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Ray Tracing

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Title: Ray Tracing


1
Ray Tracing
  • CSE167 Computer Graphics
  • Instructor Steve Rotenberg
  • UCSD, Fall 2005

2
Ray Tracing
  • Ray tracing is a powerful rendering technique
    that is the foundation of many modern photoreal
    rendering algorithms
  • The original ray tracing technique was proposed
    in 1980 by Turner Whitted, 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 lighting effects, such as accurate
    shadows and reflections/refractions from curved
    surfaces
  • Achieving these effects with the same precision
    is difficult if not impossible with a more
    traditional rendering pipeline
  • Ray tracing offers a big advance in visual
    quality, but comes with an expensive price of
    notoriously slow rendering times

3
Ray Intersections
  • Tracing a single ray requires determining if that
    ray intersects any one of potentially millions of
    primitives
  • This is the basic problem of ray intersection
  • Many algorithms exist to make this not only
    feasible, but remarkably efficient
  • Tracing one ray is a complex problem and requires
    serious work to make it run at an acceptable
    speed
  • Of course, the big problem is the fact that one
    needs to trace lots of rays to generate a high
    quality image

4
Rays
  • Recall that a ray is a geometric entity with an
    origin and a direction
  • A ray in a 3D scene would probably use a 3D
    vector for the origin and a normalized 3D vector
    for the direction
  • class Ray
  • Vector3 Origin
  • Vector3 Direction

5
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
    initial primary 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

Primary ray
Virtual image
Camera position
6
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 we
    can 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

7
Ray Intersection
  • We will assume that the results of a ray
    intersection test are put into some data
    structure which conveniently packages it together
  • class Intersection
  • Vector3 Position
  • Vector3 Normal
  • Vector2 TexCoord
  • Material Mtl
  • float Distance // Distance from ray origin to
    intersection

8
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

9
Shadow Rays
  • Shadows are an important lighting effect that can
    easily be computed with ray tracing
  • If we wish to compute the illumination with
    shadows for a point, we shoot an additional ray
    from the point to every light source
  • A light is only allowed to contribute to the
    final color if the ray doesnt hit anything in
    between the point and the light source
  • The lighting equation we looked at earlier in the
    quarter can easily be adapted to handle this, as
    clgti will be 0 if the light is blocked
  • Obviously, we dont need to shoot a shadow ray to
    a light source if the 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 dont have an infinite influence
    (bending the laws of physics)

10
Shadow Rays
11
Shadow Rays
  • Shadow rays behave slightly differently from
    primary (and secondary) rays
  • Normal rays (primary secondary) need to know
    the first surface hit and then compute the color
    reflected off of the surface
  • Shadow rays, however, simply need to know if
    something 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 closest surface hit
  • This makes them a little faster than normal rays

12
Offsetting Spawned Rays
  • We say that the shadow rays are spawned off of
    the surface, or we might say that the primary ray
    spawned off additional shadow rays
  • When we spawn new rays from a surface, it is
    usually a good idea to apply a slight adjustment
    to the origin of the ray to push it out slightly
    (0.00001) along the normal of the surface
  • This fixes problems due to mathematical roundoff
    that might cause the ray to spawn from a point
    slightly below the surface, thus causing the
    spawned ray to appear to hit the same surface

13
Reflection Rays
  • Another powerful feature often associated with
    ray tracing 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 normal equation, we just create a new
    reflection ray and trace it into the scene
  • Remember that primary rays are the initial rays
    shot from the camera. Any reflected rays (and
    others, like refracted rays, etc.), are called
    secondary rays
  • Reflected rays, like shadow rays should be moved
    slightly along the surface normal to prevent the
    ray from re-intersecting the same surface

14
Computing Reflection Direction
n
r
d
15
Reflections
  • If the reflection ray hits a normal material, we
    just compute the illumination and use that for
    the final color
  • If the reflection ray hits another mirror, we
    just recursively generate a new reflection ray
    and trace that
  • In this way, we can render complex mirrored
    surfaces that include reflections, reflections of
    reflections, reflections of reflections of
    reflections
  • To prevent the system from getting caught in an
    infinite loop, it is common to put an upper limit
    on the depth of the recursion. 10 or lower
    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

16
Reflections
17
Reflections
  • Surfaces in the real world dont act as perfect
    mirrors
  • Real mirrors will absorb a small amount of light
    and only reflect maybe 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 polished plastic, which have a
    diffuse component as well as a shiny specular
    component
  • 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 (the diffuse and specular
    components should be weighted so as not to
    violate conservation of energy)

18
Transmission Rays
  • Ray tracing can also be used to accurately render
    the light bending in transparent surfaces due to
    refraction
  • Often, this is called transmission instead of
    refraction. Transmission is a more general term
    that also includes translucency, but I think the
    real reason this word is preferred is because
    reflection and refraction look too similar
  • When a ray hits a transparent surface (like
    glass, or water), 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 (n1sin?1n2sin?2), where n1 and n2
    are the index of refraction for the two materials

19
Computing Transmission (Refraction) Direction
20
Total Internal Reflection
  • When light traveling in a material with a high
    index of refraction hits a material with a low
    index of refraction at a steep angle, we get a
    total internal reflection
  • When this happens, no refraction ray is generated
  • This effect can be visible when one is scuba
    diving and looks up at the water surface. One can
    only see rays refracting to the outside world in
    a circular area on the water surface above
  • Total internal reflection can be detected when
    the magnitude of the z vector is greater than 1,
    causing the square root operation to become
    undefined

n
d
r
?1
n1
z
n2
21
Spawning Multiple Rays
  • When light hits a transparent surface, we not
    only see refraction, but we get a reflection off
    of the surface as well
  • Therefore, we will actually generate two new rays
    and trace both of them into the scene and combine
    the results
  • The results of an individual traced ray is a
    color, which is the color of the light that the
    ray sees
  • This color is used as the pixel color for primary
    rays, but for secondary rays, the color is
    combined somehow into the final pixel color
  • In a refraction situation, for example, we spawn
    off two new rays and combine them according to
    the Fresnel equations, provided in the last
    lecture
  • The Fresnel equations describe how the
    transmitted (refracted) ray will dominate when
    the incoming ray is normal to the surface, but
    the reflection will dominate when the incoming
    ray is edge-on

22
Refraction
Reflection ray
Camera
Normal
Primary ray
Transmission ray
23
Fresnel Equations
  • The Fresnel equations can be used to determine
    the proportion of the light reflected (fr) and
    transmitted (ft) when a ray hits an interface
    between two dielectrics (like air and water)
  • They describe separate formulas for the parallel
    and perpendicularly polarized light, but these
    are usually averaged into a single set of values

24
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

25
Recursive Ray Tracing
26
Ray Intersection
27
Ray-Scene Intersection
  • One of the key components of a ray tracer is the
    system that determines 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 more complex surfaces
    such as spheres, Bezier patches, displacement
    mapped surfaces, fractals, and more
  • Sometimes, complex primitives are simply
    tessellated into triangles in 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 a ray comes
    nearby

28
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 as triangles
    and spheres, or even collections of primitives or
    other objects
  • In order to be render-able, an object must
    provide some sort of ray intersection routine
  • 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 then write
    custom ray intersection routines for them
  • The ray intersect routine takes a ray as input,
    and returns true if the object is hit and false
    if it is missed
  • If the object is hit, the intersection data is
    filled in into the isect class

29
Ray-Sphere Intersection
  • Lets see how to test if a ray intersects a
    sphere
  • The ray has an origin at point p and a unit
    length direction u, and the sphere has a center c
    and a radius r

c
u
p
r
30
Ray-Sphere Intersection
  • The ray itself is the set of points pau, where
    a0
  • We start by finding the point q which 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)u0, or (pau-c)u0
  • We can solve the value of a that satisfies that
    relationship a-(p-c)u, so qp-((p-c)u)u

31
Ray-Sphere Intersection
  • Once we have q, we test if it is inside the
    actual sphere or not, by checking if q-cr
  • If q is outside the sphere, then the ray must not
    miss
  • If q is inside the sphere, then we find the
    actual point on the sphere surface that the ray
    intersects
  • We say that the ray will hit the sphere at two
    points q1 and q2
  • q1p(a-a)u) q2p(aa)u) where
    asqrt(r2-q-c2)
  • If a-a0, then the ray hits the sphere at q1, but
    if it is less than 0, then the actual
    intersection point lies behind the origin of the
    ray
  • In that case, we check if aa0 to test if q2 is
    a legitimate intersection

q2
q1
q
u
p
c
32
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 roots
    until 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
    ray is far away from the object being tested

33
Ray-Plane Intersection
  • A plane is defined by a normal vector n and a
    distance d, which is the distance of the plane to
    the origin
  • We test our ray with the plane by finding the
    point q which is where the ray line intersects
    the plane
  • For q to lie on the plane it must satisfy
  • dqnpnaun
  • We solve for a
  • a(d-pn)/(un)
  • However, we must first check that the denominator
    is not 0, which would indicate that the ray is
    parallel to the plane
  • If a0 then the ray intersects the plane,
    otherwise, the plane lies behind the ray, in the
    wrong direction

34
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 of the triangle
  • Once we know that the ray hits the plane at point
    q, we must verify that q lies inside the 3 edges
    of the triangle

35
Ray-Triangle
  • Does segment ab intersect triangle v0v1v2 ?
  • Does segment ab intersect triangle v0v1v2 ?


36
Barycentric Coordinates
  • Reduce to 2D remove smallest dimension
  • Compute barycentric coordinates
  • q' q-v0
  • e1v1-v0
  • e2v2-v0
  • a(q'e2)/(e1e2)
  • ß(q'e1)/(e1e2)
  • Reject if alt0, ßlt0 or aß gt1

v2
ß
q
v0
a
v1
37
Acceleration Structures
  • Complex scenes can contain millions of
    primitives, and ray tracers need 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 even counting all
    of the time doing the intersection testing
  • Therefore, it is absolutely essential to employ
    some sort of acceleration structure to speed up
    the ray intersection testing
  • An acceleration structure is some sort of data
    structure that groups objects 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 of
    hierarchical subdivision of the space around the
    group of objects

38
Bounding Volume Hierarchies
  • The basic concept of a bounding volume hierarchy
    is a complex object in a hierarchy of 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 enclose the entire
    scene in one big sphere
  • Within that sphere are several other spheres,
    each containing more spheres, until we finally
    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 the triangles/primitives
    within
  • In general, a bounding volume hierarchy can
    reduce the ray intersection time from O(n) to
    O(log n), where n is the number of primitives in
    the scene
  • This reduction from linear to logarithmic
    performance makes a huge difference and makes it
    possible to construct scenes with millions of
    primitives

39
Sphere Hierarchies
  • The sphere hierarchy makes for a good example of
    the concept, but in practice, sphere hierarchies
    are not often used for ray tracing
  • One reason is that it is not clear how to
    automatically group an arbitrary set of 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 lot of redundant
    intersection tests

40
Octrees
  • The octree starts by placing a cube around the
    entire scene
  • If the cube contains more than some specified
    number of primitives (say, 10), then it is split
    equally into 8 cubes, which are then recursively
    tested and possibly resplit
  • The octree is a more regular structure than the
    sphere tree and provides a clear rule for
    subdivision and no overlap between cells
  • This makes it a better choice usually, but still
    not ideal

41
Octrees
42
KD Trees
  • The KD tree starts by placing a box (not
    necessarily a cube) around the 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 irregular geometry 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 tree itself,
    rather than testing intersections with primitives

43
KD Trees
44
BSP Trees
  • The BSP tree (binary space partitioning) is much
    like the KD tree in that it continually splits
    space into two (not necessarily equal) halves
  • Unlike the KD tree which is limited to xyz axis
    splitting, the BSP tree allows the splitting
    plane to be placed anywhere in the volume and
    aligned in any direction
  • This makes it a much more difficult problem to
    choose the 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

45
BSP Trees
46
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 size and location (which is common)
  • As a result, they are not really a practical
    general purpose acceleration structure for ray
    tracing

47
Uniform Grids
48
Hierarchical Grids
  • One can also make a hierarchical grid
  • Start with a uniform grid, but subdivide any cell
    that contains too many primitives into a smaller
    grid
  • An octree is an example of a hierarchical grid
    limited to 2x2x2 subdivision
  • A more general hierarchical grid could support
    subdivision into any number of cells
  • Hierarchical grids tend to perform very well in
    ray tracing, especially for highly detailed
    geometry of relatively uniform size (such as the
    triangles in a tessellated surface)

49
Acceleration Structures
  • All of the acceleration structures we looked at
    store some geometry and provide a function for
    intersecting a ray
  • In other words, they are really just a more
    complex type of primitive themselves
  • 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
    acceleration structure that contains a bunch of
    triangles, and then place that acceleration
    structure within a larger acceleration structure,
    etc.
  • This provides a nice, consistent way to represent
    scenes, similar to the scene graph concept we
    covered in the lecture on realtime scene
    management
  • class KDTreepublic Object
  • public
  • bool IntersectRay(Ray r,Intersection isect)

50
Distribution Ray Tracing
51
Distribution Ray Tracing
  • In 1984, an important modification to the basic
    ray tracing algorithm was proposed, known as
    distributed ray tracing
  • The concept basically involved shooting several
    distributed rays to achieve what had previously
    been done with a single ray
  • The goal is not to simply make the rendering
    slower, but to achieve a 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 in modern days, the
    distributed ray tracing technique is now called
    distribution ray tracing, and the term
    distributed is reserved for parallel ray
    tracing, which is also an important subject

52
Soft Shadows
  • One nice visual effect we can achieve with
    distribution ray tracing is soft shadows
  • Instead of treating a light source as a point and
    shooting a single ray to test for shadows, we can
    treat the light source as a sphere and shoot
    several rays to test for partial 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 umbra and the partially
    shadowed region is called the penumbra

53
Area Lights
  • The soft shadow technique enables us to define
    lights in a much more 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 the light source, distributed across
    the surface and weighted according to the surface
    area of the sample and the direction of the
    average normal
  • Larger light sources create softer, diffuse
    shadows, while smaller light sources cause sharp,
    harsh shadows
  • Larger light sources also require more rays to
    adequately sample the shadows, making area lights
    a lot more expensive than point lights.
    Inadequate sampling of the light source can cause
    noise patterns to appear in the penumbra region,
    known as shadow aliasing

54
Blurry Reflections
  • We can render blurry or glossy reflections 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
    to achieve rendering of unpolished glass

55
Antialiasing
  • For good quality antialiasing, we will can shoot
    several primary rays for each pixel
  • We covered several different antialiasing
    patterns in an earlier lecture, but recall that
    the supersampled, area-weighted, jittered
    Gaussian distribution was chosen as one of the
    best overall sampling schemes
  • As we have complete control over the direction of
    every ray, we can use any sampling pattern we
    choose, as well as any number of rays we choose

56
Depth of Field
  • The blurring caused by a camera lens being out of
    focus is due to the lens limited depth of field
  • In computer graphics, the term depth of field
    usually refers to the general process 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 distributing the 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. A pinhole camera has an
    aperture size of 0, and therefore, will not have
    any blurring due to depth of field

57
Motion Blur
  • As if soft shadows, antialiasing, depth of field,
    and blurry reflections and refractions werent
    enough we can also use distribution ray tracing
    to include the effect from motion blur
  • We do this by distributing rays in time
  • Therefore, we need to know the starting and
    ending transformations for all of the objects in
    the scene (and the lights camera)
  • In fact, we can even consider objects moving
    along more complex paths, so we can motion blur
    bounces as well

58
Distribution Ray Tracing
  • Ray tracing had a big impact on computer graphics
    in 1980 with the first images of accurate
    reflections and refractions from curved surfaces
  • Distribution ray tracing had an even bigger
    impact in 1984, as it re-affirmed the power of
    the basic ray tracing technique and added a whole
    bunch of sophisticated effects, all within a
    consistent framework
  • Previously, techniques such as depth of field,
    motion blur, soft shadows, etc., had only been
    achieved individually and by using a variety of
    complex, hacky algorithms

59
Distribution Ray Tracing
  • If ray tracing is slow, then distribution ray
    tracing must be considerably slower
  • Now, instead of one or two splits per level in
    our recursion, we are have to shoot 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 only need to shoot a
    small number of primary rays per pixel
  • For example, we can shoot 16 rays in a 4x4
    antialiasing pattern, where each 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, as the results will be
    blended with 15 other samples
  • Still, we end up with lots and lots of rays and
    potential for exponential problems in scenes with
    a lot of soft or blurry features
  • This problem is at least partially addressed with
    path tracing which is one of the techniques for
    global illumination that we will see in the next
    lecture
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