CS248 Final Review
CS248 Final
• Thurs, December 12, 7-10 pm, Gates
B01, B03
• Mainly from material in the second half
of the quarter
– will not include material from last part of last
lecture (volume rendering, image-based
rendering)
• Review session slides available from
class website
• Office hours as regularly scheduled
CS248 Final Review Contents
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Image warping, texture mapping
Perspective
Visibility
Lighting / Shading
Texture Mapping
• Coordinate systems
– [u,v,q] => [xo, yo zo, wo] => [xw, yw zw, ww]
=> [x, y, w]
– Assuming all transforms are linear, then
– [A][u, v, q]’ = [x, y, w]
• Common mappings
– forward mapping (scatter), texture->screen
– backward mapping (gather)
Texture Warps
• Rotation, translation
• perspective
• Minification (decimation)
– unweighted average: average projected
texel elements that fall within a pixel’s filter
support
– area-weighted average: average based on
area of texel support
Texture Warps
• Magnification
– Unweighted
– Area-weighted
– bilinear interpolation
= texel
= pixel
Textures
1.Mipmapping
1.multi-resolution texture
2.bilinear interpolation at 2 closest
resolutions to get 2 color values
3.linear interpolate 2 color values based on
actual resolution
2.Summed area tables
1.fast calculation of prefilter integral in
texture space
Viewing: Planar Projections
• Perspective Projection
– rays pass through center of projection
– parallel lines intersect at vanishing points
• Parallel Projection
– center of projection is at infinity
– oblique
– orthographic
How many vanishing points are there in an image
produced by parallel projection ?
Specifying Perspective Views
• Observer position (eye, center of projection)
• Viewing direction (normal to picture plane)
• Clipping planes (near, far, top, bottom, left,
right)
Viewing: OpenGL Pipeline
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Object Space
Eye Coordinates
Projection Matrix
Clipped to Frustum
Homogenize to normalized device
coordinates
• Window coordinates
Visibility
1.6 visible-surface determination
algorithms:
1.Z-buffer
2.Watkins
3.Warnock
4.Weiler-Atherton
5.BSP Tree
6.Ray Tracing
Things to know
how does it work
what are the necessary preconditions?
asymptotic time complexity
how can anti-aliasing be done?
how can shading be incorporated?
well-suited for hardware?
parallelizable?
ease of implementation
best-case/worst-case scenarios
Z-buffer
• Project all polygons to the image plane,
at each pixel, pick the color
corresponding to closet polygon
Watkins
• Scanline + depth
– progressing across scanline, if pixel is
inside two or more polygons, use depth to
pick
– process interpenetrating polygons, add
those events
Warnock Subdivision
• Start with area as original image
– subdivide areas until either:
• all surfaces are our outside the area
• only one inside, overlapping or surrounding
• a surrounding surface obscures all other
surfaces
*
Weiler-Atherton Subdivision
• Cookie-cutter algorithm:clips polygons
against polygons
– front to back sort of list
– clip with front polygon
BSP Trees
• Provides a data structure for back-tofront or front-to-back traversal
– split polygons according to specified
planes
– create a tree where edges are front/back,
leaves are polygons
Ray Tracing
• “Ray Casting”
– for each pixel, cast a ray into the scene,
and use the color of the point on the
closest polygon
– Parametric form of a line: u(t) = a+(b-a)t
a
y
(0,0)
x
t
b
Ray Tracing
• Sphere: |P-Pc|2 – r2 = 0
• Plane: N • P = -D
• Can you compute the intersection of a
ray and a plane? A ray and a sphere?
Ray Tracing
• Point in polygon tests
– Odd, even rule
• draw a line from point to infinity in one direction
• count intersections: odd = inside, even =
outside
– Non-zero winding rule
• counts number of times polygon edges wind
around a point in the clockwise direction
• winding number non zero = inside, else outside
Lighting
• Terminology
– Radiant flux: energy/time (joules/sec =
watts)
– Irradiance: amount of incident radiant flux /
area (how much light energy hitting a unit
area, per unit time)
– Radiant intensity (of point source): radiant
flux over solid angle
– Radiance: radiant intensity over a unit area
Lighting
• Point to area transport
– Computing the irradiance to a surface
– Cos falloff: N • L
– E = Fatt x I x (N • L)
Lighting
• Lambertian (diffuse) surfaces
– Radiant intensity has cosine fall off with
respect to angle
– Radiance is constant with respect toangle
– Reason: the projected unit area ALSO gets
smaller as a cosine fall off!
– Fatt x I x Kd x (N • L)
N
I  length = cos(t)
V
Radiance intensity: intensity/solid angle
N
V
Lighting
• BRDF = Bidirectional Reflectance Distribution
Function
– description of how the surface interacts with
incident light and emits reflected light
– Isotropic
• Independent of absolute incident and reflected angles
– Anisotropic
• Absolute angles matter
– Don’t forget the generalizations to the BRDF!
• Spatially/spectrally varying, florescence,
phosphorescence, etc.
Lighting
• Phong specular model
– Isn’t true to the physics, but works pretty
well
– reflected light is greatest near the
reflection angle of the incident light, and
falls off with a cosine power
– Lspec = Ks x cosn(a), a= angle between
viewer and reflected ray
R N
– how do you compute the reflected ray
vector?
V
L
Lighting
• Local vs. infinite lights
– Understand them! Know how to draw the
goniometric diagrams for various
light/viewer combinations
• N • H model
– H is the halfway vector between the viewer
and the light
HN
– What is the difference in specular R
highlight?
V
L
Shading
• Gouraud shading
– Compute lighting information (ie: colors) at
polygon vertices, interpolate those colors
– Problems?
• Misses highlights
• need high resolution mesh to catch highlights
• mach bands!
Shading
• Angle interpolation
– interpolate normal angles according to the implicit
surface
– compute shading at each point of the implicit
surface
– CORRECT! But very expensive
Shading
• Phong shading
– Compute lighting normals at all points on the
polygon via interpolation, and do the lighting
computation on the interpolated normals (of the
polygon)
– Problems? Difference with angle interpolation?
N1
Polygon approximation
N2
Implicit surface
Lighting and Shading
• Know the OpenGL 1.1, 1.2 light
equations
Exotic uses of textures
• Environment/reflection mapping
• Alphas for selecting between
textures/shading parameters
• Bump mapping
• Displacement mapping
• Object placement
• 3d textures
Good Luck!
Good Luck on the Final! 
More review questions at:
http://graphics.stanford.edu/courses/cs248-99/final_review