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Visual Imaging in the Electronic Age Perspective Transformations Lecture # 3 August 29, 2017 Prof. Donald P. Greenberg

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Page 1: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Visual Imaging in the Electronic Age

Perspective Transformations

Lecture # 3August 29, 2017

Prof. Donald P. Greenberg

Page 2: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Lab This WeekTuesdays at 2:30 PM and Thursdays at 4:30 PM: Left-brainers (blue)Tuesdays at 4:30 PM and Thursdays at 2:30 PM: Right-brainers (green)Wednesdays at 4:30 PM: Surplus and/or engineering honors section

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 3: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Simple Perspective Transformation

e

es

e

es

e

e

e

e

e

es

e

es

SzDyy

SzDxx

S.ndow size by the wican dividefraction, nsionless to a dimeTo convert

zDy

zDxx

zy

Dy

zx

Dx

==

==

==

,

y ,

,

ss

Page 4: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Pinhole Camera

Note that the entire image through the pinhole is totally in focus on a single image plane.

Page 5: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

• We have seen how a pinhole camera (or camera obscura) works

• We have also seen Brunelleschi’s experiment with his perspective panels

• Both are obviously two dimensional and exist on a planar (flat) surface

Page 6: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

End of Review

Page 7: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Perspective Depth

• We now know how to mathematically (and geometrically) construct the perspective image. (e.g., xs, ys)

• But how do we find the third dimension (zs)?

Page 8: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert
Page 9: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

David Macauley, Locating the Vanishing Point

Locating the Vanishing Point

Page 10: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

How a Camera Lens Works• Rays which pass through the focal point are bent

to leave the lens perpendicular to its principal axis.

Page 11: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 12: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 13: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Railroad Tracks in Perspective

elevation

Page 14: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

How a Camera Lens Works• Rays which pass through the focal point are bent

to leave the lens perpendicular to its principal axis.

• Rays which are perpendicular to the principal axis of the lens are bent through the focal point.

Page 15: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

How a Camera Lens Works

f

P(xe, ye, ze)

P′(xp, yp, zp)

-zp - 2f

Ze

Xe

Ye

A

A′

B

B′

f

-yp

Page 16: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

e

ep

e

ep

zx

fx

zy

fy

=−

=−

,Using triangles A and A':

fzy

fy

p

pe

2−−

−=Using triangles B and B':

e

ep

e

ep z

yfyzxfx −=−=∴ ,

How a Camera Lens Works

Page 17: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

ffz

yy

zf

yy p

e

p

ee

p 2 ,

−−=

−=

−We know that:

ep

e

p

zffz

zf

ffz 2

2 ,2

−=+=

−−∴

ep z

ffz2

2 −−=

How a Camera Lens Works

Page 18: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

How a Camera Lens Works

Page 19: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Camera Film

Plane

Page 20: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Camera Film

Plane

Page 21: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert
Page 22: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

http://upload.wikimedia.org/wikipedia/en/5/5f/DoF-sym.png

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 23: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Perspective Projections in Computer Graphics

Eye coordinate systemPlan or elevation view

Page 24: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Mapping a Viewing Frustumto a Standard Viewbox

11ze = DXe

Frustum of vision Screen coordinate system

Ze

Ye

ze = F

Xs

Ys Zs

1

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 25: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Viewing Frustum - User ParametersLocation and Orientation (vectors):

Eye location of eye pointAt a point along view direction vectorUp a vector defining the Y axis on the hither plane

Shape (scalars):fov field of view angleaspect view plane aspect ratiohither perpendicular distance to hither planeyon perpendicular distance to yon plane

AtUp

Eye

aspect = width / height

hither yon

fov

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Object Space

(One point perspective)

Page 27: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Image Space

(One point perspective)

Page 28: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Object Space

(Two point perspective)

Page 29: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Image Space

(Two point perspective)

Page 30: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Object Space

(Three point perspective)

Page 31: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Image Space

(Three point perspective)

Page 32: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

3D Perspective Image(x & y coordinates in virtual image space)

e

ep z

xfx −=e

ep z

yfy −=

e

ep z

xx constant=e

ep z

yy constant=

Page 33: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

ep z

ffz2

2 −−=

ep z

z 121 constantconstant +=

3D Perspective Image(z depth coordinate in virtual image space)

Page 34: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Equivalent Distorted Geometriesin Virtual Image Space

Page 35: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Distorted Geometry Model

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 36: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Distorted Geometry Model

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 37: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Uses of Perspective in Enlarging Architectural Space (3D)

Page 38: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Palazzo Spada

Borromini

Page 39: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 40: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 41: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 42: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 43: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 44: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 45: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 46: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Imss,Firenze 2004 - 3 Scenografia prospettica

Page 47: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Borromini

Palazzo Spada

Page 48: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Borromini

Palazzo Spada

Page 49: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

Probe Demo

® Donald P. Greenberg - Cornell Program of Computer Graphics

Page 50: CS 417 Perspective Transformation - 3D S02-1 · PDF fileSimple Perspective Transformation. e e s e e s e e e e e s e e s e. Sz Dy y Sz Dx x can divide by the window size S. To convert

End. . .

® Donald P. Greenberg - Cornell Program of Computer Graphics


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