Graphics

Document Sample

```					    159.235 Graphics & Graphical
Programming
Lecture 26 - Visible & Hidden Surface
Determination

159.235              Graphics                1
Hidden Surface Removal
• Inventing renderers led to geometric models, a call for
more realism and therefore lots of polygons.
• All this led to intense interest in finding efficient
algorithms for hidden surface removal.
• Some algorithms are more correctly called visible
surface algorithms but the two names are used
interchangeably.
• Today we’ll look at several out of many generated
during the period of intense interest (around late 60s to
late 70s).

159.235                  Graphics                      2
Visibility of primitives
• We don’t want to waste time rendering primitives
which don’t contribute to the final image.
• A scene primitive can be invisible for 3 reasons:
– Primitive lies outside field of view
– Primitive is back-facing (under certain
conditions)
– Primitive is occluded by one or more objects
nearer the viewer
• How do we remove these efficiently?
• How do we identify these efficiently?

159.235                Graphics                      3
The Visibility Problem.
Two problems remain:
(Clipping we have covered)
• Removal of faces facing away from the viewer.
• Removal of faces obscured by closer objects.

159.235                   Graphics                4
Visible Surface Algorithms
Hidden/Visible Surface/Line Elimination/Determination
• Requirements
– Handle diverse set of geometric primitives
– Handle large number of geometric primitives
Classification: Sutherland, Sproull, Schumacher (1974):
• Object Space
– Geometric calculations involving polygons
– Floating point precision: Exact
– Often process scene in object order
• Image Space
– Visibility at pixel samples
– Integer precision
– Often process scene in image order
159.235                     Graphics                     5
Back Face Culling
• We saw in modelling, that the vertices of
polyhedra are oriented in an
anticlockwise manner when viewed from
outside – surface normal N points out.
• Project a polygon.
– Test z component of surface normal.
If negative – cull, since normal
points away from viewer.
– Or if N.V > 0 we are viewing the
back face so polygon is obscured.
• Only works for convex objects without
holes, ie. closed orientable manifolds.

159.235                          Graphics   6
Back Face Culling
• Back face culling can be applied anywhere in the
pipeline: world or eye coords, NDC, image space.
• Where is the best point? What portion of the scene
is eliminated, on average?
– Depends on application
• If we clip our scene to the view frustrum, then
remove all back-facing polygons – are we done?
• NO! Most views involve overlapping polygons.

159.235                Graphics                       7
How de we handle overlapping?

How about drawing the polygons in the “right order” so that we
get the correct result ( eg. blue, then green, then peach)?
Is it just a sorting problem ? Yes it is for 2D, but in 3D we can
encounter intersecting polygons or groups of non-intersecting
polygons which form a cycle where order is impossible (later).
159.235                      Graphics                          8
Simple Z-buffering
• Simple to include in scanline algorithm.
• Interpolate z during scan conversion.
• Maintain a depth (range) image in the frame
buffer (16 or 24 bits common ).
• When drawing, compare with the currently
stored z value.
• Pixel given intensity of nearest polygon.
159.235             Graphics                9
Implementation
• Initialise frame buffer to background colour.
• Initialise depth buffer to z = max. value for
far clipping plane, ie. LHCS
• Need to calculate value for z for each pixel
– But only for polygons intersecting that pixel.
– Could interpolate from values at vertices.
• Update both frame and depth buffer.

159.235                   Graphics                      10
Determining Depth
Use plane equation :
Ax  By  Cz  D  0
If at ( x, y ), z value evaluates to z1 ,
at ( x  Δx,y) , now value of z is :
 D  Ax  By
z                                   A
C                       z1  (x)
C

-Only one subtraction needed
-Depth coherence.

159.235                    Graphics                             11
Why is z-buffering so popular ?
• Simple to implement in hardware.
– Memory for z-buffer is now not expensive
• Diversity of primitives – not just polygons.
• Unlimited scene complexity
• Don’t need to calculate object-object intersections.
• Extra memory and bandwidth
• Waste time drawing hidden objects
• Z-precision Errors
• May have to use point sampling
159.235                     Graphics                     12
Z-compositing                            Colour photograph.

Can use depth other than
from polygons.
Laser range return.
Reflected laser power

Data courtesy of UNC.

159.235                 Graphics                              13
Ray Casting
• Sometimes referred to as Ray-tracing.
• Involves projecting an imaginary ray from the centre of
projection (the viewers eye) through the centre of each
pixel into the scene.

Scene

Eyepoint

Window

159.235                    Graphics                         14
Computing Ray-Object Intersections
• The heart of ray tracing.
• E.g sphere ( the easiest ! ).
Expand, substitute for x,y & z.
Express line in parametric form.
Gather terms in t.
x  x0  tx ; y  y  ty ; z  z  tz                  Quadratic equation in t.

Solve for t.
Equation for a sphere:
-No roots – ray doesn’t intersect.
( x  a ) 2  ( y  b) 2  ( z  c ) 2  r 2
- 1 root – ray grazes surface.
- 2 roots – ray intersects sphere,
(entry and exit)

159.235                                     Graphics                                     15
Ray-Polygon Intersection
•         Not so easy !
1.      Determine whether ray intersects polygon’s plane.
2.      Determine whether intersection lies within polygon.
•         Easiest to determine (2) with an orthographic projection
onto the nearest axis and the 2D point-in-polygon test.

z

Ray

x

y
159.235                             Graphics                       16
Ray Casting
• Easy to implement for a variety of primitives – only need a
ray-object intersection function.
• Pixel adopts colour of nearest intersection.
• Can draw curves and surfaces exactly – not just triangles !
• Can generate new rays inside the scene to correctly handle
visibility with reflections, refraction etc – recursive ray-
tracing.
• Can be extended to handle global illumination.
• Can perform area-sampling using ray super-sampling.
• But… too expensive for real-time applications.

159.235                     Graphics                        17
Examples of Ray-traced
images.

159.235        Graphics   18
Painters Algorithm (object space)
• Draw surfaces in back
to front order – nearer
polygons “paint” over
farther ones.
• Supports transparency.
• Key issue is order
determination.
• Doesn’t always work
– see image at right.

159.235                 Graphics   19
BSP (Binary Space Partitioning) Tree
•One of class of “list-priority” algorithms – returns
ordered list of polygon fragments for specified view
point (static pre-processing stage).                                            5

•Choose polygon arbitrarily
2
•Divide scene into front (relative to normal) and
back half-spaces.                                                         3
1                      4
•Split any polygon lying on both sides.

•Choose a polygon from each side – split scene
again.

•Recursively divide each side until each node              View of scene from above
contains only 1 polygon.

159.235                                      Graphics                                20
BSP Tree                                                                 5
•Choose polygon arbitrarily                                            5a
2                         5b
•Divide scene into front (relative to
normal) and back half-spaces.
3
1                         4
•Split any polygon lying on both sides.

•Choose a polygon from each side – split
scene again.

•Recursively divide each side until each node
front    3            back
contains only 1 polygon.
1              4
2             5b
5a

159.235                                 Graphics                                  21
BSP Tree
5
5a
2                                 5b
•Choose polygon arbitrarily

•Divide scene into front (relative to                                        3
1                               4
normal) and back half-spaces.

•Split any polygon lying on both sides.

•Choose a polygon from each side –
split scene again.
front           3            back
•Recursively divide each side until each                         2                 4
node contains only 1 polygon.                         front
5b
5a          1

159.235                                 Graphics                                            22
BSP Tree
5
5a
2                                  5b
•Choose polygon arbitrarily
3
•Divide scene into front (relative to                     1                                4
normal) and back half-spaces.

•Split any polygon lying on both sides.

•Choose a polygon from each side – split
scene again.                                                             3
front                        back

•Recursively divide each side until                              2                4
front
each node contains only 1 polygon.
5a          1                5b

159.235                                 Graphics                                             23
BSP Tree
5

2
•Choose polygon arbitrarily
3
•Divide scene into front (relative to                  1                                  4
normal) and back half-spaces.

•Split any polygon lying on both sides.

•Choose a polygon from each side – split        5     back
scene again.                                              4       back
Alternate
•Recursively divide each side until each                           3                          formulation
front
starting at 5
node contains only 1 polygon.                                 1            back
2

159.235                                 Graphics                                                      24
Displaying a BSP Tree
• Once we have the regions – need priority list
• BSP tree can be traversed to yield a correct priority list for
an arbitrary viewpoint.
• Start at root polygon.
– If viewer is in front half-space, draw polygons behind root first,
then the root polygon, then polygons in front.
– If polygon is on edge – either can be used.
– Recursively descend the tree.
• If eye is in rear half-space for a polygon – then can back
face cull.

159.235                            Graphics                                 25
BSP Tree
• A lot of computation required at start.
– Try to split polygons along good dividing plane
– Intersecting polygon splitting may be costly
• Cheap to check visibility once tree is set up.
• Can be used to generate correct visibility for
arbitrary views.
 Efficient when objects don’t change very
often in the scene.
159.235                  Graphics                    26
Warnock’s Algorithm
• Elegant hybrid of object-space and image-space.
• Uses standard graphics solution:- if situation too
complex then subdivide problem.
– If zero or one intersecting, contained or surrounding
polygon then scan convert window
– Else subdivide window as quadtree
– Recurse until zero or one polygon, or some set depth
– Depth may be pixel resolution, display nearest polygon
159.235                      Graphics                       27
Warnock’s Example

159.235          Graphics     28
Warnock Performance Measure
• Warnock’s algorithm:
– Screen-space subdivision (screen resolution, r = w*h)
hybrid object-space & image-space algorithm good for
relatively few static primitives, precise.
– Working set size (memory requirement): O(n)
– Storage overhead (over & above model): O(n lg r)
– Time to resolve visibility to screen precision: O(n*r)
– Overdraw (depth complexity – how often a typical
pixel is written by rasterization process): none
159.235                      Graphics                       29
BSP Performance Measure
• Tree construction and traversal (object-space
ordering algorithm – good for relatively few static
primitives, precise)
• Working set size (depends on application): O(1),
O(lg n)
• Time to resolve visibility to screen precision:
O(n2)
• Overdraw: maximum

159.235                 Graphics                    30
Z-buffer Performance
• Brute-force image-space algorithm scores best for
complex scenes – not very accurate but is easy to
implement and is very general.
• Working set size: O(1)
• Time to resolve visibility to screen precision: O(n)
• Overdraw: maximum
• But even O(n) is now intolerable!
159.235                 Graphics                     31
Example Architectural scenes
Here there can be an enormous amount of occlusion

159.235                         Graphics                      32
Occlusion at various levels

159.235               Graphics          33
Cells & Portals (object-space)
D       F

Model scene as a graph:                  C       B       E

• Nodes: Cells (or rooms)                        A       G
• Edges: Portals (or doors)

Graph gives us:
• Potentially visible set
• Superset of visible polygons                       A
• Room to room visibility
• Not a complete solution !                          B

C       D       E

159.235                      Graphics                           34
Example Application : Quake game engine

• Calculates visibility separately for
environment and objects.
• Environment
– Use portals to determine potentially visible set
– Use BSP-tree to order polygons front-back.
– Scan-convert polygons maintaining order.

159.235                   Graphics                        35
Example : Quake Game Engine
• Scan converting polygons
– Maintain front-back ordering.
– Use visibility mask on scanline.

F          - Create active edge list for polygons F & B.
Scanline              - Write in front-back order.
- Mask values once written to buffer.
- Write colour to framebuffer
B
- Write z to z-buffer

159.235                    Graphics                                  36
Example : Quake Game Engine
• Calculates visibility separately for environment
and objects.
• Environment
–    Use portals to determine potentially visible set
–    Use BSP-tree to order polygons front-back.
–    Scan-convert polygons maintaining order.
–    Maintain colour and z buffers.
• Objects.
– Use Z-buffer from environment stage.

159.235                          Graphics                    37
Other Applications

• Outdoor environments:
– Urban regions, forests, natural scenes in general
• Or very complex assemblies: mechanical
CAD parts (Boeing 777 engine block)
• Molecular visualization

Very hard and still not solved problem
159.235                    Graphics                   38
Z-Buffer & BSP Not Only
Approaches
• Other approaches:
– Depth sort – Newell, Newell & Sancha
– Scan-line algorithms
– Weiler-Atherton subdivision on polygon edges

159.235                 Graphics                      39
Visible Surfaces - Summary
• Visible and Hidden Surfaces
• Z-Buffering
• BSP

• Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whose
Graphics Course some of these slides were adapted.

159.235                       Graphics                           40

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 35 posted: 4/29/2011 language: Danish pages: 40