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Hidden Surface EliminationIntroduction
Need is to eliminate ambiguity
OR
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Hidden Surface EliminationIntroduction
Wire frame
Hidden Line Elimination Hidden Surface Elimination
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Hidden Surface EliminationIntroduction
Approaches
• Image SpaceThrough pixel
• Object SpaceThrough primitive
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Hidden Surface EliminationImage Space Approach
for (each pixel in the image){
determine the object closest to the viewer that isintercepted by the projector through the pixel;
draw the pixel in the appropriate color;}
Computational effort: npn : number of objectsp : number of pixels
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Hidden Surface EliminationObject Space Approach
for (each object in the world){
determine those parts of the object whose view isunobstructed by other parts of it or any other object;
draw those parts in the appropriate color;}
Computational effort: n2
n: number of objects
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Hidden Surface EliminationFloating Horizon Algorithm
Surface FunctionF(x,y,z)=0
x
z
z1= constantz2
z3x
y
zz1= constantz2
z3z4
z5
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Hidden Surface EliminationFloating Horizon Algorithm
With z=constant plane closest to the viewpoint, the curve ineach plane is generated (for each x coordinate in image spacethe appropriate y value is found).
x
y
z1z2
z3z4
z5
Projection on z=0 planeAlgorithm:If at any given value of x the y valueof the curve in the current plane islarger than the y value for anyprevious curve at that x value,then the curve is visible,otherwise it is hidden.
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Hidden Surface EliminationFloating Horizon Algorithm
x
y
z1z2
z3z4
z5
Projection on z=0 plane
Algorithm:If at any given value of x the y valueof the curve in the current plane islarger than the y value or smallerthan the minimum y value for anyprevious curve at that x value,then the curve is visible,otherwise it is hidden.
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Hidden Surface EliminationBack Face Culling
Preprocessing to eliminate faces which are not visible
P
np
If a surface’s normal is pointing away from theeye (viewer), then this is a back face
backface then Vn If p 0<⋅
V
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Hidden Surface EliminationBack Face Culling
z
x
Conservative algorithm
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Hidden Surface EliminationBack Face Culling
z
x
Conservative algorithm
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Hidden Surface EliminationZ-Buffer Algorithm
Proposed by Catmull 1974Simple to implementZ-buffer is like a frame buffer, contains depth
Zb(x, y)C(x, y)
(x, y)
(x, y)
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Hidden Surface EliminationZ-Buffer Algorithm
Initialize all d[i,j]=1.0 (max depth), c[i,j]=background color.for (each polygon) for (each pixel in polygon’s projection) { Find depth-z of polygon at (x,y) corresponding to pixel (i,j); If z < d[i,j] d[i,j] = z; p[i,j] = color; end }
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Hidden Surface EliminationZ-Buffer Algorithm
1)x( CA
zxCA
zz
xCA
zz
CDByx)xA
z
xx At
CC
DByAxz
x AtDCzByAx
=−=−=
−=−
+++−=
+
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ÄÄ
Ä
)Ä((Ä
0,)(
0
1
1
1
Q
Computationally
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Hidden Surface EliminationZ-Buffer Algorithm
Example
∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞
∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞
Z-buffer Screen
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Hidden Surface EliminationZ-Buffer Algorithm
Example
[0,1,5]
[0,7,5] [6,7,5]
5 5 5 55 5 5 55 5 5 55 5 5 55 5 5 ∞5 5 ∞ ∞5 ∞ ∞ ∞∞ ∞ ∞ ∞
5 5 5 ∞5 5 ∞ ∞5 ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞
Z-buffer
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Hidden Surface EliminationZ-Buffer Algorithm
Example
[0,1,2]
[0,6,7]
[5,1,7]
76 75 6 74 5 6 73 4 5 62 3 4 5
76 7
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Hidden Surface EliminationZ-Buffer Algorithm
Example
5 5 5 55 5 5 55 5 5 55 5 5 54 5 5 73 4 5 62 3 4 5∞ ∞ ∞ ∞
5 5 5 ∞5 5 ∞ ∞5 ∞ ∞ ∞∞ ∞ ∞ ∞∞ ∞ ∞ ∞7 ∞ ∞ ∞6 7 ∞ ∞∞ ∞ ∞ ∞
Z-buffer Screen
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Hidden Surface EliminationZ-Buffer Algorithm
• Simple method• Complexity
Time : nxm buffer k poygonsO(nmk)
Space : b depth precisionO(nmb)
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Hidden Surface EliminationPainter’s Algorithm
Depth Sort, List PriorityPolygons are painted to the screen in the order of theirdistance from the viewer (More distant objects are paintedfirst)
Screen display
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Hidden Surface EliminationPainter’s Algorithm
• Sort polygons in order of increasing depth• Draw polygons in sequence, starting from the polygon(surface) of greatest depth
• Careful processing of depth
• Efficiency depends on sorting algorithm
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Hidden Surface EliminationPainter’s Algorithm
Z=20 Z=15 Z=10
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Hidden Surface EliminationPainter’s Algorithm
P
QR
z
P > Q > R
P
Qz
P > Q
Draw first P then Q and then R
Draw first P then Q
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Hidden Surface EliminationPainter’s Algorithm
P
Q
Cyclic overlappingMay require splitting
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Hidden Surface EliminationBinary Space Partitioning (BSP)
BSP Tree
Space partitioning using planes
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Hidden Surface EliminationBinary Space Partitioning (BSP)
BSP Tree
For hidden surface elimination: sets a display order
PB
F
P
BF
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Hidden Surface EliminationBinary Space Partitioning (BSP)
BSP Tree
For hidden surface elimination: sets a display order
PB
F
P
BF
Eye
Display order (back to front): BPF
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Hidden Surface EliminationBinary Space Partitioning
3
5
5b
5a
2
14
3
45b
front back
125a
Example
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Hidden Surface EliminationBinary Space Partitioning
3
5
5b
5a
2
14
3
45b
front back
2
5a 1
Example
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Hidden Surface EliminationBinary Space Partitioning
3
3
5
5b
5a
2
14
front back
2
5a 1
4
5b
Example
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Hidden Surface EliminationBinary Space Partitioning
3
3
5
5b
5a
2
14
front back
2
5a 1
4
5b
Example
Display order: 5a 2 1 3 5b 4V
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Hidden Surface EliminationBinary Space Partitioning
3
5
2
14
5
front
back
1
2
3
4
Example
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Hidden Surface EliminationBinary Space Partitioning
3
5
2
14
5
front
back
1
2
3
4
Example
V
Display order: 5 2 1 3 4
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Hidden Surface EliminationBinary Space Partitioning
Issues
How to select the root polygon?
Criteria: Number of split (fragmentation)
a. Arbitrary
b. Heuristic based with 4-5 polygons and consider which gives the leastnumber of split
Static vs Dynamic scene
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Hidden Surface EliminationArea Subdivision
Quad tree
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Hidden Surface EliminationArea Subdivision AlgorithmWarnock’s Algorithm
Polygon
Area of interest
Surrounding Intersecting Contained Disjoint
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Hidden Surface EliminationArea Subdivision Algorithm
Consider an area of the projected image
If it is easy to decide which polygons are visible inthe area, displayElse the area is subdivided into smallerareas and the decision is made recursively
Divide and Conquer
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Hidden Surface EliminationArea Subdivision Algorithm
No Subdivision is required if1. All the polygons are disjoint: background color in the area.
2. Only one intersecting or only one contained polygon: The area isfilled first by background color, then the polygon part contained inthe area.
3. Only one surrounding polygon (no contained and intersectingpolygons): The area is filled with the color of the surroundingpolygon.
4. More than one polygon is intersecting, contained in, orsurrounding the area, with surrounding polygon in front: Fill the areawith the color of the surrounding polygon.
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Hidden Surface EliminationWarnock’s Algorithm
Area
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Hidden Surface EliminationWarnock’s Algorithm
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Hidden Surface EliminationWarnock’s Algorithm
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Hidden Surface EliminationWarnock’s Algorithm
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Hidden Surface EliminationWarnock’s Algorithm
Maximum subdivision: pixel!
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Hidden Surface EliminationWeiler Atherton Algorithm
Subdivision along polygon boundaries
Clipping!
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Hidden Surface EliminationWeiler Atherton Algorithm
• Initial z-sort• Consider front most polygon, clip all polygons to generate
Fragments inside polygon and outside polygon• All inside polygons if behind delete from the list• If there is an inside polygon in front (offending), clip the inside list of polygons against this polygon• Process the outside polygon(s)
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Hidden Surface EliminationWeiler Atherton Algorithm
A
BinA
BoutA
A as clip polygonInside list: A, BinAOutside list: BoutA
Display A
Process BoutA
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Hidden Surface EliminationWeiler Atherton Algorithm
A
B
x
y
z
x
y
z
AB
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Hidden Surface EliminationWeiler Atherton Algorithm
A as clip polygon
Inside list: A, BinAOutside list: BoutA
BinA
BoutA
BinA as clip polygon
Inside list: BinA, AinBOutside list: BoutA, AoutB
Display BinA
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Hidden Surface EliminationWeiler Atherton AlgorithmMore polygons with more fragments
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Based on
Scan-line coherence across multiple scan-lines orspan-coherence !
– scan-conversion algorithm and a little more datastructure
Hidden Surface EliminationScan Line Algorithm
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Hidden Surface EliminationScan Line Algorithm
Spans
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• Each scan line is subdivided into several "spans"• Determine which polygon the current span belongs to• Shade the span using the current polygon’s color• Exploit "span coherence" :• For each span, a visibility test may need to be done
Hidden Surface EliminationScan Line AlgorithmOverview
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Hidden Surface EliminationScan Line Algorithm
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• A scan line is subdivided into a sequence of spans• Each span can be "inside" or "outside" polygon areas
– "outside“: no pixels need to be drawn (backgroundcolor)
– "inside“: can be inside one or multiple polygons• If a span is inside one polygon, the pixels in the span will
be drawn with the color of that polygon• If a span is inside more than one polygon, then compare
the z values of those polygons at the scan line edgeintersection point to determine the color of the pixel
Hidden Surface EliminationScan Line Algorithm
Inside/Outside Spans
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• When a scan line intersects an edge of a polygon– for the 1st time, the span becomes "inside" of the
polygon from that intersection point on– for a 2nd time, the span becomes "outside“ of the
polygon from that point on• A flag "in/out" for each polygon is used to keep track of the
current state• Initially, the in/out flag is set to be "outside" (value = 0 for
example). Invert the tag for “inside”.
Hidden Surface EliminationScan Line Algorithm
Inside/Outside Spans
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• Each polygon will have its own in/out flag• There can be more than one polygon having the in/out
flags to be "in" at a given instance• Keep track of polygons the scan line is currently in• If there are more than one polygon "in", perform z value
comparison to determine the color of the scan line span
Hidden Surface EliminationScan Line Algorithm
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x ymax ∆x poly-IDET
PT poly-ID A,B,C,D color in/out flag
Hidden Surface EliminationScan Line AlgorithmData StructureEdge Table (ET)Polygon Table (PT)
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In addition,
Use active In-Polygon List (IPL)
Active Edge Table (AET)
Hidden Surface EliminationScan Line Algorithm
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S T
a
b
c
1
2
3
X0
I
I I I
I I
I V
XN
BG
Example
Hidden Surface EliminationScan Line Algorithm
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Y AET IPL
I x0, ba, bc, xN BG, BG+S, BG
II x0, ba, bc, 32, 13, xN BG, BG+S, BG, BG+T, BG
IIIIII xx00, , baba, 32, ca, 13,, 32, ca, 13, x xNN BG, BG+S, BG+S+T, BG+T, BGBG, BG+S, BG+S+T, BG+T, BG
IV x0, ba, ac, 12, 13, xN BG, BG+S, BG, BG+T, BG
Hidden Surface EliminationScan Line Algorithm Example
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Hidden Surface EliminationScan Line Algorithm
S T
a
b
c
1
2
3
X0
I
I I I
I I
I V
XN
BG
Y AET IPL
I x0, ba, bc, xN BG, BG+S, BG
Example
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Hidden Surface EliminationScan Line Algorithm
S T
a
b
c
1
2
3
X0
I
I I I
I I
I V
XN
BG
Y AET IPL
II x0, ba, bc, 32, 13, xN BG, BG+S, BG, BG+T, BG
Example
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Hidden Surface EliminationScan Line Algorithm
S T
a
b
c
1
2
3
X0
I
I I I
I I
I V
XN
BG
Y AET IPL
IIIIII xx00,, ba ba, 32, ca, 13,, 32, ca, 13, x xNN BG, BG+S, BG+S+T, BG+T, BGBG, BG+S, BG+S+T, BG+T, BG
Example
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Hidden Surface EliminationScan Line Algorithm
S T
a
b
c
1
2
3
X0
I
I I I
I I
I V
XN
BG
Y AET IPL
IV x0, ba, ac, 12, 13, xN BG, BG+S, BG, BG+T, BG
Example
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Hidden Surface EliminationScan Line Algorithm Nonpenetrating
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S
I
T
a
2
BG
b
c
3
d
e
1
Hidden Surface EliminationScan Line Algorithm
I x0, ba , 23, ec, ad, 13, xN BG, BG+S, BG+S+T,BG+S+T+, BG+T, BG
Penetrating