viewing - cg.tuwien.ac.at

Einführung in Visual Computing186.822

Viewing

Werner Purgathofer

Viewing in the Rendering Pipeline

Werner Purgathofer 2raster image in pixel coordinates

clipping + homogenization

object capture/creation

modelingviewing

projection

rasterizationviewport transformation

shading

vertex stage(„vertex shader“)

pixel stage(„fragment shader“)

scene in normalized device coordinates

transformed vertices in clip space

scene objects in object space

From Object Space to Screen Space

Werner Purgathofer 3

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Viewport Transformation

From Object Space to Screen Space

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object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Viewport Transformation

assumption: scene is in clip space !clip space = [–1,1] × [–1,1] × [–1,1]orthographic camera looking in –z directionscreen resolution nx × ny pixels

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nx ny(–1,–1,–1)

(1,1,1)

(–1,–1) (0,0)(1,1) (nx,ny)

clip space:

z

Viewport Transformation

can be done with the matrix

this ignores the z-coordinate, but…

(–1,–1) (0,0)(1,1) (nx,ny)

xscreenyscreen

nx/2 00 ny/2

nx/2ny/2

0 0 11

xy1

= ·

Werner Purgathofer / Einf. in Visual Computing

Projection Transformation

From Object Space to Screen Space

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object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Parallel Projection (Orthographic Projection)

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3 parallel-projection views of an object, showing relative proportions from different viewing positions

top side front

Werner Purgathofer / Einf. in Visual Computing

Parallel vs. Perspective Projection

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For Now: Parallel (Orthographic) Projection

Werner Purgathofer 13

object space world space camera space

clip space screen space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

FB

Projection Transformation (Orthographic)

assumption: scene in box [L,R] × [B,T] × [F,N]orthographic camera looking in –z directiontransformation to clip space

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L

(L,B,F) (–1,–1,–1)(R,T,N) (1,1,1)

clip space:

(–1,–1,–1)

(1,1,1)

z(L,B,F)

(R,T,N)z

orthographic view volumein camera space:

R

T

N

Werner Purgathofer / Einf. in Visual Computing

Projection Transformation (Orthographic)

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(L,B,F) (–1,–1,–1)(R,T,N) (1,1,1)

0 0 0 1

R – L2

T – B2

N – F2

R – LR + L

T – BT + B

N – FN + F

0 0

0 0

0 0

–

–

–Morth =

1

L

F

B

1

–1

–1

–1· =

Werner Purgathofer / Einf. in Visual Computing

Parallel Projection Types

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orthographicprojection

obliqueprojection

different orientation of the projection vector –g

–g

viewing plane

–g

viewing plane

Werner Purgathofer / Einf. in Visual Computing

Camera Transformation

For Now: Parallel (Orthographic) Projection

Werner Purgathofer 18

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Viewing: Projection Plane

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coordinate reference for obtaining a selected view of a 3D scene

displayplane

Werner Purgathofer / Einf. in Visual Computing

Viewing: Camera Definitionsimilar to taking a photographinvolves selection of

camera positioncamera directioncamera orientation“window” (zoom) of camera

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worldcoordinates

cameracoordinates

Werner Purgathofer / Einf. in Visual Computing

Viewing: Camera Transformation (1)

view reference pointorigin of camera coordinate systemgaze direction or look-at point

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right-handed camera-coordinate system, with axes u, v, w, relative to world-coordinate scene

y

zx

vw

u

eo

Werner Purgathofer / Einf. in Visual Computing

Viewing: Camera Transformation (2)

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eg

e … eye positiong … gaze direction (positive w-axis points to the viewer)t … view-up vector

wu

t

v = w × u

w = – g|g|

u =|t × w|t × w

Werner Purgathofer / Einf. in Visual Computing

Viewing: Camera Transformation (2)

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eg

e … eye positiong … gaze direction (positive w-axis points to the viewer)t … view-up vector

vw

ut

v = w × u

w = – g|g|

u =|t × w|t × w

Werner Purgathofer / Einf. in Visual Computing

Viewing: Camera Transformation (3)

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aligning viewing system with world-coordinate axes using translate-rotate transformations

Mcam = Rz· Ry· Rx·T

x

y

z

o

uvw

e

x

y

z

uv

wx

y

z

ouv

we

Werner Purgathofer / Einf. in Visual Computing

For Now: Parallel (Orthographic) Projection

Werner Purgathofer 26

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Viewing: Camera + Projection + Viewport

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xscreenyscreen

z1

= Mvp·Morth·Mcam ·

xyz1

pixels onthe screen

world coordinates

( )

Werner Purgathofer / Einf. in Visual Computing

For Now: Parallel (Orthographic) Projection

Werner Purgathofer 28

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Perspective Projection

Perspective Projection

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perspective projection of equal-sized objects at different distances from the view plane

x

y

zview plane

projection reference

point

Werner Purgathofer / Einf. in Visual Computing

Parallel vs. Perspective Projection

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parallel projection: preserves relative proportions & parallel features (affine transform.)

perspective projection: center of projection, realistic views

Werner Purgathofer / Einf. in Visual Computing

Perspective Transform

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N F

T

B

O

Werner Purgathofer / Einf. in Visual Computing

Perspective Transformation

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fypO

view plane

yp = yfz

z

y

f … focal length

derivation of perspective

transformation

Werner Purgathofer / Einf. in Visual Computing

yp fzy =

Perspective Transformation

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fz

yp

y

view plane

yp = yfz

(z=N)

O(z=0)

N

N N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

= P

xp = xdzNanalogous:

derivation of perspective

transformation

Werner Purgathofer / Einf. in Visual Computing

Perspective Transformation

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N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

xyz1

· =

x·Ny·N

z·(N+F) –F·Nz

~>

x·N/zy·N/z

(N+F) –F·N/z1

homogenization: divide by z

yp = yz

xp = xzN

N

P =derivation of

perspective transformation

Werner Purgathofer / Einf. in Visual Computing

Example: (Right) Top Near Corner

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N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

RTN1

· =

R·NT·N

N·(N+F) –F·NN

~>

RTN1

x (B or T or L or R)

view plane

NO

F

Werner Purgathofer / Einf. in Visual Computing

Example: (Left) Top Far Corner

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N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

L·F/NT·F/N

F1

· =

L·FT·F

F·(N+F) –F·NF

~>

LTF1

x (B or T or L or R)

NO

F

view plane x·F/N

Werner Purgathofer / Einf. in Visual Computing

Perspective Transform

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N F

T

B

O

Werner Purgathofer / Einf. in Visual Computing

Nonlinear z-Behaviour

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Nonlinear z-Behaviour

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From Object Space to Screen Space

Werner Purgathofer 41

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Viewing: Camera + Projection + Viewport

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xscreenyscreen

z´1

= Mvp·Morth·P·Mcam ·

xyz1

( )

Mper

pixels onthe screen

world coordinates

Werner Purgathofer / Einf. in Visual Computing

From Object Space to Screen Space

Werner Purgathofer 43

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

z-Values Remain in Order

44Werner Purgathofer

N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

xyz1

· ~>

x·N/zy·N/z

(N+F) –F·N/z1

z1, z2, N, F < 0z1 < z2

1/z1 > 1/z2 | · (–F·N) (<0)– F·N/z1 < – F·N/z2 | + (N+F)(N+F) – F·N/z1 < (N+F) – F·N/z2

Werner Purgathofer / Einf. in Visual Computing

Perspective Projection Properties

parallel lines parallel to view plane ⇒ parallel linesparallel lines not parallel to view plane ⇒ converging lines (vanishing point)lines parallel to coordinate axis ⇒principal vanishing point (one, two or three)

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Principle Vanishing Points

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2-point perspectiveprojection

3-point perspectiveprojection

1-pointperspectiveprojection

vanishing point

Werner Purgathofer / Einf. in Visual Computing