CS 551 / CS 645
Shadow Models
Administrative
• Midterm exam in 9 days
• Today we’re finishing FvD chapter 16
Recap: Lighting Models
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•
•
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Ambient Light
Diffuse Light
Phong Lighting
Gouraud Shading
– Interpolate vertex colors along edges
• Phon
Ambient Light Sources
• A scene lit only with an ambient light
source:
Directional Light Sources
• The same scene lit with a directional and an
ambient light source
Other Light Sources
• Spotlights are point sources whose intensity
falls off directionally.
– Requires color, point
direction, falloff
parameters
– Supported by OpenGL
Computing Diffuse Reflection
• The angle between the surface normal and
the incoming light is the angle of incidence:
l
n

Idiffuse = kd Ilight cos 
• In practice we use vector arithmetic:
Idiffuse = kd Ilight (n • l)
Specular Reflection
• Shiny surfaces exhibit specular reflection
– Polished metal
– Glossy car finish
• A light shining on a specular surface causes
a bright spot known as a specular highlight
• Where these highlights appear is a function
of the viewer’s position, so specular
reflectance is view-dependent
The Optics of Reflection
• Reflection follows Snell’s Laws:
– The incoming ray and reflected ray lie in a
plane with the surface normal
– The angle that the reflected ray forms with the
surface normal equals the angle formed by the
incoming ray and the surface normal:
(l)ight = (r)eflection
Phong Lighting
• The most common lighting model in
computer graphics was suggested by Phong:
Ispecular  ksIlightcos 
nshiny
• The nshiny term is a purely
empirical constant that
varies the rate of falloff
• Though this model has no
physical basis, it works
(sort of) in practice
v
Flat Shading
• The simplest approach, flat shading,
calculates illumination at a single point for
each polygon:
• If an object really is faceted, is this
accurate?
• No:
– For point sources, the direction to light varies
across the facet
– For specular reflectance, direction to eye varies
across the facet
Gouraud Shading
• This is the most common approach
– Perform Phong lighting at the vertices
– Linearly interpolate the resulting colors over
faces
• Along edges
• Along scanlines
– This is what OpenGL does
• Does this eliminate the facets?
Phong Shading
• Phong shading is not the same as Phong
lighting, though they are sometimes mixed
up
– Phong lighting: the empirical model we’ve been
discussing to calculate illumination at a point
on a surface
– Phong shading: linearly interpolating the
surface normal across the facet, applying the
Phong lighting model at every pixel
• Same input as Gouraud shading
• Usually very smooth-looking results:
• But, considerably more expensive
Shortcomings of Shading
• Polygonal silhouettes remain
• Perspective distortion not captured in
interpolation down scanlines
• Interpolation dependent on polygon
orientation
• Shared vertices
• Bad averaging to compute vertex normals
Shadows
• Fake it
– Preprocess polygons and the shadows they cast
on the ground plane
– Add ‘shadow polygons’ to the ground plane
when necessary
• Shortcomings
– Object->Object shadows are missed
– Lights and objects cannot move
Scan-line Shadow Generation
• Project each polygon’s shadow onto every
other polygon
• Create a set of ‘shadow polygons’
• When scan-line rendering, use regular
polygons to choose color, but check in
shadow polygons for intensity
• Shortcomings
– Slow
– Light and objects cannot move
Shadow Volumes
• Create polygonal volume by creating
multiple ‘shadow’ polygons anchored at the
light and passing through faces of object
Shadow Volumes
• Use shadow polygons like clipping planes
• Shadow object if contained within a volume
• How do we compute?
B
A
Shadow Volumes
• Examples
b
a
v
c
b
a
v
c
Z-buffer Shadow Maps
• Z-buffer
– Created as a byproduct of scan conversion
– Each scan converted polygon is represented as
a location in pixel space and a depth, or
distance, from camera
– Of all polygons projecting onto a pixel, store
the z-value of closest one in z-buffer
– Image each pixel having a (r, g, b, z) tuple
– We’ll use the z-buffer again when we talk about
visibility
Z-buffer Shadow Maps
• Store the z-buffer for the image from the
viewpoint of the light
• Compute the image and z-buffer from the
viewer’s position, but…
• For each pixel in image, find its object
coordinates (x, y, z)
• Map this (x, y, z) into light space (x’, y’, z’)
Z-buffer Shadow Maps
• Project (x’, y’, z’) to light’s z-buffer and
compare z’ to the z-buffer value, zlight
• If zlight < z’ z’ is shadowed by something
• Otherwise, point is visible from light and
should be illuminated
• Beware of numerical precision
– A polygon should not shadow itself
Global Illumination
• We’ve glossed over how light really works
• And we will continue to do so…
• One step better
• Global Illumination
– The notion that a point is illuminated by more
than light from local lights; it is illuminated by
all the emitters and reflectors in the global scene
The ‘Rendering Equation’
• Jim Kajiya (Current head of Microsoft
Research) developed this in 1986


I  x, x'  g  x, x'e  x, x'   r  x, x' , x' 'I  x' , x' 'dx' '
S


 e(x, x’) is the intensity from x’ to x
 r(x, x’,x’’) is the intensity of light
reflected from x’’ to x from x’
• S is all points on all surfaces
The ‘Rendering Equation’
• The light that hits x from x’ is the direct
illumination from x’ and all the light
reflected by x’ from all x’’
• To implement:
– Must handle recursion effectively
– Must support diffuse and specular light
– Must model object shadowing
Recursive Ray Tracing
• Cast a ray from the viewer’s eye through
each pixel
• Compute intersection of this ray with
objects from scene
• Closest intersecting object determines color
Recursive Ray Tracing
• Cast a ray from intersected object to light
sources and determine shadow/lighting
conditions
• Also spawn secondary rays
– Reflection rays and refraction rays
– Use surface normal as guide
– If another object is hit, determine the light it
illuminates by recursing through ray tracing
Recursive Ray Tracing
• Stop recursing when:
– ray fails to intersect an object
– user-specified maximum depth is reached
– system runs out of memory
• Common numerical accuracy error
– Spawn secondary ray from intersection point
– Secondary ray intersects another polygon on
same object
Recursive Ray Tracing
• Still producing PhD’s after all these years
• Many opportunities to improve efficiency
and accuracy of ray tracing
– Reduce the number of rays cast
– Accurately capture shadows caused by nonlights (ray tracing from the light source)
– Expensive to recompute as eyepoint changes
Radiosity
• Ray tracing models specular reflection and
refractive transparency, but still uses an
ambient term to account for other lighting
effects
• Radiosity is the rate at which energy is
emitted or reflected by a surface
• By conserving light energy in a volume,
these radiosity effects can be traced
Radiosity
•


•
B j F j 1
Radiosity of patch ‘i’ is Bi  Ei  ri 1
j n
ei is the rate at which light is emitted
ri is patch i’s reflectivity
Fj-i is the form factor
Aj
Ai
– fraction of patch j that reaches patch i as
determined by orientation of both patches and
obstructions
• Use Aj/Ai to scale units to total light emitted
by j as received per unit area by i
Radiosity
• Compute n-by-n matrix of form factors to
store radiosity relationships between each
light patch and every other light patch
cos i cos  j
• Form factor
dF

H dA
– draw picture
di  dj
r 2
ij
j
Radiosity
• Spherical projections to model form factor
– project polygon Aj on unit hemisphere centered
at (and tangent to) Ai
• Contributes cosj / r2
– Project this projection to base of hemisphere
• Contributes cosi
– Divide this area by area of circle base
• Contributes 
Radiosity
• Analytic solution of hemisphere is expensive
• Use rectangular approximation, hemicube
• cosine terms for top and sides are simplified
Radiosity
• Radiosity is expensive to compute
– Get your PhD by improving it
• Emitted light and viewpoint can change
• Light angles and object positions cannot
• Specular reflection information is not
modeled
View-dependent vs
View-independet
• Ray-tracing models specular reflection well,
but diffuse reflection is approximated
• Radiosity models diffuse reflection
accurately, but specular reflection is ignored
• Combine the two...
Texture Mapping
• Limited ability to generate complex
surfaces with geometry
• Images can convey the illusion of geometry
• Images painted onto polygons is called
texture mapping
Texture Mapping
• Texture map is an image, two-dimensional
array of color values (texels)
• Texels are specified by texture’s (u,v) space
• At each screen pixel, texel can be used to
substitute a polygon’s surface property
• We must map (u,v) space to polygon’s (s, t)
space
Texture Mapping
• (u,v) to (s,t) mapping can be explicitly set at
vertices by storing texture coordinates with
each vertex
• How do we compute (u,v) to (s,t) mapping
for points in between
– Watch for aliasing
– Watch for many to one mappings
– Watch for perspective foreshortening effects
and linear interpolation
Bump Mapping
• Use textures to modify surface geometry
• Use texel values to modify surface normals
of polygon
• Texel values correspond to height field
– Height field models a rough surface
• Partial derivative of bump map specifies
change to surface normal