1. (c) 2002 University of Wisconsin, CS559
Last Time
Line drawing
An intro to polygon filling
03/07/02
(c) 2002 University of Wisconsin, CS559

2. (c) 2002 University of Wisconsin, CS559
Today
Polygon Fill
Some methods for hidden surface removal
Homework 4 is online
It’s due two days before the midterm, but we will grade it in time and it’s useful study material
Midterm info
03/07/02
(c) 2002 University of Wisconsin, CS559

3. (c) 2002 University of Wisconsin, CS559
Sweep Fill Algorithms
Algorithmic issues:
Reduce to filling many spans
Which edges define the span of pixels to fill?
How do you update these edges when moving from span to span?
What happens when you cross a vertex?
03/07/02
(c) 2002 University of Wisconsin, CS559

4. (c) 2002 University of Wisconsin, CS559
Spans
Process - fill the bottom horizontal span of pixels; move up and keep filling
Have xmin, xmax for each span
Define:
floor(x): largest integer < x
ceiling(x): smallest integer >=x
Fill from ceiling(xmin) up to floor(xmax)
Consistent with convention
03/07/02
(c) 2002 University of Wisconsin, CS559

5. (c) 2002 University of Wisconsin, CS559
Algorithm
For each row in the polygon:
Throw away irrelevant edges
Obtain newly relevant edges
Fill span
Update current edges
Issues:
How do we update existing edges?
When is an edge relevant/irrelevant?
All can be resolved by referring to our convention about what polygon pixel belongs to
03/07/02
(c) 2002 University of Wisconsin, CS559

6. (c) 2002 University of Wisconsin, CS559
Updating Edges
Each edge is a line of the form:
Next row is:
So, each current edge can have it’s x position updated by adding a constant stored with the edge
Other values may also be updated, such as depth or color information
03/07/02
(c) 2002 University of Wisconsin, CS559

7. When are Edges Relevant (1)
Use figures and convention to determine when edge is irrelevant
For y<ymin and y>=ymax of edge
Similarly, edge is relevant when y>=ymin and y<ymax of edge
What about horizontal edges?
m’ is infinite
03/07/02
(c) 2002 University of Wisconsin, CS559

8. When are Edges Relevant (2)
Convex polygon:
Always only two edges active 2
1,2 1
1,3 3
3,4 4
03/07/02
(c) 2002 University of Wisconsin, CS559

9. When are Edges Relevant (3)
Horizontal edges? 2 2? 1 3
1,3 4? 4
03/07/02
(c) 2002 University of Wisconsin, CS559

10. (c) 2002 University of Wisconsin, CS559
Sweep Fill Details
Maintain a list of active edges in case there are multiple spans of pixels - known as Active Edge List.
For each edge on the list, must know: x-value, maximum y value of edge, m’
Maybe also depth, color…
Keep all edges in a table, indexed by minimum y value - Edge Table
For row = min to row=max
AEL=append(AEL, ET(row));
remove edges whose ymax=row
sort AEL by x-value
fill spans
update each edge in AEL
03/07/02
(c) 2002 University of Wisconsin, CS559

11. (c) 2002 University of Wisconsin, CS559
Edge Table
Row: 6 6 5 5 4 4 5 5 6 4 3 3 2 2 1 2 5 6 6 1 6 6 1 2 3 4 5 6
ymax
xmin
1/m
03/07/02
(c) 2002 University of Wisconsin, CS559

12. Active Edge List (shown just before filling each row)6 5 6 6 6 5 2 5 4 5 5 6 6 6 4 2 5 6 6 3 2 5 6 6 2 2 5 6 6 1 1 2 3 4 5 6 6 6
ymax x
1/m
03/07/02
(c) 2002 University of Wisconsin, CS559

13. (c) 2002 University of Wisconsin, CS559
Edge Table
Row: 6 6 5 5 4 4 3 3 2 2 1 2 1 5 6 -1 5 1 6 6 1 2 3 4 5 6
ymax
xmin
1/m
03/07/02
(c) 2002 University of Wisconsin, CS559

14. (c) 2002 University of Wisconsin, CS559
Active Edge List
Row: 6 6 5 5 4 3 -1 5 5 1 5 4 3 4 1 5 4 -1 5 3 2 3 1 5 5 -1 5 2 1 2 1 5 6 -1 5 1 6 6 1 2 3 4 5 6
ymax x
1/m
03/07/02
(c) 2002 University of Wisconsin, CS559

15. (c) 2002 University of Wisconsin, CS559
Comments
Sort is quite fast, because AEL is usually almost in order
Use bubble sort or insertion sort
OpenGL limits to convex polygons, meaning two and only two elements in AEL at any time, and no sorting
Can generate memory addresses (for pixel writes) efficiently
Does not require floating point - next slide
03/07/02
(c) 2002 University of Wisconsin, CS559

16. Dodging Floating Point (for integer endpoints)
For edge, m=x/y, which is a rational number
View x as xi+xn/y, with xn<y. Store xi and xn
Then x->x+m’ is given by:
xn=xn+x
if (xn>=y) { xi=xi+1; xn=xn- y }
Advantages:
no floating point
can tell if x is an integer or not, and get floor(x) and ceiling(x) easily, for the span endpoints
03/07/02
(c) 2002 University of Wisconsin, CS559

17. (c) 2002 University of Wisconsin, CS559
Anti-Aliasing
Recall: We can’t sample and then accurately reconstruct an image that is not band-limited
Infinite Nyquist frequency
Attempting to sample sharp edges gives “jaggies”, or stair-step lines
Solution: Band-limit by filtering (pre-filtering)
What sort of filter will give a band-limited result?
But when doing computer rendering, we don’t have the original continuous function
03/07/02
(c) 2002 University of Wisconsin, CS559

18. Pre-Filtered Primitives
We can simulate filtering by rendering “thick” primitives, with , and compositing
Expensive, and requires the ability to do compositing
Hardware method: Keep sub-pixel masks tracking coverage
Filter
=1/6
1/6
=2/3
2/3
over
=1/6
=1/6
1/6
=2/3
=1/6
Ideal
Pre-Filtered and composited
03/07/02
(c) 2002 University of Wisconsin, CS559

19. Post-Filtering (Supersampling)
Sample at a higher resolution than required for display, and filter image down
Two basic approaches:
Generate extra samples and filter the result (traditional super-sampling)
Generate multiple (say 4) images, each with the image plane slightly offset. Then average the images
Requires general perspective transformations
Can be done in OpenGL using the accumulation buffer
Issues of which samples to take, and how to average them
Method used in most current hardware
Note that anti-aliasing costs 4x regular rendering for 2x2 box filtering
03/07/02
(c) 2002 University of Wisconsin, CS559

20. (c) 2002 University of Wisconsin, CS559
Where We Stand
At this point we know how to:
Convert points from local to window coordinates
Clip polygons and lines to the view volume
Determine which pixels are covered by any given line or polygon
Anti-alias
Next thing:
Determine which polygon is in front
03/07/02
(c) 2002 University of Wisconsin, CS559

21. (c) 2002 University of Wisconsin, CS559
Visibility
Given a set of polygons, which is visible at each pixel? (in front, etc.). Also called hidden surface removal
Very large number of different algorithms known. Two main classes:
Object precision: computations that decompose polygons in world to solve
Image precision: computations at the pixel level
All the spaces in the viewing pipeline maintain depth, so we can work in any space
World, View and Canonical Screen spaces might be used
Depth can be updated on a per-pixel basis as we scan convert polygons or lines
03/07/02
(c) 2002 University of Wisconsin, CS559

22. (c) 2002 University of Wisconsin, CS559
Visibility Issues
Efficiency – it is slow to overwrite pixels, or scan convert things that cannot be seen
Accuracy - answer should be right, and behave well when the viewpoint moves
Must have technology that handles large, complex rendering databases
In many complex worlds, few things are visible
How much of the real world can you see at any moment?
Complexity - object precision visibility may generate many small pieces of polygon
03/07/02
(c) 2002 University of Wisconsin, CS559

23. Painters Algorithm (Image Precision)
Choose an order for the polygons based on some choice (e.g. depth to a point on the polygon)
Render the polygons in that order, deepest one first
This renders nearer polygons over further
Difficulty:
works for some important geometries (2.5D - e.g. VLSI)
doesn’t work in this form for most geometries - need at least better ways of determining ordering
Fails zs
Which point for choosing ordering? xs
03/07/02
(c) 2002 University of Wisconsin, CS559

24. The Z-buffer (1) (Image Precision)
For each pixel on screen, have at least two buffers
Color buffer stores the current color of each pixel
The thing to ultimately display
Z-Buffer stores at each pixel the depth of the nearest thing seen so far
Also called the depth buffer
Initialize this buffer to a value corresponding to the furthest point (z=1.0 for screen and window space)
As a polygon is filled in, compute the depth value of each pixel that is to be filled
if depth < z-buffer depth, fill in pixel color and new depth
else disregard
03/07/02
(c) 2002 University of Wisconsin, CS559

25. (c) 2002 University of Wisconsin, CS559
The Z-buffer (2)
Advantages:
Simple and now ubiquitous in hardware
A z-buffer is part of what makes a graphics card “3D”
Computing the required depth values is simple
Disadvantages:
Over-renders - worthless for very large collections of polygons
Depth quantization errors can be annoying
Can’t easily do transparency or filtering for anti-aliasing (Requires keeping information about partially covered polygons)
03/07/02
(c) 2002 University of Wisconsin, CS559

26. Z-Buffer and Transparency
Must render in back to front order
Otherwise, would have to store first opaque polygon behind transparent one
Front
Partially transparent
3rd
1st or 2nd
3rd: To know what to do now
Opaque
2nd
Opaque
1st
1st or 2nd
Recall this color and depth
03/07/02
(c) 2002 University of Wisconsin, CS559