General scheme of a 2D CG application
2D Image Synthesis
modeling
Virtual world
model
Balázs Csébfalvi
world defined
in a 2D plane
image synthesis
Department of Control Engineering and
Information Technology
email: [email protected]
Web: http://www.iit.bme.hu/~cseb
Metaphor:
2D drawing or sketching
2 /25
Raster graphics systems
2D image synthesis
3 /25
Vectorization
4 /25
Modeling transformation
v
y
TM
r(t)
βv
r2
r1
v
βv
p
u
αu
o
rn
αu
[0,1]: 0, t1 , t2 , ... , tn , 1
r1 =r(0), r2 = r(t2), … , rn = r(1)
x
u
[x, y] = o +αu + βv
[x, y, 1] = [α, β, 1]
5 /25
ux uy
vx vy
ox oy
TM
0
0
1
6 /25
1
Viewport transformation
Composite transformation
[X, Y, 1] = ([α, β, 1] TM ) TV
[X, Y, 1] = [α, β, 1] (TM TV ) = [α, β, 1] TC
n
n
TV
X = (x-wx) vw/ww + vx
Y = (y-wy) vh/wh + vy
vw/ww
0
0
0
vh/wh
0
[X, Y, 1] = [x, y, 1] v -w v /w v -w v /w 1
x
x w w y y h h
n
calculation of the viewport transformation T V
for each object o
• determine TM
• TC = TM TV
• for each point of object o: transformation by TC
endfor
7 /25
Clipping
n
8 /25
Clipping a segment by a half plane
Due to the vectorization, clipping is performed
on points, segments, and polygons
xmax
• Clipping points
x > xmin ,
x1, y1
x < xmax ,
y > ymin,
x2, y2
y < ymax
n
ymax
n
ymin
xmin
xi, yi
n
xmax
x(t) = x1 + (x2 - x1)t, y(t) = y1 + (y2 - y1)t
x = xmax
• intersection point:
• xmax = x1 + (x2 - x1)t ð t = (xmax-x1)/(x2-x1)
xi = xmax
yi = y1 + (y2 - y1) (xmax-x1)/(x2-x1)
9 /25
Cohen-Sutherland algorithm
10 /25
Cohen-Sutherland algorithm
#define Code(p) ((p.x < xmin)
+ ((p.y < ymin)<<1) + \\
((p.x > xmax)<<2) + ((p.y > ymax)<<3)
1001
1000
1100
0001
0000
0100
0011
0010
bool LineClip(Vector& p1, Vector& p2) {
for( ; ; ) {
int c1 = Code(p1), c2 = Code(p2);
if (c1 == 0 && c2 == 0) return true;
if ((c1 & c2))
return false;
if ((c1
else if
else if
else if
else if
else if
else if
else if
0110
& 1))
((c1 &
((c1 &
((c1 &
((c2 &
((c2 &
((c2 &
((c2 &
2))
4))
8))
1))
2))
4))
8))
p1
p1
p1
p1
p2
p2
p2
p2
=
=
=
=
=
=
=
=
IntersectX(p1,
IntersectY(p1,
IntersectX(p1,
IntersectY(p1,
IntersectX(p1,
IntersectY(p1,
IntersectX(p1,
IntersectY(p1,
p2,
p2,
p2,
p2,
p2,
p2,
p2,
p2,
xmin);
ymin);
xmax);
ymax);
xmin);
ymin);
xmax);
ymax);
}
11 /25
}
12 /25
2
Polygon Clipping
n
n
Sutherland-Hodgman Polygon Clipping
modification of line clipping
goal: one or more closed areas
n
n
display of a polygon
processed by a lineclipping algorithm
display of a correctly
clipped polygon
processing polygon boundary as a whole against each
window edge
output: list of vertices
original
polygon
clip left
clip right
clip
bottom
clipping a polygon against successive
window boundaries
13 /25
Sutherland-Hodgman Polygon Clipping
n
clip top
14 /25
Sutherland-Hodgman Polygon Clipping
four possible edge cases
out → in
in → in
output: V’1, V2
V2
in → out
V’1
out → out
no output
successive processing of pairs of polygon
vertices against the left window boundary
clipping a polygon
against the left
boundary of a
window, starting with
vertex 1. Primed
numbers are used to
label the points in
the output vertex list
for this window
boundary
finished!
15 /25
Sutherland-Hodgman Algorithm:
Combination of the 4 Passes
16 /25
Example
n
n
n
The polygon is clipped against each of the
4 borders separately,
that would produce 3 intermediate results.
By calling the 4 tests recursively,
(or by using a clipping pipeline)
every result point is immediately
processed on, so that only one result list is
produced
17 /25
pipeline of
boundary
clippers to
avoid
intermediate
vertex lists
1st clip:
left
2
2’
1’
2”
3
3’
1
2nd clip:
bottom
Processing the vertices of the polygon through
a boundary-clipping pipeline. After all vertices
are processed through the pipeline, the vertex
list for the clipped polygon is [1’, 2, 2’, 2”]
18 /25
3
Sutherland-Hodgman Polygon Clipping
n
Sutherland-Hodgeman polygon clipping
extraneous lines for concave polygons:
• split into separate convex parts
• final check of output vertex list
or
clipping the
concave polygon
with the
SutherlandHodgeman clipper
produces two
connected areas
PolygonClip(p[n] ð q[m])
m = 0;
for( i=0; i < n; i++) {
if (p[i] internal) {
q[m++] = p[i];
if (p[i+1] is external)
q[m++] = Intersect(p[i], p[i+1], borderline);
} else {
if (p[i+1] is internal)
q[m++] = Intersect(p[i], p[i+1], borderline);
}
}
}
19 /25
Rasterization
20 /25
Line drawing
model
transformation
Pixels
10-50 nanosec / pixel
clipping
rasterization
Line equation:
y = mx + b
Geometrical
primitives:
0.1-1 microsec / primitive
Line drawing
pixel operations
frame buffer
x1
x2
for( x = x1; x <= x2; x++) {
Y = m*x + b;
y = Round( Y );
write( x, y );
}
21 /25
Incremental approach
n
n
n
22 /25
DDA line drawing
Calculate Y(X)
• Y(X) = F( Y(X-1) ) = Y(X-1) + dY/dX
Line drawing: Y(X) = mX + b
• Y(X) = F( Y(X-1) ) = Y(X-1) + m
Addition: fix-point arithmetic: y = Y 2T
• T: error < 1 for the longest segment
• N 2-T < 1: T > log2 N
• round( y ): y+0.5 is truncated
DDADrawLine(x1, y1, x2, y2) {
m = (y2 - y1)/(x2 - x1)
y = y1
+ 0.5
FOR X = x1 TO x2 {
Y = round(y)
trunc(y)
WRITE(X, Y, color)
y = y+m
}
}
23 /25
24 /25
4
Bresenham’s Line Algorithm
DDA line-drawing hardware
n
X
X counter
y register
CLK
x1
Faster than simple DDA
• incremental integer calculations
• adaptable to circles, other curves
Y
y = m.(xk + 1) + b
Σ
y1
Section of a display screen
where a straight line
segment is to be plotted,
starting from the pixel at
column 10 on scan line 11
m
25 /25
26 /25
Bresenham’s
Line-Drawing Algorithm (1/4)
Bresenham’s Line Algorithm
Section of the
screen grid
showing a pixel in
column xk on scan
line yk that is to be
plotted along the
path of a line
segment with
slope 0<m<1
y = m.(xk + 1) + b
dlower = y − yk =
= m.(xk + 1) + b − yk
dupper = (yk + 1) − y =
= yk + 1 − m.(xk + 1) − b
dlower − dupper = 2m.(xk + 1) − 2yk + 2b − 1
27 /25
28 /25
Bresenham’s
Line-Drawing Algorithm (2/4)
Bresenham’s Line-Drawing
Algorithm (3/4)
dlower − dupper =
= 2m.(xk + 1) − 2yk + 2b − 1
current decision value:
pk = ∆x.(dlower − dupper) = 2∆y. xk − 2∆x. yk + c
m = ∆y/∆x
next decision value:
∆x, ∆y: separations of endpoint positions
decision
parameter:
pk = ∆x.(dlower − dupper) =
= 2∆y. xk − 2∆x. yk + c
has the same sign as (dlower − dupper)
29 /25
pk+1 = 2∆y.xk+1 − 2∆x. yk+1 + c + 0
+ pk – 2∆y.xk + 2∆x.yk – c =
starting decision
= pkk + 2∆value:
y − 2∆x..(yk+1
k+1 − yk
k)
p0 = 2∆y − ∆x
30 /25
5
Bresenham:
Example
Bresenham’s Algorithm (4/4)
1. store left line endpoint in (x0,y0)
2. plot pixel (x0,y0)
3. calculate constants ∆x, ∆y, 2∆y, 2∆y − 2∆x,
and obtain
p0 = 2∆y − ∆x
4. At each xk along the line, perform test:
if pk <0
then plot pixel (xk +1,yk ); pk+1 = pk + 2∆y
else plot pixel (xk +1,yk +1); pk+1 = pk + 2∆y − 2∆x
5. perform step 4 ∆x − 1 times.
line from (20/10)
to (30/18)
31 /25
Flood fill
2
2 1 2
1 02 1
1
32 /25
Specific area filling
Flood(x, y) {
if (pixel[x][y] is internal) {
Write(x, y, color);
Flood(x, y-1);
Flood(x, y+1);
starting point
Flood(x-1, y);
Flood(x+1, y);
}
}
n
n
Internal point =BLUE
Based on the geometrical representation
• p1,p2,...,pn vertices
• p1-p2, p2-p3,...,pn-p1edges
Internal pixels
• cast a ray into the infinity
• the number of intersection points with the
edges has to be odd
• “infinity”: the border line of the viewport
after the clipping
33 /25
Polygon filling
34 /25
Accelerated filling
n
p2
n
Check only the active edges
Incremental intersection point calculation
Dx/ Dy
p4
x1
y
x4
x3
Dx/ Dy
x2
y
p1
p3
35 /25
x(y)
x(y+1)
x(y+2)
36 /25
6
List of active edges
AET: active edge table
AEL
ymax
Dx/ Dy
x
next
ymax
Dx/ Dy
x
next
ymin=1024
ymax Dx/ Dy
x1 next
ymax Dx/ Dy
x1 next
ymin=1
ymin=0
37 /25
7