Graphics (Output) Primitives. Chapters 3 & 4

Transcription

1 Graphics (Output) Primitives Chapters 3 & 4

2 Graphic Output ad Iput Pipelie Sca coversio coverts primitives such as lies, circles, etc. ito pixel values geometric descriptio Þ a fiite scee area Clippig the process of determiig the portio of a primitive lyig withi a regio called clip regio

3 Graphic Output Pipelie applicatio model : descriptios of objects applicatio program : geerates a sequece of fuctios to display a model graphics package : clippig, sca coversio, shadig, etc. display H/W

4 Graphic Iput Pipelie user iteractio (e.g., mouse click) graphic package (by samplig or evet-drive iput fuctios) applicatio program modify the model or the image o the scree

5 Output Pipelie i Software Whe sca coversio ad clippig happe? Clippig before sca coversio for lies, rectagles, ad polygos clippig before sca covertig each primitive (scissorig) Clippig after sca covertig the etire collectio of primitives ito a temporary cavas for text

6 Sca Covertig Lies A lie from (x 1,y 1 ) to (x 2,y 2 ) Þ a series of pixels [Criteria] Straight lies should appear straight Lie ed-poits should be costraied Uiform desity ad itesity Lie algorithms should be fast

7 Why Study Sca Coversio Algorithms? Every high-ed graphics card support this. You will ever have to write these routies yourself, uless you become a graphics hardware desiger. So why lear this stuff? Maybe you will become a graphics hardware desiger. But seriously, the same basic tricks uderlie lots of algorithms: 3-D shaded polygos Texture mappig etc.

8 Simple Sca Covertig Lies Based o slope-itercept algorithm from algebra: y = mx + b Simple approach: icremet x, solve for y Floatig poit arithmetic required

9 Digital Differetial Aalyzer(DDA) Idea 1. Go to startig ed poit 2. Icremet x ad y values by costats proportioal to x ad y such that oe of them is 1. - If 0<m 1, y k+1 = y k + m - If m>1, x k+1 = x k + 1/m 3. Roud to the closest raster positio ( i x i, y ) ( x + 1, Roud( y m)) ( i i i i + x, Roud( y )) ( x + 1, y m) i i +

10 Digital Differetial Aalyzer(DDA) Drawbacks roudig to a iteger takes time floatig-poit operatios Is there a simpler way? Ca we use oly iteger arithmetic? Easier to implemet i hardware

11 Midpoit Lie Algorithm (Breseham's Lie Algorithm) Assume a lie from (x 1, y 1 ) to (x 2, y 2 ) that 0<m < 1 ad x 1 <x 2. Use symmetry

12 Midpoit Lie Algorithm (Breseham's Lie Algorithm) Suppose that we have just fiished drawig a pixel P = (x k, y k ) ad we are iterested i figurig out which pixel to draw ext. If distace(ne,m) > distace(e,m) the select E = (x k +1, y k ) y k +1 y NE d upper else d lower select NE = (x k +1, y k +1) y k P E x k x k +1

13 Midpoit Lie Algorithm (Breseham's Lie Algorithm) y = m(x k +1)+b d lower = y y k = m(x k +1)+b y k d upper = (y k +1) y = y k +1 m(x k +1) b d lower d upper =2m(x k +1) 2y k +2b 1 Decisio parameter, p k (by substitutig m=δy/δx) p k = Δx(d lower d upper ) = 2Δy x k 2Δx y k +c, where c= 2Δy+ Δx(2b 1) The sig of p k is the same as the sig of (d lower d upper )

14 Midpoit Lie Algorithm (Breseham's Lie Algorithm) p k = Δx(d lower d upper ) If y k is closer to the lie path, d lower < d upper p k <0 Þ plot the lower pixel (x k +1, y k ) Otherwise, Þ plot the lower pixel (x k +1, y k +1 )

15 Midpoit Lie Algorithm (Breseham's Lie Algorithm) p k+1 = 2Δy x k+1 2Δx y k+1 +c p k = 2Δy x k 2Δx y k +c (subtractio) p k+1 p k = 2Δy(x k+1 x k ) 2Δx(y k+1 y k ) (x k+1 = x k +1) p k+1 = p k +2Δy 2Δx(y k+1 y k ) ( y k+1 y k is either 0 or 1, depedig o the sig of p k ) p k+1 = p k +2Δy Ü y k+1 y k = 0 p k < 0 p k+1 = p k +2Δy 2Δx Ü y k+1 y k = 1 p k ³ 0

16 Midpoit Lie Algorithm (Breseham's Lie Algorithm) p 0 = 2Δy Δx If p k < 0, plot (x k +1, y k ) ad p k+1 = p k +2Δy Otherwise, plot (x k +1, y k +1 ) ad p k+1 = p k +2Δy 2Δx Advatages Oly eed add itegers ad multiply by 2 (which ca be doe by shift operatios) Icremetal algorithm

17 Midpoit Lie Algorithm- Example Lie ed poits: (x 0,y 0 ) = (5,8); (x 1,y 1 ) = (9,11) Δx = 4; Δy = 3 p 0 = 2Δy Δx = 2 > 0 è select (5+1, 8+1) p 1 = p 0 + 2(Δy - Δx) = 0 è Select (6+1,9) p 2 = p 1 + 2Δy = = 6 è Select (7+1, 9+1)

18 Sca Covertig Lies (issues) Edpoit order S 01 is a set of pixels that lie o the lie from P 0 to P 1 S 10 is a set of pixels that lie o the lie from P 1 to P 0 Þ S 01 should be the same as S 10 Varyig itesity of a lie as a fuctio of slope For the diagoal lie, it is loger tha the horizotal lie but has the same umber of pixels as the latter Þ eeds atialiasig Outlie primitives composed of lies Care must be take to draw shared vertices of polylies oly oce

19 Sca Covertig Lies (issues) Startig at the edge of a clip rectagle Startig poit is ot the itersectio poit of the lie with clippig edge Þ Clipped lie may have a differet slope

20 Sca Covertig Circles Eight-way symmetry We oly cosider 45 of a circle

21 Midpoit Circle Algorithm Suppose that we have just fiished drawig a pixel (x p,y p ) ad we are iterested i figurig out which pixel to draw ext.

22 Midpoit Circle Algorithm f(x,y) = x 2 + y 2 r 2 < 0 iside the circle = 0 o the circle > 0 outside the circle If f (x p +1, y p ½) < 0 the select E(x p +1, y p ); else select SE(x p +1, y p 1)

23 Midpoit Circle Algorithm p k = f (x k +1, y k ½) = (x k +1) 2 + (y k ½) 2 r 2 p k <0 the midpoit is iside the circle ad y k is closer to the circle boudary Þ plot the lower pixel (x k +1, y k ) Otherwise, the midpoit is outside or o the circle ad y k 1 is closer to the circle boudary Þ plot the lower pixel (x k +1, y k 1 )

24 Midpoit Circle Algorithm p k = (x k +1) 2 + (y k ½) 2 r 2 p k+1 = f (x k+1 +1, y k+1 ½) = [(x k +1)+1] 2 + (y k+1 ½) 2 r 2 p k+1 = p k +2(x k +1)+ (y k+12 y k2 ) (y k+1 y k ) +1 ( y k+1 is either y k or y k 1, depedig o the sig of p k ) p k+1 = p k +2x k+1 +1 Ü y k+1 = y k p k < 0 p k+1 = p k + 2x k y k+1 Ü y k+1 = y k 1 p k ³ 0, where 2x k+1 =2x k +2 2y k+1 =2y k 2

25 Midpoit Circle Algorithm (x 0, y o ) = (0, r) p 0 = f (1, r ½) = 5/4 r = 1 r (for r a iteger) If p k < 0, plot (x k +1, y k ) ad p k+1 = p k +2x k+1 +1 Otherwise, plot (x k +1, y k 1 ) ad p k+1 = p k + 2x k y k+1, where 2x k+1 =2x k +2 2y k+1 =2y k 2 Advatages Oly usig iteger additios ad subtractios Icremetal algorithm

26 Midpoit Ellipse Algorithm Use symmetry of ellipse Divide the quadrat ito two regios the boudary of two regios is the poit at which the curve has a slope of -1. Process by takig uit steps i the x directio to the poit P, the takig uit steps i the y directio Apply midpoit algorithm. Slope = -1 r y P r x

27 Midpoit Ellipse Algorithm æ x - x ç è rx c ö ø 2 æ ç y - y + è ry = 1 f ellipse (x,y) = r y2 x 2 + r x2 y 2 r x2 r y 2 < 0 iside the ellipse boudary = 0 o the ellipse boudary c ö ø 2 > 0 outside the ellipse boudary r y r x p1 k = f ellipse (x k +1, y k ½) p2 k = f ellipse (x k +½, y k 1)

28 Maitaiig Geometric Properties Whe lie drawig, exclude the last poit Whe drawig a eclosed area, display the area usig oly those pixels that are iterior to the object boudaries

29 Iside-Outside Tests Odd-eve Rule (Odd-parity Rule) For each pixel, determie if it is iside or outside of a give polygo. Approach From ay poit P beig tested, cast a ray to a distat poit i a arbitrary directio If the umber of crossigs is odd, the P is a iterior poit. If the umber of crossigs is eve, the P is a exterior poit.

30 Odd-Eve Rule Be sure that the lie path does ot itersect ay lie-segmet edpoits.

31 Odd-Eve Rule P Edge Crossig Rules a upward edge icludes its startig edpoit, ad excludes its fial edpoit; a dowward edge excludes its startig edpoit, ad iclude its fial edpoit; horizotal edges are excluded;

32 Odd-Eve Rule Very fragile algorithm Ray crosses a vertex Ray is coicidet with a edge Commoly used i ECAD Suitable for H/W

33 Iside-Outside Tests Nozero Widig Number Rule A widig umber the # of times the boudary of a object wids aroud a particular poit P i the couterclockwise directio No-zero values: iterior poits Zero values : exterior poits

34 Nozero Widig Number Rule Widig um: iitialized to 0 From ay poit P beig tested, cast a ray to a distat poit i a arbitrary directio +1: edge crossig the lie from right to left -1: left to right Use the sig of the cross product of the lie ad edge vectors No-zero: iterior Zero: exterior Be sure that the lie path does ot pass through ay lie-segmet edpoits.

35 How to decide iterior Vertices are umbered:

36 Polygo Tables A object described as a set of polygo surface facets Geometric data vertex table edge table surface-fact table Other parameters Color, trasparecy, lightreflectio properties

37 Area Fillig How to geerate a solid color/pattered polygo area Sca-lie fill algorithm Which pixels? What value?

38 Sca-Lie Fill Algorithm For each sca lie (1) Fid itersectios (the extrema of spas) Use Breseham's lie-sca algorithm Note that i a lie drawig algorithm there is o differece betwee iterior ad exterior pixels (2) Sort itersectios (icreasig x order) (3) Fill i betwee pair of itersectios

39 Sca-Lie fill algorithm Fid itersectios x k+1 = x k + Δx / Δy example (left edge) m = 5/2 x mi = 3 the sequece of x values 3, 3+2/5, 3+4/5, 3+5/6=4+1/5 y x pixel /5 3+4/5 4+1/5 (3,1) (3,2) (4,3) (4,4)

40 Spa Rules itersectio at iteger coordiate leftmost : iterior rightmost: exterior shared vertices cout parity at y mi vertices oly shorte edges horizotal edges do ot cout vertices A stadard covetio is to say that a poit o a left or bottom edge is iside, ad a poit o a right or top edge is outside.

41 Sca-Lie Fill Algorithm Not quite simple A sca lie passed through a vertex, it itersects two polygo edges at that poit.

42 Sca-Lie Fill Algorithm Sca lie y itersects a eve umber of edges Two pairs of itersectio poits correctly idetify the iterior spa Sca lie y itersects a odd umber (5) of edges Must cout the vertex itersectios as oly oe poit

43 Sca-Lie Fill Algorithm How to distiguish these cases (sca lie y & y ) Sca lie y Two itersectig edges are o opposite sides of sca lie Couted as just oe boudary itersectio poit Sca lie y Two itersectig edges are both above the sca lie By tracig aroud the boudary, If three edpoit y values of two cosecutive edges mootoically icrease or decrease cout the shared vertex as a sigle itersectio Otherwise, (a local miimum or maximum) add the two edge itersectios with the sca lie

44 Sca-Lie Fill Algorithm Implemetatio To shorte some polygo edges to split those vertices that should be couted as oe itersectio While processig o-horizotal edges i (couter)clockwise Check each edge whether the edge & ext edges have either mootoically icreasig or decreasig edpoits If so, shorte the lower edge

45 Sca-Lie Fill Algorithm Coherece properties spa coherece - all pixels o a spa are set to the same value sca-lie coherece - cosecutive sca lies are idetical edge coherece - edges itersected by sca lie i are also itersected by sca lie i+1

46 Sca-Lie Fill Algorithm Use edge coherece ad the sca-lie algorithm Sorted Edge Table Cotais all the o-horizotal edges. Edges are sorted by their smaller y coordiates. Table etry : y max, x-itercept value, iverse slope For each sca lie, edges are sorted from left to right Active Edge List While processig sca lies from bottom to top, product active edge list for each sca lie. Cotais edges which itersect the curret sca lie. Edges are sorted o their x itersectio values.

47 Sca-Lie Fill Algorithm

48 Sca-Lie Fill Algorithm Sca lie 9 Sca lie 10

49 Area Fillig (Fillig Methods) Pixel Adjacecy 4-coected 8-coected Boudary-Fill Algorithm If the boudary is specified i a sigle color, fill the iterior, pixel by pixel, util the boudary color is ecoutered starts from a iterior poit ad paits the iterior i a specified color or itesity.

50 Boudary-Fill Algorithm procedure boudaryfill4( x,y : iteger // startig poit i regio bordercolor: color // color that defies boudary fillcolor : color); // fillig color var c : color begi c := readpixel(x,y); if ( c bordercolor ad c fillcolor ) the begi writepixel(x,y,ewvalue); boudaryfill4(x, y-1, bordercolor, fillcolor ); boudaryfill4(x, y+1, bordercolor, fillcolor ); boudaryfill4(x-1, y, bordercolor, fillcolor ); boudaryfill4(x+1, y, bordercolor, fillcolor ); ed ed;

51 Boudary-Fill Algorithm There is the followig problem with boudary_fill4: Solve with 8- coected Ivolve heavy duty recursio which may cosume memory ad time

52 Boudary-Fill Algo. Efficiecy i space! Startig from the iitial iterior poit, first fill i the spa o the startig sca lie Locate ad stack startig positios for spas o the adjacet sca lies i dow to up order i left to right order All upper sca lies processed ðall lower sca lies processed

53 Flood-Fill Fill Algorithm Flood-Fill Algorithm Whe fillig a area that is ot defied withi a sigle color boudary start from a iterior poit ad reassig all pixel values that are curretly set to a give iterior color with the desired fill color.

54 Flood-Fill Fill Algorithm procedure flood_fill4( x,y: iteger //startig poit i regio oldvalue: color // value that defies iterior ewvalue: color); replacemet value begi if readpixel(x,y) = oldvalue the begi writepixel(x,y,ewvalue); flood_fill4(x,y-1,oldvalue,ewvalue); flood_fill4(x,y+1,oldvalue,ewvalue); flood_fill4(x-1,y,oldvalue,ewvalue); flood_fill4(x+1,y,oldvalue,ewvalue); ed ed;

55 Pattered Lies Pattered lie from P to Q is ot same as pattered lie from Q to P. P Q P Q Patters ca be geometric or cosmetic Cosmetic: Texture applied after trasformatios Geometric: Patter subject to trasformatios Cosmetic Geometric

56 Character, Symbols Bitmap fot (Raster fot) Set up a patter of biary values o a rectagular grid The simplest to defie ad display But, more storage : each variatio (size ad format) must be saved

57 Character, Symbols Outlie fot Use straight-lie ad curve sectios More flexible method Ca be icreased i size without distortig the character shapes By maipulatig curve defiitios for the character outlies. More time, sice they must be sca coverted ito the frame buffer Less storage

58 Character, Symbols Compariso of Methods Outlie fot Bitmap fot easy to rotate rotate by multiples of 90 easy to scale scale by powers of 2 variable legth storage sca covert lies fill if polygos may be ati-aliased or smoothed via curve fittig fixed legth storage sca covert poits draw as filled or outlie may be pre-ati-aliased

59 Lie Attributes Butt cap Roud cap Projectig square cap Miter joi Roud Joi Bevel joi

60 Aliasig i CG Which is better?

61 Aliasig i CG Digital techology ca oly approximate aalog sigals through a process kow as samplig. Aliaisig : the distortio of iformatio due to lowfrequecy samplig (udersamplig). Choosig a appropriate samplig rate depeds o data size restraits, eed for accuracy, the cost per sample Errors caused by aliasig are called artifacts. Commo aliasig artifacts i computer graphics iclude jagged profiles, disappearig or improperly redered fie detail, ad disitegratig textures.