k. h. ko school of mechatronics gwangju institute of science and technology

Photo-realistic Rendering and Global Illumination in Computer Graphics

Spring 2012

Hybrid Algorithms

K. H. Ko

School of MechatronicsGwangju Institute of Science and Technology

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Metropolis Light Transport It aims for robust global illumination that can

handle light transport paths that are difficult to capture.

It demonstrates the application of Metropolis sampling to image generation by sampling the extremely high-dimensional space of all possible paths.

Basic Idea of Metropolis Sampling Technique It can generate a sequence of samples from a non-

negative function f such that the samples are distributed according to f.

There are more samples where f is large and vice-versa. This important property of Metropolis sampling is

achieved without any knowledge of f or its PDF. The only requirement is that it should be possible to

evaluate the function f at each generated sample.

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Metropolis Light Transport MLT applies the Metropolis sampling technique

to the infinite-dimensional space of paths. The key idea of MLT is that paths are sampled according to

the contribution they make to the final image. The algorithm generates a sequence of light transport

paths by applying random mutations to the previous path. Example mutations are adding a vertex in the path, deleting an existing

vertex, etc. Each proposed mutation could be accepted or rejected. The probabilities that determine acceptance/rejection are

chosen so that paths are sampled according to their contribution to the image plane.

The image is computed by sampling many paths and recording their contributions to the image plane.

Metropolis Light Transport The main advantage of the MLT algorithm is that it is

an unbiased algorithm that can handle hard-to-compute illumination situations. MLT is efficient in computing images for scenes with

strong indirect illumination that only arises through a small set of paths.

Once the algorithm finds an important, but hard to find, light transport path, it explores other paths “near” that path through mutations.

It is assumed that exploring that part of the path space will find other important light transport paths.

This local exploration of the space of paths can result in faster convergence in scenes as compared to other approaches such as bidirectional path tracing.

The fundamental framework of Metropolis sampling ensures that this faster convergence is achieved while still maintaining an unbiased technique.

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Metropolis Light Transport

Another benefit of this approach is that the contribution of a new path can be computed relatively inexpensively.Only a small part of the entire path is

changed by the mutation.The visibility information for the

unchanged segments of the path do not need to be recomputed.

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Detailed Balance Given a state space Ω, a non-negative function

f: Ω -> R+, and an initial seed x0 ∈ Ω, the Metropolis sampling algorithm generates a random walk x0, x1, …, such that the xi are eventually distributed according to f, irrespective of the choice of x0.

To achieve this steady-state distribution of samples, mutations have to be accepted or rejected carefully.

The acceptance probability a(x->y) gives the probability that a mutation from x to y is accepted.

A transition function T(x->y) gives the probability density that a mutation technique would propose a mutation from state x to y. 6


Detailed Balance For a random walk in steady-state, the

transition density between two states must be equal:

f(x)T(x->y)a(x->y) = f(y)T(y->x)a(y->x). This condition is known as detailed balance. Since f and T are given, the following choice

of a results in equilibrium being achieved the fastest.

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))yx(T)x(f

)xy(T)y(f,min()yx(a

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Algorithm The MLT algorithm starts with a set of n

random paths, constructed using bidirectional path tracing, from the lights to the image plane.

These paths are then mutated using mutation strategies.

When a path x is mutated to produce a path y, the mutation is accepted based on the probability a given.

In particular, if the new path y does not contribute to the image, then the acceptance probability will be zero and the mutation will be rejected.

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Mutation Strategies Bidirectional mutation

This mutation deletes a subpath of the path, extends the ends of the two remaining subpaths with one or more vertices, and then connects these ends together.

The paths are accepted based on the acceptance probability a.

Perturbations Perturbations try to make small changes to a path,

say by moving one or more vertices of the path, while leaving most of the path the same.

Caustic perturbations, lens perturbations, and multichain perturbations to capture different light paths more efficiently. 9


Pseudo code of the basic MLT algorithm.

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MLT() { clear pixels in image to 0; x = initialSeedPath(); // n paths are chosen for I = 1 to N { y = mutate(x); a = acceptanceProbability(x,y); if (random() < a) x = y; // accept mutation recordSample (image, x); }}


MLT is an unbiased technique to compute light transport that is efficient at computing images of scenes that include hard-to-find light transport paths.

Once a hard-to-find path is found, mutations explore that part of the path space thoroughly before going to another part of the space.

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However, the implementation of MLT is quite complicated.

Care must be taken to get several important details of the algorithm right for it to work.

It is unclear how MLT performs for scenes that include multiple important paths. It is possible that the mutations will result in

slower convergence by exploring only a few of the many important paths thoroughly.

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13Ref: www-graphics.stanford.edu/papers/metro

Irradiance Caching Monte Carlo rendering can take a long time to converge to

images of reasonable quality. Using pure Monte Carlo sampling to compute irradiance

(incoming radiosity) at a point could require hundreds of ray-tracing operations.

Each of these operations, in turn, could result in more rays being traced in the scene.

Thus, this computation could be extremely slow.

Irradiance caching is an effective technique for accelerating the computation of indirect illumination in diffuse scenes. It exploits the insight that the irradiance at diffuse surfaces,

while expensive to compute, varies smoothly in most scenes. Irradiance is cached in a data structure, and when possible,

these cached values are interpolated to approximate irradiance at nearby surfaces.

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Irradiance Caching

Interpolation Irradiance gradients are used to determine

when cached values can be interpolated to produce reasonably accurate results.

The translation and rotation gradient estimate how irradiance changes with position and direction.

An error estimate, based on the split-sphere model is used to determine which samples can be used for interpolation without introducing visible artifacts.

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Irradiance Caching

Interpolation Using this model, the error at a point P due to

a cached sample i at location Pi is given as

Ri is the mean harmonic distance of objects visible from the cached sample i.

NP and Npi are the normals at P and the sample at Pi. Note that this error term penalizes samples whose

normals differ significantly from the normal of the point whose irradiance is being approximated.

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PPi

ii NN

R

PP)P(

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Irradiance Caching

Interpolation Similarly, samples that are far away are

penalized. Also, samples that are close to other

surfaces, i.e., their mean harmonic distance is small, are penalized.

Irradiance at the point P is interpolated using the cached irradiance values of nearby samples using the weight ωi for the ith sample: ωi(P)=1/Єi(P).

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Irradiance Caching

Interpolation If a point has a large error, its weight is

small, and vice-versa. A user-specified parameter a is further

used to eliminate samples whose weights are too small.

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Irradiance Caching

InterpolationThe interpolated irradiance at point P is

then

Ei(P) is the computed illuminance at Pi extrapolated to P.

The extrapolation is computed using the rotation and translation gradients of the cached values.

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N

i i

N

i ii

)P(

)P(E)P()P(E

1

1

Irradiance Caching The cached samples are stored in an octree

constructed over the scene. This data structure permits the decoupling of

geometry from illumination values. When the irradiance at a point must be computed, the

octree is searched to find “nearby” cached samples that are accurate enough to be used to approximate irradiance. The user-specified weight cutoff a specifies a radius over

which the samples are searched. If such samples are found, they are used to interpolate

irradiance using the weighting algorithm. If such samples do not exist, a sample is computed for

the current point. This sample is then stored in the irradiance cache to be reused for interpolation later.

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Photon Mapping Photon mapping is a practical two-pass

algorithm that traces illumination paths both from the lights and from the viewpoint.

However, unlike bidirectional path tracing, this approach caches and reuses illumination values in a scene for efficiency.

It is a two-pass algorithm In the first pass, “photons” are traced from the light

sources into the scene. These photons, which carry flux information, are cached in a

data structure, called the photon map. In the second pass, an image is rendered using the

information stored in the photon map.

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Photon Mapping Photon mapping decouples photon storage

from surface parameterization. This representation enables it to handle arbitrary

geometry, including procedural geometry, thus increasing the practical utility of the algorithm.

It is also not prone to meshing artifacts. By tracing or storing only particular types of

photons, it is possible to make specialized photon maps, just for that purpose.

The caustic map: It is designed to capture photons that interact with one or more specular surfaces before reaching a diffuse surface. These light paths cause caustics.

Traditional Monte Carlo sampling can be very slow at correctly producing good caustics.

By explicitly capturing caustic paths in a caustic map, the photon mapping technique can find caustics efficiently.

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Photon Mapping Photon mapping is a biased technique.

The bias is the potentially nonzero difference between the expected value of the estimator and the actual value of the integral being computed.

Since photon maps are typically not used directly, but are used to compute indirect illumination, increasing the photons eliminates most artifacts.

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Photon Mapping Tracing Photons: Pass 1

The use of compact, point-based “photons” to propagate flux through the scene is key in making photon mapping efficient.

Photons are traced from the light sources and propagated through the scene just as rays are in ray tracing.

They are reflected, transmitted, or absorbed. Russian roulette and the standard Monte Carlo sampling

techniques are used to propagate photons. When the photons hit nonspecular surfaces, they are

stored in a global data structure called the photon map.

To facilitate efficient searches for photons, a balanced kd-tree is used to implement this data structure.

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Photon mapping can be efficient for computing caustics.

A caustic is formed when light is reflected or transmitted through one or more specular surface before reaching a diffuse surface.

To improve the rendering of scenes that include caustics, the algorithm separates out the computation of caustics from global illumination.

Thus two photon maps, a caustic photon map and a global photon map, are computed for each scene.

The caustic map includes photons that traverse the paths LS+D.

The global photon map represents all paths L(S|D)*D.

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Caustic photon maps can be computed efficiently because caustics occur when light is focused.

Not too many photons are needed to get a good estimate of caustics.

Additionally, the number of surfaces resulting in caustics in typical scenes is often very small.

Efficiency is achieved by shooting photons only towards this small set of specular surfaces.

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The reflected radiance at each point in the scene can be computed from the photon map.

The photon map represents incoming flux at each point in the scene. Therefore, the photon density at a point estimates the irradiance at that point.

The reflected radiance at a point can then be computed by multiplying the irradiance by the surface BRDF.

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Photon Storing Photons are stored only when they hit diffuse

surfaces (or, more precisely, non-specular surfaces). Storing photons on specular surface does not give

any useful information: the probability of having a matching incoming photon from the specular direction is very small (and zero for perfect specular materials).

For all photon-surface interactions, data is stored in a global data structure, the photon map.

Each emitted photon can be stored several times along its path.

Information about a photon is stored at the surface, where it is absorbed if that surface is diffuse.

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Photon Storing It is important to realize that photons represent

incoming illumination (flux) at a surface. This is a valuable optimization that enables us to

use a photon to approximate the reflected illumination at several points on a surface..

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Photon Mapping Photon Storing

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To compute the photon density at a point the n closest photons to that point are found in the photon map. This search is efficiently done using the balanced kd-

tree storing the photons. The photon density is then computed by adding

the flux of these n photons and dividing by the projected area of the sphere containing these n photons.

Thus, the reflected radiance at the point x in the direction ω is

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Photon Mapping Computing Images: Pass 2

The simplest use of the photon map would be to display the reflected radiance values computed in pass 1 for each visible point in an image.

However, unless the number of photons used is extremely large, this display approach can cause significant blurring of radiance, thus resulting in poor image quality.

Instead, photon maps are more effective when integrated with a ray tracer that computes direct illumination and queries the photon map only after one diffuse or glossy bounce from the view point is traced through the scene.

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The final rendering of images could be done as follows:

Rays are traced through each pixel to find the closest visible surface.

The radiance for a visible point is split into direct illumination, specular or glossy illumination, illumination due to caustics, and the remaining indirect illumination. Direct illumination for visible surfaces is computed

using regular Monte Carlo sampling. Specular reflections and transmissions are ray

traced. Caustics are computed using the caustic photon

map. Since caustics occur only in a few parts of the scene, they are computed at a higher resolution to permit direct high-quality display.

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The final rendering of images could be done as follows:

The remaining indirect illumination is computed by sampling the hemisphere; the global photon map is used to compute radiance at the surfaces that are not directly visible. This extra level of indirection decreases visual artifacts.

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Photon Mapping A visualization of both passes of the photon-mapping

algorithm.

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Photon Mapping

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Photon Mapping

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Photon Mapping

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Photon Mapping The use of the global photon map

for indirect illumination is reminiscent of the final gathering approaches in radiosity algorithms.

However, by storing caustic maps that can be visualized directly, this algorithm is able to capture challenging caustic paths.

k. h. ko school of mechatronics gwangju institute of science and technology

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