What is the optimal aperture for camera scanning?

Theoretical considerations

The background for this article is that I had some detailed mathematical discussions with Jeff Conrad, who also sent me some nice graphs. (Jeff is the author of Depth of Field in Depth and Optimal Aperture: Balancing Defocus and Diffraction (not on the web)).

The background material is very theoretical, but the results are interesting and illustrative since they show what theory predicts in nice graphs. However, you always need carry out practical testing to see what is best with your camera and your lens and your (bulging) slides that you intend to copy with your camera scan setup.

The following graphs show the theoretical results of the combined effect of diffraction and depth-of-field at two distances:

1. At the Plane of Focus — PoF.
2. At the planes of the Depth of Field — DoF — as a result of defocus. The total focus spread at the sensor plane is ∆v as indicated in the graphs. (Focus spread is the difference between the near and far image distances, see Depth of Field in Depth for explanation.)

The theory assumes a perfect lens with no aberrations, and no considerations to the limitations of sampling (i.e assuming a perfect anti-alias filter according to the Nyquist theorem, or assuming it is an analog recording). The object size, i.e the film we shall take a picture of, is 24x36 mm. The focal length is irrelevant.

For both of the following two graphs, the total focus spread ∆v at the sensor, 0.164 mm for APS-C and 0.368 mm for FF, corresponds to 0.368 mm at the film we are photographing. In other words, 0.368 mm is the total depth of field we are studying (0.368 mm is the difference between the near and far object distances). That value was chosen quite arbitrarily based on that it results in a Circle of Confusion—CoC, the blur spot size—of 2 pixels on a 20 Mpx sensor if we use f/5.6 on APS-C or f/7 on FF. This is arbitrary, but is chosen for the purpose of comparing APS-C and full frame.

The APS-C that is considered in these graphs has a crop factor (or format factor or focal length multiplier) of 1.5, which is what Nikon use, although I use Canon which use a crop factor of 1.6. But since this is only a discussion of the principles involved, it is not an important difference. Crop factor 1.5 is simply easier to use in calculations and it results in nice even numbers for the sensor size, namely 16x24 mm.

The y-axis shows the MTF, i.e the modulation transfer function that indicates how much contrast there is: 1.0 is perfect contrast, 0.0 is no contrast at all (gray). The 'm' in the graphs is the magnification, which is 1.0 for FF and 0.667 for APS-C. The f-number is the aperture (f-stop).

The interesting and illustrative thing that these graphs show is that there is an optimal aperture! This optimal aperture appears to be about f/5.6 for APS-C and f/7 on FF. At those apertures, the depth of field is exactly the same for APS-C and FF, namely 0.368 mm when the CoC is 2 pixels on a 20 Mpx sensor. One can show that, at these magnifications, the DoF for APS-C and FF are exactly the same if FF uses an f-number that is 1.25 times the f-number for APS-C.

The highest spatial frequency in these graphs, the yellow ones, corresponds to one cycle being about 2.4 pixels on a 20 Mpx sensor, which is a very high frequency. The Nyquist frequency (the ultimate theoretical limit) is one cycle being 2 pixels, and no sensor can resolve that because it would require infinite ideal filters. (If you see any pattern there on a test chart it is guaranteed to be only alias, i.e spurious.) In practice, you cannot get as much modulation as these graphs show at the indicated frequencies, but at the frequency of the yellow line you should just barely be able to see at least some weak resolution with a 20 Mpx sensor. More megapixels would be better since we would come further away from the theoretical Nyquist limit, but you would also have to accept that the circle of confusion for DoF would have to be more than 2 pixels.

One can argue that the DoF is too narrow. Let's see what happens if we increase the DoF so that the CoC becomes 2.5 or 3 pixels on a 20 Mpx sensor when using the same aperture as above, i.e f/5.6 on APS-C. And let's study at different frequencies so that the highest frequency corresponds to the theoretical limit of one cycle being 2 pixels. This is shown below, and the graph is expanded so that it is easier to read the f-numbers on the x-axis.

The graph for ∆V = 0.205 mm corresponds to 0.460 mm DoF, i.e the depth at the slide/film.
The graph for ∆V = 0.245 mm corresponds to 0.552 mm DoF, i.e the depth at the slide/film.

As ∆v increases, i.e targeting a wider DoF, the optimal f-number increases, but the maximum achievable MTF decreases.

The highest spatial frequency in these graphs, the yellow ones, corresponds to one cycle being only 2 pixels on a 20 Mpx sensor, which is the theoretical Nyquist limit, which no real 20 Mpx sensor can resolve. The gray, red and blue lines correspond to one cycle being 2.5, 3 and 4 pixels respectively on a 20 Mpx sensor, which are all sensible frequencies to study for any sensor with at least 20 Mpx.

In summary: for all these graphs, the theoretically optimal aperture for APS-C 1.5 crop sensor should be somewhere between f/5 and f/7, most probably f/5.6 or f/6.3. The more the film bulges, the higher is the optimal f-number. For FF, the optimal f-number should be 1.25 times higher, i.e somewhere between about f/6.3 and f/9.

Optimal aperture in Practice:

To find out the optimal aperture in practice, I made detailed tests with my equipment, which is an oldish 18 Mpx Canon EOS 7D (crop factor 1.6) and an EF-S 60/2.8 USM Macro. The optimal aperture is not only dependent on how much the film bulges, it also depends on how accurate the slide is positioned, i.e how parallell it is to the sensor. It also depends on how much the lens has to be stopped down to get good resolution in the corners, and it also depends on how much field curvature the lens has.

I used various glassless mounted slides: Kodachrome in Kodak paper frames, Kodak plastic frames, and manually mounted Gepe frames with metal masks; Fuji Sensia in plastic machine mounted frames, and manually mounted Gepe frames with metal masks.

I found that bulging slides should always be inserted so that they bulge outward from the camera. The reason is of course that the lens is not perfectly flat—there is some field curvature, and since we are dealing with extremely short DoF, it is critical. In addition, the left hand side is not as good as the right hand side (which I knew from before).

Normally, the emulsion side is concave, but some slides (most notably some in manually mounted frames with metal masks) could actually bulge in the opposite direction, or be doubly bulging. Slides that are scanned mirrored will of course have to be flipped in the program (such as Lightroom).

Scrutinizing the entire image, middle, sides and corners, I found that the optimal sharpness was obtained at f/6.3 (provided that the slide bulges outward from the camera).

If you use a camera with many more pixels in the hope of getting higher resolution, I am afraid you will have to place the film between absolutely clean glass or plastic plates. But with such plates, you increase the number of transitions between air and materials with a different refractive index, and you have many sides to clean from dust, and you risk newton rings. The optimal way is to use a wet method with a suitable solution that drives away the air between the surfaces, and if the solution has a refractive index close to 1.5, alla scratches will disappear, and transparent dust virtually disappears. For me, however, I guess the method is too cumbersome to be worthwhile to use. I haven't used it myself, but a friend of mine got fantastic results.

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