clipping and scan conversionmisha/fall20/13.pdfcohen-sutherland line clipping • associate a 4-bit...

Clipping and Scan Conversion

Michael Kazhdan

(601.357/456)

HB Ch. 3.2, 3.11, 6.7, 6.8

FvDFH Ch. 3.2, 3.6, 3.12, 3.14

Announcements• Assignment 2 posted

Start yesterday!!!

Lighting

ViewingTransformation

3D Rendering Pipeline (for direct illumination)

3D Primitives

ModelingTransformation

ProjectionTransformation

Clipping

Image

ViewportTransformation

ScanConversion

2D Image Coordinates

3D Modeling Coordinates

3D World Coordinates

3D Camera Coordinates

2D Screen Coordinates




3D Model

2D Image

Transformations





Window-to-ViewportTransformation

3D Object Coordinates




(𝑥𝑥,𝑦𝑦, 𝑧𝑧)

(𝑥𝑥′,𝑦𝑦′)

3D WorldCoordinates

3D ObjectCoordinates

Transformations










3D WorldCoordinates

Transform = 𝑀𝑀𝑀𝑀 = local to world transform



Transformations










3D WorldCoordinates

3D CameraCoordinates



Transform = 𝑀𝑀𝑀𝑀 = local to world transform

Transformations










Camera Right

Camera Up

x

y

zCamera Back

𝑇𝑇𝐶𝐶→𝑊𝑊 is the camera to world transform𝑅𝑅𝑥𝑥 𝑈𝑈𝑥𝑥 𝐵𝐵𝑥𝑥 𝐸𝐸𝑥𝑥𝑅𝑅𝑦𝑦 𝑈𝑈𝑦𝑦 𝐵𝐵𝑦𝑦 𝐸𝐸𝑦𝑦𝑅𝑅𝑧𝑧 𝑈𝑈𝑧𝑧 𝐵𝐵𝑧𝑧 𝐸𝐸𝑧𝑧0 0 0 1



Transform = 𝑇𝑇𝐶𝐶→𝑊𝑊−1 𝑀𝑀

Transformations










Transform = 𝑃𝑃𝑇𝑇𝐶𝐶→𝑊𝑊−1 𝑀𝑀𝑃𝑃 = projection transform



𝑃𝑃𝑜𝑜 =

1 0 𝐿𝐿 cos𝜙𝜙 00 1 𝐿𝐿 sin𝜙𝜙 00 0 0 00 0 0 1

𝑃𝑃𝑝𝑝 =

1 0 0 00 1 0 00 0 1 00 0 1 0

Transformations










𝑣𝑣𝑥𝑥1 𝑣𝑣𝑥𝑥2𝑣𝑣𝑦𝑦1

𝑣𝑣𝑦𝑦2

𝑤𝑤𝑥𝑥1 𝑤𝑤𝑥𝑥2𝑤𝑤𝑦𝑦1

𝑤𝑤𝑦𝑦2Window Viewport

Screen Coordinates Image Coordinates

(𝑤𝑤𝑥𝑥,𝑤𝑤𝑦𝑦) (𝑣𝑣𝑥𝑥 , 𝑣𝑣𝑦𝑦)



𝑉𝑉 = viewport transform

𝑉𝑉 =1 0 𝑣𝑣𝑥𝑥1

0 1 𝑣𝑣𝑥𝑥20 0 1

𝑣𝑣𝑥𝑥2 − 𝑣𝑣𝑥𝑥1

𝑤𝑤𝑥𝑥2 − 𝑤𝑤𝑥𝑥10 0

0𝑣𝑣𝑦𝑦2 − 𝑣𝑣𝑦𝑦1

𝑤𝑤𝑦𝑦2 − 𝑤𝑤𝑦𝑦10

0 0 1

1 0 −𝑤𝑤𝑥𝑥1

0 1 −𝑤𝑤𝑦𝑦1

0 0 1

Transform = 𝑉𝑉𝑃𝑃𝑇𝑇𝐶𝐶→𝑊𝑊−1 𝑀𝑀


3D Model











(𝑥𝑥′,𝑦𝑦′) 2D Screen

Transformations𝑁𝑁

𝐿𝐿1

𝐿𝐿2

𝑉𝑉

Viewer

𝐼𝐼 = 𝐼𝐼𝐸𝐸 + �𝐿𝐿

𝐾𝐾𝐴𝐴 ⋅ 𝐼𝐼𝐿𝐿𝐴𝐴 + 𝐾𝐾𝐷𝐷 ⋅ 𝑁𝑁, 𝐿𝐿 + 𝐾𝐾𝑆𝑆 ⋅ 𝑉𝑉,𝑅𝑅 𝑛𝑛 ⋅ 𝐼𝐼𝐿𝐿

3D Model2D Screen

Lighting


3D Primitives



Clipping

Image


ScanConversion

Lighting



3D Primitives



Clipping

Image


ScanConversion









Clipping• Avoid drawing parts of primitives outside window

Window defines part of scene being viewed Must draw geometric primitives only inside window

Screen Coordinates

Window


Points Line Segments Polygons

Screen Coordinates

Window

• Is point (𝑥𝑥,𝑦𝑦) inside the clip window?

Point Clipping

Window𝑤𝑤𝑥𝑥1 𝑤𝑤𝑥𝑥2

𝑤𝑤𝑦𝑦2

𝑤𝑤𝑦𝑦1

(𝑥𝑥,𝑦𝑦)

• Is point (𝑥𝑥,𝑦𝑦) inside the clip window?

Point Clipping

Window

(𝑥𝑥,𝑦𝑦)

inside = (x >= wx1) &&(x < wx2) &&(y >= wy1) &&(y < wy2);

𝑤𝑤𝑥𝑥1 𝑤𝑤𝑥𝑥2

𝑤𝑤𝑦𝑦2

𝑤𝑤𝑦𝑦1



Screen Coordinates

Window

Line Segment Clipping• Find the part of a line inside the clip window

Do this as efficiently as possible by identifying the easiest cases first

𝑃𝑃1

𝑃𝑃10

𝑃𝑃9

𝑃𝑃8

𝑃𝑃7

𝑃𝑃4𝑃𝑃3

𝑃𝑃6

𝑃𝑃5

𝑃𝑃2

Cohen-Sutherland Line Clipping• Associate a 4-bit outcode 𝒃𝒃𝟎𝟎𝒃𝒃𝟏𝟏𝒃𝒃𝟐𝟐𝒃𝒃𝟑𝟑 to each vertex

𝑏𝑏0 = 1 if the vertex is left of the window 𝑏𝑏1 = 1 if the vertex is right of the window 𝑏𝑏2 = 1 if the vertex is below the window 𝑏𝑏3 = 1 if the vertex is above the window

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3

0000 01001000

0001 01011001

0010 01101010

Cohen-Sutherland Line Clipping• Associate a 4-bit outcode 𝑏𝑏0𝑏𝑏1𝑏𝑏2𝑏𝑏3 to each vertex• If both outcodes are 0, line segment is inside• If AND of outcodes is not 0, line segment is outside• Otherwise clip and test

0000 01001000

0001 01011001

0010 01101010

𝑃𝑃1

𝑃𝑃10

𝑃𝑃9

𝑃𝑃8

𝑃𝑃7

𝑃𝑃6

𝑃𝑃5

𝑃𝑃2

𝑃𝑃4𝑃𝑃3

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃10

𝑃𝑃9

𝑃𝑃8

𝑃𝑃7

𝑃𝑃6

𝑃𝑃5

𝑃𝑃4𝑃𝑃3

𝑃𝑃1

𝑃𝑃2

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃10

𝑃𝑃9

𝑃𝑃8

𝑃𝑃7

𝑃𝑃6

𝑃𝑃5

𝑃𝑃4𝑃𝑃3

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃10

𝑃𝑃9

𝑃𝑃8

𝑃𝑃7

𝑃𝑃4𝑃𝑃3

𝑃𝑃6

𝑃𝑃5

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010𝑃𝑃5′

𝑃𝑃10

𝑃𝑃9

𝑃𝑃8

𝑃𝑃7

𝑃𝑃4𝑃𝑃3

𝑃𝑃6

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010𝑃𝑃5′

𝑃𝑃10

𝑃𝑃9

𝑃𝑃4𝑃𝑃3

𝑃𝑃6

𝑃𝑃8

𝑃𝑃7

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010𝑃𝑃5′

𝑃𝑃10

𝑃𝑃9

𝑃𝑃3

𝑃𝑃6

𝑃𝑃8

𝑃𝑃7

𝑃𝑃4

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃7′

𝑃𝑃5′

𝑃𝑃10

𝑃𝑃9

𝑃𝑃3

𝑃𝑃6

𝑃𝑃8𝑃𝑃4

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃8′

𝑃𝑃7′

𝑃𝑃5′

𝑃𝑃10

𝑃𝑃9

𝑃𝑃3

𝑃𝑃6

𝑃𝑃4

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃8′

𝑃𝑃7′

𝑃𝑃5′

𝑃𝑃3

𝑃𝑃6

𝑃𝑃4

𝑃𝑃10

𝑃𝑃9


0000 01001000

0001 01011001

0010 01101010𝑃𝑃9′

𝑃𝑃8′

𝑃𝑃7′

𝑃𝑃5′

𝑃𝑃3

𝑃𝑃6

𝑃𝑃4

𝑃𝑃10

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3


0000 01001000

0001 01011001

0010 01101010

𝑃𝑃8′

𝑃𝑃7′

𝑃𝑃5′

𝑃𝑃3

𝑃𝑃6

𝑃𝑃4

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3

Cohen-Sutherland Line Clipping• Associate a 4-bit outcode 𝑏𝑏0𝑏𝑏1𝑏𝑏2𝑏𝑏3 to each vertex• If both outcodes are 0, line segment is inside• If AND of outcodes isnot 0, line segment is outside• Otherwise clip and test

0000 01001000

0001 01011001

0010 01101010

𝑃𝑃8′

𝑃𝑃7′

𝑃𝑃5′

𝑃𝑃3

𝑃𝑃6

𝑃𝑃4

𝑏𝑏0 𝑏𝑏1

𝑏𝑏2

𝑏𝑏3

How many bits would you need in 3D?



Screen Coordinates

Window

Polygon Clipping• Find the part of a polygon inside the clip window

Before Clipping

Sutherland-Hodgeman Clipping• Clip to each window boundary, one at a time

Sutherland-Hodgeman Clipping• How do we clip a convex polygon with respect to

a line?

OutsideInside

WindowBoundary

𝑃𝑃1𝑃𝑃2

𝑃𝑃5

𝑃𝑃4

𝑃𝑃3

Sutherland-Hodgeman Clipping• Do inside test for each point in sequence,

Insert new points when cross window boundary, Remove points outside window boundary

OutsideInside

WindowBoundary

𝑃𝑃1𝑃𝑃2

𝑃𝑃5

𝑃𝑃4

𝑃𝑃3



OutsideInside

WindowBoundary 𝑃𝑃𝑃

𝑃𝑃1𝑃𝑃2

𝑃𝑃5

𝑃𝑃4

𝑃𝑃3



OutsideInside

WindowBoundary 𝑃𝑃𝑃𝑃𝑃𝑃𝑃

𝑃𝑃1𝑃𝑃2

𝑃𝑃5

𝑃𝑃4

𝑃𝑃3



OutsideInside


𝑃𝑃1𝑃𝑃2



OutsideInside


𝑃𝑃1𝑃𝑃2

When polygons are clipped, per-vertex properties (e.g. lighting) is interpolated to the new vertices.



OutsideInside

WindowBoundary

What happens if the polygon is not convex?

Lighting



3D Primitives



Clipping

Image


ScanConversion








2D Image Coordinates3D Model

2D Screen

2D Rendering Pipeline

ScanConversion

Clipping

2D Primitives

Image

Clip portions of geometric primitivesresiding outside the window

Fill pixels representing primitives in screen coordinates

3D Primitives

Overview• Scan conversion

Figure out which pixels to fill

• Shading Determine a color for each filled pixel

• Depth test Determine when the color of a pixel should be

overwritten

Scan Conversion• Render an image of a geometric primitive

by setting pixel colors

• Example: Filling the inside of a triangle

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

void SetPixel( int x , int y , Color rgba )

Triangle Scan Conversion• Properties of a good algorithm

Must be fast No cracks between adjacent primitives

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

𝑃𝑃4

Simple Algorithm

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

void ScanTriangle( Triangle T , Color rgba ){

for each pixel center (x,y)if( PointInsideTriangle( (x,y) , T ) )

SetPixel( x , y , rgba );}

• Color all pixels inside triangle

𝑃𝑃1

𝑃𝑃2

Line defines two halfspaces• Test: use implicit equation for a line

On line: 𝑎𝑎𝑥𝑥 + 𝑏𝑏𝑦𝑦 + 𝑐𝑐 = 0 To the right: 𝑎𝑎𝑥𝑥 + 𝑏𝑏𝑦𝑦 + 𝑐𝑐 < 0 To the left: 𝑎𝑎𝑥𝑥 + 𝑏𝑏𝑦𝑦 + 𝑐𝑐 > 0

𝐿𝐿

Inside Triangle Test• A point is inside a triangle if it is in the

positive half-space of all three boundary lines Triangle vertices are ordered counter-clockwise Point must be on the left side of every boundary line

𝑃𝑃𝐿𝐿1

𝐿𝐿2

𝐿𝐿3

Inside Triangle TestBoolean PointInsideTriangle( Point P , Triangle T ){

for each boundary line L of T{

Scalar d = L.a*P.x + L.b*P.y + L.c;if( d<0.0 ) return FALSE;

}return TRUE;

}

𝐿𝐿1

𝐿𝐿2

𝐿𝐿3

Simple Algorithm

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

void ScanTriangle( Triangle T , Color rgba ){

for each pixel center (x,y)if( PointInsideTriangle( (x,y) , T ) )

SetPixel( x , y , rgba );}

• What is bad about this algorithm?

Triangle Sweep-Line Algorithm• Take advantage of spatial coherence

• Take advantage of edge linearity

𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦

Triangle Sweep-Line Algorithmvoid ScanTriangle( Triangle T , Color rgba ){

for both edge pairs{

initialize xL, xR, y;compute dxL/dyL and dxR/dyR;until y reaches the first end-point

for( int x=xL ; x<=xR ; x++ ) SetPixel( x , y , rgba );xL += dxL/dyL;xR += dxR/dyR;y++;

}} 𝑑𝑑𝑥𝑥𝐿𝐿

𝑑𝑑𝑦𝑦𝐿𝐿𝑑𝑑𝑥𝑥𝑅𝑅

𝑑𝑑𝑦𝑦𝑅𝑅

𝑥𝑥𝐿𝐿 𝑥𝑥𝑅𝑅





}}

𝑥𝑥𝐿𝐿 𝑥𝑥𝑅𝑅𝑑𝑑𝑥𝑥𝐿𝐿

𝑑𝑑𝑦𝑦𝐿𝐿

𝑑𝑑𝑥𝑥𝑅𝑅𝑑𝑑𝑦𝑦𝑅𝑅





}}

Bresenham’s algorithmworks the same way, but uses only integer operations!

Polygon Scan Conversion• Will this method work for convex polygons?

Polygon Scan Conversion• Will this method work for convex polygons?

Yes, since each scan line will only intersect the polygon at two points.

Polygon Scan Conversion• How about these polygons?

Polygon Scan Conversion• Fill pixels inside a polygon

Triangle Quadrilateral Convex Star-shaped Concave Self-intersecting Holes

What problems do we encounter with arbitrary polygons?

Polygon Scan Conversion• Need better test for points inside polygon

Triangle method works only for convex polygons

Convex Polygon

𝐿𝐿1

𝐿𝐿2

𝐿𝐿3

𝐿𝐿4𝐿𝐿5

𝐿𝐿1

𝐿𝐿2

𝑳𝑳𝟑𝟑𝟑𝟑

𝐿𝐿4𝐿𝐿5

Concave Polygon

𝐿𝐿3𝐵𝐵

Inside Polygon Rule

• What is a good rule for which pixels are inside?

Concave Self-Intersecting With Holes

Inside Polygon Rule

Concave Self-Intersecting With Holes

• Odd-parity rule Any ray from inside 𝑃𝑃 to infinity must cross an odd

number of edges

Polygon Sweep-Line Algorithm Use incremental algorithm to find spans Determine “insideness” with odd parity rule

• Takes advantage of scan line coherence

𝑥𝑥𝐿𝐿 𝑥𝑥𝑅𝑅

Triangle Polygon

Polygon Sweep-Line Algorithmvoid ScanPolygon( Polygon P , Color rgba ){

sort edges by maxymake empty “active edge list”for each scanline ( top-to-bottom ){

insert/remove edges from “active edge list”update x coordinate of every active edgesort active edges by x coordinatefor each succesive pair of edge-points (left-to-right)

SetPixels( xi , xi+1 , y , rgba );}

}

Polygon Scan Conversion• Convert everything into triangles

Scan convert the triangles

Polygon Scan Conversion• Convert everything into triangles

Scan convert the triangles

Note:OpenGL will render polygons, but it assumes that:• The polygon is planar• The polygon is convex

Note:Even if you only pass in triangles for rendering, OpenGL may still have to render (convex planar) polygons after the triangle is clipped.

Scan Conversion• What about pixels on edges?

If we set them either “on” or “off” we get aliasing or “jaggies”

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3


If we set them either “on” or “off” we get aliasing or “jaggies”

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3This is like using a “nearest” interpolation filter!

Antialiasing Techniques• Display at higher resolution

Corresponds to increasing sampling rate Not always possible (fixed size monitors, fixed refresh

rates, etc.)

• Modify pixel intensities Vary pixel intensities along boundaries for antialiasing


Setting them either “on” or “off” we get aliasing/“jaggies” Antialias by varying pixel intensities along boundaries

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

Antialiasing• Method 1: Area sampling

Calculate percent of pixel covered by primitive Multiply this percentage by desired intensity/color

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

100%2% 25% 60%

Antialiasing• Method 1: Area sampling

Calculate percent of pixel covered by primitive Multiply this percentage by desired intensity/color

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3

100%2% 25% 60%

This is like using a “bilinear” interpolation filter!

Antialiasing• Method 2: Supersampling (aka postfiltering)

Sample as if screen were higher resolution Average multiple samples to get final intensity

» This can be done by rendering the scene multiple times with different (fractional) offsets

𝑃𝑃1

𝑃𝑃2

𝑃𝑃3𝑃𝑃3

𝑃𝑃1

𝑃𝑃2

AntialiasingNote that this makes things harder because pixels are no longer “owned” by a single triangle.

Triangles contribute color rather than set color Along edges the total contribution must sum to one. Makes depth-testing more complicated.

𝑃𝑃4

𝑃𝑃2

𝑃𝑃3

𝑃𝑃1

Scan Conversion• Example:

No Anti-Aliasing 4 x Anti-AliasingImages courtesy of NVIDIA

Lighting



3D Primitives



Clipping

Image


ScanConversion









2D Window

2D Screen

Overview• Scan conversion

Figure out which pixels to fill

• Shading Determine a color for each filled pixel

• Depth test Determine when the color of a pixel comes from the

front-most primitive

• Recall:In the “lighting” phase, we calculated the color at each vertex.

Gouraud Shading

𝑉𝑉1 𝐿𝐿1

𝑁𝑁1

𝑁𝑁2

𝑁𝑁3𝐿𝐿3

𝐿𝐿2𝑉𝑉3

𝑉𝑉2

𝐼𝐼1

𝐼𝐼2

𝐼𝐼3

Lighting


3D Primitives



Clipping

Image


ScanConversion

• Recall:In the “lighting” phase, we calculated the color at each vertex. In the “scan conversion” phase, linearly

interpolate colors at vertices

Gouraud Shading

𝐼𝐼1

𝐼𝐼2

𝐼𝐼3

𝛼𝛼

𝛽𝛽

𝛾𝛾

𝐵𝐵 = (1 − 𝛽𝛽) ⋅ 𝐼𝐼2 + 𝛽𝛽 ⋅ 𝐼𝐼3

𝐼𝐼 = (1 − 𝛾𝛾) ⋅ 𝐴𝐴 + 𝛾𝛾 ⋅ 𝐵𝐵

𝐴𝐴 𝐵𝐵𝐼𝐼

Lighting


3D Primitives



Clipping

Image


ScanConversion

𝐴𝐴 = (1 − 𝛼𝛼) ⋅ 𝐼𝐼1 + 𝛼𝛼 ⋅ 𝐼𝐼2

• Recall:In the “lighting” phase, we calculated the color at each vertex. In the “scan conversion” phase, linearly

interpolate colors at vertices

Gouraud Shading

𝐼𝐼1

𝐼𝐼2

𝐼𝐼3

𝛼𝛼

𝛽𝛽

𝛾𝛾

𝐵𝐵 = (1 − 𝛽𝛽) ⋅ 𝐼𝐼2 + 𝛽𝛽 ⋅ 𝐼𝐼3

𝐼𝐼 = (1 − 𝛾𝛾) ⋅ 𝐴𝐴 + 𝛾𝛾 ⋅ 𝐵𝐵

𝐴𝐴 𝐵𝐵

Lighting


3D Primitives



Clipping

Image


ScanConversion

Note: The values of 𝛼𝛼 and 𝛽𝛽 only need to be updated as we move to the next scan-line. The value of 𝛾𝛾 needs

to be updated as we advance along the scan-line.

𝐴𝐴 = (1 − 𝛼𝛼) ⋅ 𝐼𝐼1 + 𝛼𝛼 ⋅ 𝐼𝐼2

𝐼𝐼

Lighting



3D Primitives



Clipping

Image


ScanConversion









2D Screen

clipping and scan conversionmisha/fall20/13.pdfcohen-sutherland line clipping • associate a 4-bit...

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