Ray%20Tracing

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Ray Tracing

• Rendering Methods
• Rays
• Rays for screen images
• Shadows
• Recursion

2
Visible Surface Raytracing
ObjectCoordinates
ModelviewMatrix
Eye coordinates
IntersectionData Structure
Raytracing then utilizes this data structure to
build the image.
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Raytracing

• Invented by Arthur Appel in 1968 at the IBM T.J.
  Watson Research Center.
• Not considered a major accomplishment since it
  was deemed impractical given its enormous
  computational requirements.
• Languished in obscurity until the late 1970s.
  SIGGRAPH77 saw many significant developments,
  including the first looks at modeling reflection
  and shadows.

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More History

• A true recursive ray tracer was presented for the
  first time by Turner Whitted in the classic
  article An improved illumination model for
  shaded display in the June, 1980 Communications
  of the ACM.
• Note that the article is more concerned with the
  model for illumination than the ray tracing
  algorithm.

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That Pinhole Camera Model
After eyespace transformation Projection Plane at
z-d
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Raytracing

• If we shoot a line from the center of projection
  through the center of a pixel and off into space
• What does it hit first?
• If it hits an object, compute the color for that
  point on the object
• Thats the pixel color

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Rays

• A Ray is a vector and a point
• Point The starting point of the ray
• Vector The ray direction
• This describes an infinite line starting at the
  point and going in the ray direction
• Ray t values
• A distance along the ray

t1.5
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t values

• Any point along the ray with origin (xs,ys,zs)
  and direction(xd,yd,zd) can be described using t
• x xs t xd
• y ys t yd
• z zs t zd
• Input ray
• Output color

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Where will the viewplane beand how big is it?
We describe projection with Field of view
(angle) Aspect Ratio (x/y)
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Viewplane z location
y
-z
Z location is unimportant, field of view is what
matters. Well use z-1 Could be defined using
fov or as a frustum
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Viewplane size
y
y (in direction)
-z
1
(x,y)
y tan(fov/2) height 2y x aspecty width 2x
(-x,-y)
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What if I have a frustum?

• glFrustum(left, right, bottom, top, znear, zfar)

(r,t,n)
(l,b,n)
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What if I have a frustum?

• glFrustum(left, right, bottom, top, znear, zfar)

y bottom/znear height (top
bottom)/znear xleft left/znear width
(right-left)/znear
(r,t,n)
(l,b,n)
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The View Plane as a Pixel Grid
(x,y) (11,7)
8 tall
(-x,-y) (0,0)
12 wide
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What we want

• We want to shoot a ray
• Starting point (0,0,0) (center of projection)
• Direction through the center of a pixel
• What we need is the coordinates of the center of
  a pixel

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Pixel Centers
(x,y) (11,7)
(4,5)
8 tall
(4,5)
(-x,-y) (0,0)
12 wide
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Visible Surface Raytracing
// width Screen width, height Screen
height// fov Field of view in degrees, aspect
Aspect ratiodouble ey tan(fov / 2.
DEGTORAD) double ehit2 eydouble ex ey
aspect double ewid 2 exfor(int r0
rltheight r) for(int c0 cltwidth
c) ray.start.Set(0,0,0)
ray.dir.Set(-ex ewid (c 0.5) / width,
-ey ehit (r 0.5) /
height, -1.)
ray.dir.Normalize() // Compute color in
ray direction // Save color as image pixel

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How do we figure out what we will hit?

• Ray intersection tests
• Reduces to solving for t
• x xs t xd
• y ys t yd
• z zs t zd
• Easiest is the sphere
• (x-x0)2(y-y0)2(z-z0)2r2
• Substitute ray equation in and solve for t
• Note Text uses two points on the line

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Ray intersection with a sphere
Questions Why two t values? How do we tell if
there is an intersection?
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Intersections with a plane

• Plane equation
• AxByCzD0
• (A,B,C) is the plane normal
• We can compute the normal and, given a single
  point, compute D

Questions Can t be negative? Zero? Why is this
more difficult than a sphere?
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Plane Intersection to Polygon Intersection

• Given a plane intersection, the question is
• Is this point inside the polygon?

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Polygon Interior Tests

• In a 2D projection
• Find all edges that overlap in one dimension
• Determine intersection of a line parallel with an
  axis and the overlapping edges
• How many intersections are to the left of the
  point?

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But, what projection?

• Project the polygon for the maximum area
• What is the largest component of the normal?
• If x Use a Y/Z projection
• If y Use an X/Z projection
• If z Use an X/Y projection
• Why?

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Another Way

• Suppose you know the intersection and know a
  normal for each edge that points towards the
  inside of the polygon

How do we compute these normals? What is the
characteristic of points inside the polygon?
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Computing the Normals

• Let v1, v2 be two vertices (in counter clockwise
  order). Then the edge normal is N x (v2-v1)
• Orthogonal to the edge and the polygon surface
  normal (N).

Surface Normal
v2
v1
Edge Normal
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Interior Test

• For an edge v1,v2 with edge normal e1, the
  intersection p is on the inside side if (p-v1)e1
  is not negative.

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Why is ray tracing sometimes considered
impractical?

• Suppose you do a ray for every pixel and test
  against very polygon
• O(WHP)
• Then if we allow recursion

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Faster Intersection Tests

• Grouping objects and using bounding volumes
• Bounding boxes are easier to test for
  intersections.
• Your chair object could have a box around each
  polygon and a box around the entire object.
• We can create hierarchical levels of bounding
  boxes pretty easily (cluster tree)
• Spheres are a common bounding volume

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Bounding box intersection test

• Assumes Upright box
• Sometimes called generalized box
• Consider box to have 6 planes
• Planes are orthogonal to some axis
• Box with range (0.5, 0.7, -1) to (1.3, 2.2, -4)
• Ray R((0,0,0),(1, 1, -1))
• What is t for intersections with box planes
  orthogonal to the X axis?

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Bounding box tests

• Conditions
• Box with range (0.5, 0.7, -1) to (1.3, 2.2, -4)
• Ray R((0,0,0),(1, 1, -1))
• Remember
• x xs t xd
• y ys t yd
• z zs t zd
• So, tx1 (0.5 0) / 1 0.5
• tx2 (1.3 0) / 1 1.3
• Whats the condition that indicates we
  intersected the box?

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Bounding box tests
0.8
0.9
Max tnear 4.3
Max tnear 3.5
Min tfar 4.0
4.0
3.5
y2
4.3
4.6
Min tfar 4.6
7.5
6.3
y1
x1
x2
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Bounding box tests
0.8
Max tnear gt min tfarNo intersection
0.9
Max tnear 4.3
Max tnear 3.5
Min tfar 4.0
4.0
3.5
y2
4.3
Max tnear lt min tfarIntersection
4.6
Min tfar 4.6
7.5
6.3
y1
x1
x2
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Algorithm

• Compute tnear and tfar for the three planes
• Let tnear max(tnear1, tnear2, tnear3)
• Let tfar min(tfar1, tfar2, tfar3)
• If tnear tfar we have an intersection
• Note that we can short-circuit this test after
  only two planes in some cases

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Ray Walking Algorithms

• We can also subdivide the space into boxes. Then
  we trace the ray through the series of boxes.

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Ray Walking Algorithms

• Problem Not the same as line drawing

extras for ray walking
line draw
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Cleary and Wyvill ray walking
Dx
Dy
dx
dy
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Spatial subdivisions

• Uniform divisions
• All boxes are equal sized
• Advantages? Disadvantages?
• Non-uniform divisions
• How?
• Why?

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Octtree-based spatial subdivision
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Speed-up

• Item buffer
• A z-buffer which holds pointers to objects rather
  than colors
• How can this speed up ray tracing?
• Text calls this First-hit speedup

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How do we compute the color?

• Intersection gives a point on a polygon
• Apply the color model
• How do we get
• View direction?
• Light direction?
• Normal?

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How do we do?

• Shadows?
• Reflections?

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Terms to know

• Shadow feeler
• Casting a ray in the direction of the light
• If intersection is less than distance to the
  light, light is shadowed
• Self shadowing
• Roundoff errors can make us intersect ourselves

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Recursion

• Reflections
• Simple compute the color in the reflection
  direction
• What does this do to the running time?
• Was O(WHP)

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Adaptive Depth Control

• How deep to you recurse?
• A room with mirrors?
• A room with only specular reflection?
• Keep track of how much a ray will contribute to
  the final color
• Falls below some threshold, dont recurse anymore

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How do we do?

• Shadows?
• Penumbra?
• Depth of field?
• Motion blur?
• Antialiasing?

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Stochastic Ray Tracing

• Problem with ray tracing Point sampling
• We can introduce randomness all over the place
• Jitter for antialiasing
• Light area for penumbra
• Depth of field and motion blur

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Minimum distance Poisson distribution

• Better matches distribution in the eye
• Algorithm
• Generate a random point
• If less than specified distance to nearest point,
  discard, otherwise keep

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Distribution Comparisons
Poisson
Uniform
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Adaptive sampling

• Determine if number of samples is enough
• If not, increase the number
• Works for uniform and stochastic sampling

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Ray tracing from the light sources

• In regular ray tracing we dont see
• Lighting reflected through mirrors
• Surfaces lit by reflection from other surfaces
• Large set of rays from each light
• Computes light that hits surfaces
• General solution for diffuse reflection?
• Still requires tracing from the eye as pass 2