viewing doug james’ cg slides, rich riesenfeld’s cg slides, shirley, fundamentals of computer...

ViewingViewing

Doug James’ CG Slides, Rich Riesenfeld’s CG Slides,

Shirley, Fundamentals of Computer Graphics, Chap 7

Wen-Chieh (Steve) Lin

Institute of Multimedia Engineering

ILE5014 Computer Graphics 10F 2

Getting Geometry on the ScreenGetting Geometry on the Screen

Given geometry in the world coordinate system, how do we get it to the display?

• Transform to camera coordinate system

• Transform (warp) into canonical view volume

• Clip

• Project to display coordinates

• Rasterize


Vertex Transformation PipelineVertex Transformation Pipeline


Vertex Transformation PipelineVertex Transformation Pipeline

glMatrixMode(GL_MODELVIEW)

glMatrixMode(GL_PROJECTION)

glViewport(…)


OpenGL Transformation OverviewOpenGL Transformation Overview

glMatrixMode(GL_MODELVIEW)

gluLookAt(…)

glMatrixMode(GL_PROJECTION)

glFrustrum(…)

gluPerspective(…)

glOrtho(…)

glViewport(x,y,width,height)


Viewing and ProjectionViewing and Projection

• Our eyes collapse 3-D world to 2-D retinal image (brain then has to reconstruct 3D)

• In CG, this process occurs by projection

• Projection has two parts:

– Viewing transformations: camera position and direction

– Perspective/orthographic transformation: reduces 3-D to 2-D

• Use homogeneous transformations


Pinhole OpticsPinhole Optics

• Stand at point P, and look through the hole—anything within the cone is visible, and nothing else is

• Reduce the hole to a point - the cone becomes a ray

• Pin hole is the focal point, eye point or center of

projection

P

F


Perspective Projection of a PointPerspective Projection of a Point

• View plane or image plane - a plane behind the

pinhole on which the image is formed

– sees anything on the line (ray) through the pinhole F

– a point W projects along the ray through F to appear at I (intersection of WF with image plane)

F

Image

WorldI

W


Image FormationImage Formation

• Projecting a shape– project each point onto the image plane– lines are projected by projecting end points only

F

Image

World

F

Image

World

I

W

Note: Since we don't want the image to be inverted, from now on we'll put F behind the image plane.

Note: Since we don't want the image to be inverted, from now on we'll put F behind the image plane.

Camera lens


Orthographic ProjectionOrthographic Projection

• When the focal point is at infinity the rays are parallel and orthogonal to the image plane

• When xy-plane is the image plane (x,y,z) -> (x,y,0) front orthographic view

Image

World

F


Multiview Orthographic ProjectionMultiview Orthographic Projection

• Good model for CAD and architecture. No perspective effects.

front

sidetop


Assume view transformation is doneAssume view transformation is done

• Let’s start with a simple case where the camera is put at the origin, and looks along the negative z-axis!

• Simple “canonical” views:

orthographic projection perspective projection


Orthographic Viewing CubeOrthographic Viewing Cube

• (l,b,n) = (left, bottom, near)

• (r,t,f) = (right, top, far)


Orthographic ProjectionOrthographic Projection

• Map orthographic viewing cube to the canonical view volume

• 3D window transformation

• [l, r] x [b, t] x [f, n] [-1, 1]x[-1, 1]x[-1,1]


Orthographic Projection (cont.)Orthographic Projection (cont.)

• 3D window transform (last class)

• [l, r] x [b, t] x [f, n] [-1, 1]x[-1, 1]x[-1,1]

• zcanonical is ignored

110002

100

2010

2001

1000

02

00

002

0

0002

1

z

y

x

fn

tb

rl

fn

bt

lr

z

y

x

canonical

canonical

canonical


Map Image Plane to Screen Map Image Plane to Screen

(1,1)

(-1,-1)

x

y

Image Plane Screen

x

y

y

x

(nx, ny)


Map Image Plane to Screen Map Image Plane to Screen

• If y-axis of screen coord. points downward

• [-1,1] x [-1, 1] [-0.5, nx-0.5] x [ny-0.5, -0.5]

1100

010

001

100

02

0

002

1002

110

2

101

canonical

canonicaly

x

y

x

pixel

pixel

pixel

y

xn

n

n

n

z

y

x

reflectionscaletranslation


Orthographic Projection MatrixOrthographic Projection Matrix

• Put everything together

10002

100

2010

2001

1000

02

00

002

0

0002

1000

01002

10

20

2

100

2

fn

tb

rl

fn

bt

lr

nn

nn

yy

xx

oM

11

z

y

x

z

y

x

canonical

pixel

pixel

oM


Arbitrary Viewing PositionArbitrary Viewing Position

• What if we want the camera somewhere other than the canonical location?

• Alternative #1: derive a general projection matrix. (hard)

• Alternative #2: transform the world so that the camera is in canonical position and orientation (much simpler)


Camera Control ValuesCamera Control Values

• All we need is a single translation and angle-axis rotation (orientation), but...

• Good animation requires good camera control--we need better control knobs

• Translation knob - move to the lookfrom point

• Orientation can be specified in several ways:

– specify camera rotations

– specify a lookat point (solve for camera rotations)


A Popular View Specification ApproachA Popular View Specification Approach

• Focal length, image size/shape and clipping planes are in the perspective transformation

• In addition:

– lookfrom: where the focal point (camera) is

– lookat:the world point to be centered in the image

• Also specify camera orientation about the lookfrom-lookfat axis


Implementing lookat/lookfrom/vup viewing schemeImplementing lookat/lookfrom/vup viewing scheme

• Translate by lookfrom, bring focal point to origin

• Rotate lookfrom - lookat to the z-axis with matrix R:

– w = (lookfrom-lookat) (normalized) and z = [0,0,1]

– rotation axis: a = (w × z)/|w × z|

– rotation angle: cosθ = w•z and sinθ = |w × z|

• Rotate about z-axis to get vup parallel to the y-axis


It's not so complicated…It's not so complicated…

Translate LOOKFROM to the origin

Multiply by the projection matrix and everything will be in the canonical camera position

Rotate the view vector(lookfrom - lookat) ontothe z-axis.

Rotate about z to bring vup to y-axis

START HEREw

lookfrom

vup

x

y

z

y

x

x

x

y

y

z

z

z


Translate Camera Frame Translate Camera Frame

• Let e=(ex, ey, ez) be the “lookfrom” position


START HEREw

lookfrom

vup

x

y

z

xy

z

0000

100

010

001

z

y

x

e

e

e

tM


Rotate Camera Frame Rotate Camera Frame

• You can derive these two rotation matrices based on what you learned in the last class


Rotate the view vector(lookat -lookfrom) ontothe z-axis.

Rotate about z to bring vup to y-axis

x

y

z

yx

x

y

z

z


Rotate Camera Frame (cont.)Rotate Camera Frame (cont.)

• Alternatively, we can view these two rotations as a single rotation that aligns u-v-w-axes to x-y-z-axes!

xy

zw

v Rux Rvy Rwz

wvu

wvu

wvu

zzz

yyy

xxx

R

100

010

001

www

vvv

uuu

wvu

wvu

wvu

zyx

zyx

zyx

zzz

yyy

xxx1

Rw: lookat – lookfromv: view up directionu = v x w

wvu

wvu

wvu

zzz

yyy

xxx

wvu

wvu

wvu

zzz

yyy

xxx


Recall Global vs. Local Coordinate…Recall Global vs. Local Coordinate…

• x, y, u, v, o, e are all vectors in global systemyxop pppp yxyx ),(

vuep pppp vuvu ),(

11

0

0

100

10

01

1p

p

e

e

p

p

v

u

y

x

y

x

vu

1100

10

01

1001p

p

e

eT

T

p

p

y

x

y

x

v

u

v

u

In global coord. (xp,yp) In local coord. (up,vp)


Think about 3D case ….Think about 3D case ….

• Given two coordinate frames, how do you represent a vector specified in one frame in the other frame?

e

u

v

w

X

Y

Z

P

11p

p

p

p

p

p

w

v

u

z

y

x

11p

p

p

p

p

p

z

y

x

w

v

u

?

?


Viewing TransformationViewing Transformation

• Put translation and rotation together

10001000

100

010

001

1000

0

0

0

ewww

evvv

euuu

e

e

e

www

vvv

uuu

v zzyx

yzyx

xzyx

z

y

x

zyx

zyx

zyx

tRMM

11p

p

p

p

p

p

z

y

x

w

v

u

vM

world coordinatelocal coordinate

e

u

v

w

X

Y

Z

P


Orthographic Projection SummaryOrthographic Projection Summary

Given 3D geometry (a set of points a)

• Compute view transformation Mv

• Compute orthographic projection Mo

• Compute M = MoMv

• For each point ai, compute p = Mai

10002

100

2010

2001

1000

02

00

002

0

0002

1000

01002

10

20

2

100

2

fn

tb

rl

fn

bt

lr

nn

nn

yy

xx

oM


A Simple Perspective CameraA Simple Perspective Camera

• Canonical case:– camera looks along the z-axis (toward negative z-axis)

– focal point is the origin

– image plane is parallel to the xy-plane at distance d

– We call d the focal length, mainly for historical reasons

Image plane

Center of projection


Geometry Eq. for Perspective ProjectionGeometry Eq. for Perspective Projection

• Diagram shows y-coordinate, x-coordinate is similar

• Point (x,y,z) projects to

view

pla

nee g

z

y

ys

d

yz

dys

),,( dyz

dx

z

d

e: eye positiong: gaze direction


Perspective Projection MatrixPerspective Projection Matrix

• Projection using homogeneous coordinates:

– transform (x,y,z) to

• 2-D image point:– discard third coordinate

– apply viewport transformation to obtain physical pixel coordinates

),,( dyz

dx

z

d

z

dz

dy

dx

z

y

x

d

d

d

10100

000

000

000

),,( dyz

dx

z

d

Divide by 4th coordinate(the “w” coordinate)


View VolumeView Volume

• Pyramid in space defined by focal point and window in the image plane (assume window mapped to viewport)

• Defines visible region of space

• Pyramid edges are clipping planes


View FrustumView Frustum

• Truncated pyramid with near and far clipping planes

– Why far plane? Allows z to be scaled to a limited fixed-point value (z-buffering)

– Why near plane? Prevent points behind the camera being seen


Why Canonical View Volume?Why Canonical View Volume?

• Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume

• This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping


Taking Clipping into Account Taking Clipping into Account

• After the view transformation, a simple projection and viewport transformation can generate screen coordinate.

• However, projecting all vertices are usually unnecessary.

• Clipping with 3D volume.

• Associating projection with clipping and view volume normalization.


Normalizing the Viewing FrustumNormalizing the Viewing Frustum

• Solution: transform frustum to a cube before clipping

• Converts perspective frustum to orthographic frustum• This is yet another homogeneous transform!


Map Perspective View Volume to Orthographic View Volume Map Perspective View Volume to Orthographic View Volume

• We already know how to map orthographic view volume to canonical view volume

• Set view plane at the near plane


Map Perspective View Volume to Orthographic View VolumeMap Perspective View Volume to Orthographic View Volume

• Map [x,y,n] to [x, y, n]

• Map [xf/n,yf/n,f] to [x, y, f]

01

00

00

0010

0001

n

fn

fnpM


Projection Matrix is not uniqueProjection Matrix is not unique

• Mp is not unique

11

'

'

'

z

y

x

h

h

hz

hy

hx

z

y

x

pp MM

0100

00

000

000

01

00

00

0010

0001

nffn

n

n

n

fn

fnnn pp MM


Properties of Perspective TransformProperties of Perspective Transform

• Lines and planes are preserved

• Parallel lines (not parallel to the projection plan) won’t be parallel after transform

– vanishing point: parallel lines intersect at the vanishing point

vanishing point


Orthographic Projection SummaryOrthographic Projection Summary




• Compute M = MoMv


10002

100

2010

2001

1000

02

00

002

0

0002

1000

01002

10

20

2

100

2

fn

tb

rl

fn

bt

lr

nn

nn

yy

xx

oM


Perspective Projection SummaryPerspective Projection Summary



• Map perspective to orthographic Mp


• Compute M = MoMpMv


viewing doug james’ cg slides, rich riesenfeld’s cg slides, shirley, fundamentals of computer...

Documents

f image world f image

ile5014 computer graphics

pinhole f

image plane f image

projection projection

center of projection

image plane x

projection glviewport