Computer Graphics Shading Models

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Computer Graphics- Shading Models -

• Marcus Magnor

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Overview

• So far
• Ray tracing 101
• Today
• Recursive Ray Tracing
• Physical Quantities in Rendering
• Shading
• Empirical BRDF models

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Recap Fundamental Ray Tracing Steps

• Primary ray generation
• Rays from viewpoint along viewing directions into
  3D scene
• (At least) one ray per picture element (pixel)
• Ray tracing
• Intersection of primary ray with scene geometry
• Intersected object and intersection coordinates
  on object
• Shading
• From intersection, determine radiance along
  primary ray
• Determines pixel color
• Needed
• Local material color and reflection properties
• Object texture
• Local illumination of intersection point
• Can be impossible to determine correctly

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Primary Ray Generation

• Ray from viewpoint through pixel into scene
  pinhole camera model
• Compute intersection of ray with scene geometry
• Pixel color intersected object point color
  shading
• Hidden surface detection consider only closest
  hit surface

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Recursive Ray Tracing

• Searching recursively for paths to light sources
• Interaction of light material at intersection
  points
• Recursively trace new rays in reflection,
  refraction and light direction

shadow rays
reflected ray
refracted ray
pixel
image plane
primary ray
viewpoint
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Ray Tracing Algorithm

• Trace(ray)
• Search the next intersection point ? (hit,
  material)
• Return Shade(ray, hit, material)
• Shade(ray, hit, material)
• For each light source
• if ShadowTrace(ray to light source, distance)
• Calculate reflected radiance (i.e. Phong)
• Adding to the reflected radiance
• If mirroring material
• Calculate radiance in reflected direction
  Trace(R(ray, hit))
• Adding mirroring part to the reflected radiance
• Same for transmission
• Return reflected radiance
• ShadowTrace(ray, dist)
• Return false, if intersection point with distance
  lt dist has been found

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Recursive Ray Tracing

• Pixel color depends on material property at
  ray-surface intersection point
• Shading
• Needs illumination at intersection point
• of new rays depends on material properties
• Mirror 1 ray
• Semi-transparent 2 rays
• Diffuse ??? Rays

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Shading

• Intersection point determines primary rays
  radiancegt determines pixel color
• Diffuse object color at intersection point
• No variation with viewing angle diffuse,
  Lambertian
• Must still be illuminated
• Scales linearly with received radiosity
• No illumination in shadow black
• Non-Lambertian Reflectance
• Surface point appearance depends on illumination
  direction and viewing direction
• Local Bi-directional Reflectance Distribution
  Function (BRDF)
• Simple cases
• Mirror, glass secondary rays
• Point light source shadow ray
• Extended light sources, indirect illumination
  arbitrarily difficult

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Angle and Solid Angle

• , the angle subtended by a curve in the plane, is
  the length of the corresponding arc on the unit
  circle.
• , the solid angle subtended by an object, is the
  surface area of its projection onto the unit
  sphere,
• Solid angle unit steradians sr

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Solid Angle in Spherical Coordinates
Infinitesimally small solid angle
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Solid Angle for a Small Area

• The solid angle subtended by a small surface
  patch S with area ?A is obtained after dividing
  the area of its projection
• by the square of the distance to the origin

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Radiometry

• Radiometry is the science of measuring radiant
  energy transfers. Radiometric quantities have
  physical meaning and can be directly measured
  using proper equipment such as spectral
  photometers.
• Radiometric Quantities
• energy watt second n h?
• radiant power (total flux) watt ?
• radiance watt/(m2 sr) L
• irradiance watt/m2 E
• radiosity watt/m2 B

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Radiometric Quantities Radiance

• Radiance is used to describe radiant energy
  transfer.
• Radiance L is defined as the power (flux)
  traveling at some point x in a specified
  direction, per unit area perpendicular to the
  direction of travel, per unit solid angle.
• Thus, the differential power d2F radiated through
  the differential solid angle d?, from the
  projected differential area dA cos? is

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Spectral Properties

• Since light is composed of electromagnetic waves
  of different frequencies and wavelengths, most of
  the energy transfer quantities are continuous
  functions of wavelength. Thus, total radiance is
  expressed as the integral of spectral radiance
  over the spectrum

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Radiometric Quantities Irradiance

• Irradiance E is the total power per unit area
  (flux density) incident onto a surface with a
  fixed orientation. To obtain the total flux
  incident to dA, the incoming radiance Li is
  integrated over the upper hemisphere O above the
  surface

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Radiometric Quantities Radiosity

• Radiosity B is defined as the total power per
  unit area (flux density) leaving a surface. To
  obtain the total flux radiated from dA, the
  outgoing radiance Lo is integrated over the upper
  hemisphere O above the surface.

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Photometry

• The human eye is sensitive to a limited range of
  radiation wavelengths (from 380nm to 770nm). The
  response of our visual system is not the same for
  all wavelengths, and can be characterized by the
  luminous efficiency function V( ) which
  represents the average human spectral response.
• A set of photometric quantities can be derived
  from radiometric quantities by integrating them
  against the luminous efficiency function V( ).

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Radiometry vs. Photometry
Physics-based quantities
Perception-based quantities
Weighted with luminous efficiency function
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Surface Radiance

• Visible surface radiance
• Surface position
• Outgoing direction
• Incoming illumination direction
• Self-emission
• Reflected light
• Incoming radiance from all directions
• Direction-dependent reflectance

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Reflection Equation - Reflectance

• Reflection equation
• Reflectance
• Ratio of reflected to incident power (radiosity /
  irradiance)
• Directional-hemispherical reflectance fraction
  of incident irradiance coming from a given
  direction reflected in all possible directions

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Bidirectional Reflectance Distribution Function

• BRDF describes surface reflection for light
  incident from direction (??,f?) observed from
  direction (?i,fi)
• Bidirectional
• depends on two directions (4-D function)
• Distribution function
• Unit 1/sr

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BRDF Properties

• Helmholtz reciprocity principle
• BRDF remains unchanged if incident and reflected
  directions are interchanged
• Smooth surface isotropic BRDF
• reflectivity independent of rotation around
  surface normal
• BRDF has only 3 instead of 4 degrees of freedom

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BRDF Properties

• Characteristics
• BRDF units sr--1
• not intuitive
• Range of values
• from 0 (absorption) to
• ? (reflection, d -function)
• Energy conservation law
• No self-emission
• Possible absorption
• Reflection only at the point of entry (xi xo)
• No subsurface scattering

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Reflection Geometry

• Direction vectors (normalize)
• N surface normal
• I vector to the light source
• V viewpoint direction vector
• R(I) reflection vector
• R(I) I - 2(IN)N
• Tangential surface local plane

Top view
R(V)
N
R(I)
N
R(I)
I
I
R(V)
V
V
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Ideal Specular Reflection

• Angle of reflectance equal to angle of incidence
• Reflected vector in a plane with incident ray and
  surface normal vector

R(-I) 2 cos? N -2(I N) N R(I) I - 2(I N)
N
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Mirror BRDF

• Dirac function d(x)
• d(x) zero everywhere except at x0
• unit integral
• Specular reflectance ?s
• ratio of reflected radiance in reflection
  direction and incoming radiance
• dimensionless quantity between 0 and 1

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Law of Refraction

• Snells law
• Faster
• Total internal reflection
• Reflectance/Transmittance
• Incident-angle dependent Fresnel term

N x T ? N x I
n1
T ? I ? N
n2
? -? I N (1- ? 2(1-(I N ) 2 )) 1/2
1- ? 2(1-(I N ) 2 ) lt 0
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Fresnel Equations

• Light-matter interaction at dielectric surfaces
• Reflection/transmission percentage
• Electromagnetic wave theory
• Maxwell equations, boundary conditions
• Polarization-dependent
• Dispersion
• n wavelength-dependent

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Metal Reflection

• Complex index of refraction
• Free electron gas
• Transmission absorption above Langmuir
  frequency
• Silver (UV)
• Gold (violet)
• Copper (blue)
• Fresnel reflection term

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Wavelength-dependent Reflectance

• Grazing angle
• Total reflection
• Illumination color
• Normal incidence
• Highest absorption
• Typical metal color
• Reconstruct wavelength dependence of n

Copper reflectance
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Diffuse Reflection

• Light equally likely to be reflected in any
  output direction (independent of input direction)
• Constant BRDF
• kd diffuse coefficient, material property

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Lambertian Diffuse Reflection

• Radiosity
• Diffuse Reflectance
• Lamberts Cosine Law
• For each light source
• Lr,d kd Li cos?i kd Li (IN)

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Lambertian Surfaces
Self-Luminous Lambertian Light Source
Illuminated Lambertian Reflector
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Lambertian Objects
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Lambertian Objects II

• Absorption in photosphere
• Path length through photosphere longer from
  the Suns rim

• Surface covered with fine dust
• Dust on TV visible best from slanted viewing
  angle

? Neither the Sun nor the Moon are Lambertian
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Diffuse Reflection

• Theoretical explanation
• Multiple scattering
• Experimental realization
• Pressed magnesium oxide powder
• Almost never valid at high angles of incidence
• Paint manufacturers attempt to create ideal
  diffuse paints

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Wrap-Up

• Recursive Ray Tracing Algorithm
• Primary, secondary, shadow rays
• Shading
• Physical Quantities in Rendering
• Radiance
• Radiosity
• Irradiance
• Special BRDFs
• Mirror
• Glass
• Diffuse