1. Lecture 6 Rasterisation, Antialiasing, Texture Mapping,
Computer Graphics
Lecture 6
Rasterisation, Antialiasing, Texture Mapping,

2. Some Tutorial about the Project
I have already covered all the topics needed to finish the 1st practical
Today, I will briefly explain how to start working on it.
I have already provided you a program to import an obj file.

3. Steps 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 }
Export the frame buffer data into a PPM file
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 }

4. 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 = αz1 + β z2 + γ z3
c = αc1 + β c2 + γ c3
If (z < zbuf[pixel]) {
framebuffer[pixel] = colour
zbuf[pixel] = c }

5. 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

6. Today Anti-aliasing Texture mapping Common texture coordinates mapping

7. Rasterisation
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

8. Anti-aliasing
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

9. 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

10. 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

11. 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

12. Different Sampling Schemes

14. 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 anti- aliasing
Also useful for rendering transparent objects, motion blur (will be covered later in the course)
Edges

15. 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

16. The oversampling
rate is 1 and 2 from left to right

17. Today Anti-aliasing Texture mapping Common texture coordinates mapping

18. Texture Mapping : Why needed?
We don't want to represent all this detail with geometry

19. Texture mapping.
Method of improving surface appearance by adding details on surface.

20. 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. v u y x

21. Photo-textures

22. Texture Interpolation
Specify a texture coordinate (u,v) at each vertex
Can we just linearly interpolate the values in screen space?
(0,0)‏
(1,0)‏
(0,1)‏

23. Interpolating the uv coordinates
Again, we use baricentric coordinates u= α u1 + β u2 + γ u3 v = α v1 + β v2 + γ v3
u1 v1
u3 v3
u2 v2

24. Interpolation - What Goes Wrong?
Linear interpolation in screen space:
texture source
what we get|
what we want

25. Why does it happen?
Uniform steps on the image plane does not correspond to uniform steps along the edge

26. 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

27. Texture Mapping Examples
Linear interpolation vs. Hyperbolic interpolation
Two triangles per square

28. Computing the uv coordinates at the internal points
For three points of the triangle, get
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)‏

29. Common Texture Coordinate Mappings
Orthogonal
Cylindrical
Spherical

30. 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
Diffuse Texture Color
Constant Diffuse Color
Texture used as Label
Texture used as Diffuse Color

31. Readings Chapter 5.1-2 of Real-Time Rendering Second edition
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“Hyperbolic Interpolation” IEEE Computer Graphics and Applications, vol12, no.4, , 1992
Demoed Software