Xiaolin Wu's line algorithm
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Xiaolin Wu's line algorithm is an algorithm for line antialiasing.
Antialiasing technique
Xiaolin Wu's line algorithm was presented in the article "An Efficient Antialiasing Technique" in the July 1991 issue of Computer Graphics, as well as in the article "Fast Antialiasing" in the June 1992 issue of Dr. Dobb's Journal.
Bresenham's algorithm draws lines extremely quickly, but it does not perform anti-aliasing. In addition, it cannot handle any cases where the line endpoints do not lie exactly on integer points of the pixel grid. A naive approach to anti-aliasing the line would take an extremely long time. Wu's algorithm is comparatively fast, but is still slower than Bresenham's algorithm. The algorithm consists of drawing pairs of pixels straddling the line, each coloured according to its distance from the line. Pixels at the line ends are handled separately. Lines less than one pixel long are handled as a special case.
An extension to the algorithm for circle drawing was presented by Xiaolin Wu in the book Graphics Gems II. Just as the line drawing algorithm is a replacement for Bresenham's line drawing algorithm, the circle drawing algorithm is a replacement for Bresenham's circle drawing algorithm.
Algorithm
Like Bresenham’s line algorithm, this method steps along one axis and considers the two nearest pixels to the ideal line. Instead of choosing the nearest, it draws both, with intensities proportional to their vertical distance from the true line. This produces smoother, anti-aliased lines.
The pseudocode below assumes a line where <math>x_0 < x_1</math>, <math>y_0 < y_1</math>, and the slope <math>k = \frac{dy}{dx}</math> satisfies <math>0 \le k \le 1</math>. This is a standard simplification — the algorithm can be extended to all directions using symmetry.
The algorithm is well-suited to older CPUs and microcontrollers because:
- It avoids floating point arithmetic in the main loop (only used to initialize d)
- It renders symmetrically from both ends, halving the number of iterations
- The main loop uses only addition and bit shifts — no multiplication or division
<syntaxhighlight lang="python" line="1">
function draw_line(x0, y0, x1, y1)
N := 8 # brightness resolution (bits) M := 15 # fixed-point fractional bits I := maximum brightness value
# Compute gradient and convert to fixed-point step k := float(y1 - y0) / (x1 - x0) d := floor((k << M) + 0.5)
# Start with fully covered pixels at each end img[x0, y0] := img[x1, y1] := I
D := 0 # Fixed-point accumulator
while true:
x0 := x0 + 1
x1 := x1 - 1
if x0 > x1:
break
D := D + d
if D overflows:
y0 := y0 + 1
y1 := y1 - 1
# Brightness = upper N bits of fractional part of D
v := D >> (M - N)
img[x0, y0] := img[x1, y1] := I - v
img[x0, y0 + 1] := img[x1, y1 -1] := v
</syntaxhighlight>
Floating Point Implementation
<syntaxhighlight lang="pascal" line="1"> function plot(x, y, c) is
plot the pixel at (x, y) with brightness c (where 0 ≤ c ≤ 1)
// fractional part of x function fpart(x) is
return x - floor(x)
function rfpart(x) is
return 1 - fpart(x)
function drawLine(x0,y0,x1,y1) is
boolean steep := abs(y1 - y0) > abs(x1 - x0)
if steep then
swap(x0, y0)
swap(x1, y1)
end if
if x0 > x1 then
swap(x0, x1)
swap(y0, y1)
end if
dx := x1 - x0
dy := y1 - y0
if dx == 0.0 then
gradient := 1.0
else
gradient := dy / dx
end if
// handle first endpoint
xend := floor(x0)
yend := y0 + gradient * (xend - x0)
xgap := 1 - (x0 - xend)
xpxl1 := xend // this will be used in the main loop
ypxl1 := floor(yend)
if steep then
plot(ypxl1, xpxl1, rfpart(yend) * xgap)
plot(ypxl1+1, xpxl1, fpart(yend) * xgap)
else
plot(xpxl1, ypxl1 , rfpart(yend) * xgap)
plot(xpxl1, ypxl1+1, fpart(yend) * xgap)
end if
intery := yend + gradient // first y-intersection for the main loop
// handle second endpoint
xend := ceil(x1)
yend := y1 + gradient * (xend - x1)
xgap := 1 - (xend - x1)
xpxl2 := xend //this will be used in the main loop
ypxl2 := floor(yend)
if steep then
plot(ypxl2 , xpxl2, rfpart(yend) * xgap)
plot(ypxl2+1, xpxl2, fpart(yend) * xgap)
else
plot(xpxl2, ypxl2, rfpart(yend) * xgap)
plot(xpxl2, ypxl2+1, fpart(yend) * xgap)
end if
// main loop
if steep then
for x from xpxl1 + 1 to xpxl2 - 1 do
begin
plot(floor(intery) , x, rfpart(intery))
plot(floor(intery)+1, x, fpart(intery))
intery := intery + gradient
end
else
for x from xpxl1 + 1 to xpxl2 - 1 do
begin
plot(x, floor(intery), rfpart(intery))
plot(x, floor(intery)+1, fpart(intery))
intery := intery + gradient
end
end if
end function </syntaxhighlight>