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CS 551 / 645:
Introductory Computer Graphics
David Luebke
[email protected]
http://www.cs.virginia.edu/~cs551
David Luebke
7/27/2016
Administrivia
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Next semester: Intro Graphics again
– Look for Brogan’s animation seminar
David Luebke
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Recap: Texture Mapping
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Problem: it is impractical to explicitly model
fine surface detail with geometry
Solution: use images to capture the “texture”
of surfaces
David Luebke
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Recap: Texturing Fundamentals
Recap: Texture Coordinates
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Give each vertex of the triangle a texture
coordinate (u, v)
For other points on the triangle, interpolate
texture coordinate from the vertices
Problem: interpolating u & v in screen-space
(a la Gouraud shading) is incorrect
– Instead, interpolate uw, vw, and w, and calculate
uw/w and vw/w for each pixel
– Known as perspective-correct texture mapping
David Luebke
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Recap: Texture Map Aliasing
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Naïve texturing looks blocky and pixelated
– Problem: using only one texel to color each pixel
– Actually, each pixel maps to a region in texture
 Pixel is smaller than a texel, we should
interpolate between texel values somehow
 Pixel is larger than a texel, we should average
the contribution from multiple texels somehow
 Even if pixel size  texel size, a pixel will in
general fall between four texels
– An example of a general problem called aliasing
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Texture Map Antialiasing
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Use bilinear interpolation to average nearby
texel values into a single pixel value
– Find 4 nearest texels
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T1
T2
Round u & v up and down
– Interpolate texel values in u
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 = u - u
A = (1-)T4 + T1, B = (1-)T3 + T2
A
T4
P
B
T3
– Interpolate resulting values in v
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 = v - v
P = (1- )A + B
– Also addresses the problem of many pixels
projecting to a single texel (oversampling)
David Luebke
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Texture Map Antialiasing
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What about undersampling, when a single
pixel covers many texels?
– Problem: sampling those texels at a single point
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Produces Moire patterns in coherent texture (checkers)
Leads to flicker or texture crawling as the texture moves
– Approach: blur the image under the pixel,
averaging the contributions of the covered texels
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But calculating which texels each pixel covers is
expensive, especially as the texture is very compressed
– Solution: pre-calculate lower-resolution versions
of the texture that incorporate this averaging
David Luebke
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MIP-maps
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For a texture of 2n x 2n pixels, compute n-1
textures, each at ½ the resolution of previous:
Original Texture
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Lower Resolution Versions
This multiresolution texture is called a MIP-map
David Luebke
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Generating MIP-maps
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Generating a MIP-map from a texture is easy
– For each texel in level i, average the values of
the four corresponding texels in level i-1
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If a texture requires n bytes of storage, how
much storage will a MIP-map require?
Answer: 4n/3
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Representing MIP-maps
R G
R G
B B
R
R G
B
G
B
Trivia: MIP = Multim In Parvo (many things in a small place)
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Using MIP-maps
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Each level of the MIP-map represents a preblurred version of multiple texels
– A texel at level n represents 2n original texels
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When rendering:
– Figure out the texture coverage of the pixel (i.e.,
the size of the pixel in texels of the original map)
– Find the level of the MIP map in which texels
average approximately that many original texels
– Interpolate the value of the four nearest texels
David Luebke
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Using MIP-maps
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Even better:
– Likely, the coverage of the pixel will fall
somewhere between the coverage of texels in two
adjacent levels of the MIP map
– Find the pixel’s value in each of the two textures
using two bilinear interpolations
– Using a third interpolation, find a value in between
these two values, based on the coverage of the
pixel versus each of the MIP-map levels
– This is (misleadingly?) called trilinear interpolation
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Using MIP-maps
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How many interpolations does a texture
lookup using trilinear interpolation in a MIPmapped texture involve?
How many texel values from the MIP-map
must be fetched for such a lookup?
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MIP-map Example
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No filtering:
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MIP-map texturing:
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Can We Do Better?
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What assumption does MIP-mapping
implicitly make?
A: The pixel covers a square region of the
texture
– More exactly, the compression or oversampling
rate is the same in u and v
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Is this a valid assumption? Why or why not?
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MIP-maps and Signal Processing
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An aside: aliasing and antialiasing are
properly topics in sampling theory
– Nyquist theorem, convolution and reconstruction,
filters and filter widths
– Textures are particularly difficult because a tiled
texture can easily generate infinite frequencies
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E.g., a checkered plane receding to an infinite horizon
– Using a MIP-map amounts to prefiltering the
texture image to reduce artifacts caused by
sampling at too low a rate
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Summed-Area Tables
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A technique called summed-area tables lets
us integrate texels covered by the pixel more
exactly (but still quickly)
– Details in the book
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Example:
MIP-map texturing
David Luebke
Summed-area table texturing
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Texture Mapping Variations
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A texture can modulate any parameter in the
rendering process:
I total  ka I ambient 
Texture as
R,G,B:
David Luebke
#lights

i 1

 
I i  kd Nˆ  Lˆ  k s Vˆ  Rˆ


nshiny


Texture as
diffuse lighting
coefficients:
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Bump Mapping
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The texture map can even modulate the
surface normal used for shading
Sphere w/ diffuse texture
David Luebke
Swirly bump map
Sphere w/ diffuse texture
and swirly bump map
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More Bump Mapping
+
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=
How can you tell a bumped-mapped object from an
object in which the geometry is explicitly modeled?
Last Bump Mapping Example
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Displacement Map
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A displacement map actually displaces the
geometry
– Treats the texture as a height field to be applied to
the surface
– Hard to do in interactive graphics pipeline
– But doable in ray tracing
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Displacement Map Example
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What is the fundamental difference between
displacement mapping and bump mapping?
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Illumination Maps
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Quake uses illumination maps or light maps to
capture lighting effects:
Texture map:
Light map
Texture map
+ light map:
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Illumination Maps
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Illumination maps differ from texture maps in
that they:
– Usually apply to only a single surface
– Are usually fairly low resolution
– Usually capture just intensity (1 value) rather than
color (3 values)
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Illumination maps can be:
– Painted by hand: Disney’s Aladdin ride
– Calculated by a global illumination process:
Nintendo64 demo
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Origin of the idea
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Other Texture Applications
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Lots of other interesting applications of the
texture-map concept:
–
–
–
–
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Shadow maps
3-D textures (marble, wood, clouds)
Chrome maps
Procedural textures
For a neat explanation of the first two (with
cool applets, as usual) check out:
http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture22/Slide21.html
David Luebke
7/27/2016