Upload
others
View
19
Download
0
Embed Size (px)
Citation preview
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