Cli Clipping and i d A ti li Antialiasing i Werner Purgathofer - - PowerPoint PPT Presentation

Einführung in Einführung in Visual Computing Visual Computing

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Cli i d Clipping and A ti li i Antialiasing

Werner Purgathofer g

Clipping in the Rendering Pipeline

scene objects in object space

• bject capture/creation

modeling viewing

vertex stage ( vertex shader“)

g projection

(„vertex shader )

transformed vertices in clip space clipping + homogenization transformed vertices in clip space scene in normalized device coordinates viewport transformation rasterization shading

pixel stage

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raster image in pixel coordinates shading

p g („fragment shader“)

Overview: Clipping line clipping l li i polygon clipping triangle clipping g pp g

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Clipping partly visible or completely invisible parts must not be ignored and must not be drawn i d ll d ignored scrolled vertices edges  must be cut off (as early as possible) g

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(as early as possible)

Clipping Operations remove objects outside a clip window

clip window: rectangle polygon curved clip window: rectangle, polygon, curved boundaries applied somewhere in the viewing pipeline can be combined with scan conversion can be combined with scan conversion

• bjects to clip: points, lines, triangles,

polygons curves text polygons, curves, text, ...

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3 Principle Possibilities for Clipping analytically in world coordinates

reduces WC DC transformations reduces WC DC transformations

analytically in clip coordinates

simple comparisons simple comparisons

d i t ti during raster conversation = as part of the rasterization algorithm

may be efficient for complex primitives

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Line Clipping (1)

P9 P10 P4 P2 P2 P1 P6 P8 P10 P3 P1 P6 P´8

6

P7 P5 P3

6

P´5 P´7

before clipping after clipping

7 7

[line clipping against a rectangular clip window]

pp g pp g

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Line Clipping (2) goals

eliminate simple cases fast eliminate simple cases fast avoid intersection calculations

for endpoints (x0,y0), (xend,yend) for endpoints (x0,y0), (xend,yend) intersect parametric representation x = x0 + u·(x

• x0)

x x0 + u (xend x0) y = y0 + u·(yend - y0) with window borders: with window borders: intersection  0 < u < 1

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Cohen-Sutherland Line Clipping assignment of region codes to line endpoints

bit1: left

1001 1000 1010

Window

bit2: right bit3: below

0001 0000 0010

bit3: below bit4: above

0101 0100 0110 0101 0100 0110

binary region codes assigned to line endpoints binary region codes assigned to line endpoints according to relative position with respect to the clipping rectangle

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the clipping rectangle

Cohen-Sutherland Line Clipping “or” of codes of both

1001 1000 1010

• r of codes of both

points = 0000  li ti l i ibl

Window

0001 0000 0010

line entirely visible “and” of codes of both

0101 0100 0110

points  0000  line entirely invisible

0101 0100 0110

line entirely invisible all others intersect!

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Cohen-Sutherland Line Clipping

P2 [lines extending from one coordinate Window P2 P”2 P’2 from one coordinate region to another may pass through Window P 2 may pass through the clip window, or they may intersect they may intersect clipping boundaries without entering the P3 P’1 without entering the window] P1 P4 P’3

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Cohen-Sutherland Line Clipping remaining lines

intersection test with bounding lines of clipping intersection test with bounding lines of clipping window left, right, bottom, top discard an outside part discard an outside part repeat intersection test up to four times vertical: y = y0 + m(xwmin – x0), y = y0 + m(xwmax – x0) horiz : + ( )/m + ( )/m

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horiz.: x = x0 + (ywmin – y0)/m, x = x0 + (ywmax – y0)/m

Cohen-Sutherland Line Clipping

P passes through li i i d Wi d P2 P” P’2 clipping window Window P 2 intersects boundaries without boundaries without entering clipping window P3 P’1 window P1 P4 P’3

vertical: y = y0 + m(xwmin – x0), y = y0 + m(xwmax – x0) horiz : + ( )/m + ( )/m

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horiz.: x = x0 + (ywmin – y0)/m, x = x0 + (ywmax – y0)/m

Polygon Clipping modification of line clipping goal: one or more closed areas g

[di l f l [display of a polygon [display of a correctly clipped polygon] [display of a polygon processed by a line- clipping algorithm]

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clipping algorithm]

Sutherland-Hodgman Polygon Clipping processing polygon boundary as a whole against each window edge against each window edge

• utput: list of vertices
• riginal

polygon clip left clip right clip bottom clip top clipping a polygon against successive polygon

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window boundaries

Sutherland-Hodgman Polygon Clipping four possible edge cases

V1 V2 V2 V1 V1 V2 Vnew V V

2 1

• ut  in

in  in in  out

• ut  out

V1 V2 V1 V2 Vnew

• ut  in
• utput: Vnew, V2

in  in V2 in  out Vnew

• ut  out

no output successive processing of pairs of polygon vertices against the left window boundary

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vertices against the left window boundary

V2:= 1st vertex

for 1 edge:

Sutherland-Hodgman

V1:=V2 V1: V2 V2:=next vertex V2 result list V2 visible?

no yes

V1 visible?

yes

V1 visible?

no yes no V1 visible?

V1 visible? V’1 = clip edge  V1V2 result list

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Sutherland-Hodgman Polygon Clipping

Window 3 [clipping a polygon against the left 2’ 2 against the left boundary of a window, starting with vertex 1. 1’ 2’ 1 4 starting with vertex 1. Primed numbers are used to label the points 3’ 6 1 4 used to label the points in the output vertex list for this window 5’ 3’ 5 for this window boundary]

finished!

5 4’ 5

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Polygon Clip: Combination of 4 Passes the polygon is clipped against each of the 4 borders separately borders separately, that would produce 3 intermediate results. by calling the 4 tests recursively by calling the 4 tests recursively, (or by using a clipping pipeline) every result point is immediately processed every result point is immediately processed

• n, so that only one result list is produced

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Sutherland-Hodgman Clipping Example

2

pipeline of boundary

Window

2

boundary clippers to avoid 1st clip: left

2’

avoid intermediate li left

2 1’ 3

vertex lists 2nd clip:

2” 1’ 1

2 clip: bottom

3’

[Processing the polygon vertices through a boundary-clipping

• pipeline. After all vertices are processed through the pipeline,

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the vertex list for the clipped polygon is {1’, 2, 2’, 2”} ]

Sutherland-Hodgman Polygon Clipping extraneous lines for concave polygons:

split into separate parts

• r

split into separate parts or final check of output vertex list

[clipping the concave polygon with the Sutherland-

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[ pp g p yg Hodgeman clipper produces three connected areas]

Clipping of Triangles

• ften b-reps are “triangle soups”

li i t i l  t i l ( ) clipping a triangle  triangle(s) 4 possible cases: p

inside

• tside

(inside)

• utside

triangle quadrilateral  2 triangles

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Clipping of Triangles corner cases need no extra handling!

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From Object Space to Screen Space

modeling transformation camera transformation

• bject space

world space camera space

projection p j transformation viewport transformation

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clip space screen space

Clipping in Clip-Space clipping against x = ± 1, y = ± 1, z = ± 1 ( ) i id ? (x,y,z) inside?  only compare one value! y p is done before homogenization:

• ge

at o x = ± h, y = ± h, z = ± h clips points that are behind the camera! reduces homogenization divisions

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reduces homogenization divisions

Antialiasing Antialiasing

Antialiasing in the Rendering Pipeline

scene objects in object space

• bject capture/creation

modeling viewing

vertex stage ( vertex shader“)

g projection

(„vertex shader )

transformed vertices in clip space clipping + homogenization transformed vertices in clip space scene in normalized device coordinates viewport transformation rasterization shading

pixel stage

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raster image in pixel coordinates shading

p g („fragment shader“)

Aliasing and Antialiasing h t i li i ? [' ili i ] what is aliasing? ['eiliæsiη] what is the reason for aliasing? what can we do against it? what can we do against it?

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What is Aliasing?

errors that are caused by the discretization

too bad resolution

• f analog data to digital data

too bad resolution too few colors too few images / sec geometric errors

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g numeric errors

Aliasing: Staircase Effect

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Various Aliasing Effects

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Aliasing from too few Colors

artificial color borders can appear artificial color borders can appear

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Aliasing in Animations jumping images "worming“

t

backwards rotating wheels

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Solutions against Aliasing?

• 1. improve the devices

p

higher resolution more color levels

expensive

• r

more color levels faster image sequence

incompatible

2 improve the images = antialiasing

• 2. improve the images = antialiasing

postprocessing

software

prefiltering !

software

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Nyquist-Shannon Sampling Theorem a signal can only be reconstructed without information loss if the sampling frequency is at least twice the highest frequency is at least twice the highest frequency of the signal this border frequency is called "Nyquist Limit" q y yq

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Nyquist-Shannon Sampling Theorem

• riginal signal

g g reconstructed signal

Nyquist Nyquist sampling interval

sampling rate 



interval

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Antialiasing: Nyquist Sampling Frequency a signal can only be reconstructed without information loss if the sampling without information loss if the sampling frequency is at least twice the highest f f th i l frequency of the signal

max

2 f fs 

Nyquist sampling frequency:

max cycle

/ 1 x with f x x

cycle s

    

max cycle

2 f

s

i.e. sampling interval  one-half cycle interval

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i.e. sampling interval  one half cycle interval

Antialiasing Strategies supersampling straight-line segments subpixel weighting masks area sampling straight-line segments area sampling straight-line segments filtering techniques compensating for line-intensity differences antialiasing area boundaries antialiasing area boundaries

(adjusting boundary pixel positions) adjusting boundary pixel intensity

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Antialiasing: Supersampling Lines

0 0 1 0 0 4 0 0 1 0 2 2 0 0 4 1 6 8 0 2 2 3 1 0 1 6 8 8 5 0 3 1 0 8 5 0

3 i t it 9 i i mathematical line line of finite width 3 = max. intensity ... 0 = min intensity 9 = max. intensity ... min intensity

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0 = min. intensity 0 = min. intensity

Antialiasing

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Antialiasing: Pixel Weighting Masks more weight for center subpixels center subpixels must be divided by y sum of weights subpixel grids can subpixel grids can also include some i hb i i l neighboring pixels relative weights for a grid of 3x3 subpixels

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grid of 3x3 subpixels

Antialiasing: Area Sampling Lines calculate the pixel coverage exactly

0% 0% 0% 0% 43% 43%

coverage exactly can be done with

0% 0% 0% 0% 43% 43% 15% 15% 71% 71% 84% 84%

incremental schemes

15% 15% 71% 71% 84% 84% 90% 90% 52% 52% 3% 3% 90% 90% 52% 52% 3% 3%

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Antialiasing: Filtering Techniques continuous overlapping weighting functions to calculate the antialiased values with integrals to calculate the antialiased values with integrals

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box filter cone filter Gaussian filter

Antialiasing: Intensity Differences unequal line lengths q g displayed with the same number of same number of pixels in each line/row have line/row have different intensities proper antialiasing compensates for compensates for that!

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Antialiasing Area Boundaries

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Antialiasing Area Boundaries (1) alternative 1: supersampling supersampling adjusting pixel adjusting pixel intensities along an area boundary area boundary

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Antialiasing Area Boundaries (2) alternative 2: similar to Bresenham algorithm ´ [ ( 1) b] ( 0 5) p´ = y  ymid = [m(xk + 1) + b]  (yk + 0.5) p´<0  y closer to y p <0  y closer to yk p´>0  y closer to yk+1 p = p´+(1m) : y y

id

p p +(1 m) : p<1m  closer to yk p>1 m  closer to y ymid p>1m  closer to yk+1 ( and p  [0,1] )

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( and p  [0,1] )

Antialiasing Area Boundaries (3) p = p´+(1m)= [m(xk + 1) + b]  (yk + 0.5) + (1  m) = mx + b y + 0 5 = = mx + b (y 0 5) = mxk + b  yk + 0.5 = mxk + b  (yk  0.5) yk+ 0 5 p for next pixel yk+ 0.5 p for next pixel = overlap area for current pixel p´ yk m for current pixel p p x x +1 p 1-m p p´

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xk xk+1 p

Antialiasing Area Boundaries (4)

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Antialiasing Examples

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