cs 376 introduction to computer graphics 02 / 05 / 2007 instructor: michael eckmann
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Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Today’s Topics• Questions?• a little about an Ellipse drawing algorithm• Antialiasing• Polygons• Determining if a point is in the interior of a polygon• Polygon filling algorithm• tiling• halftoning• dithering• antialiasing polygons
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Initial decision variable value• The initial value of p in the Bresenham line algorithm
is calculated by using the equation for p with (x0, y
0)
• Derivation:
pk = 2dyx
k – 2dxy
k + c
where c = 2dy + dx(2b-1)
and since yk=mx
k+b, b = y
k–mx
k and m = dy/dx
pk = 2dyx
k – 2dxy
k + 2dy + dx(2(y
k– (dy/dx)x
k)– 1)
multiplied out:
pk = 2dyx
k – 2dxy
k + 2dy + 2dxy
k– 2dyx
k – dx
= 2dy – dx
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Circle Algorithms• There's a figure in the text book (figure 3-20) and it looks to me like the red 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 red circle, the algorithm would “turn on” (3,9) not (3,10), because the midpoint is clearly outside the red circle there.
• I didn't verify it, but I trust the calculations in the table which says that p2 = -1 which
would be inside the circle.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
– When keeping fixed addressability --- then we can change the intensity of the pixels being turned on. And 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 at some intermediate location between the two pixel centers.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Antialiasing
• Any ideas on how we could adapt Bresenham's line algorithm to
do antialiased lines?
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Antialiasing
• Any ideas on how we could adapt Bresenham's line algorithm to
do antialiased lines?
• dlower
+ dupper
= 1
• The upper pixel's intensity
can be set to be dlower
• The lower pixel's intensity
can be set to be dupper
• When the line passes exactly through the center of a pixel, that
pixel's intensity is 1 and the other is 0 (off).
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Antialiasing
• Alternately, we could use a weighting mask that would give more
weight to the center subpixel of the 3x3 grid1 2 1
2 4 2
1 2 1
• 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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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 guassian. 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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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?
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Antialiasing helps total intensity• See figures 4-51 & 4-52 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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygons• Degenerate polygons are
– Those with 3 or more collinear vertices– Those with repeated vertices– Etc.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon interior points• What happens when the horizontal line crosses a vertex. Should it
be counted as 1 or 2?
• Example on board.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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 it's vertices are in
counterclockwise order. We'll know if we're seeing the back face
of a polygon if they are in clockwise order.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon fill areas• To do a scanline fill of a polygon (see figure 4-20).
– 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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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 ( y
1>y
2 => f(y
1) >= f(y
2))or
all 3 are monotonically decreasing then the two edges are on different sides.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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
(xk+1
, yk+1
)
– m = (yk+1
– yk) / (x
k+1 – x
k )
– (yk+1
– yk) = 1, so, m = 1 / (x
k+1 – x
k ) and solve for x
k+1
– So, scan line yk+1
crosses that edge at (xk + 1/m, y
k )
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon fill areas• Section 10.9 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.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon fill areas• How is halftoning a tradeoff between intensities and addressability
(resolution) ?
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon fill areas• Guidelines to creating halftoning patterns
– With halftoning, one thing to be careful about is that the patterns can become apparent. That is undesirable. 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 10-40.
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon fill areas• Dithering
– Dithering refers to techniques for approximating halftones but without reducing addressability (resolution).
– An nn dither matrix Dn is used to detemine if a pixel should
be on or off. – The matrix is filled with the numbers from 0 to n2 – 1– on page 589 (equation 10-49) there's a mistake, it should be the
dither matrix D4
15 7 13 5 3 11 1 912 4 14 6 0 8 2 10
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Polygon fill areas• Example of dither matrix D
4
15 7 13 5 3 11 1 912 4 14 6 0 8 2 10
– For some pixel (x, y) an intensity level I(x,y) between 0 and n2 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) > Dn(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?
Michael Eckmann - Skidmore College - CS 376 - Spring 2007
Antialiasing Polygons• Edges of polygons are lines. Lines (and therefore edges) have aliasing
problems when made discrete. Antialiasing edges of polygons can be
done using similar techniques as were used with lines. One way
described below is the way I described antialiasing Bresenham lines.
• During scanline fill of a polygon, if the edge falls between two pixels on
the scanline, then the intensities of the pixels depend on how close to the
centers of the two pixels the line falls.
• The pixel at (xj, y
k): could be set to have x
j+1 – x and x
j+1
• The pixel at (xj+1
, yk) would then be set to have x – x
j
• Pixels that fall fully within the interior of a polygon are set to full
intensity.