Raster Graphics Algorithms

Transcription

1 Overview of Grahics Pieline Raster Grahics Algorithms D scene atabase traverse geometric moel transform to worl sace transform to ee sace scan conversion Line rasterization Bresenham s Mioint line algorithm Mioint circle algorithm cli transform to D screen sace rasterize Mioint ellise algorithm an more Fille rimitives D image (frame-buffer values) Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes Frame-buffer Moel D Scan Conversion Raster Disla: D arra of icture elements (iels) Piel can be set (grascale/color) Winow coorinates: iels centere at integers Geometric Primitives: D: oint, line, olgon, circle, D: oint, line, olheron, shere, Problem: Primitives are continuous, screen is iscrete Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes D Scan Conversion () Line Rasterization Solution: comute iscrete aroimation Scan Conversion: Algorithms for efficient generation of the samles comrising this aroimation Scan converting D line segments Inut: segment en oints as integer values: (, ), (, ) Outut: set of iels lighten u Set 9-Oct, CMPT-6 : Hami Younes 5 Set 9-Oct, CMPT-6 : Hami Younes 6

2 Line Rasterization: Basic Math Review Line Rasterization: Basic Math Review Length of line segment between P an P Sloe = = Solving for or = m + b = + ( + ) P = (, ) P = (, ) P = (, ) L = ( ) + ( ) Mioint of a line segment between P an P + + P mi = (, ) Two lines are erenicular iff m = m Parametric form of line equation: = + t( ) = + t( ) Set 9-Oct, CMPT-6 : Hami Younes 7 Set 9-Oct, CMPT-6 : Hami Younes 8 Basic Line Algorithms A Naïve Algorithm Must: Comuter integer coorinates of iels which lie on or near a line Be efficient Create visuall satisfactor images: lines shoul aear straight lines shoul terminate accuratel lines shoul have constant ensit lines ensit shoul be ineenent of length an angle Alwas be efine increment in each ste Comute as a function of Roun (wh?) Set Piel (, Rn(()) ) = + ( + or = m + b ) m = = Set 9-Oct, CMPT-6 : Hami Younes 9 Set 9-Oct, CMPT-6 : Hami Younes A Naïve Algorithm () Basic Incremental Algorithm Problems: Comuting value is eensive, for ever : floating oint multilication floating oint aition call to rouning function cetion aroun vertical lines Gets isconnecte (strung out) for m > Also calle: Digital Differential Analzer (DDA) Base on arametric equation of line: Comute an so that one of them is an the other is less than Start from (, ) in each ste, increment b an b Set Piel (Rn(), Rn()) Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes

3 Basic Incremental Algorithm () Bresenham s Mioint Algorithm voi DDA (int, int, int, int ) int length; float,,, ; if abs(-) > abs(-) length = abs(-) else length = abs(-) Kenote: At each ste along the line, there are onl two choices for the net iel to be fille in. = (-)/length; = (-)/length; = ; = ; for (int i = ; i <=length; i++) setiel(rn(), rn()) += ; += ; Creates goo lines, but still there are roblems The set of which two iels shoul be fille in eens on the slo of the line Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes Bresenham s Mioint Algorithm () Bresenham s Mioint Algorithm () For a line with a slo between an, the two ossible iels that might be fille in, are the an N neighbors Let s consier < sloe < We can etermine the net iel b utting a ecision oint half-wa between the two caniate oints: What are the ossible net iels for a line with: - < slo < slo > The net iel is selecte base on which sie of the ecision oint, the true line is on, so we can efine a ecision variable (). Set 9-Oct, CMPT-6 : Hami Younes 5 Set 9-Oct, CMPT-6 : Hami Younes 6 Bresenham s Mioint Algorithm () Bresenham s Mioint Algorithm (5) + + / + / choices for net ste (base on the ecision variable that will be comute in the current ste) + + / + / = m + B = + B F(, ) =.. + B. + + choices for revious current ste iel (base on the ecision variable comute in the revious ste) In each ste of the algorithm: etermine the coorinates of current base on the ecision variable comute in revious ste comute ecision variable for net ste revious iel + + F (, ) = : [, ] is on the line F (, ) > : [, ] is below the line F (, ) < : [, ] is on to of the line ecision variable: = F ( m, m ) where [ m, m ] is the coorinates of the mioint So, How to efine ecision variable ()? e.g. the value of in current ste: +, + / ) = ( +). ( + / ).+B. Set 9-Oct, CMPT-6 : Hami Younes 7 Set 9-Oct, CMPT-6 : Hami Younes 8

4 Bresenham s Mioint Algorithm (6) Bresenham s Mioint Algorithm (7) + / ol = ( +). ( + / ).+B. + / ol = ( +). ( + / ).+B / case ) ol < : net = case ) ol > : net = N + / case ) ol < : net = case ) ol > : net = N revious iel + + case ) +, + ) = ( + ). ( + ). + B. = [( + ). + ] ( + ). + B. = + ol < net = ol revious iel + + case) > net = N +, + ) = ( + ). ( + ). + B. = [( + ). + ] [( + ). + ] + B. = + ( ) ol ol Set 9-Oct, CMPT-6 : Hami Younes 9 Set 9-Oct, CMPT-6 : Hami Younes Bresenham s Mioint Algorithm (8) Bresenham s Mioint Algorithm (9) First mioint So far, We comute an incremental function for the ecision variable. What about the ecision value? How to get rie of the ½ fraction? Solution: We onl nee the sign of, so multil b! +, + ) = ( + ). ( + ). + B. = [. + ] [. + ] + B., ) + F(, ) = = ( ) ( N ) = + ol = + ( ) ol = Oh, Noooo!!! floating oint Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes Bresenham s Mioint Algorithm (Finall the coe!!!) Lines with more general sloes? voi MiointLine(int, int, int, int ) int = ; int = ; int = * // value of int = ; int = ; int inc = * ; // increment when choose int incn = * ( ); // increment when choose N ) > - ) > SetPiel (, ); while ( < ) if ( <= ) += inc ; ++; else += incn; ++; ++; SetPiel (, ); // the start iel // choose // choose N ) - > 5) - > - 6) - > - ) > 8) > - 7) - > Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes

5 Lines with more general sloes? () Scan Converting Circles Solution : comute the roer ecision variable for each 8 octants: e.g. o): N an N o5): W an SW Assume center of circle is at (, ): + = R = F (, ) = + R mmm no, thanks! we have alrea ha enough trouble for that one octant! Solution : convert all octants to octant same Mioint algorithm convert to original octant in SetPiel e.g. o): Mioint (, ) SetPiel (, ) F (, ) = : [, ] is on the circle F (, ) > : [, ] is outsie the circle F (, ) < : [, ] is insie the circle What if center is not at (, )? R Set 9-Oct, CMPT-6 : Hami Younes 5 Set 9-Oct, CMPT-6 : Hami Younes 6 Scan Converting Circles () Mioint Circle Algorithm Take avantage of smmetries to minimize cases (sloes) an amount of curve rawing Onl one eighth of a circle nees to be comute! M M ol +, -½) = ( +) + ( -½) -R -< sloe < : voi CirclePoints (,, center, center) SetPiel ( + center, + center); sloe = S MS case ) ol < : net = case ) ol > : net = S SetPiel ( + center, + center); SetPiel (- + center, + center); SetPiel ( + center, - + center); SetPiel (- + center, - + center); SetPiel (- + center, - + center); SetPiel ( + center, - + center); SetPiel (- + center, + center); sloe = - Previous Piel Current Piel net Piel case) ol < net = +, ) = ( ) ( ) + + R = [( + ) + + ] + ( ) R = + + ol Set 9-Oct, CMPT-6 : Hami Younes 7 Set 9-Oct, CMPT-6 : Hami Younes 8 Mioint Circle Algorithm () Mioint Circle Algorithm () ol +, -½) = ( +) + ( -½) -R M M M M S MS case ) ol < : net = case ) ol > : net = S Previous Piel Current Piel net Piel case) +, ) = ( + ) + ( ) R = [( + ) + + ] + [( ) + ] R = ol > net = S ol First Piel Current Piel = [ + + ] + [ + ] R =, = R +, ) = ( + ) + ( ) R 5 = R Set 9-Oct, CMPT-6 : Hami Younes 9 Set 9-Oct, CMPT-6 : Hami Younes 5

6 Mioint Circle Algorithm () Mioint Circle Algorithm (5) Floating Point: to be or not to be? ( ) ( S) int ' = = ol + + = ol = R ' < < an ' > > so wh not use '?!! ' = R int int ( ) ( S) = ol = ol + + int elta int eltas elta = eltaol eltas = eltasol + elta = eltaol + S + +, eltas = eltasol + Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes Mioint Circle Algorithm (6) Scan Conversion llises voi MiointCircle(int center, int center, int R) int = ; int = R; int = R // value for ecision variable int elta = ; int eltas = -*R + 5; CirclePoints (,, center, center); b + a = a b = F (, ) = b + a a b (-, ) b (, ) while ( > ) // sloe is between - an if ( < ) // Select += elta; elta += ; eltas += ; else // Select S += eltas; elta += ; eltas += ; --; ++; CirclePoints (,, center, center); // while F (, ) = : [, ] is on the ellise F (, ) > : [, ] is outsie the ellise F (, ) < : [, ] is insie the ellise (-, -) What if not centere at (, )? a (, -) Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes Mioint llise Algorithm Mioint llise Algorithm () F(, ) = b + a a b Another metho to test the sloe : Graient F F Graient( F) = i + j Graient( F) = b i + a j G(F) j > G(F) i : a > b region G(F) i > G(F) j : b > a region Region G(F) j >G(F) i Region graient vector G(F) j =G(F) i Set 9-Oct, CMPT-6 : Hami Younes i 5 j graient vector G(F) i >G(F) j Region + S Region - S +, ) region + + S +, ) S +, ) region - - S +, ) S Set 9-Oct, CMPT-6 : Hami Younes 6 6

7 Mioint llise Algorithm () Rotate llises? The rest is more straightforwar: calculate as a function of ol calculate the values then we have two main loos: while a > b, scan along ++ while >, scan along -- + = K (constant) ist( P, C ) + ist( P, C ) = K C C P ( P C ) + ( P C ) + ( P C ) + ( P C ) = K F(, ) = ( P C ) + ( P C ) + ( P C ) + ( P C ) K Pleeese, sto right here!!! Set 9-Oct, CMPT-6 : Hami Younes 7 Set 9-Oct, CMPT-6 : Hami Younes 8 Wh sto? We want more! General roceure to evelo a mioint metho for an arbitrar shae: formulate the shae (arabola, sinus, ) Fille Primitives Rectangle (Bo) for ( = ; <= ; ++) for ( = ; <= ; ++) SetPiel (,, fillcolor); etermine the ecision variable etermine the scanning ath(s) so that the ecision in each ste is limite tr to formulate the equations as incremental an avoi floating oint oerations as much as ossible, or at least convert them to easier ones (mult to a, sin to mult, )? Set 9-Oct, CMPT-6 : Hami Younes 9 Set 9-Oct, CMPT-6 : Hami Younes Fille Primitives Circle (Dist) Fille Primitives Triangle voi CirclePointsFill (,, center, center) int i; for (i = +center; i <= - + center; i++) SetPiel (i,+center, fillcolor); SetPiel (i,-+center, fillcolor); for (i = +center; i <= - + center; i++) SetPiel (i,+center, fillcolor); SetPiel (i,-+center, fillcolor); The most frequentl use rimitive in D grahics! Just, Kee track of two Bresenham lines in each ste an otimization? u: ski stes when oes not change! Set 9-Oct, CMPT-6 : Hami Younes Set 9-Oct, CMPT-6 : Hami Younes 7