It all started when I shot the sunset panorama below from one of the locations where I work. It shows a horizontal field of view comparable to a 60mm lens on a 35mm film camera. There is nothing exceptional about this panorama except maybe that it shows -- on the righthand side -- the Zugspitze which happens to be Germany's highest mountain.
The Zugspitze is 2962 m (or 9718 Ft) high and from the point where the photo was taken, exactly 83.39 km (or 52 miles) away. So, I thought it may be a good idea to crop into that image and look what guys are doing up there on mountain top ;)
Theoretically, such a crop should be feasible because the image actually was stitched from several images taken with the FA* 300mm f/4.5 lens on the Pentax K-7 each. So, a pixel corresponds to as little as 1.4 m up there on mountain top and this excellent lens clearly outresolves the sensor.
Below is what the crop (100% pixel level and corresponding to a 2000mm lens on 35mm film) looks like:
Really looks blurry as if it were out of focus or having motion blur. Actually, razor-sharp trees in 300 m distance point out that this may play a role as well. Because 300 m isn't within hyperfocal distance for 5µm at f/8. But as we'll see, we can actually ignore this little detail.
Lesson #1: Focus on something about 1-2 miles away (i.e., which
is within hyperfocal distance for a 5µm circle of confusion) because infinity may be too blurry to focus at and more nearby objects may be, well too nearby.
As it turns out, the
real problem actually is that large distance objects look blurry indeed (no wonder the auto focus didn't properly lock on them).
So, I ask myself the question how sharp to expect an object at a given distance to appear?
There is scientific literature about this but I couldn't find anything accessible to photographers. So, I decided to compile a little How To guide. Starting now ...
1. Possible optical resolutions for long range tele photographs taken in the atmosphereOf course, photos taken outside the atmosphere aren't the most important category for most people reading this ;)
Another category are photos taken from the atmosphere into outer space (astro photography) and a common figure one is finding is to expect resolutions of up to 1 arcsec resolution but no better. In nights with very low atmospheric turbulence aka as excellent "Seeing". When the stars blink less than they usually do ;) This is related to the Fried parameter
r0 which is about 5cm (sea level) to 20cm (in the mountains at a very good night) large. It isn't possible to achieve better resolution than with a diffraction-limited lens with diameter
r0.
A lens with diameter 300mm/8 or 38mm (<
r0) isn't limited by atmospheric turbulences. However, the turbulences vary at a time scale of
t0 = 0.3 r0 / v_wind and with typical values of
v_wind = 2 m/s (10 m above ground) we obtain
t0 ~ 1/125s.
For anything slower than
t0, we effectively smear out the turbulent perturbations and decrease the resolution.
Lesson #2: Shoot at 1/125s or less, even when on a tripod :)
Of course, I wasn't aware of this and used 1/25s. But as we shall see, this isn't a big problem either. Because for excellent results, we need extremely low noise (lower than at ISO 100) and will need a long effective exposure time.
One way would be to adaptively restore turbulent distortion using a parameterized grid and stacking many restored image frames. Which is nothing but applying adaptive optics.
Another and more practically feasible way is to accept the loss in resolution due to atmospheric turbulence. But how large is it?
Well, I managed to find a formula in the scientific literature and adapt it for an optical path with constant atmospheric conditions:
MTF_turbulence(f) = exp (-21.57 f^(5/3) lambda^(-1/3) Cn^2 L)
where:
MTF: Atmospheric modular transfer function
for turbulent distortions along a horizontal path [%].
f: angular spatial frequency [cycles/rad].
Cn^2: (refractive-index structure coefficient),
typically between 10^-15 and 10^-13 [m^(-2/3)].
lambda: wavelength e.g. 0.55 [µm].
L: pathlength e.g. 83.39 [km].
Source:
R. E. Hufnagel and N. R. Stanley, "Modulation transfer function through turbulent media", J. Opt. Soc. Am. 54, 52–61 (1964).A public online source (i.e., free of charge) discussing this formula is
I. Dror and N. S. Kopeika, "Experimental comparison of turbulence modulation transfer function and aerosol modulation transfer function through the open atmosphere", (1995).
Be
f = L / (2 x) where
x is the size of the smallest resolved detail. Then I derive that the limiting resolving power (i.e. where MTF drops to 5.0%) is reached where
x = (L / L1)^1.6
where L1 = (1.633 * (Cn^2)^0.6 * lambda^-0.2)^(-5/8) is the distance where a 1m-sized detail can be resolved. Typical values are:
L1 = 50,000 [m^0.375] for weak turbulences (good seeing),
L1 = 20,000 [m^0.375] for normal turbulences,
L1 = 10,000 [m^0.375] for strong turbulences.
This formula as it stands is my own work and I hope it may be of good use for fellow photographers. The L1 values are slightly rounded (~10%) from the results using the typically rounded values for Cn^2 as given above. However, my table below uses
L1 values as computed from the rounded
Cn^2 values.
MTF jumps from 5% to about 40% for details twice as large (
2x) and drops to below 0.01% for details half the size (
x/2). So, there really isn't a reason to use more than one pixel per detail
x and the maximum useful focal length can be computed from the above. The following table does so and assumes 5µm large pixels:
Distance [m] |
max. focal [mm] (5 µm pixel) |
Seeing: |
good
|
normal
|
bad
|
1 |
171.410 |
43.056 |
10.815 |
2 |
113.088 |
28.406 |
7.135 |
5 |
65.261 |
16.393 |
4.118 |
10 |
43.056 |
10.815 |
2.717 |
20 |
28.406 |
7.135 |
1.792 |
50 |
16.393 |
4.118 |
1.034 |
100 |
10.815 |
2.717 |
682 |
200 |
7.135 |
1.792 |
450 |
500 |
4.118 |
1.034 |
260 |
1.000 |
2.717 |
682 |
171 |
2.000 |
1.792 |
450 |
113 |
5.000 |
1.034 |
260 |
65 |
10.000 |
682 |
171 |
43 |
20.000 |
450 |
113 |
28 |
50.000 |
260 |
65 |
16 |
100.000 |
171 |
43 |
11 |
200.000 |
113 |
28 |
7 |
This means that you don't have to care about atmospheric turbulence if you shoot shorter than 200 m only (assuming your longest lens is 500mm).
In all other cases, turbulence may be of concern. Typically, it may not be useful to shoot more far than 1 mile away. Because you would be tempted to use your longer than 500mm which then resolves worse than 500mm.
Lesson #3: Don't shoot your 20+ Gigapixel panorama at a day with just "normal" atmospheric turbulences.
Lesson #4: Wildlife photographers wanting to resolve 1mm at a bad Seeing condition day (like in Africa) either approach to at least 100m or use 1000mm f/22 (
r0!) 1/250s (
t0!) which applying the Sunny 16 rule, means a tripod
and ISO 400...
2. Possible optical contrast for long range tele photographs taken in the atmosphereSo far, we looked at a loss of resolution due to atmospheric turbulences. While being the strongest enemy for astro photography (besides light pollution), it isn't for long range tele photographs. While crystal-clear days exist where it is possible to view 200 km far away (on a mountain), other days clearly exist where vision is limited to a few meters only (fog).
The normal is somewhere in between where aerosol particles (due to condensed water, smoke etc.) scatter light along its path thru the atmosphere and dramatically lower the MTF with distance. The effect is much more dependent on distance than on detail size which is why we tend to not even see the object at all. Nevertheless, if we see a distant object it may be at very low contrast only. Formulas exist for MTF due to atmospheric aerosol scattering. They only mean that the useful range of tele photo lenses is limited even more.
To make things more fun, turbulence and aerosol scattering counteract each other. Dry air normally means less aerosol scattering but more turbulences too due to the heat which dried the air in the first place.
Low contrast is of double concern. Because we may wish to reconstruct missing detail which is only possible for high signal to noise ratios.
3. Improving long range tele photographsWe will apply a three step procedure to improve our tele photographs. Note that this will only be applicable for static subjects, though.
Step 1: Improve the signal to noise ratioI took 16 images and selected the best 10 of them. Then, I used PhotoAcute to align and stack them into a "superresolved" image with a signal to noise ratio corresponding to ISO 10:
If you click onto the photo and select "
Original size", you'll see that the image is twice as large. But not sharp. PhotoAcute's superresolution technique actually works for images which are sharp in the first place. Here, we only used it to boost the signal to noise ratio. Parameters used are a Nikon D40 camera with Sigma 30mm/1.4 lens, a combination I found particlarly neutral, i.e., PhotoAcute doesn't try to deconvolve for lens aberrations too much ;)
Step 2: SharpeningThe next step is a restauration of image sharpness using a deconvolution technique. FocusMagic seems to deliver best results, even in the case here where the defocus' point spread function doesn't strictly apply.
The sharpness is clearly improved. I used a blur radius of 6 pixels (after scaling the image back to 50% size). And "Forensic" regularization, made feasible by the stacking in the prior step. There is a window reflecting the sun. And because being 83 km away, it should be a perfect point. The ring artefact is a sign that a different deconvolution kernel would have yielded better results.
Step 3: Contrast enhancementThe last step is boosting the contrast within the given area. All tone values are typically within just a small range and the first step is clipping. The remaining tone mapping may be done using a gamma correction and a dose of clarity.
This resulting image may not be the most beautiful image of the top of the Zugspitze mountain. Nevertheless, from 83 km away, it not only shows a radio emmitter pole which is 4 m wide at its base and 2 m wide at its middle portion. It even shows (in the background on the right side at a 45° angle) the steel cables of the Austrian side funicular. It doesn't resolve the individual cables. But imaging them at all from more than 50 miles away is ... well, interesting ;)
Lesson #5: Burst enough images to be able to boost contrast.
Lesson #6: Stock up on a bunch of capable post-processing tools.
I hope you enjoyed the read.
4. Links
- I. Dror and N. S. Kopeika, "Experimental comparison of turbulence modulation transfer function and aerosol modulation transfer function through the open atmosphere", (1995)
- PhotoAcute
- FocusMagic
