Texture Mapping
Aaron Bloomfield
CS 445: Introduction to Graphics
Fall 2006
(Slide set originally by David Luebke)
Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Texture Mapping: Motivation
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Scenes created with diffuse lighting look
convincingly three-dimensional, but are flat, chalky,
and “cartoonish”
Phong lighting lets us simulate materials like
plastic and (to a lesser extent) metal, but scenes
still seem very cartoonish and unreal
Big problem: polygons are too coarse-grained to
usefully model fine surface detail
Solution: texture mapping
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Texture Mapping: Motivation
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Adding surface detail helps keep CG images from
looking simple and sterile
Explicitly modeling this detail in geometry can be
very expensive
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Zebra stripes, wood grain, writing on a whiteboard
Texture mapping pastes images onto the surfaces
in the scene, adding realistic fine detail without
exploding the geometry
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Texture Mapping: Examples
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Texture Mapping: Examples
Doom III (ID Software)
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Texture Mapping
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In short: it is impractical to explicitly model fine
surface detail with geometry
Solution: use images to capture the “texture” of
surfaces
Texture maps can modulate many factors that
affect the rendering of a surface
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Color or reflectance (diffuse, ambient, specular)
Transparency (smoke effects)
What else?
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Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Texture Mapping: Fundamentals
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A texture is typically a 2-D image
Can also be
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A 2-D procedural texture
A 3-D procedural texture
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See video
A 3-D volumetric texture (rarely used)
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Texture Mapping: Fundamentals
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A texture is typically a 2-D image
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Image elements are called texels
Value stored at a texel affects surface appearance in
some way
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Example: diffuse reflectance, shininess, transparency…
The mapping of the texture to the surface determines
the correspondence, i.e., how the texture lies on the
surface
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Mapping a texture to a triangle is easy (why?)
Mapping a texture to an arbitrary 3-D shape is more complicated
(why?)
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Texturing Fundamentals
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A texture is typically a 2-D array of texels
Mapping the texture to an arbitrary 3-D shape is
complex:
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Texturing Fundamentals
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A texture is typically a 2-D array of texels
Mapping the texture to an arbitrary 3-D shape is
complex:
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Texturing Fundamentals
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A texture is typically a 2-D array of texels
Mapping the texture to an arbitrary 3-D shape is
complex:
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Texturing Fundamentals
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A texture is typically a 2-D array of texels
Mapping the texture to an arbitrary 3-D shape is
complex!
http://www.cs.virginia.edu/~gfx/Courses/2003/Intro.spring.03/animations/unfold.mo
v
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How to map a texture
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There are a number of possibilities:
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Flat
Cube
Tube (i.e. cylinder)
Sphere
The slides on the next slide are from
http://mediawiki.blender.org/index.php/Manual/Map_Input#2D_to_3D_Mapping
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Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Texture Mapping: Rendering
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Rendering uses the mapping:
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Find the visible surface at a pixel
Find the point on that surface corresponding to that pixel
Find the point in the texture corresponding to that point
on the surface
Use the parameters associated with that point on the
texture to shade the pixel
Using triangulated meshes reduces the problem to
mapping a portion of the image to each triangle:
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Texture Mapping: Rendering
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Texture Mapping:
User-Generated Mappings
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For complex 3-D objects, mapping textures is still
something of an art…so we often let the user do it
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Texture Mapping: Rendering
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We typically parameterize the texture as a function
in (u, v)
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For simplicity, normalize u & v to [0, 1]
Associate each triangle with a texture
Give each vertex of the triangle a texture
coordinate (u, v)
For other points on the triangle, interpolate texture
coordinate from the vertices
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Much like interpolating color or depth
But there’s a catch...
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Naïve Texture Mapping
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A first cut at a texture-mapping rasterizer:
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For each pixel:
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Interpolate u & v down edges and across spans
Look up nearest texel in texture map
Color pixel according to texel color (possibly modulated by
lighting calculations)
McMillan’s demo of this is at
http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide05.html
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What artifacts do you see in this demo?
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Naïve Texturing Artifacts
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Another serious artifact is warping at the edges of
triangles making up the mesh
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A more obvious example:
http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide06.html
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To address this, need to consider the geometry of
interpolating parameters more carefully
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Interpolating Parameters
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The problem turns out to be fundamental to
interpolating parameters in screen-space
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Uniform steps in screen space  uniform steps in world coords
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Interpolating Parameters
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Perspective foreshortening is not getting applied to
our interpolated parameters
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Is this a problem with Gouraud shading?
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Parameters should be compressed with distance
Linearly interpolating them in screen-space doesn’t do
this
Is everything I taught wrong?
A: It can be, but we usually don’t notice (why?)
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Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Perspective-Correct Interpolation
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Skipping a bit of math to make a long story
short…
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Rather than interpolating u and v directly, interpolate
u/z and v/z
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These do interpolate correctly in screen space
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Also need to interpolate z and multiply per-pixel
Problem: we don’t know z anymore
Solution: we do know w  1/z
So…interpolate uw and vw and w, and compute
u = uw/w and v = vw/w for each pixel
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This unfortunately involves a divide per pixel (Just 1?)
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Perspective-Correct Texturing
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Known as perspective-correct texture mapping
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Early PC cards and game consoles didn’t support it
So how did they avoid the warping problem?
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http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide15.html
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As mentioned, other interpolation schemes really
ought to use perspective correction
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E.g., Gouraud shading
Generally get away without it because it is more
important to be smooth than correct
Java code fragment from McMillan’s edgeequation triangle rasterizer:
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Perspective-Correct Texturing: Code
...
PlaneEqn(uPlane, (u0*w0), (u1*w1), (u2*w2));
PlaneEqn(vPlane, (v0*w0), (v1*w1), (v2*w2));
PlaneEqn(wPlane, w0, w1, w2);
...
for (y = yMin; y <= yMax; y += raster.width) {
e0 = t0;
e1 = t1;
e2 = t2;
u = tu;
v = tv;
w = tw;
z = tz;
boolean beenInside = false;
for (x = xMin; x <= xMax; x++) {
if ((e0 >= 0) && (e1 >= 0) && (e2 >= 0))) {
int iz = (int) z;
if (iz <= raster.zbuff[y+x]) {
float denom = 1.0f / w;
int uval = (int) (u * denom + 0.5f);
uval = tile(uval, texture.width);
int vval = (int) (v * denom + 0.5f);
vval = tile(vval, texture.height);
int pix = texture.getPixel(uval, vval);
if ((pix & 0xff000000) != 0) {
raster.pixel[y+x] = pix;
raster.zbuff[y+x] = iz;
}
}
beenInside = true;
} else if (beenInside) break;
e0 += A0;
e1 += A1;
e2 += A2;
z += Az;
u += Au;
v += Av;
w += Aw;
}
t0 += B0;
t1 += B1;
t2 += B2;
tz += Bz;
tu += Bu;
tv += Bv;
tw += Bw;
}
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Texture Tiling
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It is often handy to tile a repeating texture pattern
onto a surface
The previous code does this via tile():
int uval = (int) (u * denom + 0.5f);
uval = tile(uval, texture.width);
int vval = (int) (v * denom + 0.5f);
vval = tile(vval, texture.height);
int pix = texture.getPixel(uval, vval);
int tile(int val, int size) {
while (val >= size)
val -= size;
while (val < 0)
val += size;
}
See http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide18.html
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Overview
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
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Texture Transparency
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McMillan’s code also includes a “quick fix” for
handling transparent texture:
if ((pix & 0xff000000) != 0) {
raster.pixel[y+x] = pix;
raster.zbuff[y+x] = iz;
}
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Note that this doesn’t handle partial transparency (How
might such partial transparency arise?)
Demo at:
http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide19.html
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Texture Map Aliasing
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Naive texture mapping looks blocky, pixelated
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Problem: using a single texel to color each pixel:
int uval = (int) (u * denom + 0.5f);
int vval = (int) (v * denom + 0.5f);
int pix = texture.getPixel(uval, vval);
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Actually, each pixel maps to a region in texture
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If the pixel is larger than a texel, we should average the
contribution from multiple texels somehow
If the pixel is smaller than a texel, we should interpolate
between texel values 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 (Draw it)
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Find 4 nearest texture samples
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Round u & v up and down
Interpolate texel values in u
Interpolate resulting values in v
Also addresses the problem of many pixels
projecting to a single texel (Why?)
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Texture Map Antialiasing
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What if a single pixel covers many texels?
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Problem: sampling those texels at a single point (the
center of the pixel):
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Produces Moire patterns in coherent texture (checkers)
Leads to flicker or texture crawling as the texture moves
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Moire Patterns
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Texture Map Antialiasing
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What if a single pixel covers many texels?
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Approach: blur the image under the pixel, averaging the
contributions of the covered texels
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But calculating which texels every pixel covers is way too
expensive, especially as the texture is compressed
Solution: pre-calculate lower-resolution versions of the
texture that incorporate this averaging
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Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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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
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Generating MIP-maps
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Generating a MIP-map from a texture is easy
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For each texel in level i, average the values of the four
corresponding texels in level i-1
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 = Multum 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
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A texel at level n represents 2n original texels
When rendering:
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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
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Using MIP-maps
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Even better:
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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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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
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More exactly, the compression or oversampling rate is
the same in u and v
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
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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)
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Details in the book
Example:
MIP-map texturing
Summed-area table texturing
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Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Texture Mapping Variations
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In addition to the texture, we add the lighting
model also:
I total  ka I ambient 
Texture as
R,G,B:
#lights
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i 1

 
I i  kd Nˆ  Lˆ  k s Vˆ  Rˆ


nshiny

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Texture with
lighting:
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Bump Mapping
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The texture map can modulate the surface normal
used for shading
Sphere w/ diffuse texture
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?
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Another Bump Mapping Example
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From Wikipedia
+
=
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Displacement Map
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A displacement
geometry
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actually
displaces
the
Treats the texture as a height field to be applied to the
surface
Starting to appear in the interactive graphics pipeline
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map
First supported in Matrox Parhelia card
Can sort of implement with beta drivers in ATI & NVIDIA cards
Will soon appear in all cards
Implemented by recursive subdivision of triangles/quads
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Displacement Map Example
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What is the biggest visual difference between
displacement mapping and bump mapping?
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Overview
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Motivation and Examples
Fundamentals
Algorithms
Perspective-Correct Texturing
Transparency and Anti-Aliasing
MIP-Maps
Bump and Displacement Maps
Illumination Maps
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Illumination Maps
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Quake introduced illumination maps or light maps to
capture lighting effects in video games
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:
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Usually apply to only a single surface
Are usually fairly low resolution
Usually capture just intensity (1 value) rather than color
(3 values)
Illumination maps can be:
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Painted by hand: Disney’s Aladdin ride
Calculated by a global illumination process: Nintendo64
demo, modern level builders
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Other Texture Applications
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Lots of other interesting applications of the texturemap concept (we’ll return to some):
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Shadow maps
3-D textures (marble, wood, clouds)
Procedural textures
Environment maps & cube maps
For a neat explanation of the first three (with cool
applets, as usual) check out:
http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture22/Slide21.html
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