Download pdf - Raster Algorithms - CAIG Lab - National Chiao Tung …caig.cs.nctu.edu.tw/course/CG2007/slides/raster.pdfDCP4516 Introduction to Computer Graphics 11 Bresenham’s Algorithm •DDA

Raster AlgorithmsRaster Algorithms

I-Chen Lin’s CG slides, Doug James’s CG slides

Shirley, Fundamentals of Computer Graphics, Chap 3.5, 3.7, 4.4

Hearn and Baker, Computer Graphics, Chap 3.14, 3.15

林文杰, Wen-Chieh (Steve) Lin

Department of Computer Science &

Institute of Multimedia Engineering

DCP4516 Introduction to Computer Graphics 2

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Rasterization (Scan Conversion)Rasterization (Scan Conversion)

•Final step in pipeline

•From screen coordinates (float) to pixels (int)

•Writing pixels into frame buffer


OutlineOutline

•2D graphics primitives

–Line drawing

–Circle drawing

–………

•Area filling

–Polygons

–………

•Antialiasing


Line-Drawing AlgorithmsLine-Drawing Algorithms

•Start with line segment in windowcoordinates with integer values for endpoints

xy

m

y = mx + h


Discrete LinesDiscrete Lines

•Lines vs. Line Segments

•What is a discrete line segment?

•How to generate a discrete line?


“Good”Discrete Lines“Good”Discrete Lines

•No gaps in adjacent pixels

•Pixels close to ideal line

•Consistent choices; same pixels in samesituations

•Smooth looking

•Even brightness in all orientations

•Same line for P0 P1 as for P1 P0


DDA AlgorithmDDA Algorithm

•Digital Differential Analyzer

–Line y = mx + h satisfies differential equation

•Along scan line Δx = 1For(x=x1; x<=x2,ix++) {

y+=m;

write_pixel(x, round(y), line_color)

}


ProblemProblem

•DDA = for each x plot pixel at closest y.

–Problems for steep lines


Using SymmetryUsing Symmetry

•Use for 1 ≥m ≥0

–For m > 1, swap roles of x and y

–For each y, plot closest x


Bresenham’s AlgorithmBresenham’s Algorithm

•DDA requires one floating point addition per step.

•Bresenham’s algorithm eliminates all floating pointoperations

•Consider only 1 ≥m ≥0

–Other cases by symmetry

•Assume pixel centers are at half integers

•If we start at a pixel that has been written, there areonly two candidates for the next pixel


Key to Bresenham AlgorithmKey to Bresenham Algorithm

•“Reasonable assumptions”have reduced theproblem to making a binary choice at eachpixel:

(Previous)

NE (next)

E (next)


Candidate PixelsCandidate Pixels

•1 ≥m ≥0

Last pixel

candidates


Go NE if M is below the lineGo NE if M is below the line

ideal linemidpoint

E

NE

previous

Q

1xx

y

1y

M

),(211 yx


Go E if M is above the lineGo E if M is above the line

ideal line

midpoint

E

NE

previous

Q

),( yx

),(211 yx

)1,1( yx

),( 1 yx

M


Decision Variable dDecision Variable d

•Define a logical decision variable d

•linear in form

•incrementally updated (with addition)

•tells us whether to go E or NE


Implicit Line EquationImplicit Line Equation

•Explicit: y = mx + h

•Implicit: f(x,y) = ax + by + c = 0

•Line passing (x0, y0) and (x1, y1):

f(x,y) = (y0–y1)x + (x1–x0)y + x0y1 –x1y0= 0


Recall thatRecall that

•f(x,y) = (y0–y1)x + (x1–x0)y + x0y1 –x1y0= 0

•For 1 ≥m ≥0 and x1 > x0

Above line: consider f(x,+∞)

Below line

f(x,y) > 0

f(x,y) < 0


Bresenham’s AlgorithmBresenham’s Algorithm

y = y0

For x=x0 to x1 do

draw(x,y)

If f(x+1, y+0.5) < 0 theny = y + 1

f(x+1,y+0.5) < 0

f(x+1,y+0.5) > 0


Bresenham’s Algorithm (cont.)Bresenham’s Algorithm (cont.)

•Speed up by replacing function evaluationwith incremental update

•f(x,y) = (y0–y1)x + (x1–x0)y + x0y1 –x1y0= 0

•f(x+1, y+1) = f(x, y) + (y0–y1) + (x1–x0)

•f(x+1, y) = f(x, y) + (y0–y1)



y = y0

d = f(x0+1, y0+0.5)

For x=x0 to x1 do

draw(x,y)

If d< 0 theny = y + 1

d = d + (x1-x0) + (y0-y1)

else

d = d + (y0-y1)

f(x+1,y+0.5) < 0

f(x+1,y+0.5) > 0



y = y0

d = f(x0+1, y0+0.5)

For x=x0 to x1 do

draw(x,y)


d = d + (x1-x0) + (y0-y1)

else

d = d + (y0-y1)

Now, we want to remove thelast floating point operationin the code!



•2*d = 2*f(x0+1, y0+0.5)

•f(x0+1,y0+0.5) = (y0–y1) (x0+1) + (x1–x0)(y0+0.5)+ x0y1 –x1y0= 0

•2f(x0+1,y+0.5) = 2(y0–y1) (x0+1) + (x1–x0)(2y0+1)+ 2x0y1 –2x1y0= 0


Code for Bresenham’s AlgorithmCode for Bresenham’s Algorithm

y = y0

d = 2*(y0–y1)(x0+1) + (x1–x0)(2*y0+1) + 2*(x0*y1–x1*y0)

For x=x0 to x1 do

draw(x,y)


d = d + 2*(x1-x0) + 2*(y0-y1)

else

d = d + 2*(y0-y1)


Other cases for line drawingOther cases for line drawing

•m=0; m=1 trivial cases

•0 > m > -1 flip about x-axis

•m > 1 flip about x = y


Flip about x-axis if 0>m>-1Flip about x-axis if 0>m>-1

), 00( yx

x

y

), 11( yx), 11( yx

), 00( yx

0 0 0 0

1 11 1

, ) ( ,

, ) , )

( );

( (

x y x y

x y x y


How do slopes relate?How do slopes relate?

0

1 0

0

1 0

1

1

;

by definition

y y

y y

mx x

mx x

)0

1 0

(1i iSince ,

y ymyy

x x


How do slopes relate?How do slopes relate?

1010 mm

i.e.,

)0

1 0

1( y ym

x x

m m


Flip about line y=x if m>1Flip about line y=x if m>1

), 11( yx

), 00( yx

y

x

xy ), 11( yx

), 00( yx ), 00( yx

), 11( yx


Flip about line y=x if m>1 (cont.)Flip about line y=x if m>1 (cont.)

,swap and prime them ,

,

y mx Bx yx my B

my x B

1,

1and,

1 0 1

y x Bm

mm

m m


Circle-drawing AlgorithmCircle-drawing Algorithm


Circle-drawing AlgorithmCircle-drawing Algorithm

for each x, yif | x2 + y2 –r2 | <= εSetPixel ( x, y )

for θ in [0~360 degree ]x = r cos(θ)y = r sin(θ)SetPixel ( x, y )


Midpoint Circle AlgorithmMidpoint Circle Algorithm

•Can we utilize the similar idea inBresenham’s line-drawing algorithm ?

–Check only the next candidates

–Use symmetry and simple decision rules

Symmetry of a Circle


Midpoint Circle Algorithm (cont.)Midpoint Circle Algorithm (cont.)

f(x,y) = x2 + y2 - R2f(x,y) > 0 => point outside circlef(x,y) < 0 => point inside circle

Pk = fcirc(xk + 1, yk –½)


Midpoint Circle Algorithm (cont.)Midpoint Circle Algorithm (cont.)

•Given the starting point (0,r), the computation ismore efficient.P0 = 5/4 – r

At each x position,

if(pk < 0)

the next point is (xk+1, yk)

pk+1 = pk + 2xk+1 + 1

else

the next point is (xk+1, yk-1)

pk+1 = pk + 2xk+1 + 1 – 2yk+1


2D Polygon Filling2D Polygon Filling

•In computer graphics, we usually usepolygons to approximate complex surfaces.

•Let’s focus on the polygon filling!


General PolygonsGeneral Polygons

•Inside or Outside are not obvious

–It’s not obvious when the polygon intersects itself.


Concave vs. ConvexConcave vs. Convex

•We prefer dealing with “simpler”polygons.

•Convex (easy to break into triangles)

convex concave

θ< 180o

θ > 180o


Filling Convex PolygonsFilling Convex Polygons

•Find top and bottom vertices

•List edges along left and right sides

•For each scan line from top to bottom–Find left and right endpoints of span, xl and xr

–Fill pixels between xl and xr

–Can use Bresenham’s alg. to update xl and xr

xl xr


Concave Polygons: Odd-Even TestConcave Polygons: Odd-Even Test

•Approach 1: odd-even test

•For each scan line

–Find all scan line/polygon intersections

–Sort them left to right

–Fill the interior spans between intersections

•Parity rule: inside after

an odd number of crossings


Concave Polygons: Winding RuleConcave Polygons: Winding Rule

•Approach 2: winding rule

•Orient the lines in polygon

•For each test line (not passing a vertex)

–Winding number = right-hdd –left-hdd crossings

–Interior if winding number non-zero

211

1

1 1

A

BE

DC

0+1+0+1+0=2

0+0+0+1+0=1Starting from A:

Test line


Even-odd Rule vs. Winding RuleEven-odd Rule vs. Winding Rule

•Different only for self-intersecting polygons

Even-odd rule

211

1

1 1

Winding rule


AliasingAliasing

•Artifacts created during scan conversion

•Inevitable (going from continuous to discrete)

•Aliasing (name from digital signal processing):we sample a continuous image at grid points

•Effects

–Jagged edges

–Moiré patterns


Sampling and ReconstructionSampling and Reconstruction

•An image is a 2D array of discrete samplesfrom real-world continuous signal


Sampling and AliasingSampling and Aliasing

•Artifacts due to undersampling or poorreconstruction

•Formally, high frequencies masquerading as low

•e.g. high frequency line as low freq jaggies


(Spatial) Aliasing(Spatial) Aliasing


(Spatial) Aliasing(Spatial) Aliasing

•Jaggies probably biggest aliasing problem


Moiré Patterns due to Bad DownsamplingMoiré Patterns due to Bad Downsampling

Original Image

Downsampled without filtering

Downsampled after filtering


More AliasingMore Aliasing


Antialiasing for Line SegmentsAntialiasing for Line Segments

•Use area averaging at boundary

•bottom is aliased, magnified

•top is antialiased, magnified


Antialiasing by SupersamplingAntialiasing by Supersampling

•Traditionally for off-line rendering

•Render, say, 3x3 grid of mini-pixels

•Average results using a filter

•Can be done adaptively

–Stop if colors are similar

–Subdivide at discontinuities


Supersampling ExampleSupersampling Example

•Other improvements

–Stochastic sampling (avoiding repetition)

–Jittering (perturb a regular grid)