Ray Casting

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Ray Casting
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The infamous gamma curve

• A gamma curve x-gtxg is used for many reasons
• CRT response
• Color quantization
• Perceptual effect
• Sometimes with ? gt1, sometimes ? lt1
• These issues are often oversimplified/confused,
  including in prominent textbooks
• i.e. they are explained wrong

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Cathode Ray Tube gamma

• The relationship between voltage and light
  intensity is non linear
• Can be approximated by an exponent 2.5
• Must be inverted to get linear response

From Pontons FAQhttp//www.poynton.com/
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Color quantization gamma

• The human visual system is more sensitive to
  ratios is a grey twice as bright as another one?
• If we use linear encoding, we have tons of
  information between 128 and 255, but very little
  between 1 and 2!
• This is why a non-linear gamma remapping of about
  2.0 is applied before encoding
• True also of analog signal to optimize
  signal-noise ratio
• It is a nice coincidence that this is exactly the
  inverse of the CRT gamma

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Stevens effect

• Perceived contrast increases with luminance

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Perceptual effect

• We perceive colors in darker environment less
  vivid
• Must be compensated by boosting colors

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At the end of the day

• At the camera or encoding level, apply a gamma of
  around 1/2.2
• The CRT applies a gamma of 2.5
• The residual exponent 2.2/2.5 boosts the colors
  to compensate for the dark environment
• Seehttp//www.poynton.com/GammaFAQ.html
  http//www.poynton.com/notes/color/GammaFQA.html
  http//www.poynton.com/PDFs/Rehabilitation_of_gamm
  a.pdf

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Gamma is messy

• Because its poorly understood
• Because its poorly standardized
• Half of the images on the net are linear, half
  are gamma-compressed
• Because it might make your image processing
  non-linear
• A weighted average of pixel values is not a
  linear convolution! Bad for antialiasing
• But it is often desirable for other image
  processing, because then it corresponds more to
  human perception of brightness

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Questions?
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Recap

• Modeling
• splines, hierarchy, transformation
• Animation
• quaternions, skinning, ODEs, mass spring
• Color the atom of images
• Now we need to see how to generate an image given
  a scene description

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Overview of Today

• Ray Casting Basics
• Camera and Ray Generation
• Ray-Plane Intersection
• Ray-Sphere Intersection

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Ray Casting

• For every pixel
• Construct a ray from the eye
• For every object in the scene
• Find intersection with the ray
• Keep if closest

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Shading

• For every pixel
• Construct a ray from the eye
• For every object in the scene
• Find intersection with the ray
• Keep if closest
• Shade depending on light and normal vector

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A Note on Shading

• Surface/Scene Characteristics
• surface normal
• direction to light
• viewpoint
• Material Properties
• Diffuse (matte)
• Specular (shiny)
• Much more soon!

N
L
V
Diffuse sphere
Specular spheres
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Ray Tracing

• Secondary rays (shadows, reflection, refraction)
• In a couple of weeks

reflection
refraction
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Ray Tracing
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Ray Casting

• For every pixel Construct a ray from the eye
  For every object in the scene
• Find intersection with the ray
• Keep if closest
• Shade depending on light and normal vector

Finding the intersection and normal is the
central part of ray casting
N
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Ray Representation?

• Two vectors
• Origin
• Direction (normalized is better)
• Parametric line
• P(t) origin t direction

P(t)
direction
origin
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Durers Ray Casting Machine

• Albrecht Durer, 16th century

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Durers Ray Casting Machine

• Albrecht Durer, 16th century

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Durers Ray Casting Machine

• Albrecht Durer, 16th century

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Questions?
Henrik Wann Jensen, Stephen Duck, Fredo
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Overview of Today

• Ray Casting Basics
• Camera and Ray Generation
• Ray-Plane Intersection
• Ray-Sphere Intersection

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Cameras

• For every pixel
• Construct a ray from the eye
• For every object in the scene
• Find intersection with ray
• Keep if closest

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Pinhole Camera

• Box with a tiny hole
• Inverted image
• Similar triangles

• Perfect image if hole infinitely small
• Pure geometric optics
• No depth of field issue

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Oldest Illustration

• From. R. Gemma Frisius, 1545

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Camera Obscura
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Camera Obscura Today
Abelardo Morell www.abelardomorell.net
See http//www.imdb.com/title/tt0118556/goofs
Addicted to Love
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Camera Obscura in Art
Johannes Vermeer 1665
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Simplified Pinhole Camera

• Eye-image pyramid (frustum)
• Note that the distance/size of image are arbitrary

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Camera Description?

• Eye point e (center)
• Orthobasis u, v, w (horizontal, up, -direction)
• Field of view angle
• Image rectangle height, width

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Perspective vs. Orthographic
perspective
orthographic

• Parallel projection
• No foreshortening
• No vanishing point

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Orthographic Camera

• Ray Generation?
• Origin center (x-0.5)sizehorizontal
  (y-0.5)sizeup
• Direction is constant

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Other Weird Cameras

• E.g. fish eye, omnimax, panorama

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Questions?
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Overview of Today

• Ray Casting Basics
• Camera and Ray Generation
• Ray-Plane Intersection
• Ray-Sphere Intersection

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Ray Casting

• For every pixel Construct a ray from the eye
  For every object in the scene
• Find intersection with the ray
• Keep if closest
• First we will study ray-plane intersection

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Recall Ray Representation

• Parametric line
• P(t) Ro t Rd
• Explicit representation

P(t)
direction
origin
Rd
Ro
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3D Plane Representation?

• Plane defined by
• Po (x,y,z)
• n (A,B,C)
• Implicit plane equation
• H(P) AxByCzD 0 nP D 0
• Point-Plane distance?
• If n is normalized, distance to plane, d H(P)
• d is the signed distance!

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Explicit vs. Implicit?

• Ray equation is explicit P(t) Ro t Rd
• Parametric
• Generates points
• Hard to verify that a point is on the ray
• Plane equation is implicit H(P) nP D 0
• Solution of an equation
• Does not generate points
• Verifies that a point is on the plane
• Exercise Explicit plane and implicit ray

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Ray-Plane Intersection

• Intersection means both are satisfied
• So, insert explicit equation of ray into
  implicit equation of plane solve for t
• P(t) Ro t Rd
• H(P) nP D 0
• n(Ro t Rd) D 0
• t -(D nRo) / nRd

P(t)
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Additional Housekeeping

• Verify that intersection is closer than previous
• Verify that it is not out of range (behind eye)

P(t) lt tcurrent
P(t) gt tmin
P(t)
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Normal

• For shading
• diffuse dot product between light and normal
• Normal is constant

normal
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A moment of mathematical beauty

• Duality points and planes are dual when you use
  homogeneous coordinates
• Point (x, y, z, 1)
• Plane (A, B, C, D)
• Plane equation ? dot product
• You can map planes to points and points to planes
  in a dual space.
• Lots of cool equivalences
• e.g. intersection of 3 planes define a point
• ? 3 points define a plane!

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Questions?

• Image by Henrik Wann Jensen

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Overview of Today

• Ray Casting Basics
• Camera and Ray Generation
• Ray-Plane Intersection
• Ray-Sphere Intersection

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Sphere Representation?

• Implicit sphere equation
• Assume centered at origin (easy to translate)
• H(P) PP - r2 0

Rd
Ro
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Ray-Sphere Intersection

• Insert explicit equation of ray into implicit
  equation of sphere solve for t
• P(t) Ro tRd H(P) PP - r2
  0
• (Ro tRd) (Ro tRd) - r2 0
• RdRdt2 2RdRot RoRo - r2
  0

Rd
Ro
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Ray-Sphere Intersection

• Quadratic at2 bt c 0
• a 1 (remember, Rd 1)
• b 2RdRo
• c RoRo r2
• with discriminant
• and solutions

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Ray-Sphere Intersection

• 3 cases, depending on the sign of b2 4ac
• What do these cases correspond to?
• Which root (t or t-) should you choose?
• Closest positive! (usually t-)

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Ray-Sphere Intersection

• It's so easy that all ray-tracing images have
  spheres!

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Geometric Ray-Sphere Intersection

• Shortcut / easy reject
• What geometric information is important?
• Ray origin inside/outside sphere?
• Closest point to ray from sphere origin?
• Ray direction pointing away from sphere?

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Geometric Ray-Sphere Intersection

• Is ray origin inside/outside/on sphere?
• (RoRo lt r2 / RoRo gt r2 / RoRo r2)
• If origin on sphere, be careful about
  degeneracies

O
r
Ro
Rd
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Geometric Ray-Sphere Intersection

• Is ray origin inside/outside/on sphere?
• Find closest point to sphere center, tP - Ro
  Rd
• If origin outside tP lt 0 ? no hit

O
r
Ro
Rd
tP
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Geometric Ray-Sphere Intersection

• Is ray origin inside/outside/on sphere?
• Find closest point to sphere center, tP - Ro
  Rd.
• Find squared distance, d2 RoRo - tP2
• If d2 gt r2 ? no hit

O
r
d
Ro
Rd
tP
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Geometric Ray-Sphere Intersection

• Is ray origin inside/outside/on sphere?
• Find closest point to sphere center, tP - Ro
  Rd.
• Find squared distance d2 RoRo - tP2
• Find distance (t) from closest point (tP) to
  correct intersection t2 r2 - d2
• If origin outside sphere ? t tP - t
• If origin inside sphere ? t tP t

O
r
d
Ro
Rd
tP
t
t
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Geometric vs. Algebraic

• Algebraic is simple generic
• Geometric is more efficient
• Timely tests
• In particular for rays outside and pointing away

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Sphere Normal

• Simply Q/Q
• Q P(t), intersection point
• (for spheres centered at origin)

O
Ro
Rd
Q
normal
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Questions?
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Precision

• What happens when
• Origin is on an object?
• Grazing rays?
• Problem with floating-point approximation

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The evil e

• In ray tracing, do NOT report intersection for
  rays starting at the surface (no false positive)
• Because secondary rays
• Requires epsilons

reflection
shadow
refraction
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The evil e a hint of nightmare

• Edges in triangle meshes
• Must report intersection (otherwise not
  watertight)
• No false negative

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Assignment 4 5 Ray Casting/Tracing

• Write a basic ray caster
• Orthographic camera
• Sphere Intersection
• Main loop rendering
• 2 Display modes color and distance
• We provide
• Ray origin, direction
• Hit t, Material, (normal)
• Scene Parsing

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Object-Oriented Design

• We want to be able to add primitives easily
• Inheritance and virtual methods
• Even the scene is derived from Object3D!

Object3D bool intersect(Ray, Hit, tmin)
Plane bool intersect(Ray, Hit, tmin)
Sphere bool intersect(Ray, Hit, tmin)
Triangle bool intersect(Ray, Hit, tmin)
Group bool intersect(Ray, Hit, tmin)
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Books

• Peter Shirley Fundamentals of Computer
  GraphicsAK Peters
• Ray Tracing

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Questions?
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Next Time More Ray Casting

• Other primitives
• Boxes
• Polygons
• Triangles
• IFS?