10/26/04 © University of Wisconsin, CS559 Fall 2004

Last Time

• Drawing lines

• Polygon fill rules

• Midterm Oct 28

10/26/04 © University of Wisconsin, CS559 Fall 2004

Today

• Filling polygons

• Anti-aliasing

• Hidden Surface Removal– Painter’s Algorithm

– Z-buffer

– A-buffer

– Depth Sorting?

• Homework 4 due

10/26/04 © University of Wisconsin, CS559 Fall 2004

Sweep Fill Algorithms

• Algorithmic issues:– Reduce to filling many

spans

– Which edges define the span of pixels to fill?

– How do you update these edges when moving from span to span?

– What happens when you cross a vertex?

10/26/04 © University of Wisconsin, CS559 Fall 2004

Algorithm

• For each row in the polygon:– Throw away irrelevant edges

– Obtain newly relevant edges

– Fill span

– Update current edges

• Issues:– How do we update existing edges?

– When is an edge relevant/irrelevant?

– What are the endpoints of a span

• All can be resolved by referring to our convention about what polygon a pixel belongs to

10/26/04 © University of Wisconsin, CS559 Fall 2004

When are Edges Relevant (1)

• Use figures and convention to determine which edges are needed for which scanlines (rows of pixels)

• Edges from ymin to ymax are relevant for what y values?

• What about horizontal edges?

ymin

ymax

10/26/04 © University of Wisconsin, CS559 Fall 2004

When are Edges Relevant (2)

1

2

3

43,4

1,3

1,2

Convex polygon:Always only two edges active

10/26/04 © University of Wisconsin, CS559 Fall 2004

Spans

• Fill rows from bottom to top, one at a time

• Have pixels xmin, xmax for each span

• Fill [xmin, xmax)

– Include xmin, exclude xmax

xmin,xmax

10/26/04 © University of Wisconsin, CS559 Fall 2004

Tracking xmin and xmax

• Use a variant of the midpoint method, but not quite the same– Which pixel, with respect to the edge, do we wish to choose for xmin

and xmax?

– What is the decision variable?

10/26/04 © University of Wisconsin, CS559 Fall 2004

yi

yi+1

xi+1

Midpoint Method For Polygon Edges

• Consider a left edge with slope [0-1]

• Always want to choose point on line or below

• Consider the point (xi+1,yi+1). Is it above or below the line?

xi

Choose (xi+1,yi+1)

yi

yi+1

xi+1xi

Choose (xi+1,yi)

10/26/04 © University of Wisconsin, CS559 Fall 2004

Sweep Fill Details

• For convex polygons there are always 2 edges, a left and a right– Fixes the amount of memory required, good for hardware

• Can generate memory addresses (for pixel writes) efficiently– You know address of leftmost pixel, and address of rightmost, can fill

all in between quickly

• Other values may also be updated, such as z, w or color information– Use a midpoint-like rule for these too– Modern hardware, with floating point frame buffers, just interpolates

10/26/04 © University of Wisconsin, CS559 Fall 2004

Extending to Arbitrary Polygons

• Can be more than one span in any row

• Keep sorted list of edges across row, fill between pairs of edges

10/26/04 © University of Wisconsin, CS559 Fall 2004

Anti-Aliasing

• Recall: We can’t sample and then accurately reconstruct an image that is not band-limited– Infinite Nyquist frequency

– Attempting to sample sharp edges gives “jaggies”, or stair-step lines

• Solution: Band-limit by filtering (pre-filtering)– What sort of filter will give a band-limited result?

• But when doing computer rendering, we don’t have the original continuous function

10/26/04 © University of Wisconsin, CS559 Fall 2004

Alpha-based Anti-Aliasing

• Set the of a pixel to simulate a thick line– The pixel gets the line color, but with

<=1

• This supports the correct drawing of primitives one on top of the other– Draw back to front, and composite

each primitive over the existing image

– Only some hidden surface removal algorithms support it

1/8

1/8

.914

.914

.914

1/8

1/8

1/4

1/4

1/41/40 0

00

0000

0

0

0

0 0 0

10/26/04 © University of Wisconsin, CS559 Fall 2004

Calculating

• Consider a line as having thickness (all good drawing programs do this)

• Consider pixels as little squares

• Set according to the proportion of the square covered by the line

• The sub-pixel coverage interpretation of

1/8

1/8

.914

.914

.914

1/8

1/8

1/4

1/4

1/41/40 0

00

0000

0

0

0

0 0 0

10/26/04 © University of Wisconsin, CS559 Fall 2004

Weighted Sampling

• Instead of using the proportion of the area covered by the line, use convolution to do the sampling– Equivalent to filtering the line then

point sampling the result

• Place the “filter” at each pixel, and integrate product of pixel and line

• Common filters are cones (like Bartlett) or Gaussians

10/26/04 © University of Wisconsin, CS559 Fall 2004

Post-Filtering (Supersampling)

• Sample at a higher resolution than required for display, and filter image down– Easy to implement in hardware

– Typical is 2x2 sampling per pixel, with simple averaging to get final

• What kind of filter?

• More advanced methods generate different samples (eg. not on regular grid) and filter properly– Issues of which samples to take, and how to

filter them

10/26/04 © University of Wisconsin, CS559 Fall 2004

Where We Stand

• At this point we know how to:– Convert points from local to window coordinates

– Clip polygons and lines to the view volume

– Determine which pixels are covered by any given line or polygon

– Anti-alias

• Next thing:– Determine which polygon is in front

10/26/04 © University of Wisconsin, CS559 Fall 2004

Visibility

• Given a set of polygons, which is visible at each pixel? (in front, etc.). Also called hidden surface removal

• Very large number of different algorithms known. Two main classes:– Object precision: computations that operate on primitives– Image precision: computations at the pixel level

• All the spaces in the viewing pipeline maintain depth, so we can work in any space– World, View and Canonical Screen spaces might be used– Depth can be updated on a per-pixel basis as we scan convert

polygons or lines– Actually, run Bresenham-like algorithm on z and w before

perspective divide

10/26/04 © University of Wisconsin, CS559 Fall 2004

Visibility Issues

• Efficiency – it is slow to overwrite pixels, or rasterize things that cannot be seen

• Accuracy - answer should be right, and behave well when the viewpoint moves

• Must have technology that handles large, complex rendering databases

• In many complex environments, few things are visible– How much of the real world can you see at any moment?

• Complexity - object precision visibility may generate many small pieces of polygon

10/26/04 © University of Wisconsin, CS559 Fall 2004

Painters Algorithm (Image Precision)

• Algorithm:– Choose an order for the polygons

based on some choice (e.g. depth to a point on the polygon)

– Render the polygons in that order, deepest one first

• This renders nearer polygons over further

• Difficulty: – works for some important

geometries (2.5D - e.g. VLSI)– doesn’t work in this form for most

geometries - need at least better ways of determining ordering

zs

xs

Fails

Which point for choosing ordering?

10/26/04 © University of Wisconsin, CS559 Fall 2004

The Z-buffer (1) (Image Precision)

• For each pixel on screen, have at least two buffers– Color buffer stores the current color of each pixel

• The thing to ultimately display– Z-buffer stores at each pixel the depth of the nearest thing seen so

far• Also called the depth buffer

• Initialize this buffer to a value corresponding to the furthest point (z=1.0 for canonical and window space)

• As a polygon is filled in, compute the depth value of each pixel that is to be filled– if depth < z-buffer depth, fill in pixel color and new depth– else disregard

10/26/04 © University of Wisconsin, CS559 Fall 2004

The Z-buffer (2)

• Advantages:– Simple and now ubiquitous in hardware

• A z-buffer is part of what makes a graphics card “3D”

– Computing the required depth values is simple

• Disadvantages:– Over-renders – rasterizes polygons even if they are not visible

– Depth quantization errors can be annoying

– Can’t easily do transparency or filter-based anti-aliasing (Requires keeping information about partially covered polygons)