lecture 16
Texture mapping
Aliasing (and anti-aliasing)
Texture
(images)
Texture Mapping
Q: Why do we need texture mapping ?
A: Because objects look fake and boring without it.
Adding texture improves realism.
recall mapping
for bicubic patch
recall mapping
for barycentric coordinates
of a triangle
We would like to
map the texture image
coordinates to surface
coordinates.
We do so, we use
an intermediate
map to normalized
coordinates.
normalized texture texture image
coordinates (not necessarily square )
world to camera to normalized texture coordinates
clip coordinates to world coordinates
(x and y only)
display NDC
coordinates (w x, wy, w)
What is texture mapping?
for each pixel in the image projection of the polygon{
compute corresponding texel position
copy texture RGB to image pixel RGB
}
Let's think about the matrices that are used for this mapping.
Here we simplify: assume camera coords = world coords.
display clip coordinates
coordinates (w x, wy, w)
display clip coordinates
coordinates (w x, wy, w)
Homography
A homography is an invertible 3x3 matrix that maps between 2D
spaces that are represented in homogeneous coordinates.
for each pixel in the image projection of the polygon{
Use (inverse) homography to compute corresponding texel position
Use texture RGB to determine image pixel RGB
}
Details
How to construct the homography ?
What are the sampling issues and how to deal with them ?
How to construct the homography ?
image
texture image
(display)
coordinates
coordinates
This is the usual projection
from world coordinates to
clip coordinates, but now the
z row has been deleted.
Why? Because we can
ignore hidden surface
removal in this mapping.
[ADDED April 25: I have
ignored the normalization
transformation in this
example. This is only
clip coordinates
allowed in the special case
(w x, wy, w) that the normalization matrix
with z deleted is the identity matrix.]
Example: slanted floor
Example: slanted floor
Take z = 0 plane, rotate it by degrees around x axis,
and translate it by z0.
Rotate by degrees around x axis, and translate by z 0.
But we only apply this to points on the z=0 plane.
Calculating H and its inverse gives....
Exercise :
H maps which points in R^2 to points at infinity (2D) ?
H maps points at infinity to which image points ?
OpenGL computes the homographies for you.
clip coordinates
(w x, w y, w)
Exercise: where in the pipeline does this occur ?
Details
How to construct the homography ?
What are the sampling issues and how to deal with them ?
Texture magnification: a pixel in texture image
('texel') maps to an area larger than one pixel in image
Texture minification: a pixel in texture image
('texel') maps to an area smaller than a pixel in image
minification
magnification
e.g. 8 x 8
e.g. 300 x 300
It can happen that inverse mapping is outside the range
of the texture image. We need a policy in this case.
e.g. use (x mod Nx, y mod Ny)
Details
How to construct the homography ?
Q: What are the sampling issues ...
A: "Aliasing"
Q: ... and how to deal with them ?
A: "Anti-aliasing"
I will just give a sketch. A proper treatment would take
several weeks.
"Aliasing" in computer graphics:
For any RGB image defined on a discrete grid of pixels,
there are infinitely many images defined on the 2D
continuum, that have the same RGB values at the
discrete pixels.
"Aliasing" in programming languages:
Two variables reference the same memory location.
x = new Dog()
y = x
Aliasing in scan conversion (lecture 6)
Line Segment Polygon
for x = round(x0) to round(x1) { for y = ymin to ymax {
writepixel(x, Round(y) ) compute intersection of polygon
y=y+m edges with row y
} fill in pixels between adjacent
pairs of edges
}
Aliasing and Anti-aliasing
note "big pixels" here
(magnification)
Textures ("regular") and aliasing
Suppose we sample the stripe image on the left using the
intersection points (pixels) on the grid on the right.
Q: Will we also get regular vertical stripes in the
sampled image?
A: No, unless the distance between pixels happens to
correspond exactly to the stripe width.
e.g. Moiré patterns
Caused by camera pixel frequency being higher than that
of the grid pattern on the big central door (minification).
Somehow this slide was dropped from the lecture...
Too bad, because its a classic.
from Paul Heckbert, "Survey of Texture Mapping"
How to reduce aliasing in texture mapping ?
How to choose I( xp, yp ) ?
magnification
minification
Change in notation for upcoming slides
Recall that a pixel can be thought of in two ways: as a little
square, and as a point with integer coordinates. Texture
mapping is a good example of why we need this flexibility.
Up to now in this lecture, the pixels have been little
squares. But in the following few slides, pixels in the
texture image will defined as a grid of points, namely the
intersections of the horizontal and vertical lines of the grid.
In particular, each square in the texture grid is no longer a
pixel. Rather, the corner points of the square are the pixels.
This should make more sense once you see the arguments.
Case 1: magnification
T( i, j+1 ) T( i+1, j+1 )
inverse mapped center of
square pixel I(xp,yp)
T( x, y) = ?
inverse mapped
square pixel I(xp,yp)
T( i, j )
T( i+1, j )
Solution 1a: Linear interpolation
Partition square into two triangles. Use linear interpolation
within the triangle that (x,y) lies.
T( i, j+1 ) T( i+1, j+1 )
T( x, y) = ?
T( i, j ) T( i+1, j )
Exercise: What is the problem with this method ?
Solution 1b: "Bilinear interpolation"
T( i, j+1 ) T( i+1, j+1 )
T( i, y ) T( i+1, y )
T( x, y )
T( i, j ) T( i+1, j )
Exercise: Write out the formula for T(x, y)
Case 2: texture minification
Solution: (not used by OpenGL) Take average of
intensities within the quad (inverse map of square pixel).
OpenGL used MIP mapping (next lecture).
Here is an example of what these two solutions can
produce. For minification, averaging produces grey
pixels, which is appropriate. For magnification, the
interpolation blurs the intensities, which is in this case
doesn't work so well because there is too much of it.
(We only wanted to blur the edge enough to hide the
jaggies!)
minification
(solution:
averaging)
magnification
(solution:
interpolation)
How to texture map a quadric (or bicubic) ?
Discretize sphere into polygons (or
use parametric surface patch).
Texture mapping in OpenGL
(what is required?)
- a texture image
glTexImage2D( GL_TEXTURE_2D, ...., size, .., data )
- a correspondence between polygon vertices and texture
coordinates
glBegin(GL_POLYGON)
glTexCoord2f(0, 0);
glVertex( .....)
glTexCoord2f(0, 1);
glVertex( .....)
glTexCoord2f(1, 0);
glVertex( .....)
glEnd()
