Myhrvold and I made claims and
provided supportive links, and I suspect that both of us are reasonably happy
with what we wrote. However, I have
noticed that debates prompted by the essay and the appended comments are still continuing
in the forums. There is still some
confusion and misunderstanding about what was said and not said. Accordingly, I would like to give a brief
summary of our debate and to report a simple experiment.
1. My essay on equivalent images stands
alone. Assumptions were made and the
consequences were discussed. It was all
classic optics presented for the situation where photographers encounter a
variety of sensor sizes. Pixel size
(pixel pitch) was not considered other that to assume that pixels were too small
to limit resolution. Myhrvold did not
attack the article. In fact he made kind
comments about it.
2. Myhrvold wrote to provide another
perspective. Instead of selecting a
circle of confusion (CoC) to control both the depth-of-field (DoF) and the
diffraction broadening, he was concerned with a choice of CoC that would permit
the maximum resolution that his sensor could provide. That is to say, he wanted to obtain the
maximum resolution in the plane of focus rather than to control the DoF. His conclusion was that the diffraction
broadening of the image could be sufficiently limited by selecting an f-stop
about equal to the pixel size in microns.
He also expressed the opinion that pixel sizes much smaller than the
diffraction spot diameter would be a waste of sensor capabilities.
I was reasonably happy with
Myhrvoldâs comments; and, considering the practical problems associated with
experimental tests of the determinants of resolution, I could/should have let
it stand. After all, I do agree with the
arguments and illustrations in the cited links including the cambridgeincolor web
site. Instead, I raised an esoteric
point. Basically, for a monochrome sensor if the pixel size is smaller than the
diameter of the diffraction spot, the resolution might be better than if the
pixel size matches the diffraction spot size.
This point may only matter to astronomers who photograph stars and must
select optimum pixel sizes for their sensors.
If the image spot is broadened for any reason to the size of a pixel, then
with multiple exposures each image acquired will be probably be slightly different
because of small shifts in position and one time the star image will fit in one
pixel, the next time the image will overlap parts of two pixels, etc.
Of course, even if the star image fits in one pixel, the image will be
square, not circular. The argument is a
little different for color sensors with color filter arrays.
I still think that, in principle,
oversampling can help. With my limited
knowledge of image processing, I believe that the same resolution enhancement effect
can be achieved by combining serial images.
Myhrvold suggested the use of super-resolution methods based on
combining serial images, so there may not be much disagreement. The web link for matching pixel size to the
capabilities of optical instruments is given below, and the link to
super-resolution was previously given by Myhrvold.
So where does this leave real
world (terrestrial) photographers? If
ones wishes to adjust DoF for a print, the ideas in my essay can be used as a
starting point. If one wishes to obtain the
ultimate resolution from their lens and sensor, then trial and error will
probably be required. As Myhrvold and I
have said repeatedly, our analyses assume diffraction limited lenses. Your lenses and especially your zoom lenses
are unlikely to be diffraction limited.
Therefore, diffraction will be only one of the contributors to the
broadening function. The lower the
quality of the lens, the more difficult it will be to isolate the contribution
of diffraction. I am reminded of the
early box cameras where aberrations were so severe that there was no definite
plane of focus, and over a wide range of distances objects were in âpretty
goodâ focus.
Anyway, I have performed a simple
experiment to see what difference pixel size makes in the presence of
diffraction broadening with typical equipment.
I tested a Canon 10D (pixel size = 7.4 microns) and a Rebel XTi (pixel
size = 5.7 microns), each with the same Sigma 105 mm macro lens. The cameras were mounted in turn on a solid
tripod about 10 ft from a resolution chart (see link below) that contained a
series of vertical black lines with spacings that decreased continuously from
left to right. The average distance
between centers of the lines was measured for each 10% of the range. When projected on the sensor the line
spacings were found to be 13.9, 15.2, 17.3, 20.0, 23.3, and 28.6 microns for
the groups of lines starting from the right end. The experiment consisted of photographing the
chart with f-stops ranging from 4 to 32 with mirror lockup selected.
With the 10D, the resolution was
found to be virtually identical from f/5.6 to f/11, and stopping down f/16
showed only a small decrease in resolution.
In particular, the resolution of 17.3 and 20.3 micron spacings was lost
at f/16 and f/22, respectively. In no case
was there any resolution of 13.9 micron spacings, and contrast was close to
zero for 15.2 micron spacings. Resolution loss means that the contrast in
that part of the pattern vanishes. With
the XTi, maximum resolution was obtained from f/5.6 to f/8.0, and there was an almost
imperceptible decrease at f/11. Amazingly,
at f/16 there was still resolution, albeit with low contrast, of 13.9 micron
separations. All the way to f/22 the XTi
clearly showed more resolution than the 10D, but at f/32 the patterns were
fuzzy and essentially identical for both cameras. Resolution was completely lost for any
spacings below 23.5 microns at f/32.
It is interesting to note that at
f/16 the XTi could resolve lines at spacings of 17.3 microns while the 10D
could not. Similarly, the XTi could
resolve 20.0 micron spacings at f/22 with much better contrast than the 10D. The diameters of the diffraction spots are
21.5 and 25.9 microns at f/16 and f/22, respectively. So even with diffraction broadening (spot
size) four times larger that the pixel size, we see that smaller pixels still
give an advantage.
What can we conclude? Resolution is limited by lens aberrations,
digital sampling (digital resolution), and diffraction broadening. At the larger apertures (small f-stops), aberrations
become important, and resolution improves as the lens is stopped down. This was not mentioned above, but resolution
does improve with both cameras in going from f/4 to f/5.6. At smaller apertures, digital resolution and
diffraction both contribute with diffraction dominating at the smallest
apertures (large f-stops). The best
resolution for the Sigma macro lens is found at f-stops close to those
predicted by Myhrvold, though the resolution is roughly constant over a
considerable range of apertures. When diffraction
broadening is evident, oversampling can help; but when the diffraction spot
size reaches six or seven times the pixel size, the pixel size no longer
influences the resolution.
I end with a note of
caution. You camera may have a different
anti-aliasing filter or even no filter.
Also, your lenses are sure to have a differ levels of aberration. However, there is no way to escape the
resolution destroying effect of diffraction at large f-numbers. Users of DSLRâs should think twice before
shooting at f/32 and should be aware of perhaps unacceptable resolution loss
even at f/16.
Brad D. Wallis, âSome notes on
matter of matching CCD camera pixel size to the capabilities of an instrument.â
(Especially, Figure 3)
http://geogdata.csun.edu/~voltaire/pixel.html
Resolution Chart:
http://math.berkeley.edu/~ilya/software/tmp/KM_A200-resolution-chart-ACRraw-quadratic-58percent-quartic-60percent.jpg
©2007 Charles Sidney Johnson, Jr.
