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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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Assignment 6 notes
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David Luebke
Viewport <= framebuffer (don’t stretch pixels)
Drawing lines: round to integer pixel values
Perspective projection: don’t need to preserve Z
Any other questions?
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Lighting: Transforming Normals
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Irritatingly, the matrix for transforming a
normal vector is not the same as the matrix
for the corresponding transformation on points
– In other words, don’t just treat normals as points:
David Luebke
7/27/2016
Lighting: Transforming Normals
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What is homogeneous representation of a
vector (as opposed to a point?)
Some not-too-complicated affine analysis
shows :
– If A is a matrix for transforming points,
then (AT)-1 is the matrix for transforming normals
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When is this the same matrix?
Can use this to simplify the problem: only
upper 3x3 matrix matters, so use only it
More detail in F&vD Appendix A.5
David Luebke
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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 coarsegrained 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
– Zebra stripes, wood grain, writing on a whiteboard
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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
David Luebke
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Texture Mapping: Fundamentals
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A texture is typically a 2-D image
– 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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David Luebke
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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Texture Mapping: Rendering
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Rendering uses the mapping:
– 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
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Texture Mapping: Basics
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We typically parameterize the texture as a
function in (u, v)
– For simplicity, normalize u & v to [0, 1]
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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
– Much like interpolating color or depth
– But there’s a catch...
David Luebke
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Naïve Texture Mapping
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A first cut at a texture-mapping rasterizer:
– 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
– What artifacts do you see in this demo?
David Luebke
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Naïve Texturing Artifacts
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Probably the most obvious artifact is the
blocky pixelated look of the texture
– Basic problem: using a single texel to color each
pixel
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If the pixel is larger than a texel, we should average the
contribution from multiple texles 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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David Luebke
More on this later…
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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
– A more obvious example:
http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide06.html
– 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
– 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
– Parameters should be compressed with distance
– Linearly interpolating them in screen-space
doesn’t do this
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Is this a problem with Gouraud shading?
A: It can be, but we usually don’t notice
(why?)
– http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide17.html
David Luebke
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Perspective-Correct
Interpolation
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Skipping a bit of math to make a long story
short…
– 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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David Luebke
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
– Some APIs and game consoles don’t support it
– So how can they avoid the warping problem?
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As mentioned, other interpolation schemes
really ought to use perspective correction
– E.g., Gouraud shading
– Generally get away without it because it is more
important to be smooth than correct
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Java code fragment from McMillan’s edgeequation triangle rasterizer:
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Perspective-Correct Texturing:
Code Fragment
...
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;
}
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
David Luebke
7/27/2016
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;
}
– 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
David Luebke
7/27/2016