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Ray Tracing

CSE167: Computer Graphics

Instructor: Steve Rotenberg

UCSD, Fall 2005

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Ray Tracing

Ray tracing is a powerful rendering technique that is the foundationof many modern photoreal rendering algorithms

The original ray tracing technique was proposed in 1980 by TurnerWhitted, although there were suggestions about the possibility in

scientific papers dating back to 1968 Classic ray tracing shoots virtual view rays into the scene from the

camera and traces their paths as they bounce around

With ray tracing, one can achieve a wide variety of complex lightingeffects, such as accurate shadows and reflections/refractions fromcurved surfaces

Achieving these effects with the same precision is difficult if notimpossible with a more traditional rendering pipeline

Ray tracing offers a big advance in visual quality, but comes with anexpensive price of notoriously slow rendering times

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Ray Intersections

Tracing a single ray requires determining if that rayintersects any one of potentially millions of primitives

This is the basic problem of ray intersection

Many algorithms exist to make this not only feasible, butremarkably efficient

Tracing one ray is a complex problem and requiresserious work to make it run at an acceptable speed

Of course, the big problem is the fact that one needs totrace lots of rays to generate a high quality image

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Rays

Recall that a rayis a geometric entity with anorigin and a direction

A ray in a 3D scene would probably use a 3D

vector for the origin and a normalized 3D vectorfor the direction

class Ray {

Vector3 Origin;Vector3 Direction;

};

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Camera Rays

We start by shooting rays from the camera out into the scene

We can render the pixels in any order we choose (even in random order!),

but we will keep it simple and go from top to bottom, and left to right

We loop over all of the pixels and generate an initialprimary ray (also called

a camera ray or eye ray)

The ray origin is simply the cameras position in world space

The direction is computed by first finding the 4 corners of a virtual image in

world space, then interpolating to the correct spot, and finally computing a

normalized direction from the camera position to the virtual pixel

Camera

position

Virtual image

Primary ray

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Ray Intersection

The initial camera ray is then tested for intersection with the 3D

scene, which contains a bunch of triangles and/or other primitives

If the ray doesnt hit anything, then we can color the pixel to some

specified background color Otherwise, we want to know the first thing that the ray hits (it is

possible that the ray will hit several surfaces, but we only care about

the closest one to the camera)

For the intersection, we need to know the position, normal, color,

texture coordinate, material, and any other relevant information wecan get about that exact location

If we hit somewhere in the center of a triangle, for example, then this

information would get computed by interpolating the vertex data

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Ray Intersection

We will assume that the results of a ray intersection testare put into some data structure which convenientlypackages it together

class Intersection {Vector3 Position;

Vector3 Normal;

Vector2 TexCoord;

Material *Mtl;

float Distance; // Distance from ray origin to intersection

};

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Lighting

Once we have the key intersection information (position,

normal, color, texture coords, etc.) we can apply any

lighting model we want

This can include procedural shaders, lighting

computations, texture lookups, texture combining, bump

mapping, and more

Many of the most interesting forms of lighting involve

spawning off additional rays and tracing them recursively

The result of the lighting equation is a color, which is

used to color the pixel

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Shadow Rays

Shadows are an important lighting effect that can easily becomputed with ray tracing

If we wish to compute the illumination with shadows for a point, weshoot an additional ray from the point to every light source

A light is only allowed to contribute to the final color if the raydoesnt hit anything in between the point and the light source

The lighting equation we looked at earlier in the quarter can easilybe adapted to handle this, as clgtiwill be 0 if the light is blocked

Obviously, we dont need to shoot a shadow ray to a light source ifthe dot product of the normal with the light direction is negative

Also, we can put a limit of the range of a point light, so they donthave an infinite influence (bending the laws of physics)

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Shadow Rays

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Shadow Rays

Shadow rays behave slightly differently from primary(and secondary) rays

Normal rays (primary & secondary) need to know thefirst surface hit and then compute the color reflected offof the surface

Shadow rays, however, simply need to know ifsomething is hit or not

In other words, we dont need to compute any additional

shading for the ray and we dont need to find the closestsurface hit

This makes them a little faster than normal rays

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Offsetting Spawned Rays

We say that the shadow rays are spawned off of thesurface, or we might say that the primary ray spawnedoff additional shadow rays

When we spawn new rays from a surface, it is usually agood idea to apply a slight adjustment to the origin of theray to push it out slightly (0.00001) along the normal ofthe surface

This fixes problems due to mathematical roundoff thatmight cause the ray to spawn from a point slightly belowthe surface, thus causing the spawned ray to appear tohit the same surface

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Reflection Rays

Another powerful feature often associated with raytracing is accurate reflections off of complex surfaces

If we wanted to render a surface as a perfect mirror,instead of computing the lighting through the normalequation, we just create a new reflection ray and trace itinto the scene

Remember thatprimary raysare the initial rays shotfrom the camera. Any reflected rays (and others, likerefracted rays, etc.), are called secondary rays

Reflected rays, like shadow rays should be movedslightly along the surface normal to prevent the ray fromre-intersecting the same surface

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Computing Reflection Direction

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nnddr 2

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Reflections

If the reflection ray hits a normal material, we just compute theillumination and use that for the final color

If the reflection ray hits another mirror, we just recursively generatea new reflection ray and trace that

In this way, we can render complex mirrored surfaces that includereflections, reflections of reflections, reflections of reflections ofreflections

To prevent the system from getting caught in an infinite loop, it iscommon to put an upper limit on the depth of the recursion. 10 orlower works for most scenes, except possibly for ones with lots of

mirrored surfaces In any case, most pixels will only require a few bounces, as they are

likely to hit a non-mirrored surface sooner or later

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Reflections

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Reflections

Surfaces in the real world dont act as perfect mirrors

Real mirrors will absorb a small amount of light and only reflectmaybe 95%-98% of the light

Some reflecting surfaces are tinted and will reflect different

wavelengths with different strengths This can be handled by multiplying the reflected color by the mirror

color at each bounce

We can also simulate partially reflective materials like polishedplastic, which have a diffuse component as well as a shiny specularcomponent

For a material like this, we would apply the normal lighting equation,including shooting shadow rays, to compute the diffuse component,then add a contribution from a reflection ray to get the final color (thediffuse and specular components should be weighted so as not toviolate conservation of energy)

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Transmission Rays

Ray tracing can also be used to accurately render thelight bending in transparent surfaces due to refraction

Often, this is called transmissioninstead ofrefraction.Transmission is a more general term that also includestranslucency, but I think the real reason this word ispreferred is because reflection and refraction look toosimilar

When a ray hits a transparent surface (like glass, orwater), we generate a new refracted ray and trace

that, in a similar way as we did for reflection We will assume that the transmitted ray will obey Snells

law (n1sin1=n2sin2), where n1and n2are the index ofrefraction for the two materials

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Computing Transmission (Refraction) Direction

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Total Internal Reflection

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n

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n1

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When light traveling in a material with a high index of refraction hits amaterial with a low index of refraction at a steep angle, we get a totalinternal reflection

When this happens, no refraction ray is generated

This effect can be visible when one is scuba diving and looks up at thewater surface. One can only see rays refracting to the outside world in acircular area on the water surface above

Total internal reflection can be detected when the magnitude of the z vectoris greater than 1, causing the square root operation to become undefined

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Spawning Multiple Rays

When light hits a transparent surface, we not only see refraction, butwe get a reflection off of the surface as well

Therefore, we will actually generate two new rays and trace both ofthem into the scene and combine the results

The results of an individual traced ray is a color, which is the color ofthe light that the ray sees

This color is used as the pixel color for primary rays, but forsecondary rays, the color is combined somehow into the final pixelcolor

In a refraction situation, for example, we spawn off two new rays

and combine them according to the Fresnel equations, provided inthe last lecture

The Fresnel equations describe how the transmitted (refracted) raywill dominate when the incoming ray is normal to the surface, butthe reflection will dominate when the incoming ray is edge-on

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Refraction

Transmission ray

Reflection ray

Primary ray

Camera

Normal

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Fresnel Equations

The Fresnel equations can beused to determine theproportion of the light reflected

(fr) and transmitted (ft) when aray hits an interface betweentwo dielectrics (like air andwater)

They describe separate

formulas for the parallel andperpendicularly polarized light,but these are usually averagedinto a single set of values rt

perpparr

perp

par

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Recursive Ray Tracing

The classic ray tracing algorithm includes

features like shadows, reflection, refraction, and

custom materials A single primary ray may end up spawning

many secondary and shadow rays, depending

on the number of lights and the arrangement

and type of materials These rays can be thought of as forming a tree

like structure

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Recursive Ray Tracing

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Ray Intersection

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Ray-Scene Intersection

One of the key components of a ray tracer is the system thatdetermines what surface the ray hits

A typical 3D scene may have well over 1,000,000 primitives

As usual, triangles tend to be the primitive of choice, but one

advantage of a ray tracer is that one can intersect rays with morecomplex surfaces such as spheres, Bezier patches, displacementmapped surfaces, fractals, and more

Sometimes, complex primitives are simply tessellated into trianglesin a pre-rendering phase, and then just ray traced as triangles

Alternately, it is possible to ray trace complex surfaces directly, or to

use demand-based schemes that dont tessellate an object until aray comes nearby

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Ray-Object Intersection

We will say that our scene is made up of several individual objects

For our purposes, we will allow the concept of an object to include primitives such astriangles and spheres, or even collections of primitives or other objects

In order to be render-able, an object must provide some sort of ray intersectionroutine

We will define a C++ base class object as:

class Object {

public:

virtual bool IntersectRay(Ray &r,Intersection &isect);

};

The idea is that we can derive specific objects, like triangles, spheres, etc., and thenwrite custom ray intersection routines for them

The ray intersect routine takes a ray as input, and returns true if the object is hit andfalse if it is missed

If the object is hit, the intersection data is filled in into the isect class

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Ray-Sphere Intersection

Lets see how to test if a ray intersects a sphere

The ray has an origin at point pand a unit length

direction u, and the sphere has a center cand aradius r

crp

u

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Ray-Sphere Intersection

The ray itself is the set of points p+u, where 0

We start by finding the point qwhich is the point on the

ray-line closest to the center of the sphere

The line qc must be perpendicular to vector u, in other

words, (q-c)u=0, or (p+u-c)u=0

We can solve the value of that satisfies that

relationship: =-(p-c)u, so q=p-((p-c)u)u

p u c

q

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Ray-Sphere Intersection

Once we have q, we test if it is inside the actual sphere or not, by checking if |q-c|r

If qis outside the sphere, then the ray must not miss

If qis inside the sphere, then we find the actual point on the sphere surface that theray intersects

We say that the ray will hit the sphere at two points q1and q2:

q1=p+(-a)u) q2=p+(+a)u) where a=sqrt(r2-|q-c|2)

If -a0, then the ray hits the sphere at q1, but if it is less than 0, then the actualintersection point lies behind the origin of the ray

In that case, we check if +a0 to test if q2is a legitimate intersection

p u c

q

q2q1

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Ray-Sphere Intersection

There are several ways to formulate the ray-sphere intersection test

This particular method is the one provided in the

book As a rule, one tries to postpone expensive

operations, such as division and square rootsuntil late in the algorithm when it is likely that

there will be an intersection Ideally, quick tests can be performed at the

beginning that reject a lot of cases where the rayis far away from the object being tested

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Ray-Plane Intersection

A plane is defined by a normal vector nand a distance d, which isthe distance of the plane to the origin

We test our ray with the plane by finding the point qwhich is wherethe ray line intersects the plane

For q to lie on the plane it must satisfyd=qn=pn+un

We solve for :

=(d-pn)/(un)

However, we must first check that the denominator is not 0, whichwould indicate that the ray is parallel to the plane

If 0 then the ray intersects the plane, otherwise, the plane liesbehind the ray, in the wrong direction

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Ray-Triangle Intersection

To intersect a ray with a triangle, we must first check if

the ray intersects the plane of the triangle

If we are treating our triangle as one-sided, then we can

also verify that the origin of the ray is on the outside ofthe triangle

Once we know that the ray hits the plane at point q, we

must verify that qlies inside the 3 edges of the triangle

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Ray-Triangle

Does segment abintersect triangle v0v1v2?

0v

q

pu

1v

2v

Does segment abintersect triangle v0v1v2?

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Barycentric Coordinates

Reduce to 2D: remove smallest dimension

Compute barycentric coordinates

q' =q-v0e1=v1-v0

e2=v2-v0

=(q'e2)/(e1e2)

=(q'e1)/(e1e2)

Reject if

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Acceleration Structures

Complex scenes can contain millions of primitives, and ray tracersneed to trace millions of rays

This means zillions of potential ray-object intersections

If every ray simply looped through every object and tested if it

intersected, we would spend forever just doing loops, not evencounting all of the time doing the intersection testing

Therefore, it is absolutely essential to employ some sort ofacceleration structureto speed up the ray intersection testing

An acceleration structure is some sort of data structure that groupsobjects together into some arrangement that enables the ray

intersection to be sped up by limiting which objects are tested There are a variety of different acceleration structures in use, but

most of the successful ones tend to be based on some variation ofhierarchical subdivision of the space around the group of objects

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Bounding Volume Hierarchies

The basic concept of a bounding volume hierarchy is a complex object in a hierarchyof simpler ones

This works much like the hierarchical culling we looked at in the scene graph lecture

For example, if one were using spheres as their bounding volume, we could enclosethe entire scene in one big sphere

Within that sphere are several other spheres, each containing more spheres, until wefinally get to the bottom level where spheres contain actual geometry like triangles

To test a ray against the scene, we traverse the hierarchy from the top level

When a sphere is hit, we test the spheres it contains, and ultimately thetriangles/primitives within

In general, a bounding volume hierarchy can reduce the ray intersection time fromO(n) to O(log n), where nis the number of primitives in the scene

This reduction from linear to logarithmic performance makes a huge difference andmakes it possible to construct scenes with millions of primitives

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Sphere Hierarchies

The sphere hierarchy makes for a good example of the concept, but inpractice, sphere hierarchies are not often used for ray tracing

One reason is that it is not clear how to automatically group an arbitrary setof triangles into some number of spheres, so various heuristic options exist

Also, as the spheres are likely to overlap a lot, they end up triggering a lotof redundant intersection tests

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Octrees

The octree starts by placing a cube around theentire scene

If the cube contains more than some specified

number of primitives (say, 10), then it is splitequally into 8 cubes, which are then recursivelytested and possibly resplit

The octree is a more regular structure than thesphere tree and provides a clear rule forsubdivision and no overlap between cells

This makes it a better choice usually, but still notideal

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Octrees

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KD Trees

The KD tree starts by placing a box (not necessarily a cube) aroundthe entire scene

If the box contains too many primitives, it is split, as with the octree

However, the KD tree only splits the box into two boxes, that need

not be equal The split can take place on the x, y, or z place at some arbitrary

point within the box

This makes the KD tree a little bit more adaptable to irregulargeometry and able to customize a tighter fit

In general, KD trees tend to be pretty good for ray tracing

Their main drawback is that the tree depth can get rather deep,causing the ray intersection to spend a lot of time traversing the treeitself, rather than testing intersections with primitives

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KD Trees

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BSP Trees

The BSP tree (binary space partitioning) is much like theKD tree in that it continually splits space into two (notnecessarily equal) halves

Unlike the KD tree which is limited to xyz axis splitting,the BSP tree allows the splitting plane to be placedanywhere in the volume and aligned in any direction

This makes it a much more difficult problem to choosethe location of the splitting plane, and so many heuristics

exist In practice, BSP trees tend to perform well for ray

tracing, much like KD trees

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BSP Trees

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Uniform Grids

One can also subdivide space into a uniform grid,instead of hierarchically

This is fast for certain situations, but gets too expensive

in terms of memory for large complex scenes It also tends to loose its performance advantages in

situations where primitives have a large variance in sizeand location (which is common)

As a result, they are not really a practical generalpurpose acceleration structure for ray tracing

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Uniform Grids

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Hierarchical Grids

One can also make a hierarchical grid

Start with a uniform grid, but subdivide any cell thatcontains too many primitives into a smaller grid

An octree is an example of a hierarchical grid limited to2x2x2 subdivision

A more general hierarchical grid could supportsubdivision into any number of cells

Hierarchical grids tend to perform very well in raytracing, especially for highly detailed geometry ofrelatively uniform size (such as the triangles in atessellated surface)

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Acceleration Structures

All of the acceleration structures we looked at store some geometry andprovide a function for intersecting a ray

In other words, they are really just a more complex type of primitivethemselves

We can derive acceleration structures off of our base Object class, just like

we did for Spheres and Triangles Also, acceleration structures can be designed so that they store a bunch of

generic Objects themselves, and so one could build an accelerationstructure that contains a bunch of triangles, and then place that accelerationstructure within a larger acceleration structure, etc.

This provides a nice, consistent way to represent scenes, similar to thescene graph concept we covered in the lecture on realtime scene

management

class KDTree:public Object {

public:

bool IntersectRay(Ray &r,Intersection &isect);

};

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Distribution Ray Tracing

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Distribution Ray Tracing

In 1984, an important modification to the basic ray tracing algorithmwas proposed, known as distributed ray tracing

The concept basically involved shooting several distributed rays toachieve what had previously been done with a single ray

The goal is not to simply make the rendering slower, but to achievea variety of soft lighting effects such as antialiasing, camera focus,soft-edge shadows, blurry reflections, color separation, motion blur,and more

As the term distributed tends to refer to parallel processing inmodern days, the distributed ray tracing technique is now calleddistribution ray tracing, and the term distributed is reserved for

parallel ray tracing, which is also an important subject

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Soft Shadows

One nice visual effect we can achieve with distributionray tracing is soft shadows

Instead of treating a light source as a point and shootinga single ray to test for shadows, we can treat the light

source as a sphere and shoot several rays to test forpartial blocking of the light source

If 15% of the shadow rays are blocked, then we get 85%of the incident light from the light source

In lighting terminology, the completely shadowed region

is called the umbraand the partially shadowed region iscalled thepenumbra

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Area Lights

The soft shadow technique enables us to define lights in a muchmore complex way than we have previously

We can now use any geometry to define a light, including triangles,patches, spheres, etc.

To determine the incident light, we shoot several rays towards thelight source, distributed across the surface and weighted accordingto the surface area of the sample and the direction of the averagenormal

Larger light sources create softer, diffuse shadows, while smallerlight sources cause sharp, harsh shadows

Larger light sources also require more rays to adequately samplethe shadows, making area lights a lot more expensive than pointlights. Inadequate sampling of the light source can cause noisepatterns to appear in the penumbra region, known as shadowaliasing

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Blurry Reflections

We can render blurry or glossyreflections by creating

several reflection rays instead of just one

The rays can be distributed around the ideal reflection

direction Blurry surfaces will causes a wider distribution (and

require more rays), while more polished surfaces will

have a narrow distribution

The same concept can apply to refraction in order toachieve rendering of unpolished glass

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Depth of Field

The blurring caused by a camera lens being out of focus is due to the lenslimited depth of field

In computer graphics, the term depth of field usually refers to the generalprocess of rendering images that include a camera blurring effect

A lens will typically be set to focus on objects at some distance away,

known as the focal distance Objects closer or farther than the focal distance will be blurry, and the

blurriness increases with the distance to the focal plane

Depth of field can be rendered with distribution ray tracing by distributingthe primary rays shot from the camera

Rays area distributed across a virtual aperture, which represents the(usually circular) opening of the lens

The larger the aperture, the more pronounced the blurring effect will be. Apinholecamera has an aperture size of 0, and therefore, will not have anyblurring due to depth of field

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Distribution Ray Tracing

Ray tracing had a big impact on computer graphics in1980 with the first images of accurate reflections andrefractions from curved surfaces

Distribution ray tracing had an even bigger impact in1984, as it re-affirmed the power of the basic ray tracingtechnique and added a whole bunch of sophisticatedeffects, all within a consistent framework

Previously, techniques such as depth of field, motion

blur, soft shadows, etc., had only been achievedindividually and by using a variety of complex, hackyalgorithms

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Distribution Ray Tracing

If ray tracing is slow, then distribution ray tracing must be considerablyslower

Now, instead of one or two splits per level in our recursion, we are have toshoot dozens or even hundreds of rays to achieve some of these effects

This can cause an exponential expansion in the number of rays

The good news is that we can combine these features so that we still onlyneed to shoot a small number of primary rays per pixel

For example, we can shoot 16 rays in a 4x4 antialiasing pattern, whereeach ray has a random distribution in time and in the camera aperture

Each of these rays only needs to spawn a few reflection or shadow rays, asthe results will be blended with 15 other samples

Still, we end up with lots and lots of rays and potential for exponentialproblems in scenes with a lot of soft or blurry features

This problem is at least partially addressed withpath tracingwhich is one ofthe techniques for global illumination that we will see in the next lecture