Realistic RaytracingZack Waters 
Introduction  
For image synthesis we need to have some model of light. The prevalent models of light are geometric optics and physical optics. Geometric optics models light propagation in terms of rays that travel in straight lines and their paths are described through a series of reflections and refractions. Physical optics treats light propagation as a wave phenomenon where effects of polarization, interference and diffraction can be modeled. 

Classical raytracing, an application of geometric optics, point samples the incoming radiance from a scene by tracing rays from the eye through the scene to the lights. The images generated by an implementation of the classical raytracing algorithm can look wonderful but also unrealistic and often times suffer from severe aliasing problems. The problem is that the algorithm severely undersamples the domains for the integral equations that describe the complex optical interations of light. Distribution raytracing, originally called
distributed raytracing, extends classical raytracing by incorporating 


For realistic image synthesis we also need to
solve the rendering integrals for the light paths that account for global
illumination and caustics. However,
these integrals can be deeply nested and seemingly impossible to solve.
Using Path tracing was developed as a solution to
the complete rendering equation and is heavily based on In the following sections I will discuss the rendering equation, distribution raytracing, and will briefly touch upon path tracing. This writeup reflects my understanding of these topics and there may be errors. If there are any please let me know. 

The Rendering Equation 

Introduced by James Kajiya in 1986, the rendering equation describes the transport of light from one surface point to another as the sum of emitted radiance and reflected radiance. 



where: 



Radiance tells us how much light energy is arriving or leaving a surface in a particular direction within some time frame. In a vacuum radiance is constant along a line of sight which is an important property that makes raytracing possible. The term that is the most interesting is the reflected radiance term, Lr, and can be described as the following: 



This integral takes into account all of the incoming light and computes the reflected light. It takes into account all light paths such as those that contribute to caustics and global illumination. The next several sections contains the details of this integral. 





The bidirectional reflectance distribution function (BRDF) describes how a surface reflects light energy. Reflectance is the fraction of incident light that is reflected. The simplified form of the BRDF is a 4D function of incident and outgoing directions. The BRDF is given by the following ratio.  




where:  


For a BRDF to be physically plausible it must follow the law of energy conservation and must obey the Helmholtz reciprocity principle. Following the law of energy conservation the BRDF should evaluate to a real number in the range [0, 1]. To be more specific the differential BRDF integrated over a hemisphere must be less then or equal to one. This means we can not reflect more light then we have received. This can be expressed mathematically as follows:  


This expression is also known as the directional hemispherical reflectance and describes the total amount of incident light energy that is reflected which must be less then or equal to 1. Helmholtz reciprocity principle means that the sampling incident and reflected directions of the BRDF can be switched and the result will be the same. 



Obtaining a BRDF for a material can be done in several ways. One can obtain a BRDF through empirical measurements and then fit a mathematical function to the data. Some examples of these empirical models are the popular Lambert, Phong, BlinnPhong and Ward models. The BRDF for a perfectly lambertian (diffuse) surface is just a constant as light is equally reflected in all directions. Physically based models can be developed analytically and are rooted in physics. Examples of these are CookTorrance model and a model presented by Kajiya in 1985 that handles anisotropic reflection.  




The function Li describes the incident radiance at x. The function may be a deeply nested integral equation as incident light may be indirectly reflected from other surfaces in the environment including itself.  




For this term we are taking the dot product of the incident direction and normal to project the differential area subtended by the solid angle onto the base of the hemisphere as seen by x. This term is necessary to account for Lambert’s cosine law. For a better understanding we have the following:  


Surface A is equally illuminated by surfaces B and C which both have the same surface area. However, A can see more of B then it does of C because the projected area of C onto A is smaller then that of B.  




The differential solid angle is the differential quantity of choice when integrating over a hemisphere. The solid angle generalizes a range of directions through a differential area on the hemisphere.  


Solving the Rendering Equation 

Two main approaches to solving the rendering equation are finite element
methods and point sampling techniques. For
the rest of this paper we will only discuss point sampling techniques using 

Distribution Raytracing 

Even before Kajiya formalized the rendering equation,
Cook et al recognized that rendering is just the process of solving a set of
nested integrals. Some examples
of these integrals are the integral over the pixel area, over the lens area,
over time, and over the hemispheres for reflection and transmission.
These integrals do not have an analytical solution that can be
computed in finite time, so instead we use Basic idea behind 



where:  


There are several forms of 

At surface locations distributed raytracing does not completely integrate over the entire domain, the hemisphere, but instead only over light sources and about reflection and transmission rays. Because of this distributed raytracing does not take into account all light paths such as indirect illumination and caustic effects. Never the less distributed raytracing is computationally expensive.  
Sampling Pixels 

Pixels have a finite area and represent a single color. In classical raytracing we are sampling the incoming radiance from the scene once per pixel.  


From the start classical raytracing is grossly undersampling the scene. Distributed raytracing extends classical raytracing to use stochastic sampling, a Monte Carlo method, to approximate the integral over the area of the pixel.  


A common technique to approximate this nested integral is to use a jittered grid, a form of stratified sampling, to subsample the pixel. The basic idea is to divide the pixel area into a grid of equal size cells or stata and the sample points are generated by jittering the center point of each cell.  


The pixel color is calculated by firing rays from the eye point through the each of the jittered sample locations and then these samples are averaged.  
click on images to enlarge 



Soft Shadows 

In a basic raytracer implementation shadows are very sharp and typically this is because point light sources are used. In order to create soft shadows we need an area light source. The figure below shows the anatomy of a shadow for both a point light source and an area light source.  


The umbra is the region where the light is entirely occluded and the penumbra is the region where the light is partially occluded. Soft shadows can be approximated in a standard raytracer by just placing several point light source near each other where each one has Nth the intensity of the base light they are approximating. Using this technique sharp transitions occur in the penumbra because we are still dealing with individual point light sources. Here is an example where this technique is used. 



Distributed raytracing resolves the problem by stochastically sampling area light sources. To implement this we can add a shadow factor to our rendering equation that is a number between 0 and 1. A shadow factor of 0 means we are completely occluded (umbra) or 1 if we are completely unoccluded. A factor in between 0 and 1 means we are partially occluded (penumbra). The shadow factor is calculated for each light source by generating a set of random sample points that are well distributed across the surface of the light source and then distributing a set of shadow feeler rays to these sample point locations. 



Each shadow feeler ray, Si, either hits the light source in which case it is assigned a value of 1 or it doesn’t in which case it is assigned a value of 0. To calculate the light’s contribution we just take the average of the shadow feelers and multiply it by the light’s intensity.  


Generating the sample points over the area of the light source is done by generating a jittered grid for n samples and then mapping the grid sample locations to a set of locations on the surface of the light source. The mapping for a square area light is straight forward.  
click on images to enlarge 



For other complex light types the mapping is not as simple.
However, 

Gloss 

Glossy surfaces are specular surfaces that are somewhat rough. Distribution raytracing tackles the problem by distributing a set of reflection rays by randomly perturbing the ideal specular reflection ray. The spread of the distribution determines the glossiness where a wider distribution spread models a rougher surface.  


To implement gloss you just need to setup a square that is perpendicular to the specular reflection vector and then map a jittered grid of sampling locations in which reflection rays are fired through. The width of this square determines the glossiness of the surface. In addition the ray contributions should be weighted using the BRDF or the same specular weighting function used on the ideal specular reflection rays.  


Alone this technique can only model very shallow bumpy surfaces such as a kitchen countertop where the distribution of the bumps is uniform. However, we can incorporate a texture map that we look up into to determine the pattern of glossiness where each texel in the map is a gloss factor. The gloss map can be used in addition to a bump map to model deeper groves.  


Translucency 

Transluceny is implemented in a similar manner as gloss. Once the bulk of the implementation is complete for gloss it is straight forward to slightly modify it to include translucency. The only difference is that instead of perturbing the ideal specular reflection ray you are now perturbing the ideal specular transmission ray in order to generate a set of secondary transmission rays. A wider distribution of transmission rays will make the glass like material appear rougher. Effects such as frosted gloss can be achieved. For a wider range of effects this technique can be mixed with translucent and refraction maps.  
Depth of Field 

The depth of field is the distance that objects appear in focus. It is necessary to model a camera with a lens system to achieve depth of field effects. For basic camera operation you need an opaque contain that contains a single aperture (opening). This aperture controls the amount of light that may strike the image or film. Light passing through the aperture is the only light allowed to strike the film. The amount of time that the film is exposed to the light is controlled by a shutter. In computer graphics the pinhole camera model is the most popular. The film is enclosed in a box containing a pinhole aperture. Everything in the scene is sharply focused onto the image plane. 



The pinhole which is infinitely small becomes the focal point and the scene lies in front of it and is projected to the film which lies behind the pinhole. In computer graphics the typical setup is that the image and scene lies in front of the focal point or point of projection. Controlling the focal distance, the distance from the image plane to the point of projection, or the image dimensions effects the field of view but does not affect the focus of objects. They will always appear in sharp focus. A simple camera model for implementing depth of field is the thin lens camera model. In this model the camera lens is a double convex lens of negligible thickness. This means that light passing through the lens is refracted on a single plane called the principle plane instead of two refractions. The lens has two focal points at equal distances in front and behind the lens. 



A phenomenon called the circle of confusion occurs when light from a single point is focused through a lens onto an image as a circle. The reason that this occurs is because the focal distance needed to project the light as a single point is either in front or behind the image plane. Remember light leaves a surface in all directions about the hemisphere and thus the circle is formed because different parts of the lens are focusing all light that it receives from the single surface point. In the figure below light from the point on the focal plane is projected as a circle on the image plane with distance C.  


Given a focal length, f, and a distance, i, which is the distance from the pixel to the center of the lens, we can compute the depth of field effects by first computing the distance, s, from the center of the lens to a focal plane where all points will be in focus. Using the thinlens approximation we can find s.  




Motion Blur 

Motion blur occurs because the shutter allows the film in the camera is exposed to light for a period of time. While the shutter is opened any animation in the scene is captured on a single piece of film. The result is an image where objects in motion are blurred. To implement this multiple rays from the pixel are fired into the scene at different times and the contribution is averaged. When taking into account depth of field, the computation can become more costly. To reduce the computation the samples in time can be paired with pixel and lens samples. Then the rays traced through these dimensions are then averaged for each pixel. 

Path Tracing 

Path tracing was introduced by Kajiya in the same paper that he formalized the "The Rendering Equation". He developed path tracing as a solution to the rendering the equation. It handles indirect lighting as well as caustics. In a path tracer you are tracing a single path from the eye back to the light source. Instead of distributing mulitple secondary rays at each intersection, a single new ray path is chosen probabilistically.  
Kajiya points out that first generation rays as well as light source rays are most important in terms of variance that they contribute to the pixel integral. Secondary rays effect contribute less. When you are only tracing a single path there needs to be a heuristic that tells the algorithm where the important sampling directions are. Typically path tracers incorporate an importance function that basically describe what parts of the domain are important. Sample paths are weighted according to their importance or contribution. We do not need to solve for the entire domain as the overall result will not look much different. For example, we only need an accurate solution of the radiance equation for visible surfaces with significant contribution. Importance needs to be propagated through the environment. If a surface is important then any surfaces that contributes significantly to that surface is also important.  
One major problem with path tracing is that variance shows up as noise in the final render. The main contributor to noise is caustics and can be alleviated by using other techniques such as photon mapping.  
click to enlarge 



References 

