1. © University of Wisconsin, CS559 Spring 2004
Last Time
Drawing Polygons
Fill rules: non-exterior, parity, non-zero winding number
Sweep fill algorithm (also called scanline algorithm)
Fill one row at a time
Keep track of left and right edges, update incrementally from one row to the next
Bottom of edges is in, top of edges is out, remove horizontal edges
Simplest for convex, can be extended to concave
Antialiasing
Smooth objects before drawing – pre-filtering
Supersample and smooth results – post-filtering
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2. © University of Wisconsin, CS559 Spring 2004
Today
More anti-aliasing
Visibility (what’s in front)
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© University of Wisconsin, CS559 Spring 2004

3. Alpha-based Anti-Aliasing
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Set the  of a pixel to simulate a thick line
The pixel gets the line color, but with <=1
This supports the correct drawing of primitives one on top of the other
Draw back to front, and composite each primitive over the existing image
Only some hidden surface removal algorithms support it
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4. © University of Wisconsin, CS559 Spring 2004
Calculating 
Consider a line as having thickness (all good drawing programs do this)
Consider pixels as little squares
Set  according to the proportion of the square covered by the line
The sub-pixel coverage interpretation of 
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5. © University of Wisconsin, CS559 Spring 2004
Weighted Sampling
Instead of using the proportion of the area covered by the line, use convolution to do the sampling
Equivalent to filtering the line then point sampling the result
Place the “filter” at each pixel, and integrate product of pixel and line
Common filters are cones (like Bartlett) or Gaussians
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6. © University of Wisconsin, CS559 Spring 2004
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
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7. © University of Wisconsin, CS559 Spring 2004
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 operate on primitives
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
Actually, run Bresenham-like algorithm on z and w before perspective divide
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8. © University of Wisconsin, CS559 Spring 2004
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 environments, 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
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9. 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
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10. 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 canonical 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
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11. © University of Wisconsin, CS559 Spring 2004
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 – rasterizes polygons even if they are not visible
Depth quantization errors can be annoying
Can’t easily do transparency or filter-based anti-aliasing (Requires keeping information about partially covered polygons)
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12. 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
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13. © University of Wisconsin, CS559 Spring 2004
OpenGL Depth Buffer
OpenGL defines a depth buffer as its visibility algorithm
The enable depth testing: glEnable(GL_DEPTH_TEST)
To clear the depth buffer: glClear(GL_DEPTH_BUFFER_BIT)
To clear color and depth: glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
The number of bits used for the depth values can be specified (windowing system dependent, and hardware may impose limits based on available memory)
The comparison function can be specified: glDepthFunc(…)
Sometimes want to draw furthest thing, or equal to depth in buffer
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© University of Wisconsin, CS559 Spring 2004

14. The A-buffer (Image Precision)
Handles transparent surfaces and filter anti-aliasing
At each pixel, maintain a pointer to a list of polygons sorted by depth, and a sub-pixel coverage mask for each polygon
Matrix of bits saying which parts of the pixel are covered
Algorithm: When drawing a pixel:
if polygon is opaque and covers pixel, insert into list, removing all polygons farther away
if polygon is transparent or only partially covers pixel, insert into list, but don’t remove farther polygons
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15. © University of Wisconsin, CS559 Spring 2004
The A-buffer (2)
Algorithm: Rendering pass
At each pixel, traverse buffer using polygon colors and coverage masks to composite:
Advantage:
Can do more than Z-buffer
Coverage mask idea can be used in other visibility algorithms
Disadvantages:
Not in hardware, and slow in software
Still at heart a z-buffer: Over-rendering and depth quantization problems
But, used in high quality rendering tools
over =
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© University of Wisconsin, CS559 Spring 2004

16. Scan Line Algorithm (Image Precision)
Assume polygons do not intersect one another
Except maybe at edges or vertices
Observation: across any given scan line, the visible polygon can change only at an edge
Algorithm:
fill all polygons simultaneously, using concave-handling version of sweep-fill algorithm
at each scan line, have all edges that cross scan line in Active Edge List
keep record of current depth at current pixel - use to decide which is in front in filling span
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17. © University of Wisconsin, CS559 Spring 2004
Scan Line Algorithm (2) zs zs xs
Spans xs zs
Where polygons overlap, draw front polygon
Spans xs
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18. © University of Wisconsin, CS559 Spring 2004
Scan Line Algorithm (3)
Advantages:
Simple
Potentially fewer quantization errors (more bits available for depth)
Don’t over-render (each pixel only drawn once)
Filter anti-aliasing can be made to work (have information about all polygons at each pixel)
Disadvantages:
Invisible polygons clog Active Edge List, Edge Table
Non-intersection criteria may be hard to meet
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© University of Wisconsin, CS559 Spring 2004

19. Depth Sorting (Object Precision, in view space)
An example of a list-priority algorithm
Sort polygons on depth of some point
Render from back to front (modifying order on the fly)
Rendering: For surface S with greatest depth
If no overlap in depth with other polygons, scan convert
Else, for overlaps in depth, test for overlaps in the image plane
If none, scan convert and go to next polygon
If S, S’ overlap in depth and in image plane, swap order and try again
If S, S’ have been swapped already, split and reinsert
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20. © University of Wisconsin, CS559 Spring 2004
Depth Sorting (2)
Testing for overlaps: Start drawing when first condition is met:
x-extents or y-extents do not overlap
S is behind the plane of S’
S’ is in front of the plane of S
S and S’ do not intersect in the image plane S S S’ or S’ z S S’ x S z S’ x S S’
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21. © University of Wisconsin, CS559 Spring 2004
Depth sorting
Advantages:
Filter anti-aliasing works fine
Composite in back to front order with a sequence of over operations
No depth quantization error
Depth comparisons carried out in high-precision space, before rasterization
Disadvantages:
Over-rendering
Potentially very large number of splits - (n2) fragments from n polygons
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22. © University of Wisconsin, CS559 Spring 2004
Area Subdivision
Exploits area coherence: Small areas of an image are likely to be covered by only one polygon
Three easy cases for determining what’s in front in a given region:
a polygon is completely in front of everything else in that region
no surfaces project to the region
only one surface is completely inside the region, overlaps the region, or surrounds the region
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© University of Wisconsin, CS559 Spring 2004

23. Warnock’s Area Subdivision (Image Precision)
Start with whole image
If one of the easy cases is satisfied (previous slide), draw what’s in front
Otherwise, subdivide the region and recurse
If region is single pixel, choose surface with smallest depth
Advantages:
No over-rendering
Anti-aliases well - just recurse deeper to get sub-pixel information
Disadvantage:
Tests are quite complex and slow
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24. © University of Wisconsin, CS559 Spring 2004
Warnock’s Algorithm
Regions labeled with case used to classify them:
One polygon in front
Empty
One polygon inside, surrounding or intersecting
Small regions not labeled
Note it’s a rendering algorithm and a HSR algorithm at the same time
Assuming you can draw squares 2 3 2 2 3 3 3 2 3 3 3 1 3 1 1 1 1 3 3 3 3 3 2 3 3 3 2 2 2 2
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© University of Wisconsin, CS559 Spring 2004

25. BSP-Trees (Object Precision)
Construct a binary space partition tree
Tree gives a rendering order
A list-priority algorithm
Tree splits 3D world with planes
The world is broken into convex cells
Each cell is the intersection of all the half-spaces of splitting planes on tree path to the cell
Also used to model the shape of objects, and in other visibility algorithms
BSP visibility in games does not necessarily refer to this algorithm
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26. © University of Wisconsin, CS559 Spring 2004
BSP-Tree Example A A C 4 3 - + B C - B + - + 1 3 2 4 1 2
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27. © University of Wisconsin, CS559 Spring 2004
Project 2 Intro
You are given the following:
Rooms, defined in 2D by the edges that surround the room
The height of the ceiling
Each edge is marked opaque or clear
For each clear edge, there is a pointer to the thing on the other side
You know where the viewer is and what the field of view is
The viewer is given as (cx,cy,cz) position
The view frustum is given as a direction vector (dx,dy,dz) and an angle for the field of view
(dx,dy,dz)
(cx,cy,cz)
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28. © University of Wisconsin, CS559 Spring 2004
Project 2 (2)
You don’t have everything needed to do this project yet, but you can do the hardest part
Represent the frustum as a left and right clipping line
You don’t have to worry about the top and bottom
Each clip line starts at the viewer’s position and goes to infinity in the viewing direction
Write a procedure that clips an edge to the view frustum
This takes a frustum and returns the endpoints of the clipped edge, or a flag to indicate that the edge is not visible
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29. © University of Wisconsin, CS559 Spring 2004
Project 2 (3)
Write a procedure that takes a room and a frustum, and draws the room
Clip each edge to the frustum
If the edge is visible, draw the wall that the edge represents
Create the 3D wall from the 2d piece of edge
Project the vertices
Draw the polygon in 2D
If the edge is clear, do nothing right now
Draw the floor and ceiling first, because they will be behind everything
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© University of Wisconsin, CS559 Spring 2004