Introduction to aberrations
Lecture #1
College of Optical Sciences
University of Arizona
Prof. Jose Sasian
Photo credit:
Gary Mackender
Prof. Jose Sasian
What is imaging?
Imaging with rays: theories
Maxwell’s Ideal:
1) Each point is imaged stigmatically
2) The images of all points in an object plane lie on an image plane
3) The ratio of image to heights to object heights is the same for all points.
Collinear transformation
X' 
a1 X  b1Y  c1 Z  d1
a0 X  b0 Y  c0 Z  d0
Prof. Jose Sasian
Y' 
a2 X  b2 Y  c2 Z  d2
a0 X  b0 Y  c0 Z  d0
Z' 
a3 X  b3Y  c3 Z  d3
a0 X  b0 Y  c0 Z  d0
Central Projection 1D
X’
OBJECT LINE
PROJECTION POINT
IMAGE
POINT
OBJECT
POINT
X
Prof. Jose Sasian
IMAGE LINE
a1 X
X' 
a0 X  b0
Central Projection 2D
X’
X
OBJECT
LINE
Y
PROJECTION
POINT
OBJECT
PLANE
X' 
Prof. Jose Sasian
a1 X
a0 X  b0 Y  c0
IMAGE
LINE
Y’
IMAGE
PLANE
a2 Y
Y' 
a0 X  b0 Y  c0
Central Projection 3D
a1 X  b1Y  c1 Z  d1
X' 
a0 X  b0 Y  c0 Z  d0
a2 X  b2 Y  c2 Z  d2
Y' 
a0 X  b0 Y  c0 Z  d0
a3 X  b3Y  c3 Z  d3
Z' 
a0 X  b0 Y  c0 Z  d0
Point into points, lines into lines, and planes into
planes.
Prof. Jose Sasian
Axial symmetry I
  x2  y2
‘
 '  x '2  y '2
and Z’ must be only a function of  and Z, leads to:
a1 X
X' 
c0 Z  d 0
a1Y
Y' 
c0 Z  d 0
c3 Z  d3
Z' 
c0 Z  d 0
•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
X' 
c0 Z  1
m
Prof. Jose Sasian
1
c0 Z  1
Y' 
Y
c0 Z  1
c3
Z'  f ' 
c0
Z' 
c3 Z
c0 Z  1
Z f 
1
c0
Gaussian equations
1
m
1 Z \ f
1
Z
 1
f
m
Z'
 1 m
f'
Z'
m  1
f'
f' f
 1
Z' Z
.
Prof. Jose Sasian
Newtonian Equations
Z
1
= f
m
Z'
= - m
f'
Prof. Jose Sasian
ZZ'  ff '
Scheimpflug condition
(Plane symmetry)
tan(’) – (Z’\Z) tan() = 0
object
plane
image
plane
theta
P
P’
theta’
Scheimpflug construction
X
X'  m
1  KY
Prof. Jose Sasian
Y
Y'  m
1  KY
K  m
tan ( )
tan (  ')
=
f
f '
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
f  68 cm
 
D  f  tan    f tan 45 
Prof. Jose Sasian
However…
• Image parity issues
• Needed a flat mirror or
• To copy again the image
Prof. Jose Sasian
Central projection
Prof. Jose Sasian
By Don Cowen
Other “perspectives”
Classical
Impressionism
A plurality of new and exciting
new concepts in imaging
Prof. Jose Sasian
Cubist
Imaging on the sphere
Prof. Jose Sasian
IPAD
Imaging
Prof. Jose Sasian
Class summary
• Imaging with rays
• Central projection and collinear
transformation
• Art application
• Homework on symmetry
Prof. Jose Sasian