Understanding image sharpness part 1A:
Resolution and MTF curves in film and lenses
by Norman Koren

---

In this page we illustrate how MTF is used to characterize the performance of film and lenses.

Film

---

.

f_50a = sqrt( f₅₀²-2f₅₀ f_boost - f_boost²) so that f₅₀ (as entered in MTFcurve) is still the 50% MTF frequency.

[‰æ‘œ:MTF illustration for f50 = 40 lpmm, fboost = 13 (Fuji Velvia)]
This plot on the right is generated from Matlab by entering,
MTFcurve2 45 13

The striking feature of the response, shown in the red curve below the image, is the exaggerated contrast boundaries. Tones overshoot their low frequency (steady state) levels, and the result is plainly visible. The overall effect is that this image appears slightly sharper than the image above for Ektachrome E100VS or Fuji Provia 100F, even though it has slightly lower response at 100 line pairs/mm.

This effect is similar to the sharpen or unsharp mask functions in image editing programs.

There is a fairly elementary explanation of how this happens in film. It it particularly noticeable with black & white films developed in highly diluted (one-shot) developers. Visualize a contrast boundary between two areas, one side of which is heavily exposed (a highlight) and the other lightly exposed (a shadow). As the film is developed (particularly in the interval between between agitations), the developer in the heavily exposed area becomes depleted more rapidly than the developer in the lightly exposed area, slowing down the development. Some of the developer diffuses across the boundary, so that the developer in the heavily exposed area adjacent to the boundary is less depleted than in the rest of the region, hence develops more rapidly, and the developer in the lightly exposed area adjacent to the boundary is more depleted, hence develops more slowly. The result is an exaggerated contrast boundary, as illustrated above. This is known as the adjacency effect, and film/developer combinations that exhibit it are said to have high acutance. In Velvia, the diffusion probably takes place inside the film.

The adjacency effect increases the perceived image sharpness, though it can become irritatingly obvious if it's overdone. It boosts the the MTF 50% frequency, but it has relatively little effect on the 10% level, which is related to perceived line pairs per mm resolution. Developing technique affects MTF, particularly for B&W films, where there is a large choice of developers and agitation procedures. Development options are more limited for color films.

..

Film links and MTF data

---

Since manufacturers keep altering their websites, I'll present a guide to finding the data rather than the exact URL.

Kodak publishes MTF data for some color reversal and B&W negative films. Go to Kodak Professional Products or Kodak Consumer Products- Film and navigate from there. Searching for MTF on Kodak's main page also works well. Kodak Germany publishes MTF curves for Gold films, called Farbwelt in German. Ektachrome E200 has f₅₀ = 34 lp/mm. Kodak T-MAX 100 has f₅₀ = 125 lp/mm and T-MAX 400 has f₅₀ = 100 lp/mm with 20% boost. T-MAX P3200 has f₅₀ = 85 lp/mm. These MTF's are quite striking-- far beyond any color reversal film! Photodo's page on 35 mm, medium format, or large format? indicates that T-MAX 100 is indeed a remarkable film. 35 mm T-MAX 100 outperformed medium format Tri-X using very fine equipment. Photodo's articles are worthy of serious study. Kodak also uses a measurement called print grain index for prints.

Fujifilm publishes MTF data for all their films. Go to Consumer Film Product Line-up or Professional Film Product Line-up and navigate from there. Fujicolor NHGII ASA 800 and NPH 400 color negative films have f₅₀ = 45 lp/mm.

There is one troubling aspect to some manufacturer's MTF curves, for example Fujichrome Provia 100F. MTF should be 100% at very low frequencies, but it remains flat at 120% from 10 lp/mm down to 1 lp/mm. There are three possible reasons. (1) The adjacency effect extends to extremely low spatial frequencies. Doubtful. 1 lp/mm is awfully low. (2) The graphic arts department took liberties with the plot before sending it out for publication. This happened to me in the old days before I had access to computer tools. (3) The marketing department cheated. It happens.

Lenses

---

[‰æ‘œ:MTF curves for Canon 28-70mm f/2.8L from http://www.photodo.com]MTF for film is rather simple-- it's the same in all directions and uniform throughout the film surface. Not so lenses. MTF is a function of the distance from the image center, the aperture (f-stop), the spectrum of the light, the focal length (for zooms), and even the focusing distance (macro lenses are designed to maintain good MTF at close focus). Not only that, but there are two MTF's at each point: one along the radial (or sagittal) direction (pointing away from the image center) and one in the tangential direction (along a circle around the image center), at right angles to the radial direction. Because of this, the MTF curve cannot be displayed in the same way as for film.

The MTF curves on the right are from the Photodo (old site) test of the Canon 28-70mm f/2.8L lens. Photodo is an outstanding Internet lens test resource, with 458 lenses optically MTF tested as of June 2000. Photodo curently uses Imatest for its lens tests. In their page, Understanding the MTF graphs, numbers and grades, they state,

"The graphs show MTF in percent for the three line frequencies of 10 lp/mm, 20 lp/mm and 40 lp/mm, from the center of the image (shown at left) all the way to the corner (shown at right). The top two lines represent 10 lp/mm, the middle two lines 20 lp/mm and the bottom two lines 40 lp/mm. The solid lines represent sagittal MTF (lp/mm aligned like the spokes in a wheel). The broken lines represent tangential MTF (lp/mm arranged like the rim of a wheel, at right angles to sagittal lines). On the scale at the bottom 0 represents the center of the image (on axis), 3 represents 3 mm from the center, and 21 represents 21 mm from the center, or the very corner of a 35 mm film image. Separate graphs show results at f8 and full aperture. For zoom lenses, there are graphs for each measured focal length."

They state elsewhere that performance at 10 line pairs/mm is indicative of the lens contrast while 40 line pairs/mm is indicative of its sharpness.

Lens performance is typically limited by aberrations at large apertures and diffraction at small apertures. Aberrations depend on lens design and manufacturing quality; they differ markedly for different lenses. Diffraction is a fundamental physical effect; it depends on the aperture alone. A lens is sharpest between the two extremes, near its optimim aperture, which tends to be around f/8 or f/11 for the 35mm format. It is smaller (as low as f/4) for compact digital cameras and larger (f/11 to f/22) for large format cameras.

Unlike film, the MTF of lenses don't necessarily match the second order 1/(1+( f / f₅₀)²) equation. MTF rolloff can vary widely, depending on lens design and manufacturing tolerance. To accommodate this, MTFcurve has an input variable for the order of the lens's MTF rolloff, lord, which defaults to 2 if not entered or if entered as 0. The lens MTF equation becomes,

MTF_lens(f) = 1/(1+|f/f_lens|^lord) (lord defaults to 2 in MTFcurve if not entered)

F_lens is the frequency where lens MTF = 0.5 = 50%, corresponding to f₅₀ in film. We use lord = 2, which appears to be an adequate approximation for typical lenses, until better information is available. It's not easy to derive lord from Photodo's data because most of the curves are above MTF = 50%; you need to look at MTF below 30% to get a clear picture of the rolloff. Flens can be estimated by interpolating between curves if MTF at 40 line pairs/mm (MTF₄₀) is below 50%. Or it can be estimated from MTF₄₀ using the following table, which is based on the inverse of the second order equation, flens = 40/sqrt(1/MTF₄₀-1), where MTF₄₀ is expressed as a fraction rather than a percentage. Sqrt can be replaced by exponentiation to the 1/lord power ( flens = 40/(1/MTF₄₀-1)^(1/lord) ) when lord is not 2. .

Lens links

---

Diffraction

---

Lenses are sharpest between about two stops down from maximum aperture and the aperture where diffraction, an unavoidable consequence of physics, starts to dominate. For 35mm lenses, this is typically between f/5.6 (f/8 for slow zooms) and f/11. At large apertures, resolution is limited by aberrations (astigmatism, coma, etc.), which lens designers work valiantly to overcome. MTF wide open is almost always poorer than MTF at f/8.

Diffraction worsens as the lens is stopped down (the f-stop is increased). The equation for the Rayleigh diffraction limit, adapted from R. N. Clark's scanner detail page, is,

Rayleigh limit (line pairs per mm) = 1/(1.22 Nω)

N is the f-stop setting and ω = the wavelength of light in mm = 0.0005 mm for a typical daylight spectrum. (0.00055 mm is the wavelength of green light, where the eye is most sensitive, but 0.0005 mm may be more representative of daylight situations.) I've seen a simple rule of thumb, Rayleigh limit = 1600/N, which corresponds to ω = 0.000512 mm. The light circle formed by diffraction, known as the Airy disk, has a radius equal to1/(Rayleigh limit).

The MTF at the Rayleigh limit is about 9%. Significant Rayleigh limits are 149 lp/mm @ f/11, 102 lp/mm @ f/16, 74 lp/mm @ f/22, and 51 lp/mm @ f/32. Larry, an experienced lens designer, finds these numbers to be somewhat conservative because the Rayleigh limit is based on a spot, which has lower resolution than a band. His numbers of 125 lp/mm @ f/16 and 64 lp/mm @ f/32 are derived from a Kodak chart he contributed to Robert Monaghan's Lens Resolution Testing page. Most lenses are aberration-limited (relatively unaffected by diffraction) at f/8 and below. The OTF (optical transfer function) curve in David Jacobson's Lens Tutorial shows how MTF (the magnitude of OTF) varies with spatial frequency for a purely diffraction-limited lens at f/22.

We can derive some interesting relationships from David Jacobson's graph. At the Rayleigh diffraction limit of 68 lp/mm (for f/22, ω = 555 nm = 0.000555 mm), MTF is approximately 9%. It is 10% at about 64 lp/mm and 50% at 32 lp/mm. The following relationships therefore hold for diffraction-limited lenses:

f₁₀ = 0.77/(Nω) ; N = F-stop; ω = the wavelength of light in mm = 0.0005mm
f₅₀ = 0.38/(Nω) = 0.46*Rayleigh limit

The diffraction curve is somewhat difficult to express mathematically, but it can be approximated-- matched at the 10% and 50% MTF points-- by the equation used in the MTFcurve program ( f₅₀ is the same as flens; f₅₀ and lord are the third and fourth input arguments to MTFcurve),

MTF_{diffraction-ltd}(f) ~= 1/(1+| f/f₅₀|^lord) ( f₅₀ = 0.38/(Nω); lord = 3.17)

The question remains, at what f-stop does a lens become diffraction limited? The best estimate is the f-stop where the diffraction-limited f₅₀ (0.38/(N ω)) equals flens-- the 50% MTF frequency at the sharpest aperture. For the excellent 35mm lens, f_lens = 61 lp/mm. The f-stop with the same diffraction-limited f₅₀ is N = 0.38/(61*0.0005) = 12.5. We can therefore say with some confidence that good 35mm lenses are relatively unaffected by diffraction at f/8 and below, moderately affected at f/11, and diffraction-limited at f/16 and beyond. Such lenses should only be stopped down beyond f/11 (larger f-stops for larger formats) when extreme depth of field is required. Diffraction in digital cameras is discussed here.

Large format

---

[‰æ‘œ:Published data for Schneider Apo-Symmar 150mm f/5.6 at f/22]Thanks to Schneider's large format lens data, we can compare large format image quality to 35mm. The MTF curve on the left is for the 150mm f/5.6 Schneider Apo-Symmar lens, focused at infinity at f/22. It could be a little better than actual lens performance because it's derived from computer simulation. But it's almost certainly worse than optimum because the lens is diffraction-limited at f/22; it is almost certainly sharper at f/11 and f/16. Its image circle is u' = 110mm, more than sufficient to cover the 4x5 (inch) format with room for camera movements. The principal difference between this curve and the Photodo curves (above) is that the three lines (top to bottom) represent 5, 10, and 20 lines per mm (instead of 10, 20, and 40). This is a first class view camera lens; hence we'll refer to it as an "excellent 4x5 lens."

We can find the lens's 50% MTF value, flens, by modifying the above equation to flens = 20/sqrt(1/MTF₂₀-1), or we can also use the above table by substituting MTF₂₀ for MTF₄₀ and dividing flens by 2. Since MTF₂₀ ~= 66%, we estimate flens to be 28 line pairs per mm. This is just under half the value for the excellent 35mm lens and just slightly under the diffraction limit ( f₅₀ = 32 lp/mm for f/22). Even at optimum aperture (around f/11) a view camera lens is not likely to be as sharp as the excellent 35mm lens; it has to cover about 4 times the image circle (16 times the area). Next we run MTFcurve 45 13 28, (Velvia film + lens) and we find that the 50% and 10% MTF values of the film + lens are 27.1 and 49.9 line pairs/mm. Sharpness is almost entirely limited by the lens; the film hardly plays a role.

Assuming a 35mm frame is cropped for an 8x10 print, a 4x5 frame is 4 times larger. The ratio of total detail at the 50% level is 4*27.1/36.8 = 2.94. The ratio at the 10% level is 4*49.9/68.6 = 2.91. A 4x5 image can therefore resolve approximately three times the linear detail of 35mm, assuming both employ good technique: excellent lenses and film, optimum aperture, correct focus, sturdy camera support, good atmospheric conditions, etc. Since the passing of the Speed Graphic era, such good technique has been standard practice in large format photography; it's less common with 35mm. A 24x30 inch print from 4x5 would have the same detail as an 8x10 from 35mm. It can be extremely sharp! This result is in substantial agreement with R. N. Clark's scanner detail page. If we are to believe Kodak's T-MAX 100 data, the ratio would be lower: 35mm images would be phenomenally sharp, limited only by the lens.

If you are interested in large format photography, Large Format Photography . Info, edited by Quang-Tuan Luong and Bjorn Nilsson, is an outstanding resource.