Prof. Jose Sasian

Introduction to aberrationsLecture #1

College of Optical SciencesUniversity of Arizona

Prof. Jose Sasian

What is imaging?Photo credit:Gary Mackender

Prof. Jose Sasian

Imaging with rays: theories

X a X b Y c Z da X b Y c Z d

'

1 1 1 1

0 0 0 0

Y a X b Y c Z da X b Y c Z d

'

2 2 2 2

0 0 0 0

Z a X b Y c Z da X b Y c Z d

'

3 3 3 3

0 0 0 0

Collinear transformation

Maxwell’s Ideal:

1) Each point is imaged stigmatically2) The images of all points in an object plane lie on an image plane3) The ratio of image to heights to object heights is the same for all points.

Prof. Jose Sasian

Central Projection 1D

X’

OBJECT LINE

IMAGE LINE

PROJECTION POINT

OBJECTPOINT

IMAGEPOINT

X

X a Xa X b

'

1

0 0

Prof. Jose Sasian

Central Projection 2D

X X’

Y Y’OBJECT

LINEIMAGELINE

OBJECTPLANE

IMAGEPLANE

PROJECTIONPOINT

X a Xa X b Y c

'

1

0 0 0

Y a Ya X b Y c

'

2

0 0 0

Prof. Jose Sasian

Central Projection 3D

X a X b Y c Z da X b Y c Z d

'

1 1 1 1

0 0 0 0

Y a X b Y c Z da X b Y c Z d

'

2 2 2 2

0 0 0 0

Z a X b Y c Z da X b Y c Z d

'

3 3 3 3

0 0 0 0

Point into points, lines into lines, and planes into planes.

Prof. Jose Sasian

X a Xc Z d

'

1

0 0

Y a Yc Z d

'

1

0 0

Z c Z dc Z d

'

3 3

0 0

.

22 yx

Axial symmetry I22 ''' yx

‘ and Z’ must be only a function of and Z, leads to:

•Origins transform into the origins: d3=0•Origins located at the planes of unit magnification implies: a1 equal to d0

Prof. Jose Sasian

Axial Symmetry II

X Xc Z

' 0 1

Y Yc Z

' 0 1

Z c Zc Z

'

3

0 1

mc Z

110

Z f cc

' ' 3

0

Z fc

1

0

Prof. Jose Sasian

Gaussian equations

mZ f

11 \

Zf m 1 1

Zf

m'' 1 m Z

f 1 '

'

fZ

fZ

'' 1

.

Prof. Jose Sasian

Newtonian Equations

Zf

= - 1m

Zf''

= - m

ZZ ff' '

Prof. Jose Sasian

Scheimpflug condition(Plane symmetry)

P P’

theta

theta’

objectplane

imageplane

Scheimpflug construction

tan(’) – (Z’\Z) tan() = 0

X m XKY

' 1

Y m YKY

' 1

tan( ) tan( ')=

'K m

f f

Prof. Jose Sasian

Civil War 1859

Prof. Jose Sasian

And much more…•Gaussian equations•Newtonian equations•Anamorphic systems•Focal and afocal systems•Scheimpflug condition•Cardinal points•Keystone distortion•…and much more!

Prof. Jose Sasian

Collinear transformation, camera obscura and lenses

Camera Obscura, Athanasius Kircher, 1646

Prof. Jose Sasian

The secret in Vermeer’s paintings

Johannes Vermeer, XVII century

Prof. Jose Sasian

Perspective of the painting

45tantan ffD

cmf 68

Prof. Jose Sasian

However…

• Image parity issues• Needed a flat mirror or• To copy again the image

Prof. Jose Sasian

Central projection

By Don Cowen

Prof. Jose Sasian

Other “perspectives”

A plurality of new and excitingnew concepts in imaging

Classical Impressionism Cubist

Prof. Jose Sasian

Imaging on the sphere

Prof. Jose Sasian

IPADImaging

Prof. Jose Sasian

Class summary

• Imaging with rays• Central projection and collinear

transformation• Art application• Homework on symmetry