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Project proposal (2 pages): 1st of June
Project idea presentation: 8th of June
Optics
Final Project presentation: 20th of July
Project report
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Real Lens
Hendrik Lensch, Summer 2007
Optics
Outline
Refraction, focusing, formulas
Field of view, sensor format
Aperture and depth of field
Aberrations
Acknowledgements for slides
Steve Marschner, Bennett Wilburn, Pat Hanrahan,
Marc Levoy
Cutaway section of a Vivitar Series 1 90mm f/2.5 lens
Cover photo, Kingslake, Optics in Photography
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Pinhole Camera
Hendrik Lensch, Summer 2007
Pinhole camera
Large pinhole gives
geometric blur
Small pinhole gives
diffraction blur
Optimal pinhole gives
very little light
for 35mm format is
around f/200
image: Hecht
image: Wandell
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 1
Hendrik Lensch, Summer 2007
Diffraction
The Reason for Lenses
Huygens: every point on a wavefront can be
considered as a source of spherical wavelets
diffraction from a
circular aperture:
Airy rings
Fresnel: the amplitude of the optical field is the superposition
of these waves, considering amplitude and phase
Fraunhofer: resulting far-field diffraction pattern
images: Hecht 1987
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Purpose of lens
Purpose of lens
Produce bright but still sharp image
Produce bright but still sharp image
Focus rays emerging from a point to a point
Focus rays emerging from a point to a point
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Paraxial Refraction
Paraxial Refraction
“First order” (or Gaussian) optics
Refraction governed by Snell’s Law
1. assume e = 0
n sin i = n’ sin i’
2. assume sin a = tan a ~ a
n i ≈ n’ i’ (Gaussian optics for small angles)
i
i’
a
(n)
e
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 2
(n’)
Hendrik Lensch, Summer 2007
Paraxial Refraction
Paraxial Refraction
What is z’?
i
h
i
h
u
i’
r
u
i’
r
a
P
P'
z
a
P
z’
P'
z
z’
i=u+a
a = u’ + i’
u=h/z
u’ = h / z’
a =h/r
n i = n’ i’
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Paraxial Refraction
Hendrik Lensch, Summer 2007
Focal length
i
h
i’
r
u
rr
a
P
P'
z
z’
z’
focal length
i=u+a
a = u’ + i’
n ( u + a) = n’ ( u’ – a )
u=h/z
u’ = h / z’
n (h/z + h/r) = n’ (h/z’ – h/r)
a =h/r
z = inf
n/r = n’/z’ – n’/r
n/z + n/r = n’/z’ – n’/r
z’ = f = focal length = r/2(n-1)
n i = n’ i’
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Focal Points and Focal Lengths
Hendrik Lensch, Summer 2007
Gauss’ Ray Tracing Construction
To focus: move lens relative to backplane
Parallel Ray
1 1 1
= +
z′ z f
Focal Ray
Object
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 3
Chief Ray
Image
Hendrik Lensch, Summer 2007
Real Image
Magnifying Glass
Virtual Image
Parallel Ray
Focal Ray
Object
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Thick lenses
Hendrik Lensch, Summer 2007
The “center of perspective”
Complex optical system is characterized
by a few numbers
In a thin lens, the chief ray traverses the lens (through its optical
center) without changing direction
In a thick lens, the intersections of this ray with the optical axis are
called the nodal points
For a lens in air, these coincide with the principal points
The first nodal point is the center of perspective
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
image: Hecht 1987
Computational Photography
Focal length and magnification
Hendrik Lensch, Summer 2007
Lens-makers Formula
Refractive Power
1
1  1
P = ( n′ − n )  −  =
 R1 R2  f
1

 m = diopters 


Biconvex
Pos. Meniscus
Plano concave
Plano-convex
Biconcave
Neg. meniscus
Convex = Converging
image: Kingslake 1992
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 4
Concave = Diverging
image: Smith 2000
Hendrik Lensch, Summer 2007
Convex and Concave Lenses
Focal length and field of view
positive vs. negative focal length
Changing the magnification lets us move back from a
subject, while maintaining its size on the image
Moving back changes perspective relationships
From (a) to (c), we’ve moved back from the subject
and employed lenses with longer focal lengths
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Field of View
Field of View
images: London and Upton
Computational Photography
Hendrik Lensch, Summer 2007
images: London and Upton
Computational Photography
Effects of image format
Smaller formats have...
fov
filmsize
tan
=
2
2f
shorter focal length for same field of view, as
we’ve seen
smaller aperture size for same f-number
lighter, smaller lens for same design
Types of lenses
Film camera
36mm x 24mm filmsize
50mm focal length = 40º field of view
leads to larger depth of field
enables use of bulkier designs
Beware: diffraction does not scale down!
Digital camera
Hendrik Lensch, Summer 2007
Effects of image format
Field of view
image: Kingslake 1992
Hendrik Lensch, Summer 2007
smaller apertures suffer more from diffraction
field of view is 2/3 of film for given focal length
images: dpreview.com
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 5
Hendrik Lensch, Summer 2007
Aperture: Stops and Pupils
Aperture
Irradiance on sensor is proportional to
square of aperture diameter A
inverse square of sensor distance (~ focal length)
Aperture N therefore specified relative to focal length
f
A
numbers like “f/1.4” – for 50mm lens, aperture is
~35mm
exposure proportional to square of F-number, and
independent of actual focal length of lens!
Doubling series is traditional for exposure
therefore the familiar (rounded) sqrt(2) series
1.4,
2.0, 2.8, 4.0, 5.6, 8.0, 11, 16, 22,
32,Lensch,
… Summer 2007
Computational
Photography
Hendrik
N =
• Principal effect: changes exposure
• Side effect: depth of field
Computational Photography
Hendrik Lensch, Summer 2007
How low can N be?
Depth of Field
Canon EOS 50mm f/1.0
(discontinued)
Principal planes are the paraxial approximation of a
spherical “equivalent refracting surface”
N =
1
2 sin θ '
Lowest N (in air) is f/0.5
Lowest N in SLR lenses is f/1.0
images: London and Upton
image: Kingslake 1992
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Depth of focus
Depth of Field
(in image space)
(in object space)
tolerance for placing the focus plane
the range of depths where the object will be in focus
C’ - circle of confusion
Note that distance from (in-focus) film plane
to front versus back of depth of focus differ
www.cambridgeincolour.com
image: Kingslake 1992
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 6
Hendrik Lensch, Summer 2007
Depth of field
Numerical Aperture
(in object space)
total depth of field (i.e. both sides of in-focus plane)
Dtot =
NA = n sin θ
2 N CU2
f2
where
The size of the finest detail that can be resolved
is proportional to λ/NA.
larger numerical aperture resolve finer detail
(from Goldberg)
N = F-number of lens
C = size of circle of confusion (on image)
U = distance to focused plane (in object space)
f = focal length of lens
hyperfocal distance
back focal depth becomes infinite when U = f 2 / C N
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Numerical Aperture vs. F-Number
f /# ≈
f /# w =
Examples
1
2 NA
Dtot =
1
≈ (1 − m) f /#
2 NA
2 N CU2
f2
N = f/4, C = 8µ,
U = 1m, f = 50mm
working f-number:
Hendrik Lensch, Summer 2007
Dtot = 13mm
f /# w
N = f/16, C = 8µ, U = 9mm, f = 65mm
distance-related magnification: m
relevant for systems with high magnification
(microscopes or marco lenses)
Canon MP-E at 5:1 (macro lens)
use N’ = (1+M)N at short distances (M=5 here)
Dtot = 0.05mm !
image: Charles Chien
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Tilt and Shift Lens
Diffraction Limit
Lens shift simply moves the optical axis with regard to
the film.
Diameter d of 70% radius of the Airy disc
d = 1.22λ
change of perspective (sheared perspective)
f
a
Tilt allows for applying Scheimpflug principle
all points on a tilted plane in focus
single spot
barely resolved
no longer resolved
image: wikipedia
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 7
Hendrik Lensch, Summer 2007
Camera Exposure
Aperture vs Shutter
H = E ×T
Exposure overdetermined
Aperture: f-stop - 1 stop doubles H
Interaction with depth of field
Shutter: Doubling the effective time doubles H
Interaction with motion blur
f/16
1/8s
f/4
1/125s
f/2
1/500s
images: London and Upton
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Describing sharpness
Describing sharpness
Modulation transfer function (MTF)
Hendrik Lensch, Summer 2007
Modulus of Fourier transform of PSF
image: Smith 2000
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Lens Aberrations
Chromatic Aberration
Spherical aberration
Index of refraction varies with wavelength
Coma
For convex lens, blue focal length is shorter
Astigmatism
Can correct using a two-element “achromatic doublet”,
with a different glass (different n’) for the second lens
Curvature of field
Distortion
Achromatic doublets only correct at two wavelengths…
Why don’t humans see chromatic aberration?
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 8
Hendrik Lensch, Summer 2007
Chromatic aberrations
Chromatic aberrations
Longitudinal chromatic aberration
(change in focus with wavelength)
Lateral color (change in magnification with wavelength)
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
Spherical Aberration
Oblique Aberrations
Focus varies with position on lens.
Spherical and chromatic aberrations occur on the lens
axis. They appear everywhere on image.
images: Forsyth&Ponce
and Hecht 1987
Oblique aberrations do not appear in center of field and
get worse with increasing distance from axis.
• Depends on shape of lens
• Can correct using an aspherical lens
• Can correct for this and chromatic aberration by combining
with a concave lens of a different n’
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Aberrations
Astigmatism
Coma
The shape of the lens for an of center point might look
distorted, e.g. elliptical
off-axis will focus to different locations
depending on lens region
(magnification varies with ray height)
different focus for tangential and sagittal rays
images: Smith 2000
and Hecht 1987
Computational Photography
Hendrik Lensch, Summer 2007
image: Smith 2000
Computational Photography
Page 9
Hardy&Perrin
Hendrik Lensch, Summer 2007
Astigmatic Lenses
Curvature of Field
focus “plane” is actually curved
Image
Object
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Distortion
Geometric distortion
Ratios of lengths are no longer preserved.
Change in magnification with image position
Object
Image
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Hendrik Lensch, Summer 2007
Flare
Artifacts and contrast reduction caused by stray
reflections
image: Curless notes
image: Kingslake
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Page 10
Hendrik Lensch, Summer 2007
Flare
Ghost Images
Artifacts and contrast reduction caused by stray
reflections
Minimize artifacts, maximize
flexibility
Artifacts
Can be reduced by antireflection coating (now universal)
Spherical Aberration
Chromatic Aberration
Distortions
Lens Flare
image: Kingslake 1992
images: Curless notes
Computational Photography
Hendrik Lensch, Summer 2007
Computational Photography
Ghost Images
Hendrik Lensch, Summer 2007
Vignetting – your lens is basically a long tube.
Cos^4 falloff.
image: Kingslake 1992
Computational Photography
Hendrik Lensch, Summer 2007
At an angle, area of aperture reduced by cos(a)
1/r^2: Falls off as 1/cos(a)^2 (due to increased
distance to lens)
Light falls on film plane at an angle, another
cos(a) reduction.
Computational Photography
Real lens designs
Real lens designs
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
Hendrik Lensch, Summer 2007
image: Smith 2000
Computational Photography
Page 11
Hendrik Lensch, Summer 2007
Real lens designs
Real lens designs
image: Smith 2000
Computational Photography
Hendrik Lensch, Summer 2007
image: Kingslake 1992
Computational Photography
Bibliography
Hecht, Optics. 2nd edition, Addison-Wesley, 1987.
Smith, W. J. Modern Optical Engineering. McGraw-Hill,
2000.
Kingslake, R. A History of the Photographic Lens.