8.2-rendering.ppt

Triangles Rasterization

Final pixels

Triangles Rasterization

Computer Graphics

Scan-line Algorithm

Basic Idea:

1. Sort edges on y

2. For every row

1. Find the first fragment A inside

2. Find the first fragment B outside

3. Output all fragments from A included to B

Triangles Rasterization

Scan-line Algorithm

Basic Idea:

1. Sort edges on y

2. For every row

1. Find the first fragment A inside

2. Find the first fragment B outside

Bresenham

• For every row:

– Finding hte first fragment inside and the first one outside is similar to Bresenham, but:

• Only one case (iterate on rows)

• Rounding to the smallest integer larger

Terminology in triangle rasterization

• "Scan-Conversion" = Rasterization

• "to scan-convert" = to Rasterize

• "Span" = interval from the first-in to the first-out

• "Line-scan" = processed line

• "Active edges" = edges taken into consideration

• "Edge tables" = edges (precomputed)

3

2

In the case of triangles!

Problems with the scan-line algorithm

• Fragments are produced one at the time

• Not suited for a parallel implementation

Solution

• Produce the fragments in groups

– E.g., groups of 2x2 or 4x4 or 4x1 or 8x1...

• Not all members of a group of fragments will belong to the triangle

• Test each fragment:

Inclusion test for a triangle

• Inclusion test for an half plane:

YES

NO

k

n

x

n

p

x











)

0

(

n

p

k





where TEST:

Line defined by a point p

Inclusion test for a triangle

• The halfplane is defined by:

NO

v₀=(x_0, y₀ )

n

v₁₌(x_1, y₁ )

q

The

so-called

EDGE

FUNCTIO

N

n = ( - ( y₁ -y₀ ) , x₁ -x₀)

p = (y₀ , x₀)

f(q) = n‧q - n‧p

Inclusion test for a triangle

• Triangle=intersection of 3 half-planes

Inclusion test for a triangle

• Triangle=intersection of 3 half-planes

NO

v₀ YES

NO YES YES YES NO v₁ v₂ YES YES YES YES NO NO YES NO NO NO YES NO YES

Inclusion test for a triangle

• _Swap

v

v

v₂

v₁

v₀

Back-facing triangles (counterclockwise)

Inclusion test for a triangle

Three Edge Functions: (one per edge)

• All YES:

– Fragment inside a front-facing triangle

• All NO:

– Fragment inside a back-facing triangle

• Mixed results:

Example, using a bounding box

1. Compute the bounding box of the triangle

How to compute the bounding box

Example, using a bounding box

Example, using a bounding box

1. Compute the bounding box of the triangle

2. Process each block (e.g. 2x2) 1. Test each fragment in the

block

Triangles Rasterization

SETUP: compute the bounding box

test a group of fragments (in parallel)

Final pixels

Clipping

• Clipping!

Outside: CULLED !

 no prob.

Screen

Inside: Rasterize (simple case)

Partly

overlappin g

Clipping: old approach

• Clipping!

Screen

1. Compute intersections 2. Connect the intersections

A

B

3. Split the polygon into triangles 4. Rasterize each triangle

Clipping: old approach

• Clipping!

Screen

1. Compute intersections 2. Connect the intersections

3. Split the polygon into triangles 4. Rasterize each triangle

B

Clipping: old approach

• Clipping!

Screen

B

C

A

D

Worst case, very complex

Very bad for HW implementations

We cannot ignore this case

1. Compute intersections 2. Connect the intersections

Clipping: modern implementation

• Clipping!

Screen

1. Compute the bounding box 2. Intersect the bounding box

with the screen

3. Rasterize wrt the bounding box, as usual

Watch out!

• What about clipping against the far and near planes?

bottom plane right plane

left plane

near plane

view frustum

top plane

Triangles Rasterization

• The method based on computing the bounding box and the inclusion test (edge functions):

– Pros

• easily parallelizable

– (groups 4x4)

• clipping is easy

– For the planes UP, DOWN, LEFT e RIGHT

– Cons

• Large Overheads

A terrible situation

Screen

Triangles:

• Long and thin

• Along the diagonal



Very unpleasant situation in computer graphics:

(not just for rasterization)

Long and thin= bad

Many algorithms perform poorly

Samples

•

most things in the real world are

continuous

•

everything in a computer is

discrete

•

_{the process of mapping a continuous function to a}

discrete one is called

sampling

•

_{the process of mapping a discrete function to a}

continuous one is called

reconstruction

•

_{the process of mapping a continuous variable to a}

discrete one is called

quantization

•

_{rendering an image requires sampling and}

quantization

Jaggy Line Segments

• we tried to sample a line segment so it would map to a 2D raster display

• we quantized the pixel values to 0 or 1

Less Jaggy Line Segments

• better if quantize to many shades

– image is less visibly jaggy

• find color for area, not just single point at center of pixel

30

Supersample and Average

• supersample: create image at higher resolution

– e.g. 768x768 instead of 256x256

– shade pixels wrt area covered by thick line/rectangle

• average across many pixels

– e.g. 3x3 small pixel block to find value for 1 big pixel

– rough approximation divides each pixel into a finer grid of pixels

6/9 9/9

5/9 9/9

31

Supersample and Average

•

_{supersample: jaggies less obvious, but still there}

– small pixel center check still misses information – unweighted area sampling

• equal areas cause equal intensity, regardless of distance from pixel center to area

• aka box filter

6/9 9/9

5/9 9/9

0/9 4/9

x Intensity

Supersampling Example: Image

no supersampling 3x3 supersampling with

Weighted Area Sampling

•

intuitively, pixel cut through the center should be

more heavily weighted than one cut along corner

•

weighting function, W(x,y)

– specifies the contribution of primitive passing

through the point (x, y) from pixel center

– Gaussian filter (or approximation) commonly used

x Intensity

•

some objects missed entirely, others poorly sampled

– could try unweighted or weighted area sampling

– but how can we be sure we show everything?

•

need to think about entire class of solutions!

– brief taste of signal processing

Image As Signal

• image as spatial signal

• _{2D raster image}

– discrete sampling of 2D spatial signal

• 1D slice of raster image

– discrete sampling of 1D spatial signal

Examples from Foley, van Dam, Feiner, and Hughes

Pixel position across scanline

In

te

ns

Sampling Frequency

• if don’t sample often enough, resulting signal misinterpreted as lower-frequency one

– we call this aliasing

Sampling Theorem

continuous signal can be completely recovered from its samples

iff

sampling rate greater than twice maximum frequency present in signal

Nyquist Rate

• lower bound on sampling rate

Aliasing

• incorrect appearance of high frequencies as low frequencies

• to avoid: antialiasing

– supersample

• sample at higher frequency

– low pass filtering

Low-Pass Filtering

Low-Pass Filtering

Filtering

• low pass

– blur

• high pass

Texture Antialiasing

Temporal Antialiasing

• subtle point: collision detection about algorithms for finding collisions in time as much as space

• temporal sampling

– aliasing: can miss collision completely with point samples!

• temporal antialiasing