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Depth of Field and Pinhole Cameras:
The Depth of Field (DOF) and depth of focus (dof)1 are strictly geometric quantities,
outside of the region where the diffraction PSF dominates.2 As such, they are relevant
only to optical systems that are undersampled – that is, the Nyquist cutoff frequency of
the detector is significantly less than the diffraction limit of the optics. This is extremely
common for most digital imaging systems (with the exception of microscopes), since the
lens diameter is sized to allow sufficient light capture for convenient imaging and often
results in a diffraction limit far beyond the ability of the detector to register.
MTF
Diffraction-Limited
Optics
“Typical”
Camera
Optics
diffraction
limit
spatial
frequency
Nyquist
limit
For a pixel spacing of p , the Nyquist frequency is N F 
1
, where the diffraction
2p
D
1

, where D is the exit pupil diameter (essentially the lens
f  FN
diameter), f is the focal length, and FN is the F-number. Typical values for a digital
camera would be:
limit is d F 
p = 10m  NF = 50 lp/mm
 = 0.55m, FN = 2.8  dF = 650 lp/mm
This is the reason that many camera lenses seen in the literature fail to reach the
diffraction limit – not only is there no need, too much resolution (beyond the detector
limit) can result in aliasing with digital detectors. (This wasn’t a problem with film.)
Aliasing is when a discretely-sampled frequency, that is sampled too rarely is detected as
indistinguishable from a lower frequency:
1
2
My own notation – not standard.
The exception to this is the pinhole camera.
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2
There is actually a penalty for using a lens with too much resolution with a digital
detector. Hence, modern “digital lenses” are usually lower resolution than older lenses
for film camera, where excess resolution didn’t have any ill effects. Often, higher quality
film camera lenses did approach the diffraction limit, as the designer could not know
what resolution film the user would choose.
Aliasing considerations affect the wardrobe choice in movies. In the digitized version of
Hitchcock’s “To Catch a Thief”, Cary Grant wears a striped shirt with a pattern that is
often (at sufficient camera distances) beyond the Nyquist limit of the digitizer. The result
is that low-frequency patterns race back and forth across the shirt, distracting the viewer.
This wasn’t a problem with the film version, as any frequencies too high were simply not
recorded.
Time-aliasing in movies (wagon wheels moving “backwards”, etc.) are a separate issue
and affect digital and film equally.
Typical Camera Lens from a famous designer:
MTF plotted to the diffraction limit:
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MTF plotted to a reasonable detector limit (for 10µm pixels):
Microscope objectives are challenging to design, because microscopists usually size the
detector and camera magnification to detect at (critically sampled) or beyond (oversampled) the diffraction limit.
Question: What possible value is there in sampling beyond the diffraction limit?
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Geometric DOF:
For an undersampled system, the DOF and dof are strictly a function of the geometry of
the optical system and the pixel size (or, alternatively, the “acceptable blur diameter”) of
the detector. The following illustrates the determination of the geometric dof and DOF:
DOF
dof
u
u’
b
B
The drawing is not to scale – both the DOF and dof would be much smaller than shown
in a practical system. Given the approximation that the magnification and conjugate
relationship are approximately constant over the DOF and dof (that is, the DOF is
imaged onto the dof), we have (assuming absolute values of all quantities):
B
b
b u
, u 
, M 
DOF
dof
B u
b
B
1 b dof
hence : DOF   M  2   2
u M u M u M
u
Also, in the approximation where sin  tan, we have:
b
b
dof  
 2b FNO
u  NA
Hyperfocal Distance:
The hyperfocal distance is defined as the distance to focus a camera such that objects at
infinity will still be “acceptably” in focus (i.e., the PSF will not be larger than the
acceptable blur diameter). The following drawing illustrates an optical system focused
at the hyperfocal distance:
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l’
dZ
b
F
Objects at  focus at F, with the maximum blur diameter, b. The lens is focused at the
conjugate of l’ = F+dZ/2, and the minimum in focus distance is the conjugate of l’+dZ.
Hence, the hyperfocal distance (point of focus) is:
1
1 1
 
l F
F  z
2
2F 2
F2
 l 
F 
F
z
b   f #
and the near conjugate is:
1
1 1
 
F  z l F
F2
F2
 l 
F 
F
z
2b   f #
where f# is the focal length divided by the aperture.
(Note that Wikipedia gets this formula wrong.)
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Pinhole Camera:
A pinhole camera is a camera where the lens has been replaced by a small aperture.
The aperture sorts incoming rays such that the rays that reach each part of the detector
come from “almost” one direction, thus forming an image. Pinhole cameras have a
virtually unlimited FOV, and can be used to make pictures that would otherwise be
impossible:
A pinhole camera setting on a piano keyboard
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Try making this image with a lensed camera and you get:
There is no way to get all the keys in focus.
Pinhole cameras can make all kinds of novelty pictures. Here is what a pinhole camera
setting on a vinyl record sees as the record plays:
Perhaps this should be titled “Pictures are Strange…”
The questions arise: What is the resolution of a pinhole camera? What is the optimum
size of the pinhole? Does is vary depending on the “focal length”? The Air Force once
converted an empty hanger into a “pinhole” camera to make a publicity image of a WWII
base, before it was re-developed. The “film” was a huge sheet of canvas coated with 500
gallons of emulsion. The image was developed in specially built tanks on the floor of the
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hanger. Hundreds of pounds of black plastic and duct tape were used to light-proof the
hanger.
The “film”
The Image
If you were consulting on this project, could you specify the pinhole size and exposure?
The optimum pinhole size is not a simple task – Optics Infobase has several dozen
technical papers on calculating the imaging properties of a pinhole. We are going to use
a simple one.
The geometric PSF of a pinhole is just a spot the same size as the pinhole. There is
diffraction to take into account, however.
The diffraction pattern at  (the intensity of the Plane Wave Spectrum) of a uniformly
illuminated pinhole is:
 2 J ka sin(  
I   I 0  1
,
 ka sin(  ) 
where a is the pinhole radius and J1 is the first-order Bessel function:
2
4
6

x
1  x
1  x
1  x
J1 ( x)  1 


 
 
   
2  1! 2!  2 
2! 3!  2  3! 4!  2 

(Question: Where and how does this function arise?)
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This function is known as the “Airy Pattern” (not, “Airy Function”), and is the familiar
PSF for a circularly symmetric, uniformly illuminated optical system:
I() has a first zero at kasin() = 3.83, so the Airy Pattern has a 1st zero at:
ka sin(  )  3.83
3.83

  sin(  ) 
 0.61
2
a
a
F

to be the “radius” of the diffraction component of the pinhole,
a
and a to be the geometric radius, we can plot the two “radii” as a function of increasing
pinhole size:
If we consider 0.61
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We could guess that where the two were equal might be close to the optimum. This
would give
F
0.61
a
a
 a  0.78 F
or, D  2a  1.56 F
This value is somewhat “fuzzy” as it depends on the perception of sharpness, which is
hard to quantify. Values in the literature range from 1.5(F) (modern) to 1.9(F)
(Rayleigh).
As a check, we calculate the slopes of the geometric and diffraction spot sizes:
The slope of the geometric spot size is 1 everywhere. The slope of the diffraction spot
dr
0.61F
a2



 1
size, evaluated at r = a is:
da r a
a 2 r a
a2
So, the two functions are orthogonal at their meeting place, further strengthing the
assumption that the minimum spot size is there or nearby.
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