Computer Graphics
Computer Graphics
Lecture 6
Rasterisation, Antialiasing, Texture
Mapping,
Inf4/MSc
Computer Graphics
Inf4/MSc
Some Tutorial about the Project
• I have alreadystcovered all the topics needed
to finish the 1 practical
• Today, I will briefly explain how to start
working on it.
• I have already provided you a program to
import an obj file.
2
Computer Graphics
Steps
Inf4/MSc
•Apply transformations to all vertices
•Prepare the frame buffer and Z-buffer
•For each triangle
• Project it to the screen space
• Find the 2D bounding box
• For each pixel in the bounding box
• Check if it is inside the triangle by computing its
barycentric coordinates
• If yes, use barycentric coordinates to compute the depth
and colour at the pixel
• If (depth < zbuf[pixel]) {
framebuffer[pixel] = colour
zbuf[pixel] = depth
}
3
•Export the frame buffer data into a PPM file
Computer Graphics
Inf4/MSc
• For each pixel in the bounding box
• Check if it is inside the triangle by computing
its barycentric coordinates
• If yes, use barycentric coordinates to compute
the depth and colour at the pixel
z = αz + β z2 + γ z3
c = αc + β c2 + γ c3
• If (z < zbuf[pixel]) {
1
1
framebuffer[pixel] = colour
zbuf[pixel] = c
}
Computer Graphics
Inf4/MSc
Computing the baricentric
coordinates of the interior pixels
• The triangle is composed of 3 points p0 (x0,y0), p1 (x1, y1), p2(x2,y2)
(α,β,γ) : barycentric coordinates
Only if 0<α,β,γ<1, (x,y) is inside the triangle
Depth can be computed by αZ0 + βZ1 +γZ2
Can do the same thing for color, normals, textures
5
Computer Graphics
Inf4/MSc
Today
• Anti-aliasing
• Texture mapping
– Common texture coordinates mapping
– Texture coordinates
6
Computer Graphics
Rasterisation
Inf4/MSc
• Converts the vertex information output by
the geometry pipeline into pixel information
needed by the video display
– Anti-aliasing
– Z-buffer
– Texture mapping
– Bump mapping
– Transparent objects
– Drawing lines
7
Computer Graphics
Anti-aliasing
Inf4/MSc
• Aliasing: distortion artifacts produced when representing a
high-resolution signal at a lower resolution.
• Anti-aliasing : techniques to remove aliasing
Aliased polygons
(jagged edges)
Anti-aliased polygons
8
Computer Graphics
Inf4/MSc
Nyquist Limit
• The signal frequency (fsignal) should be no greater than half the sample
frequency (fsample)
• fsignal <= 0.5 fsample
• In the top, fsignal = 0.8 fsample -> cannot reconstruct the original signal
• In the bottom fsignal =0.5 fsample -> the original signal can be reconstructed
by slightly increasing the sampling rate
Computer Graphics
Inf4/MSc
Screen-based Anti-aliasing
Each pixel is subdivided (sub-sampled) into n regions,
and each sub-pixel has a color;
Compute the average color value
n
p(x, y) wic(i, x, y)
i1
p(x, y):pixel
color
at(x,y)
c(i, x, y):sample
color
wi : weight
10
Computer Graphics
Inf4/MSc
Accumulation Buffer (A-Buffer)
• Use a buffer that has the same resolution as
the original image
• To obtain a 2x2 sampling of a scene, 4
images are made by shifting the buffer
horizontally/vertically for half a pixel
• The results are accumulated and the final
results are obtained by averaging
• Various sampling schemes are available
Pixel center
Subsampled point
Different Sampling Schemes
Inf4/MSc
Computer Graphics
Computer Graphics
Inf4/MSc
Computer Graphics
Inf4/MSc
Accumulation Buffer (A-Buffer)
• The lighting computation is usually done
only once per vertex
• Not doing the lighting computation at each
sample point
• The A-buffer’s focus is on the edge antialiasing
• Also useful for rendering transparent
objects, motion blur (will be covered later in
the course)
Edges
Computer Graphics
Inf4/MSc
Stochastic Sampling
• A scene can be produced of objects that are
arbitrarily small
• A regular pattern of sampling will always
exhibit some sort of aliasing
• One approach to solve this is to randomly
sample over the pixel
– Jittering : subdivide into n regions of equal size
and randomly sample inside each region
Computer Graphics
The oversampling
rate is 1 and 2 from left to right
Inf4/MSc
Computer Graphics
Inf4/MSc
Today
• Anti-aliasing
• Texture mapping
– Common texture coordinates mapping
– Texture coordinates
17
Computer Graphics
Inf4/MSc
Texture Mapping : Why needed?
• We don't want to represent all this detail
with geometry
Computer Graphics
Inf4/MSc
Texture mapping.
• Method of
improving surface
appearance by
adding details on
surface.
19
Computer Graphics
Inf4/MSc
Texture mapping.
• Image is ‘pasted’ onto a polygon.
• Image is called a Texture map, it’s pixels are often
referred as a Texels and have coordinates (u,v)
• Texture coordinates are defined for each vertex of
the polygon and interpolated across the polygon.
y
v
v
u
u
x
20
Computer Graphics
Inf4/MSc
Photo-textures
Computer Graphics
Inf4/MSc
Texture Interpolation
• Specify a texture coordinate (u,v) at each vertex
• Can we just linearly interpolate the values in screen
space?
(0,1)
(0,0)
(1,0)
Computer Graphics
Inf4/MSc
Interpolating the uv coordinates
Again, we use baricentric
coordinates
u1 v1
u3 v3
u= α u + β u2 + γ u3
v = α v + β v2 + γ v3
1
1
u2 v2
Computer Graphics
Interpolation - What Goes
Wrong?
Inf4/MSc
• Linear interpolation in screen space:
texture source
what we get|
what we want
Computer Graphics
Inf4/MSc
Why does it happen?
• Uniform steps on the image plane does not
correspond to uniform steps along the edge
Computer Graphics
Inf4/MSc
How do we deal with it?
• Use hyperbolic interpolation
– (u,v) cannot be linearly interpolated, but 1/w and
(u/w, v/w) can
– w is the last component after the canonical view
transformation
rl
 2n

0

0
x
rl
xpw
 rl
 
2n
tb
y w
 
0 y
 p  0 tb tb
z
zpw
 
f n 2fn
 
  0
0
f n f n1
 w 
0

0
1
0


Computer Graphics
Inf4/MSc
Texture Mapping Examples
• Linear interpolation vs. Hyperbolic interpolation
• Two triangles per square
Computer Graphics
Inf4/MSc
Computing the uv coordinates at the
internal points
• For three points of the triangle, get
1u
i v
i
r

[
p
,
,
,
]
i
i
w
i w
i w
i
1 ui vi
[ , , ]
wi wi wi
• Compute
at the internal points of the
triangle by interpolation
• Compute wi by inverting 1/wi, and multiply them
to (ui /wi, ,vi /wi) to compute (ui,vi)
Computer Graphics
Inf4/MSc
Common Texture Coordinate
Mappings
• Orthogonal
• Cylindrical
• Spherical
Computer Graphics
Inf4/MSc
Texture Mapping & Illumination
• Texture mapping can be used to alter some or all
of the constants in the illumination equation:
– pixel color, diffuse color ….
Phong’s Illumination Model
Constant Diffuse Color
Diffuse Texture Color
Texture used as Label
Texture used as Diffuse Color
Computer Graphics
Inf4/MSc
Readings
• Chapter 5.1-2 of Real-Time Rendering
Second edition
– http://books.google.co.uk/books?id=mOKEfBT
w4x0C&printsec=frontcover&source=gbs_v2_
summary_r&cad=0#v=onepage&q=&f=false
• “Hyperbolic Interpolation” IEEE Computer
Graphics and Applications, vol12, no.4, 8994, 1992
• Demoed Software
http://www-ui.is.s.u-tokyo.ac.jp/~takeo/java/smoothteddy/index.html