geometries in which the light path is scattered first by a specular reflection, then a diffuse reflection and then another specular reflection.