Hidden Surface Elimination

Hidden Surface EliminationHidden Surface Elimination

Wen-Chieh (Steve) Lin

Institute of Multimedia Engineering

I-Chen Lin’ CG Slides, Rich Riesenfeld’s CG Slides,

Shirley, Fundamentals of Computer Graphics, Chap 8

ILE5014 Computer Graphics 10F 3

Hidden Surface Elimination (Removal)Hidden Surface Elimination (Removal)

• In a 3D wireframe displayer, the line segments between projected point pairs are drawn.

• To fill projected polygons, we have to remove “hidden surfaces”.


Hidden Surface Elimination (Removal)Hidden Surface Elimination (Removal)

• Object (Model) Space Algorithms

– Work in the model data space

• Image Space Algorithms

– Work in the projected space

– Most common domain for hidden surface removal


Hidden Surface RemovalHidden Surface Removal

• Object-space approach: use pairwise testing between polygons (objects)

• Worst case complexity O(n2) for n polygons


Backface CullingBackface Culling

View


Back Face Culling: Object SpaceBack Face Culling: Object Space

nn

n

n

v

v

v

v

z


Back Face Culling: Object SpaceBack Face Culling: Object Space

zn n

v

v

z

0nvVisible:


Back Face Culling TestBack Face Culling Test

• For Object Space look at sign of

• For Image Space look at sign ofz

n

nv


Back Face Culling: Image SpaceBack Face Culling: Image Space

nn

n

n

z


Back Face Culling: Image SpaceBack Face Culling: Image Space

n n

z

0z

nVisible:


Painter’s AlgorithmPainter’s Algorithm

• Render polygons in a back to front order so that polygons behind others are simply painted over

• Depth sort method


Depth Sort MethodDepth Sort Method

• Requires ordering of polygons first

– O(n log n) calculation for ordering

– Not every polygon is either in front or behind all other polygons

• Sort polygons and deal with easy cases first, harder later


Easy CasesEasy Cases

• A polygon lies behind all other polygons

– Can render

• Polygons overlap in z but not in either x or y

– Can render independently

z

z


Difficult CasesDifficult Cases

cyclic overlap penetration


Space PartitioningSpace Partitioning

• Avoid rendering an object when it’s unnecessary

– In many real-time applications, we want to eliminate as many objects as possible within the application.

– Reduce burden on pipeline

– Reduce traffic on bus

• Octree

• BSP tree


OctreeOctree

http://www.imagico.de/fast_iso/patch.html

http://www.imagico.de/fast_iso/patch.html


Why do we use BSP trees?Why do we use BSP trees?

• Hidden surface removal

– A back-to-front painter’s algorithm

• Partition space with Binary Spatial Partition (BSP) Tree


A Simple ExampleA Simple Example

top view

The plane of A separates B and C from D, E and F


Binary Space Partitioning TreeBinary Space Partitioning Tree

• Can continue recursively

– Plane of A separates C from D

– Plane of D separates E and F

• Can put this information in a BSP tree

– Use for visibility and occlusion testing


Key Ideas of BSPKey Ideas of BSP

• Assume T2 not cross the plane of T1

• If e and T2 on the same side, T1 won’t block

• If e and T2 on different sides, T1 may block


What if a triangle cross a plane of another triangle?What if a triangle cross a plane of another triangle?

• Break the triangle into three!


Creating a BSP treeCreating a BSP tree

0

0

11

2

2

33

4 -> 4f,4b

4f

4b 4f 4b5f

5b

5f

5b


Back-to-Front RenderBack-to-Front Render

Render(node, view){

if node is a leaf

draw this node to the screenelse

if the viewpoint is in back of the dividing line {

render(front subtree) draw node to screen render(back subtree) } else the viewpoint is in front of the dividing line {

render (back subtree) draw node to screen render (front subtree) }


0

0

11

2

2

33

4f

4b 4f 4b5f

5b

5f

5b

5b 1 4f 3 5f 4b 0 2



0

0

11

2

2

33

4f

4b 4f 4b5f

5b

5f

5b

2 0 5b 1 4f 3 5f 4b



Optimizing BSP TreeOptimizing BSP Tree

• Efficiency of tree traversal is more important than tree construction

• Speed determined by the number of nodes

• Fewer nodes if less node splitting

• Order of adding triangles determines the number of nodes in a BSP

• Choose a better ordering from a random collection of permutations

2

1b1f

1

2


BSP-based CullingBSP-based Culling

• Pervasively used in first person shooting games.

– Doom, quake….etc.

• Visibility test

• Skip objects that are “occluded”.a screen shot from Doom.


Image Space ApproachImage Space Approach


Image Space ApproachImage Space Approach

• Look at each projector (nm for an n m frame buffer) and find the k closest polygons

• Complexity O(nmk)

• z-buffer

• Ray casting


z-Buffer Algorithmz-Buffer Algorithm

• The z or depth buffer

– stores the depth of the closest object at each pixel found so far

• As we render each polygon, compare the depth of each pixel to depth in z buffer

– If less, place the shade of pixel in the color buffer and update z buffer


z-Buffer Algorithmz-Buffer Algorithm

Function setpixel(int i, int j, rgb c, real z)

If z > z-buffer(i, j) then

z-buffer(i, j) = z

screen(i,j) = c


A

C

BA’

C’ B

’

focus

Projection plane

Interpolation of Z valuesInterpolation of Z values


• To fill the polygon on the screen, we only fill the color and estimate the z value “pixel by pixel”.

• How to estimate z of in-between pixels ?

z1

z2

z3

za zbzt

Interpolation of Z valuesInterpolation of Z values


• Interpolation in screen space

– P(m) = P1 + m(P2 – P1)

• Interpolation in 3D space

– V(n) = V1 + n(V2 – V1)

– Py(n) / f = Vy(n) / Vz(n) P1

P2

V1

V2

m

n 212

1

zzmz

mzn

n ≠ m

Screen Space vs. 3D SpaceScreen Space vs. 3D Space

f


Suppose plane equation for P is

0 DzCyBxA

yBxADC

z 1

Solving for z,

Scan Conversion of Polygon PScan Conversion of Polygon P


y

x

1z 2

z

Scan Conversion of Polygon P (2)Scan Conversion of Polygon P (2)


1 111z D Ax ByC

is first z-value on scanline,1

z


1 121 ( )z D A x x ByC

Consider , next z-value on scanline,2z

),(112

yxxz


1 121 ( )z D A x x ByC

2 1A

Cz z x

In incremental terms,

1 111z D Ax ByC



Kzz 12

Assume ,1

x

And note ,KC

A

Therefore,

One add (subtract) generates the next iz



Incremental ApproachIncremental Approach

• Can efficiently generate z-values incrementally

• Can use Bresenham to grow the edges

efficiently after direct evaluation at vertices


Integer Z-bufferInteger Z-buffer

• If B bins are used for depth range (n, f)

• Bin size △z = (n-f)/B

• Smaller △z is desirable

– Depth of two objects are indistinguishable if △z is larger than the difference between their real z-values

– Increase B

– Move n and f closer


Bin size for z-buffer in world spaceBin size for z-buffer in world space

• Take perspective into account

• Differentiating both side

• Bin sizes vary in depth

• The bin size in world space is

wzfnfnz

ww

zz

fnz 2

zfnzz w

w

2


Bin size for z-buffer in world spaceBin size for z-buffer in world space

• Bin size in the world space increases as depth increases

• Objects farther away is more likely to be indistinguishable in depth (same integer z-values)

• Maximize n and minimize f for small △zw

zfnzz w

w

2

nzfz

w max

zw=f

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Hidden Surface Elimination