CS 325 Introduction to Computer Graphics

Transcription

1 CS 325 Introduction to Computer Graphics 01 / 29 / 2014 Instructor: Michael Eckmann

2 Today s Topics Comments/Questions? Bresenham-Line drawing algorithm Midpoint-circle drawing algorithm Ellipse drawing Antialiasing Polygons Polygon filling algorithm

3 Bresenham's Line Algorithm Bresenham's Algorithm is efficient (fast) and accurate. The key part of Bresenham's algorithm lies in the determination of which pixel to turn on based on whether the line equation would cause the line to lie more near one pixel or the other.

4 Bresenham's Line Algorithm Think of a line whose slope is >0, but < 1. These are first octant lines (the area of the coordinate plane between the positive x-axis and the line y=x.) Assume we have the pixel at (x k, y k ) turned on and we're trying to determine between the following: Should (x k+1, y k ) or (x k+1, y k+1 ) be on? That is, the choice is between either the pixel one to the right of (x k, y k ) or the pixel one to the right and one up. The decision can be made based on whether d lower - d upper is positive or not. The goal is to come up with an inexpensive way to determine the sign of (d lower d upper )

5 Bresenham's Line Algorithm The y coordinate on the line is calculated by: y = m(x k + 1) + b and d lower = y y k d upper = y k+1 y d lower d upper = 2m(x k + 1) 2y k + 2b 1 Since m is < 1 (recall it's in the 1 st octant) it is a non-integer. That's not good so, replace it with dy/dx, where dy is the change in y endpoints and dx is the change in x endpoints these will be integer values divided --- but we'll get rid of the division. d lower d upper = 2 ( dy / dx )(x k + 1) 2y k + 2b 1 Multiply both sides by dx and get: dx (d lower d upper ) = 2 dy (x k + 1) 2 dx y k + 2 dx b dx dx (d lower d upper ) = 2 dy x k 2 dx y k + 2 dy + 2 dx b dx 2 dy + 2 dx b dx is a constant which is independent of each iteration

6 Bresenham's Line Algorithm So we end up with dx (d lower d upper ) = 2 dy x k 2 dx y k + c We only need to determine the sign of (d lower d upper ) dx is positive so it doesn't have an effect on the sign (so now there's no need to divide) --- does that make sense? Let p k = dx (d lower d upper ) p k is the decision parameter So, we check if 2 dy x k 2 dx y k + c < 0 and if it is we turn on the lower pixel, so (x k+1, y k ) should be the pixel turned on otherwise (x k+1, y k+1 ) is turned on As the algorithm iterates though it needs to then calculate the next value of p, which is p k+1

7 Bresenham's Line Algorithm Recall that to calculate p k we had: p k = 2 dy x k 2 dx y k + c So, p k+1 = 2 dy x k+1 2 dx y k+1 + c If we subtract p k+1 p k = 2dy ( x k+1 x k ) 2dx ( y k+1 y k ) But, x k+1 = x k + 1 because the first octant slope always increment x by 1 So, we have p k+1 p k = 2dy 2dx ( y k+1 y k ) Then, p k+1 = p k + 2dy 2dx ( y k+1 y k ) Where ( y k+1 y k ) is either 1 or 0 (why?) depending on the sign of p k. So, either p k+1 = p k + 2dy (when p k < 0) p k+1 = p k + 2dy 2dx (when p k >= 0)

8 Initial decision variable value The initial value of p in the Bresenham line algorithm is calculated by using the equation for p with (x 0, y 0 ) Derivation: p k = 2dyx k 2dxy k + c where c = 2dy + dx(2b-1) and since y k =mx k +b, b = y k mx k and m = dy/dx p k = 2dyx k 2dxy k + 2dy + dx(2(y k (dy/dx)x k ) 1) multiplied out: p k = 2dyx k 2dxy k + 2dy + 2dxy k 2dyx k dx = 2dy dx

9 Bresenham's Line Algorithm /* Bresenham line-drawing procedure for m < 1.0. */ void linebres (int x0, int y0, int xend, int yend) { int dx = fabs (xend - x0), dy = fabs(yend - y0); int p = 2 * dy - dx; int twody = 2 * dy, twodyminusdx = 2 * (dy - dx); int x, y; /* Determine which endpoint to use as start position. */ if (x0 > xend) { } x = xend; y = yend; xend = x0; else { } x = x0; y = y0; setpixel (x, y); while (x < xend) { x++; if (p < 0) } } p += twody; else { } y++; p += twodyminusdx; setpixel (x, y); /* From Hearn & Baker's Computer Graphics with OpenGL, 3 rd Edition */

10 Circle Algorithms

11 Circle Algorithms So, we only need to calculate the coordinates of an eighth of the circle and plot the following: (-x, -y), (-x, y), (x, -y), (x, y), (y, x), (y, -x), (-y, x), (-y, -x) Bresenham's Line Algorithm has been adapted to circles by deciding which pixel is closest to the actual circle at each sampling step. It should be able to draw circles with any center, but to make calculations easier, compute the pixel coordinates based on the same radius but centered at the origin. Then when you plot the pixel, add in the center coordinates. Consider only the octant between the lines x=0 and y=x. The slope of the curve varies from 0 (at the top) to -1 at the 45 degree angle. This implies that we should do unit steps on the x-axis and calculate y (I leave this for you to convince yourself that this is the right approach.) At each x, the decision is between which y is correct among two possible y's, the one to the right or the one to the right and down.

12 Circle Algorithms

13 Circle Algorithms The decision is always between the pixel one to the right or one to the right and one down. Respectively, y k or y k -1. The midpoint circle algorithm is a way to determine it using only integer arithmetic. Define a circle function as f(x,y) = x 2 + y 2 r 2 A point x,y on the circle with radius r will cause f(x,y) = 0 A point x,y in the circle's interior will cause f(x,y) < 0. A point x,y outside the circle will cause f(x,y) > 0. The point we use is the midpoint between the two pixel choices. And f(x,y) as defined above is our decision variable p k The point is (x k + 1, y k ½) If p k is < 0, it is inside the circle and closer to y k Otherwise choose y k -1

14 Circle Algorithms p k = f(x k + 1, y k ½) = (x k + 1) 2 + (y k ½) 2 r 2 If p k is < 0, it is inside the circle and closer to y k Otherwise choose y k -1 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 = ((x k + 1) + 1) 2 + (y k+1 ½) 2 r 2 (p k+1 can be expressed in terms of p k ) Similar to how we determined the recursive decision variable computation in Bresenham's Line Algorithm, we do for the Circle Algorithm as well: skipping a bunch of steps leads to: p k+1 = p k + 2x k (when p k < 0) p k+1 = p k + 2x k y k+1 (otherwise) Assuming we start at point (0,r), we'll have p 0 = f(0 + 1, r ½) = f(1, r ½) which yields 5/4 r (or 1 r for integer radii because of rounding) Let's plot some points as an example for a circle of radius 6, and center (-2, 4) We do calculations with center 0,0 and for each point we plot, we add in the center.

15 Circle Algorithm Example p 0 = 1 r p k+1 = p k + 2x k (when p k < 0) p k+1 = p k + 2x k y k+1 (otherwise) Center = (-2, 4) K P k (x k+1, y k+1 ) Center + (x k+1, y k+1 ) 2x k+1 2y k (1, 6) (-1, 10) (2, 6) (0, 10) (3, 5) (1, 9) 6 10 Plot: (-1, 10), (0, 10), (1, 9), (2, 8) Also, plot points in the other 7 octants for each of these four. e.g. The other 7 points for (-1, 10) to plot are: (-3, 10), (-1,-2), (-3,-2), (4,5), (4, 3), (-8, 5), (-8, 3) These were calculated by adding the center (-2,4) to (-1,6), (1,-6), (-1,-6), (6, 1), (6,-1), (-6,1), and (-6,-1)

16 Circle Algorithms There's a figure in the text book (figure 6-15) and it looks to me like the circle line is drawn incorrectly but the pixels are correct. It is showing the center of the circle at the bottom left corner of the 0,0 pixel box, but it should be in the center of the 0,0 pixel box. I noticed it because when x=3, based on the drawn circle, the algorithm would turn on (3,9) not (3,10), because the midpoint is clearly outside the circle there. I didn't verify it, but I trust the calculations in the table which says that p 2 = -1 which would be inside the circle.

17 Ellipse Algorithm We can generalize the circle algorithm to generate ellipses but with only 4-way symmetry. The slope of the ellipse changes from > -1 to < -1 in one quadrant so that we will have to switch from unit samples along the x-axis and calculate y to unit samples along the y-axis and calculate x I'm not as concerned about the details of ellipses as I am with you understanding the details of the line and circle algorithms.

18 Ellipse Algorithms

19 Ellipse Algorithms

20 Antialiasing Aliasing is the distortion we see when we undersample. The jaggedness stairstep quality of lines is an example of aliasing. Solutions: Higher addressability e.g. Move from a 800x600 grid to a 1280x1024 grid and your lines will look smoother. If keep addressability fixed then we can change the intensity of the pixels being turned on. Instead of turning on one pixel at full intensity, we appropriately weight the intensities of two pixels at each step to cause the illusion of a point living at some intermediate location between the two pixel centers.

21 Antialiasing Example on the board.

22 Antialiasing Any ideas on how we could adapt Bresenham's line algorithm to do antialiased lines?

23 Antialiasing Any ideas on how we could adapt Bresenham's line algorithm to do antialiased lines? d lower + d upper = 1 The upper pixel's intensity can be set to be d lower The lower pixel's intensity can be set to be d upper When the line passes exactly through the center of a pixel, that pixel's intensity is 1 and the other is 0 (off).

24 Antialiasing Supersampling is a technique to treat the screen as a finer grid than is actually addressable and plot the points by turning on multiple points in the finer grid. Then when ready to display --- use the finer grid points to determine the intensities of the pixels of the actual screen. Example --- divide each actual addressable pixel into 9 (3x3) finer subpixels. Draw a Bresenham line on the subpixels. Then base the intensity of actual pixel on how many of the 9 subpixels would be on. 3 subpixels is max, so that's full intensity 2 subpixels could cause the pixel to be at 2/3 intensity 1 subpixel on could cause the pixel to be at 1/3 intensity 0 would be off

25 Antialiasing Alternately, we could use a weighting mask that would give more weight to the center subpixel of the 3x3 grid Another alternative is to treat the line as a rectangle of width 1 pixel & draw it on the finer grid (w/ 9 subpixels per pixel). Then how many of the 9 subpixels are on (on = bottom left corner of subpixel is inside the rectangle) determines the intensity of pixel (9 = full intensity, 8 = 8/9 intensity, etc.) This is better than the three intensity levels of the other supersampling method.

26 Antialiasing Area sampling is a technique that computes areas of overlap of each pixel with the object (e.g. line or polygon) to be displayed. This is similar to the supersampling method described last (treating the line as a rectangle), but instead of just 9 different intensities, let the intensity of a pixel be the area of the rectangle that overlaps the pixel / the area of the whole pixel.

27 Antialiasing There are other techniques to do antialiasing --- filtering techniques. These are similar to the discrete mask two slides ago, except they work with a continuous function to determine the weightings of the pixels. Examples on the next slide show a box, a cone and a gaussian. The volumes of these are normalized to 1 and to determine the weighting of a pixel we integrate the function over the pixel surface. Since integrating is expensive (in time) lookup tables can be used.

28 Antialiasing

29 Antialiasing also helps with... Notice: without antialiasing, compare a line with slope = 1 made up of n pixels and a line with slope = 0 made up of n pixels. The slope = 1 line is SQRT(2) longer but is made up of the same number of pixels. What do you think is the visual effect of this?

30 Antialiasing helps total intensity See figures 6-63 & 6-64 in text. In addition to lines looking straighter, when using the antialiasing technique that treats lines as having some finite width, lines of different angles will not have different total intensities.

31 Polygons 3 or more vertices describe a polygon. The vertices are connected with line segments (called an edge, or a side). Each edge can only have endpoints in common with any other edge. Definition of convex polygon Def 1: All interior angles are < 180 degrees Def 2: The interior lies completely on 1 side of the infinite extension of any edge. Def 3: If we select any two interior points, the line segment joining them is also fully in the interior of the polygon Definition of concave polygon Not convex.

32 Polygons Degenerate polygons are Those with 3 or more collinear vertices Those with repeated vertices Etc.

33 Polygon interior points A common operation is to determine if a point is in the interior of a polygon or not. One way to do it is, for some point, draw a horizontal line from that point to a distant point outside the possible area of the polygon and count how many times the horizontal line crosses edges of the polygon. If the line crosses an odd number of edges, then it is an interior point. If it crosses an even number of edges, then it is exterior. Example on board.

34 Polygon interior points What happens when the horizontal line crosses a vertex. Should it be counted as 1 or 2? Example on board.

35 Polygon interior points What happens when the horizontal line crosses a vertex. Should it be counted as 1 or 2? If the vertex is a local maximum or a local minimum then count it as 2. Otherwise count it as 1.

36 Polygon fill areas Polygons in OpenGL are specified by vertices. It is good to get into the habit now that you specify the vertices in counterclockwise order. The reason for this is, when we do some things later this semester like displaying 3D objects, we're going to care about which side (face) of the polygon is visible. The front face of a polygon is visible when its vertices are in counterclockwise order. We'll know if we're seeing the back face of a polygon if they are in clockwise order.

37 Polygon fill areas To do a scanline fill of a polygon (see figure 6-46, 47, 48). Determine the intersections (crossings) of the horizontal line (y=c) with the edges (y=mx + b) of the polygon. Sort the crossings (x coordinates) from low to high (left to right). There will be an even number of crossings, so draw horizontal lines between the first two crossings, then the next 2 and so on. If the scanline goes through a vertex Add the crossing to the list twice if it's a local max or min. Add the crossing only once otherwise. Horizontal edges of a polygon can be ignored (not turned on.) Example on board.

38 Polygon fill areas Another way to determine how many to count if the scanline goes through a vertex If the two edges are on the same side of the scanline, then add it twice. If the two edges are on different sides of the scanline, then add it once. To check for this, we can compare the y coordinates of the 3 vertices in question (draw on board) in either clockwise or counterclockwise fashion If all 3 are monotonically increasing or all 3 are monotonically decreasing then the two edges are on different sides.

39 Polygon fill areas Edge coherence properties can be used to speed up the calculation of the crossings of the scanline and edges. The crossing x-coordinate of one scanline with an edge differs from the crossing of the x-coordinate of the next scanline only by 1/m as seen below. Assuming we're processing the fill from bottom to top and the y's increase as we go up, Assume scan line y k crosses an edge at (x k, y k ) and y k+1 crosses at (x k+1, y k+1 ) m = (y k+1 y k ) / (x k+1 x k ) (y k+1 y k ) = 1, so, m = 1 / (x k+1 x k ) and solve for x k+1 So, scan line y k+1 crosses that edge at (x k + 1/m, y k+1 ) This is less processing than doing the intersection of y = y k+1 with y = mx + b which is x = (y k+1 +b)/m.

40 Polygon fill areas Efficient polygon filling 1) proceed around edges in a clockwise or counterclockwise fashion 2) store each edge in an edge table 3) sort the edges based on the minimum y in the edge 4) process the scanlines in a bottom to top order 5) when the current scanline reaches the lower endpoint (with the min y) of the edge, that edge becomes active 6) active edges are sorted by their x coordinates (left to right) During any given scanline non-active edges can be marked as either finished, or not yet active. Horizontal edges are ignored. Example on board.

41 Polygon fill areas Masks Besides being filled with a solid color, polygons are sometimes filled with patterns. The process of filling an area with a pattern is called tiling. Fill patterns can be stored as rectangular arrays. The arrays are called masks and they are applied to the fill area and are usually smaller than the area to be filled. The mask needs a starting position within the area to be filled. Starting at this position the mask is replicated both vertically and horizontally. If the mask is an m n array, and starting position is (0,0) a pixel at (x, y) will be drawn with the color in mask((x mod n), (y mod m)). Example on board.

42 Polygon fill areas Section 17.9 Halftoning and dithering When there are very few intensity levels available (e.g. 2, black & white) to be displayed, halftoning is a technique that can provide more intensities with the tradeoff in addressability. Newspapers show grey level photos by displaying black circles. The black circles vary in size according to how dark that part of the photo should be. In graphics systems this halftoning technique is approximated with halftone patterns. Halftone patterns are typically n n squares of pixels where the number of pixels turned on in that pattern correspond to the intensity desired.

43 Polygon fill areas How is halftoning a tradeoff between intensities and addressability (resolution)?

44 Polygon fill areas Guidelines to creating halftoning patterns With halftoning, one thing to be careful about is that unintended patterns can become apparent. The intent is to have the viewer see the increase in intensities without noticing unintentional patterns. Avoid only horizontal or vertical or diagonal pixels on in the patterns to reduce the possibility of streaks. Further it is good to approximate the way that newspapers do it --- that is, concentrate on turning on pixels in the center of the pattern at each successive intensity. To minimize unintentional patterns to be displayed, we can evolve each successive grid pattern by copying it and turning on an additional pixel. See next slide figure

45 Polygon fill areas

46 Polygon fill areas Dithering Dithering refers to techniques for approximating halftones but without reducing addressability (resolution). An n n dither matrix D n is used to detemine if a pixel should be on or off. The matrix is filled with the numbers from 0 to n 2 1 on page 534 (equation 17-49) there's a mistake, it should be the dither matrix D

47 Polygon fill areas Example of dither matrix D For some pixel (x, y) an intensity level I(x,y) between 0 and n 2 is desired. Compute i = x mod n and j = y mod n to find which element (i,j) of the matrix D n to compare to the desired intensity. If I(x,y) > D n (i,j) then the pixel is turned on, otherwise it is not. This has the effect of different intensities, without reducing resolution. But what gets lost?