Post on 14-Dec-2015
From Vertices to Fragments
Software College, Shandong University
Instructor: Zhou Yuanfeng E-mail: yuanfeng.zhou@gmail.com
Review
•Shading in OpenGL;•Lights & Material;•From vertex to fragment:
•Cohen-Sutherland2
ProjectionFragmentsClippingShading
Black box Surface hiddenTextureTransparency
3
Objectives
•Geometric Processing:
Cyrus-Beck clipping algorithm
Liang-Barsky clipping algorithm• Introduce clipping algorithm for polygons•Rasterization: DDA & Bresenham
Cohen-Sutherland
• In case IV:
•o1 & o2 = 0
• Intersection: Clipping Lines by Solving Simultaneous Equations
4
Solving Simultaneous Equations
•Equation of a line: Slope-intercept: y = mx + h difficult for vertical line
Implicit Equation: Ax + By + C = 0
Parametric: Line defined by two points, P0 and P1
• P(t) = P0 + (P1 - P0) t
• x(t) = x0 + (x1 - x0) t
• y(t) = x0 + (y1 - y0) t
6
Parametric Lines and Intersections
For L1 :x=x0l1 + t(x1l1 – x0l1)y=y0l1 + t(y1l1 – y0l1)
For L2 :x=x0l2 + t(x1l2 – x0l2)y=y0l2 + t(y1l2 – y0l2)
The Intersection Point:x0l1 + t1 (x1l1 – x0l1) = x0l2 + t2 (x1l2 – x0l2) y0l1 + t1 (y1l1 – y0l1) = y0l2 + t2 (y1l2 – y0l2)
•Cyrus-Beck algorithm (1978) for polygons Mike Cyrus, Jay Beck. "Generalized two- and three-dimensional
clipping". Computers & Graphics, 1978: 23-28.
•Given a convex polygon R:
7
Cyrus-Beck Algorithm
P1
P2
A
N
R
para tspara te
112 )()( PtPPtP 0≤t≤1
0))(( AtPN )(tP,then is inside of R;
0))(( AtPN )(tP
0))(( AtPN )(tP
,then is on R or extension;
,then is outside of R.
cos)(
))((
AtPN
AtPN
0cos 90
0cos 90
0cos 90
How to get ts and te
Cyrus-Beck Algorithm
• Intersection:
•NL ● (P(t) – A) = 0
•Substitute line equation for P(t)
P(t) = P0 + t(P1 - P0)
•Solve for t
t = NL ● (P0 – A) / -NL
● (P1 - P0)
A
NL
P(t)
Inside
OutsideP0
P1
Cyrus-Beck Algorithm
•Compute t for line intersection with all edges;•Discard all (t < 0) and (t > 1);•Classify each remaining intersection as
Potentially Entering Point (PE)
Potentially Leaving Point (PL) (How?)
NL●(P1 - P0) < 0 implies PL
NL●(P1 - P0) > 0 implies PE
Note that we computed this term in when computing t
Cyrus-Beck Algorithm
•For each edge:
10
L1
L2L3
L4
L5
P0
P1
t1
t2
t5
t3t4
para tspara te
0 1 0( ) ( ) 0, 0 1i i i i iP A P P t t N N
1 0
1 0
max{0,max{ | ( ) 0}}
min{1,min{ | ( ) 0}}s i i
e i i
t t P P
t t P P
N
NCompute PE with largest tCompute PL with smallest tClip to these two points
Cyrus-Beck Algorithm
•When ; then • if
• if
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1 0( ) 0i P P N
0( ) 0i iP A N
Then P0P1 is invisible;
0( ) 0i iP A N
Then go to next edge;
1 0( )i P P N
Programming:
12
for ( k edges of clipping polygon ){ solve Ni·(p1-p0); solve Ni·(p0-Ai); if ( Ni·(p1-p0) = = 0 ) //parallel with the edge { if ( Ni·(p0-Ai) < 0 ) break; //invisible else go to next edge; } else // Ni·(p1-p0) != 0 { solve ti; if ( Ni·(p1-p0) < 0 )
else
} }
1 0min{1,min{ | ( ) 0}}e i it t P P N
1 0max{0,max{ | ( ) 0}}s i it t P P N
Output:if ( ts > te ) return nil;else return P(ts) and P(te) as the true clip intersections;
Input:If (P0 = P1 ) Line is degenerate so clip
as a point;
Liang-Barsky Algorithm (1984)
13
The ONLY algorithm named for Chinese people in Computer Graphics course books
Liang, Y.D., and Barsky, B., "A New Concept and Method for Line Clipping", ACM Transactions on Graphics, 3(1):1-22, January 1984.
• Because of horizontal and vertical clip lines: Many computations reduce• Normals• Pick constant points on edges
• solution for t:tL=-(x0 - xleft) / (x1 - x0)
tR=(x0 - xright) / -(x1 - x0)
tB=-(y0 - ybottom) / (y1 - y0)
tT=(y0 - ytop) / -(y1 - y0)
Liang-Barsky Algorithm (1984)
PE
PLP1
PL
PE
P0
(-1, 0)(1, 0)(0, -1)
(0, 1)
15
)(
)(
12
1
PP
APt
N
N
)(
)(
12
1
xx
XLx
12
1 )(
xx
XRx
)(
)(
12
1
yy
YBy
12
1 )(
yy
YTy
EdgeInner
normalA P1-A
Leftx=XL
( 1 ,0 )
( XL ,y )
( x1-XL , y1-
y )
Rightx=XR
( -1 ,0 )
( XR ,y ) ( x1-XR , y1-y )
Bottomy=YB
( 0 ,1 )
( x , YB ) ( x1-x , y1-YB )
Topy=YT
( 0 , -1 )
( x , YT ) ( x1-x , y1-YT )
Liang-Barsky Algorithm (1984)
• When rk<0, tk is entering point; when rk>0, tk is leaving point. If rk=0 and sk<0, then the line is invisible; else process other edges
16
Liang-Barsky Algorithm (1984)
Let ∆x=x2 - x1 ,∆ y=y2 - y1:
,
,
,
,
4
3
2
1
yr
yr
xr
xr
,
,
,
,
14
13
12
11
yys
yys
xxs
xxs
T
B
R
L
TBRLkrst kkk , , , , /
Comparison
• Cohen-Sutherland: Repeated clipping is expensive
Best used when trivial acceptance and rejection is possible for most lines
• Cyrus-Beck: Computation of t-intersections is cheap
Computation of (x,y) clip points is only done once
Algorithm doesn’t consider trivial accepts/rejects
Best when many lines must be clipped
• Liang-Barsky: Optimized Cyrus-Beck• Nicholl et al.: Fastest, but doesn’t do 3D
18
Clipping as a Black Box
•Can consider line segment clipping as a process that takes in two vertices and produces either no vertices or the vertices of a clipped line segment
19
Pipeline Clipping of Line Segments
•Clipping against each side of window is independent of other sides
Can use four independent clippers in a pipeline
20
Clipping and Normalization
•General clipping in 3D requires intersection of line segments against arbitrary plane
•Example: oblique view
21
Plane-Line Intersections
0 1
2 1
( )
( )
n p pt
n p p
0 1 2 1(( ( ( ))) n 0P P t P P
Point-to-Plane Test
•Dot product is relatively expensive 3 multiplies
5 additions
1 comparison (to 0, in this case)
•Think about how you might optimize or special-case this?
23
Normalized Form
before normalization after normalization
Normalization is part of viewing (pre clipping)but after normalization, we clip against sides ofright parallelepiped
Typical intersection calculation now requires onlya floating point subtraction, e.g. is x > xmax
top view
Clipping Polygons
•Clipping polygons is more complex than clipping the individual lines
Input: polygon
Output: polygon, or nothing
25
Polygon Clipping
•Not as simple as line segment clipping Clipping a line segment yields at most one line
segment
Clipping a polygon can yield multiple polygons
•However, clipping a convex polygon can yield at most one other polygon
26
Tessellation and Convexity
• One strategy is to replace nonconvex
(concave) polygons with a set of
triangular polygons (a tessellation)
• Also makes fill easier (we will study later)
• Tessellation code in GLU library, the best is Constrained Delaunay Triangulation
27
Pipeline Clipping of Polygons
• Three dimensions: add front and back clippers
• Strategy used in SGI Geometry Engine
• Small increase in latency
Sutherland-Hodgman Clipping
Ivan Sutherland, Gary W. Hodgman: Reentrant Polygon Clipping. Communications of the ACM, vol. 17, pp. 32-42, 1974
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
Sutherland-Hodgman Clipping
•Basic idea: Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all planes, the polygon is fully clipped
•Will this work for non-rectangular clip regions?•What would 3-D clipping involve?
Sutherland-Hodgman Clipping
• Input/output for algorithm: Input: list of polygon vertices in order
Output: list of clipped poygon vertices consisting of old vertices (maybe) and new vertices (maybe)
•Note: this is exactly what we expect from the clipping operation against each edge
Sutherland-Hodgman Clipping
•Sutherland-Hodgman basic routine: Go around polygon one vertex at a time
Current vertex has position p
Previous vertex had position s, and it has been added to the output if appropriate
Sutherland-Hodgman Clipping
•Edge from s to p takes one of four cases:(Gray line can be a line or a plane)
inside outside
s
p
p output
inside outside
s
p
no output
inside outside
sp
i output
inside outside
sp
i outputp output
Sutherland-Hodgman Clipping
•Four cases: s inside plane and p inside plane
• Add p to output• Note: s has already been added
s inside plane and p outside plane• Find intersection point i• Add i to output
s outside plane and p outside plane• Add nothing
s outside plane and p inside plane• Find intersection point i• Add i to output, followed by p
Sutherland-Hodgman Clipping
42
Point-to-Plane test
•A very general test to determine if a point p is “inside” a plane P, defined by q and n:
(p - q) • n < 0: p inside P
(p - q) • n = 0: p on P
(p - q) • n > 0: p outside P
P
np
q
P
np
q
P
np
q
44
Bounding Boxes
• Rather than doing clipping on a complex polygon, we can use an axis-aligned bounding box or extent
Smallest rectangle aligned with axes that encloses the polygon
Simple to compute: max and min of x and y
45
Bounding Boxes
Can usually determine accept/reject based only on bounding box
reject
accept
requires detailed clipping
Ellipsoid collision detection
46
Rasterization
•Rasterization (scan conversion) Determine which pixels that are inside primitive
specified by a set of vertices
Produces a set of fragments
Fragments have a location (pixel location) and other attributes such color, depth and texture coordinates that are determined by interpolating values at vertices
•Pixel colors determined later using color, texture, and other vertex properties.
47
Scan Conversion of Line Segments
•Start with line segment in window coordinates with integer values for endpoints
•Assume implementation has a write_pixel function
y = mx + h
x
ym
Scan Conversion of Line Segments
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One pixel
49
DDA Algorithm
• Digital Differential Analyzer (1964) DDA was a mechanical device for numerical
solution of differential equations
Line y=mx+h satisfies differential equation dy/dx = m = y/x = y2-y1/x2-x1
• Along scan line x = 1
For(x=x1; x<=x2, x++) { y += m; //note:m is float number write_pixel(x, round(y), line_color);}
50
Problem
•DDA = for each x plot pixel at closest y Problems for steep lines
51
Using Symmetry
•Use for 1 m 0•For m > 1, swap role of x and y
For each y, plot closest x
52
Bresenham’s Algorithm
• DDA requires one floating point addition per step
• We can eliminate all fp through Bresenham’s algorithm
• Consider only 1 m 0 Other cases by symmetry
• Assume pixel centers are at half integers (OpenGL has this definition)
Bresenham, J. E. (1 January 1965). "Algorithm for computer control of a digital plotter". IBM Systems Journal 4(1): 25–30.
Bresenham’s Algorithm
•Observing:
If we start at a pixel that has been written, there are only two candidates for the next pixel to be written into the frame buffer
53
54
Candidate Pixels
1 m 0
last pixel
candidates
Note that line could havepassed through anypart of this pixel
55
Decision Variable
-
d = x(a-b)△
d is an integerd < 0 use upper pixeld > 0 use lower pixel
How to compute a and b?
b-a =(yi+1–yi,r)-( yi,r+1-yi+1) =2yi+1–yi,r–(yi,r+1) = 2yi+1–2yi,r–1
ε(xi+1)= yi+1–yi,r–0.5
=BC-AC=BA=B-A
= yi+1–(yi,r+ yi,r+1)/2
AB
C
Incremental Form
56
if ε(xi+1) ≥ 0, yi+1,r= yi,r+1, pick pixel D, the next pixel is
( xi+1, yi,r+1)
yi,r
yi,r+1
AB
xixi+1
D
C
d1
d2
yi,r
yi,r+1
A
xi xi+1
D
Cd1
d2
if ε(xi+1) < 0, yi+1,r= yi,r, pick pixel C, the next pixel is
( xi+1, yi,r)
Improvement
anew = alast – m anew = alast – (m-1)
bnew = blast + m bnew = blast + (m-1)
57
- -
- -
d = x(a-b)△
Improvement
58
•More efficient if we look at dk, the value of the decision variable at x = k
dk+1= dk – 2 y, if d△ k > 0dk+1= dk – 2(△y - △x), otherwise
•For each x, we need do only an integer addition and a test•Single instruction on graphics chips multiply 2 is simple.
59
BSP display
• Type Tree Tree* front; Face face; Tree *back;
• End• Algorithm DrawBSP(Tree T; point: w)
//w 为视点 If T is null then return; endif If w is in front of T.face then
• DrawBSP(T.back,w);• Draw(T.face,w);• DrawBSP(T.front,w);
Else // w is behind or on T.face• DrawBSP(T.front,w);• Draw(T.face,w);• DrawBSP(T. back,w);
Endif• end
60
Hidden Surface Removal
•Object-space approach: use pairwise testing between polygons (objects)
•Worst case complexity O(n2) for n polygons
partially obscuring can draw independently
61
Image Space Approach
•Look at each projector (nm for an n x m frame buffer) and find closest of k polygons
•Complexity O(nmk)
•Ray tracing • z-buffer
62
Painter’s Algorithm
•Render polygons a back to front order so that polygons behind others are simply painted over
B behind A as seen by viewer Fill B then A
63
Depth Sort
•Requires ordering of polygons first O(n log n) calculation for ordering
Not every polygon is either in front or behind all other polygons
• Order polygons and deal with
easy cases first, harder later
Polygons sorted by distance from COP
64
Easy Cases
• (1) A lies behind all other polygons Can render
• (2) Polygons overlap in z but not in either x or y
Can render independently
65
Hard Cases
(3) Overlap in all directionsbut can one is fully on one side of the other
cyclic overlap
penetration
(4)
66
Back-Face Removal (Culling)
•face is visible iff 90 -90equivalently cos 0or v • n 0
•plane of face has form ax + by +cz +d =0but after normalization n = ( 0 0 1 0)T
•need only test the sign of c
•In OpenGL we can simply enable cullingbut may not work correctly if we have nonconvex objects
67
z-Buffer Algorithm
• Use a buffer called the z or depth buffer to store 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 shade of pixel in color buffer and update z buffer
68
Efficiency
• If we work scan line by scan line as we move across a scan line, the depth changes satisfy ax+by+cz=0
Along scan line
y = 0z = - x
c
a
In screen space x = 1
69
Scan-Line Algorithm
•Can combine shading and hsr through scan line algorithm
scan line i: no need for depth information, can only be in noor one polygon
scan line j: need depth information only when inmore than one polygon
70
Implementation
•Need a data structure to store Flag for each polygon (inside/outside)
Incremental structure for scan lines that stores which edges are encountered
Parameters for planes
71
Visibility Testing
• In many realtime applications, such as games, we want to eliminate as many objects as possible within the application
Reduce burden on pipeline
Reduce traffic on bus
•Partition space with Binary Spatial Partition (BSP) Tree
72
Simple Example
consider 6 parallel polygons
top view
The plane of A separates B and C from D, E and F
73
BSP Tree
•Can continue recursively Plane of C separates B from A
Plane of D separates E and F
•Can put this information in a BSP tree Use for visibility and occlusion testing
74
Polygon Scan Conversion
•Scan Conversion = Fill•How to tell inside from outside
Convex easy
Nonsimple difficult
Odd even test• Count edge crossings
Winding numberodd-even fill
75
Winding Number
•Count clockwise encirclements of point
•Alternate definition of inside: inside if winding number 0
winding number = 2
winding number = 1
76
Filling in the Frame Buffer
•Fill at end of pipeline Convex Polygons only
Nonconvex polygons assumed to have been tessellated
Shades (colors) have been computed for vertices (Gouraud shading)
Combine with z-buffer algorithm• March across scan lines interpolating shades• Incremental work small
77
Using Interpolation
span
C1
C3
C2
C5
C4scan line
C1 C2 C3 specified by glColor or by vertex shadingC4 determined by interpolating between C1 and C2
C5 determined by interpolating between C2 and C3
interpolate between C4 and C5 along span
78
Flood Fill
• Fill can be done recursively if we know a seed point located inside (WHITE)
• Scan convert edges into buffer in edge/inside color (BLACK)flood_fill(int x, int y) { if(read_pixel(x,y)= = WHITE) { write_pixel(x,y,BLACK); flood_fill(x-1, y); flood_fill(x+1, y); flood_fill(x, y+1); flood_fill(x, y-1);} }
79
Scan Line Fill
• Can also fill by maintaining a data structure of all intersections of polygons with scan lines
Sort by scan line
Fill each span
vertex order generated by vertex list desired order
80
Data Structure
81
Aliasing
• Ideal rasterized line should be 1 pixel wide
•Choosing best y for each x (or visa versa) produces aliased raster lines
82
Antialiasing by Area Averaging
• Color multiple pixels for each x depending on coverage by ideal line
original antialiased
magnified
83
Polygon Aliasing
•Aliasing problems can be serious for polygons
Jaggedness of edges
Small polygons neglected
Need compositing so color
of one polygon does not
totally determine color of
pixel
All three polygons should contribute to color