CS 450: COMPUTER GRAPHICS RASTERIZING LINES SPRING 2016 DR. MICHAEL J. REALE

Transcription

1 CS 45: COMPUTER GRAPHICS RASTERIZING LINES SPRING 6 DR. MICHAEL J. REALE

2 OBJECT-ORDER RENDERING We going to start on how we will perform object-order rendering Object-order rendering Go through each OBJECT find out what piels are affected Advantages: Much more efficient for large scenes with lots of objects although how ou access data will mae this more or less efficient Also called rendering b rasterization or scanline renderering

3 RASTERIZATION Rasterization = finding all piels in an image that are occupied b a geometric primitive Also called scan conversion Eample: have nice, continuous formula for a line pic reasonable piels discrete positions to fill in to represent line, f

4 GRAPHICS PIPELINE Graphics pipeline We ve got this and we want this Generates or renders a D image given a 3D scene of objects I.e., whole sequence of operations that taes us from 3D OBJECTS PIXELS IN IMAGE Also called graphics rendering pipeline or just the pipeline Composed of both hardware and software We ll tal about the other steps in the pipeline in more detail later For now, let s assume that we have D representations of the primitives we want to draw lines, circles, triangles, etc. Basicall woring in screen space Picing the right piels job of the rasterization stage

5 RASTERIZATION STAGE Rasterization stage Enumerates piels covered b primitive Interpolates values called attributes across the primitive Color would be an attribute Outputs fragments One fragment for each piel covered b primitive Each fragment lives at a particular piel Each fragment has its own set of attribute values Fragments then need to be processed and possibl blended to get each piel s final color E.g., given two fragments from different objects but in the same location which one do ou draw? Do ou blend them together? Some include the fragment processing and blending stages as part of the rasterization stage

6 RASTERIZING LINES OVERVIEW

7 A POINT OF CLARITY In this case, we are taling about drawing/rasterizing line segments That is, the lines have a defined beginning point and end point

8 RASTERIZING LINES OVERVIEW To rasterize lines, we ll discuss several algorithms: DDA Digital differential analzer Midpoint Bresenham

9 A NOTE ABOUT EFFICIENCY As we discuss these algorithms, we will tal about their relative efficienc HOWEVER, hardware has changed what is most efficient has changed Still true: Should do fewer operations if ou can Multipl/divide more comple than add/subtract USED to be true: Integer calculations were much faster than floating-point operations on older hardware Bresenham most efficient in this respect because it onl uses integers However, modern hardware doesn t reall have this problem anmore End of the da: need to actual profile our code on our hardware to see where the bottlenecs are

10 DDA LINE DRAWING ALGORITHM

11 SLOPE-INTERCEPT FORM As we ve seen before, a straight line can be mathematicall defined using the Cartesian slope-intercept equation: m We re dealing with line segments, so these have specified starting and ending points: So, we can compute the slope m and the intercept b as follows: m b end end m b, end, end

12 INTERVALS Let s sa we have: change change in in interval interval in in "run" "rise" That means we can get δ from the slope m and δ: rise m* * run run rise Similarl, we can get δ from δ: m

13 DDA ALGORITHM Digital differential analzer DDA Scan-conversion line algorithm based on calculating either δ or δ Sample one coordinate at unit intervals find nearest integer value for other coordinate Eample: < m <=. slope positive, with δ > δ Increment in unit intervals δ = Compute successive values as follows: Round value to nearest integer m

14 DDA ALGORITHM: M >. Problem: If slope is positive AND greater than. m >., then we increment b sip piels in! Solution: swap roles of and! Increment in unit intervals δ = Compute successive values as follows: Round value to nearest integer m

15 DDA ALGORITHM: WHICH DO WE STEP IN? So, to summarize so far, which coordinate should be increment? Remember: If absδ > absδ: Step in X Otherwise: Step in Y m end end

16 DDA ALGORITHM: LINES IN REVERSE We ve been assuming that the ending point has a coordinate value greater than the starting point: Left to right, if incrementing Bottom to top, if incrementing However, we could be going in reverse. If so, then: If right to left, δ = - If top to bottom, δ = - Right-to-left Top-to-bottom

17 DDA ALGORITHM: CODE // Get d and d int d = - ; int d = - ; int steps, ; float Increment, Increment; // Set starting point float = ; float = ;

18 DDA ALGORITHM: CODE // Determine which coordinate we should step in if absd > absd steps = absd; else steps = absd; // Compute increments Increment = floatd / floatsteps; Increment = floatd / floatsteps;

19 DDA ALGORITHM: CODE // Let s assume we have a magic function called setpiel, that sets a piel at, to the appropriate color. // Set value of piel at starting point setpielround, round; // For each step for = ; < steps; ++ { // Increment both and += Increment; += Increment; } // Set piel to correct color // NOTE: we need to round off the values to integer locations setpielround, round;

20 DDA ALGORITHM: PROS AND CONS Advantage: Faster than using the slope-intercept form directl no multiplication, onl addition Caveat: initial division necessar to get increments Wors for lines in all directions b default Disadvantages: Accumulation of round-off error line can drift off true path Rounding procedure time-consuming

21 MIDPOINT ALGORITHM

22 MIDPOINT ALGORITHM DDA uses eplicit form Midpoint algorithm uses implicit form f, Avoids rounding Basic idea: test whether the point f+, + ½ is inside or outside choose closest point Can be etended to other inds of curves beond lines We ll tal about some of these later

23 CHOOSING THE NEXT PIXEL For starters, let s assume our slope m is, ] We ll tal about the other three cases later Thus, we re incrementing in Basicall, if we ve plotted a piel,, we need to choose between: +, +, + Question: which one is closer to the true line?

24 CHOOSE WISELY Pseudocode-wise, this becomes: = For = to draw, if some condition: = + So, what is some condition?

25 THE MIDPOINT PART f If we have an implicit equation for the line f,:, Then: f, > ABOVE line f, = ON line f, < BELOW line SO, we tae a loo at the midpoint between our two choices: f +, +.5 f +, +.5 > midpoint ABOVE line BOTTOM point closer choose +, f +, +.5 < midpoint BELOW line TOP point closer choose +, +

26 SOME CONDITION Thus, the pseudocode becomes: = For = to draw, if f+, +.5 < : = +

27 INCREMENTAL VERSION One problem: we are evaluating the function f, ever iteration We can mae an incremental version of the approach that will mae it more efficient Let s sa our net point is going to be, or,+ need to evaluate f, +.5 Observation: we must have evaluated ONE of the following in the previous iteration: f,.5 OR f, +.5 depending on whether we chose to increment last time or not

28 INCREMENTAL VERSION Two interesting things to note about our implicit line equation:,, f f,, f f

29 INCREMENTAL VERSION This means that, if we alread have f,, we can compute both f+, and f+,+ b adding on certain constant values: SO, that means, if we need f, +.5: DIDN T increment last time have f, +.5 DID increment last time have f,.5,,,, f f f f.5,.5, f f.5,.5, f f

30 INCREMENT VERSION: ADDITIONAL EFFICIENCY Can mae faster b precomputing and storing:.5,.5, f f.5,.5, f f

31 INCREMENTAL VERSION: CODE To eep trac of this, we have a decision variable d: = d = f +, +.5 For = to draw, if d < : = + d = d + + else d = d +

32 MIDPOINT ALGORITHM: PROS AND CONS Advantages: Lie DDA onl using addition in main loop UNLIKE DDA: No rounding No initial division needed Etendable to other inds of curves circles, ellipses, etc. Disadvantages: If lines are VERY long can accumulate floating point errors over time Decision variable = still floating-point number Usuall not a big problem in practice Still using floating point calculations slower on older hardware

33 BRESENHAM S ALGORITHM

34 INTRODUCTION The Midpoint algorithm still uses floating-point calculations for its decision variable Bresenham s algorithm uses all integers More efficient in das of ore Slightl more involved derivation Uses distances between candidate points and true line to mae decision

35 BRESENHAM S LINE ALGORITHM: THE IDEA We ll start b assuming the slope is, ] lie before Similar to midpoint loo at a decision variable to help us mae the choice at each step However, derivation of variable is different Our previous position:, d d lower upper distance distance of of,, from true line coordinate from true line coordinate,,

36 BRESENHAM S LINE ALGORITHM: MATH The value of for the mathematical line at + is given b: Ergo: b m b m d b m d upper lower

37 BRESENHAM S LINE ALGORITHM: MORE MATH To determine which of the two piels is closer to the true line path, we can loo at the sign of the following: Positive d upper is smaller choose + Negative d lower is smaller choose b m b m b m b m b m d d upper lower

38 BRESENHAM S LINE ALGORITHM: EVEN MORE MATH Remember that: Substituting with our current equation: We will let our decision variable p be the following m end end b d d upper lower c b b b b d d p upper lower

39 BRESENHAM S LINE ALGORITHM: DECISION VARIABLE p c Multipling b Δ won t affect the sign of p, since Δ > Note that constant c does not depend on the current position at all, so can compute it ahead of time: So, as before: c b p positive d upper is smaller choose + p negative d lower is smaller choose

40 BRESENHAM S LINE ALGORITHM: UPDATING P K We can get the net value of the decision variable i.e., p + using p c c c c p p c p c p

41 BRESENHAM S LINE ALGORITHM: UPDATING P K However, we now: Therefore: So, we need to determine what + was: If p was positive + = If p was negative + = p p p p p p

42 BRESENHAM S LINE ALGORITHM: SUMMARIZED. Input two line endpoints and store LEFT endpoint in, Allows us to deal with lines going in the OPPOSITE direction!. Plot first point, 3. Compute constants Δ, Δ, Δ, and Δ - Δ. Also compute first value of decision variable: p = Δ Δ 4. At each, test p : If p plot p plot p : Otherwise 5. Perform step 4 Δ times : p p,, NOTE: Effectivel assuming line starts at, and thus b = p b NOTE: This version ONLY wors with < m <.!!! Ergo, Δ and Δ are positive here!

43 COMPARISON TO MIDPOINT Bresenham s decision variable: Starting value: Increment if p > Increment cases: Incremented last time: Did NOT increment last time: p p p p p Bresenham: - Everthing multiplied b removes floating point - Decision variable and increments flipped Midpoint decision variable: Starting value: Increment if d < Increment cases: Incremented last time: Did NOT increment last time: f,.5 f, *.5 *.5 d d d d d

44 BRESENHAM S LINE ALGORITHM: CODE // NOTE: d and d are ABSOLUTE VALUES in this code int d = fabs - ; int d = fabs - ; int p = *d - d; int twod = *d; int twodminusd = *d - d; int,;

45 BRESENHAM S LINE ALGORITHM: CODE // Determine which endpoint to use as start position if > { = ; = ; = ; } else { = ; = ; } // Plot first piel setpiel,;

46 BRESENHAM S LINE ALGORITHM: CODE while < { ++; } ifp < p += twod; else { ++; p += twodminusd; } setpiel,;

47 DRAWING LINES IN ALL DIRECTIONS

48 DRAWING LINES IN ALL DIRECTIONS For Midpoint and Bresenham, we have thus far assumed slope m is within, ] Minor eception: our current version of Bresenham allows us to draw lines in reverse in, but the slope is still positive To draw lines in ALL directions, we ll have to emplo a few trics We will be discussing the trics with Bresenham in mind, but ou can use similar strategies for Midpoint as well

49 LINE DRAWING GENERALIZED Abschange in Y > abschange in X? Swap roles of X and Y Is X going in negative direction? Swap endpoints Set starting Y coordinate Is Y going in negative direction? Set Y increment to - Swap Y coordinates of endpoints Basic idea: We will start at the original Y and we will decrement Y as necessar. HOWEVER, we will PRETEND lie we are drawing a line that is going up decisions will be the same e.g., eep current Y, change Y Calculate first decision variable and decision variable increments

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51 SPECIAL CASES IN LINE DRAWING To save time, if ou have a line that is: Δ = vertical Δ = horizontal Δ = Δ diagonal ou can just draw it directl without going through the entire algorithm.

52 ALGORITHM COMPARISON

53 ALGORITHM COMPARISON DDA Digital differential analzer Not ver efficient rounding Ver straightforward Can drift because of round-off error Midpoint Efficient No rounding, no initial division lie in DDA Mostl straightforward Uses minimal floating-point calculations Can drift, but onl for ver long lines Bresenham Efficient More involved to derive Uses all integers