Hidden Surface Elimination

1
Hidden Surface Elimination
Visible Surface Determination
2
Approaches

• Back-Face Removal
• z-Buffer (Depth-Buffer)
• Scanline Algorithm
• Depth-Sort
• BSP-Tree

3
Back-Face Removal (Culling)

• Used to remove unseen polygons from convex,
  closed polyhedron (Cube, Sphere)
• Does not completely solve hidden surface problem
  since one polyhedron may obscure another

4
Back Face Algorithm

• Before the Viewing Transformation
• 1. List the vertices for each polygon
  counter-clockwise
• with respect to the front face.
• P1, P2, P3, P4
• P5, P8, P7, P6
• 2. Compute the equation of the plane for each
  polygon such that the normal vector points out of
  the front face.
• 3. If A eyex B eyey C eyez D polygon is a backface.
• The transform T(-PRP)RT(-VRP) takes the PRP to
  the origin.
• Therefore, the inverse will take the origin to
  PRP (the eye)
• (T(-PRP)RT(-VRP))-10,0,0,1 (eyex, eyey,
  eyez, 1)
• After the Perspective Projection
• 1. Look at the z-coordinate of the Normal vector

5
z- Buffer (Depth-Buffer)

• Z-Buffer has memory corresponding to each pixel
  location.
• Usually 16 to 20 bits/location.
• Initialize
• Each z-buffer location ? Max z value
• Each frame buffer location ? background
  color
• For each polygon
• Compute z(x,y), polygon depth at the pixel
  (x,y)
• If z(x,y)
• z buffer(x,y) ? z(x,y)
• pixel(x,y) ? color of polygon at (x,y)

6
z-Buffer (cont.)

• Advantages/Disadvantages
• LOTS of memory
• Modifications needed to implement antialiasing,
  transparency, translucency effects
• Linear performance
• Polygons may be processed in any order
• Commonly implemented in firmware/hardware ? very
  fast.

7
What z????

• Z-buffer is done after the perspective
  transformation since two different points in
  3-space may map to the same pixel.
• Example (100, 100, 100) and (200, 200, 200)
  with d 10 both map to pixel (10, 10).
• In deriving the perspective transformation we
  assumed either z 0 or zd!
• We need a z-value that preserves the depth
  relationship between objects after the
  perspective transformation.
• If A is in front of B, the projected_A should be
  in front of projected_B.
• Transforming points on a straight line should
  result in co-linear points.
• Transforming coplanar points in a polygon should
  not warp the polygon.

8
Old Value of z will not work

• We need a projection that takes the z-coordinate
  into a projected z-coordinate space
  that maintains lines and planes.

9
Scaling Z
10
Scaling Z (cont.)

• -1 ? x ? 1, -1 ? y ? 1, -1 ? z ? 0

(x, y, (z -zmin) /( 1 zmin), -z) Projected
3-D Point is ( x/(-z), y/(-z), (z -zmin )
/ (-z)( 1 zmin) )
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Computing Pixel z-values
The perspective projection gives you z-values for
the vertices of polygons. To find the z-values
for the boundary and interior pixels you do a
linear interpolation.

• When moving vertically zi1 zi ?zv, ? zv
  (z1-z3)/(y1-y3)
• When moving horizontally zi1 zi ? zh, ? zh
  (z1-z3)/(x1-x3)
• This could also be done by using the equation of
  the plane that transformed polygon lies in, but
  first you have to compute the equation of the
  plane. (The old one will not do!)

12
Scan Line Algorithm

• Idea is to intersect each polygon with a
  particular scanline. Solve hidden surface
  problem for just that scan line.
• Requires a depth buffer equal to only one scan
  line
• Maintain an active polygon and active edge list
• Can implement antialiasing as part of the
  algorithm

13
BSP Trees

• Binary Space Paritition is a relatively easy way
  to sort the polygons relative to the eyepoint
• To Build a BSP Tree
• 1. Choose a polygon, T, and compute the equation
  of the plane it defines.
• 2. Test all the vertices of all the other
  polygons to determine if they are in front of,
  behind, or in the same plane as T. If the plane
  intersects a polygon, divide the polygon at the
  plane.
• 3. Polygons are placed into a binary search tree
  with T as the root.
• 4. Call the procedure recursively on the left
  and right subtree.

14
BSP Tree Example
15
Traversing the BSP-Tree

• Traverse the BSP tree such that the branch
  descended first is the side that is away from the
  eyepoint. This can be determined by substituting
  the eye point into the plane equation for the
  polygon at the root.
• When there is no first branch to descend, or that
  branch has been completed then render the polygon
  at this node.
• After the current node's polygon has been
  rendered, descend the branch that is closer to
  the eyepoint.

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Traversing the BSP Tree Example
EYE 1
A
F2
C
B
Z
D
F1
D
E1
C
F1
E2
F2
E2
A
E1
B
X
-X
EYE 2
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Splitting Triangles
If all our polygons are triangles then we always
divide a triangle into more triangles when it is
intersected by the plane.

• It is possible for the number of triangles to
  increase exponentially but in practice it is
  found that the increase may be as small as two
  fold.
• A heuristic to help minimize the number of
  fractures is to enter the triangles into the
  tree in order from largest to smallest.