Xiaolin Wu's line algorithm: Difference between revisions

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{{Short description|Line algorithm with antialiasing}}

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{{third-party|date=April 2018}}

{{no footnotes|date=January 2013}}

{{Cleanup|reason=Implementation does not provide explanation|date=November 2023}}

}}

[[fileFile:LineXiaolinWu.gif|thumb|336px|Demonstration of Xiaolin Wu's algorithm]]

'''Xiaolin Wu's line algorithm''' is an [[algorithm]] for line [[spatial anti-aliasing|antialiasing]].

[[File:Xiaolin anti-aliased line comparison.png|thumb|Anti-Aliased Lines (blue) generated with Xiaolin Wu's line algorithm alongside standard lines (red) generated with Bresenham's line algorithm]]

==Antialiasing technique==

Xiaolin Wu's line algorithm was presented in the article "An Efficient Antialiasing Technique" in the July 1991 issue of ''[[Computer Graphics (publicationnewsletter)|Computer Graphics]]'', as well as in the article "Fast Antialiasing" in the June 1992 issue of ''[[Dr. Dobb's Journal]]''.

[[Bresenham's line algorithm|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|Bresenham’s line algorithm]], this method steps

<syntaxhighlight lang="pascal">

along one axis and considers the two nearest pixels to the ideal line. Instead of

function plot(x, y, c) is

choosing the nearest, it draws both, with intensities proportional to their vertical

plot the pixel at (x, y) with brightness c (where 0 ≤ c ≤ 1)

distance from the true line. This produces smoother, anti-aliased lines.

[[File:Wu-line-animation.gif|thumb|Animation showing symmetry of Wu's line algorithm ]]

// integer part of x

function ipart(x) is

return floor(x)

The pseudocode below assumes a line where <math>x_0 < x_1</math>, <math>y_0 < y_1</math>,

function round(x) is

and the slope <math>k = \frac{dy}{dx}</math> satisfies <math>0 \le k \le 1</math>. This

return ipart(x + 0.5)

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:

gradient x0 := dyx0 /+ dx1

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

Line 47 ⟶ 96:

dx := x1 - x0

dy := y1 - y0

gradient := dy / dx

if dx == 0.0 then

gradient := 1.0

else

gradient := dy / dx

end if

// handle first endpoint

xend := roundfloor(x0)

yend := y0 + gradient * (xend - x0)

xgap := rfpart1 - (x0 +- 0.5xend)

xpxl1 := xend // this will be used in the main loop

ypxl1 := ipartfloor(yend)

if steep then

plot(ypxl1, xpxl1, rfpart(yend) * xgap)

Line 68 ⟶ 119:

// handle second endpoint

xend := roundceil(x1)

yend := y1 + gradient * (xend - x1)

xgap := fpart1 - (x1xend +- 0.5x1)

xpxl2 := xend //this will be used in the main loop

ypxl2 := ipartfloor(yend)

if steep then

plot(ypxl2 , xpxl2, rfpart(yend) * xgap)

Line 85 ⟶ 136:

for x from xpxl1 + 1 to xpxl2 - 1 do

begin

plot(ipartfloor(intery) , x, rfpart(intery))

plot(ipartfloor(intery)+1, x, fpart(intery))

intery := intery + gradient

end

Line 92 ⟶ 143:

for x from xpxl1 + 1 to xpxl2 - 1 do

begin

plot(x, ipartfloor(intery), rfpart(intery))

plot(x, ipartfloor(intery)+1, fpart(intery))

intery := intery + gradient

end

Line 112 ⟶ 163:

| url = http://portal.acm.org/citation.cfm?id=122734

| title = An efficient antialiasing technique

| journal=[[ACM SIGGRAPH Computer Graphics (publication)]]

| date=July 1991 | volume=25 | issue=4 | pages=143–152

| doi = 10.1145/127719.122734

| isbn=0-89791-436-8

| url-access=subscription

}}

* {{cite book

Line 131 ⟶ 183:

==External links==

* [http://www.ece.mcmaster.ca/~xwu/ Xiaolin Wu's homepage]

* [https://www.eng.mcmaster.ca/ece/faculty/dr-xiaolin-wu Xiaolin Wu's homepage at McMaster University]

{{DEFAULTSORT:Xiaolin Wu's Line Algorithm}}

[[Category:Anti-aliasing algorithms]]