Understanding image sharpness part 2:
Resolution and MTF curves in scanners and sharpening
by Norman Koren

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Table of contents

for the
image sharpness
series

Part 1: Introduction
Part 1A: Film and lenses
Part 2: Scanners and sharpening
Scanners | Nyquist theorem and aliasing
Sharpening
Digital vs. conventional darkroom prints
Summary | Download MTFcurve
Equations | Zeiss statement on MTF
4000 vs. 8000 dpi scans
Part 3: Printers and prints
Part 4: Epson 1270 results
Part 5: Lens testing
Part 6: Depth of field and diffraction
Digital cameras vs. film, part 1 | part 2
Part 8: Grain and sharpness: comparisons
In this page we examine the MTF of scanners and sharpening algorithms, and we address the question, "What scanner resolution (pixels per inch or dpi) is required for a digital print to appear sharper than a conventional darkroom print?"
Green is for geeks. Do you get excited by an elegant equation? Were you passionate about your college math classes? Then you're probably a math geek a member of a maligned and misunderstood but highly elite fellowship. The text in green is for you. If you're normal or mathematically challenged, you may skip these sections. You'll never know what you missed.

Scanners

A scanner samples an image at a spatial resolution specified by pixels or dots per inch (dpi) and digitizes each sample to a certain number of bits (which defines tonal resolution), typically between 8 and 14. It outputs the image to a file with 24 or 48 bits per pixel (8 or 16 bits per color). Since good quality scanner sensors are capable of more than 8 bits per pixel, some detail is lost in 24 bit color files. 48 bit files retain all the tonal detail captured by the scanner. Not all scanner control and image editing programs work with 48 bit files; Photoshop and Picture Window Pro are two that do. You can learn more about film scanners— specifications and a product list— on the Scanners page.

Now, let's look at actual scanner resolution, starting with the dpi specification. We estimate scanner MTF using mathematical approximations that closely match available data.

A typical dedicated film scanner has a characteristic MTF that can be approximated by,

MTFscan( f ) = |sinc( f /dscan)|3
where   sinc(x) = sin(πx)/(πx)   if x is nonzero ;     sinc(0) = 1
|...| denotes absolute value.
Dscan is the scanner resolution in pixels per mm; samples are spaced 1/dscan mm apart. MTFscan( f ) is the magnitude of the transfer function at spatial frequency f. Since sinc(x) has nulls at x =1, 2, 3, ..., MTFscan(f/dscan) has nulls where they are expected, at f = dscan, 2*dscan, etc. You can visualize the null at f = dscan if you realize that there is one complete cycle of the signal, which averages to zero, over sampling length 1/dscan.

|Sinc( f /dscan)|3 is one of a class of functions of the form |sinc( f /dscan)|n that can be used to approximate MTF for scanners and digital camera sensors because they are simple, have nulls at the correct frequencies, and closely approximate measured data. |sinc( f /dscan)|n is shown in the graph on the upper right for n = 1 through 4. The corresponding spatial sensitivity functions are shown below. The spatial sensitivity function for sinc( f /dscan) is a rectangle of length 1/dscan (the length of a sensor element with a fill factor of 1). An "ideal" rectangular sensor with a 100% fill factor would have a sinc( f /dscan) response. For n > 1, the spatial sensitivity is the rectangle convolved with itself (n-1) times.  For sinc2 it is a triangle of length 2/dscan. For large n, both sincn and its corresponding spatial sensitivity function approach a gaussian curve (exp(-k*x2/2)).

Much of the data for my approximations comes from two papers from the Imaging Technology Research Group of the University of Westminster in the UK: "An Evaluation of MTF Determination Methods for 35mm Film Scanners," by S. Triantaphillidou, R. E. Jacobson, and R. Fagard-Jenkin, and "A Novel Approach to the Derivation of Expressions for Geometrical MTF in Sampled Systems", by R. B. Fagard-Jenkin, R. E. Jacobson, N. R. Axford, both published in PICS 1999: Image Processing, Image Quality, Image Capture, Systems Conference. In the latter paper, they discuss how sensor response is sensitive to signal phase. For signal peaks in the center of an ideal sensor, the MTF approximates |sinc( f /dscan)|. But when an average of possible phases is taken, the MTF curve is a good match to |Sinc( f /dscan)|2. Sinc( f /dscan)|3 is a better fit to dedicated film scanners because of the additional degradation from the scanner optical system (resolution, focus and diffraction) and movement. Don Williams, author of What is an MTF ... and Why Should You Care?, has authored papers on scanner MTF with data that fits the sinc3 approximation. Inexpensive flatbed scanners such as the Epson 2450 and the Canon D2400UF are not as sharp as dedicated film scanners; their response approximates sinc4.
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The Nyquist sampling theorem and aliasing
The Nyquist sampling theorem states that if a signal is sampled at a rate dscan and is strictly band-limited at a cutoff frequency  fC no higher than dscan/2, the original analog signal can be perfectly reconstructed. The frequency fN = dscan/2 is called the Nyquist frequency. For example, in a digital camera with 5 micron pixel spacing, dscan = 200 pixels per mm or 5080 pixels per inch. Nyquist frequency fN = 100 line pairs per mm or 2540 line pairs per inch.

The first sensor null (the frequency where a complete cycle of the signal covers one sample, hence must be zero regardless of phase) is twice the Nyquist frequency. The sensor's average response (the average of all sampling phases) at the Nyquist frequency can be quite large.

Signal energy above  fN is aliased— it appears as artificial low frequency signals in repetitive patterns, typically visible as Moiré patterns. In non-repetitive patterns aliasing appears as jagged diagonal lines— "the jaggies." Aliasing is visible in some of the small boxes in this article where bands of high spatial frequency interact with the low sampling rate of the monitor screen, roughly 80 pixels per inch.  The figure below illustrates how response above the Nyquist frequency leads to aliasing.
 

Example of aliasing
  Signal  (3fN /2)                                                
  Pixels   1     2     3     4     5     6     7     8  
  Response  (fN /2)                                                
In this simplified example, sensor pixels are shown as alternating white and cyan zones in the middle row. By definition, the Nyquist frequency is 1 cycle in 2 pixels. The signal (top row; 3 cycles in 4 pixels) is 3/2 the Nyquist frequency, but the sensor response (bottom row) is half the Nyquist frequency (1 cycle in 4 pixels)— the wrong frequency. It is aliased.

The sensor responds to signals above Nyquist— MTF is nonzero, but because of aliasing, it is not good response.

Many digital camera sensors have anti-aliasing or low-pass filters (the same thing) to reduce response above Nyquist. Anti-aliasing filters blur the image slightly; they reduce resolution. Kodak publishes MTF curves for a low cost anti-aliasing filter. Sharp cutoff filters don't exist in optics as they do in electronics, so some residual aliasing remains. Lens MTF losses also reduce aliasing. Analog reconstruction is not shown.

Aliasing from hell. The 14 megapixel Kodak DCS 14n, and its successors, the Pro N and Pro C, have no antialiasing filter. MTF response is outstanding, but remains high (0.31) at the Nyquist frequency. The 14n behaves very badly in the vicinity of Nyquist ( 0.5/(7.9 um) = 63 lp/mm. This image of my MTF test chart was supplied by Sergio Lovisolo. Some of the problem is due to the Bayer sensor: the effective horizontal and vertical Nyquist frequency for red and blue pixels is half that of green. (Diagonals are something else...) The Foveon sensor in the Sigma SD9 and SD10 cameras is sensitive to all three colors at each pixel site. It also has no anti-aliasing filter and high MTF at Nyquist, but aliasing is much less visible because it is monochrome, not color.

The Moiré visible above the Nyquist frequency is evidence that an excellent lens was used to make this image. A little tip: you can achieve outstanding resolution with little Moiré with the 14n or its successors by stopping down to around f/16, where diffraction acts as an effective anti-aliasing filter.

In all fairness, this aliasing is rarely objectionable in real images, though it could show up on fabrics. Kodak has some software solutions The sensor manufacturer, Fill Factory, has an interesting list of publications including an excellent paper on this sensor.

To accurately reconstruct the original analog signal using the Nyquist theorem, the samples must be convolved with the strongly oscillating sinc(x*dscan) function (sinc(x) = 1 for x = 0; sin(π x) / (π x) otherwise). This is routinely done in electronic signal processing, where high-order sharp cutoff filters are employed. The best known example is the CD player, with 44,100 samples per second; Nyquist frequency = 22.05 kHz. Response extends to 20 kHz. Unfortunately this can't be done so easily with optics.

How the Nyquist theorem actually applies to imaging belongs (in my view) to the realm of Car Talk's "Murky Research." As far as I can tell, the sinc(x*dscan) function is rarely, if ever, applied. The original signal is never strictly band-limited and there is no single means of analog reconstruction. (Printers differ from monitors, etc.) Since filters with sharp cutoff don't exist in optics, there is a severe tradeoff between resolution and aliasing. The original signal cannot be reconstructed with the same precision as electronic systems.

The MTF at the Nyquist frequency ( f = dscan/2) for the sincn( f /dscan) functions used to approximate the response of digital sensors is sincn(1/2) = (2/pi)n = {0.637, 0.405, 0.258, 0.164 } for n = {1, 2, 3, 4}, respectively. In imaging systems with losses from the film and lens, most aliasing is eliminated in scanners with n >= 3. The best flatbed scanners in Don Williams' paper had MTF = 0.3 at the Nyquist frequency— close to the response for n = 3. Many digital cameras have optical "anti-aliasing" or "low-pass" filters in front of their sensors. These filters spread the light, increasing the value of n in the sensor's response. The reduced aliasing comes at the expense of resolution.

Given the difficulty that optical systems have in reconstructing the original analog signal, the rule of thumb that the number of line pairs per mm a scanner can resolve is less than half its pixel (or dots) per mm resolution is pretty good. If the Nyquist theorem could be applied precisely (using sinc(x*dscan) reconstruction), it would be very close to half. But in reality you need more than two pixels to resolve one line pair of actual image detail (for example, a window screen). I've seen the number four used as a rough estimate, but that seems overly pessimistic in light of the results below. Three seems to be a pretty good number, especially when sharpening is applied. Judge for yourself.

Imatest is a program that lets you measure MTF and other factors that contribute to image quality in digital cameras and digitized film images. It is a valuable tool for learning about scanners and sharpening.
Imatest: affordable software for measuring sharpness and image quality

Scanner simulations

The curve on the right is for a 2400 pixel per inch scan (equivalent to the HP Photosmart S20) of Velvia film with the excellent 35mm lens, generated by,

MTFcurve2  45  13  61  2  2400/25.4 3

You'll notice a difference from the previous images. The bottom part is full contrast and slanted so we can see all sampling phases. If only one phase were shown, the result might be better or worse than average. Slanting the bands allows us to observe aliasing, the low frequency artifacts that appear when energy above the Nyquist frequency— half the sampling rate— is present. Aliasing between the bands in the small box and the video screen (roughly 75 pixels per inch) is quite apparent. The top of each slanted band has the same tone as the bottom of the next band.

The thin magenta curve is the spatial response of the film + lens prior to scanning. The kinky red curve is the spatial response of the scanned image. The thin blue line is the MTF of the film + lens. The black line is the MTF of the film + lens + scanner. The blue dashed line is the MTF of the scanner only (the |sinc|3 function).

The 50% and 10% points for MTF are now 27.9 and 44.6 lines/mm, down significantly (24% and 35%) from 36.8 and 68.6 lines/mm for the unscanned image.

The curve on the right is the result of a 4000 pixel per inch scan (equivalent to the Polaroid Sprintscan 4000, Nikon 4000 ED or CanoScan FS4000US) of Velvia film with the excellent 35mm lens.

MTFcurve2  45  13  61  2  4000/25.4 3

The 50% and 10% points for MTF are now 32.7 and 55.4 lines/mm, down somewhat less (11% and 19%) from 36.8 and 68.6 lines/mm for the unscanned image. Even though the 4000 dpi scanner has 67% more resolution than the 2400 dpi scanner, the actual scanner resolution is only about 17% to 24% higher. The image is sharper than the 2400 dpi scan, but the difference is not dramatic.

But we're not done yet. There's more to the tale!

Scanner links

R. N. Clark's scanner detail page is required reading for anyone interested in image sharpness. It presents much of the material covered here from a different viewpoint: real images.
How Many Pixels Are There In A Slide?  Ian Porteous' fascinating comparison of a very sharp slide scanned at 2438 dpi, 4000 dpi, and photographed through a microscope. He also has some very interesting lens comparisons (Canon 28-70L, 28-105 and primes).
The Research Libraries Group, a nonprofit coalition of universities, libraries, and archives, maintains some excellent pages. Selecting a Scanner by Don Williams is extremely informative. What is an MTF ... and Why Should You Care? by Don Williams is a valuable and quite technical introduction to MTF. To probe even deeper, look at the paper, Diagnostics for Digital Capture using MTF by Don Williams and Peter D. Burns.
I3A (the International Imaging Industry Association; formerly PIMA) establishes standards for electronic still picture image quality. The fascinating document, "film scanner spatial resolution measurements standard (ISO 16067-2)" is no longer available online.
Film grain and aliasing  P. L. Andrews' impassioned discourse on a much neglected subject that causes unexpected image quality degradation.
Electronic image sensors vs. film: beyond state-of-the-art by B. Dierickx—  A fascinating technical paper that compared idealized film and digital sensors.
Kodak has some interesting technical articles on image sensors for scanners and digital cameras. Their main image sensor page as links to product descriptions, specifications and application notes.
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Sharpening

Sharpening increases image contrast at boundaries. This enhances perceived sharpness and detail, although it doesn't increase actual information content. Sharpening can be overdone: too much results in edge artifacts ("halos") and excessive grain. But the right amount definitely enhances an image. Although sharpening can be performed during the scan or in the digital camera, it is usually best done late in the image editing process.

A simple sharpening algorithm subtracts a fraction of two neighboring pixels from each pixel, as illustrated on the right. The thin black curve is the input to the sharpening algorithm: it is the response of the lens + film to a sharp line (impulse response). The two thin dashed blue curves are replicas of the input multiplied by -ksharp/2 and shifted by ±1 pixel. The thin red curve the impulse response that results from sharpening— the sum of the black curve and the two blue curves, normalized to the same amplitude as the input. The thick black and red curves (shifted up and left) are the corresponding edge responses before and after sharpening.
 

The formula for the simple sharpening algorithm is,
Lsharp(x) = (L(x) - ksharp/2 * (L(x-V) + L(x+V))) / (1-ksharp)
L(x) is the input pixel level and Lsharp(x) is the sharpened pixel level. Ksharp is the sharpening constant (related to the slider setting scanning or editing program). V is the shift used for sharpening. V = R/dscan, where R is the sharpening radius (the number of pixels between original image and shifted replicas) in pixels. 1/dscan is the spacing between pixels.

The sharpening algorithm has its own MTF.

MTFsharp( f ) = (1-ksharp * cos(2 π f V ))/(1-ksharp)
This equation boosts response at high spatial frequencies. Maximum boost takes place where cos(2 π f V ) = cos(π) = -1, or f = 1/(2 V) = dscan/(2 R). This is equal to the Nyquist frequency,..fN = dscan/2, for R = 1 and lower for R > 1.  Actual sharpening is a two dimensional operation, where the formula for Dsharp can be expressed as a matrix. Click on this Photo.net thread for some additional details. The actual sharpening matrix is a bit more complex when R is greater than one: it includes additional terms for anti-aliasing.

An unpleasant side effect of sharpening is that it exaggerates grain and defects such as scratches. Unsharp mask is an operation that sharpens the image when the difference between adjacent pixels is greater than a specfied threshold. It is used to avoid exaggerating grain in smooth areas like skies (masks are also valuable for this function). The threshold must be set by the user, and involves an inevitable compromise between enhancing detail and boosting grain.

Sharpening simulations

The image on the right illustrates the results of sharpening a 2400 line per inch scan with ksharp = 0.44, generated by,

MTFcurve2  45 13  61  2  2400/25.4 3  .44
 

The thin magenta curve is the spatial response of the film + lens prior to scanning. The kinky red curve is the spatial response of the scanned + sharpened image. The thin blue line is the MTF of the film + lens. The black line is the MTF of the film + lens + scanner + sharpening. The dashed blue line is the MTF of the scanner + sharpening, for R = 1 (V = 1/2400 inch = 0.0106 mm = 10.6 microns).

More detail is visible in the sharpened image. This is particularly clear in the image sequence in the Summary below. Sharpening results in some overshoot at contrast boundaries. This isn't a problem as long as it's not overdone. It tends to disappear when the image is printed.

The MTF for sharpening (from the above equation) has maximum boost at half the scanner resolution ( fN= dscan/2 = 47.2 lp/mm). Because of the scanner's MTF rolloff, the total boost is pretty much gone by dscan/2 (50 lp/mm). In the frequency domain, sharpening works by boosting response attenuated by the film, lens, and scanner. A reasonable amount of response must remain for it to be effective. In a fuzzy image, there won't be much response left at dscan/2, but you can still sharpen (though not as much) by setting R (the number of pixels between the original image and the replica used for sharpening) larger than 1, which drops the frequency of maximum boost.


The image on the right shows the results of sharpening a 4000 dot per inch scan with ksharp = 0.54. The final result is much sharper than anything we've seen so far! The difference between the 2400 and 4000 dpi scans is much greater after sharpening. The MTF for the scanner + sharpening (the blue dashed line) is 50% at approximately 87lp/mm, significantly better than you'd expect from an enlarging lens (around 50% at 60 lp/mm— about the same as an excellent 35mm camera lens).

A reason for the excellent result is that there is still significant response for the film, lens, and scanner around the frequency of maximum sharpening boost (dscan/2 = 78 lp/mm). Since response for the film + lens drops off rapidly above this frequency, increasing the scanner resolution above 4000 dpi does little to improve image sharpness.

There is a price to be paid for sharpening: grain and the effects of dust and scratches are exaggerated. For this reason, sharpening is usually performed with care in the image editing program. "Unsharp mask," which sharpens only near contrast boundaries, is often employed to avoid exaggerating grain.
 

[The sharpening function is identical to the pulse slimming equalization used in disk and tape drives. Pulses are sharpened to maximize recording density, but there is an optimum. Too much sharpening increases the noise from the disk or tape and electronics, degrading performance. Grain is the counterpart of noise in film.]

Sharpening can be performed during scanning or image editing. If you scan at less than the maximum scanner resolution, you may wish to do sharpening during scanning. But sharpening during scanning offers little control. The HP Photosmart S20 scanner has a slider that controls sharpening, but the preview scan image has insufficient resolution to judge its precise effect— it can be quite misleading. The default setting is "15." (Could it mean 2*ksharp expressed in %?) Based on experience I sometimes increase it for older, less sharp images. But most of the time, if I'm scanning with the S20's maximum 2400 dpi resolution, I just leave it at the default. I almost always do additional sharpening (using Unsharp Mask) during image editing.

The Sharpen transformation control box from Picture Window Pro provides a good example. When I perform this function I enlarge the input and Preview windows to show portions of the images at 1:1 magnification. In this way I can clearly see the effects of sharpening as I work. Amount is proportional to ksharp; Radius is R from the equation, v  = R/dscan— the number of pixels over which sharpening takes place. For 2400 dpi scans I typically set Radius to 1 for images that are reasonably sharp to begin with. 4000 dpi scans show up much more detail; an image has to be exceptionally sharp for me to set Radius to 1. I usually use 2; occasionally 3. Radius needs to be larger for fuzzy images. I prefer Unsharp Mask to the simple Sharpen transformation because it allows a decision threshold to be set: Sharpening only takes place when the difference between adjacent pixels is greater than Threshold. This keeps smooth areas like skies from becoming excessively grainy. I examine different portions of the preview image for grain and edges. I cut back on Amount if edges are oversharpened and increase Threshold if grain is exaggerated in skies and smooth areas. I usually make a mask to eliminate sharpening the sky and other tonally smooth regions.

Digital vs. conventional darkroom prints

From Schneider's enlarging lens MTF data,  f50 for a good enlarging lens appears to be around 60 lp/mm at useable apertures of f/8 and f/11— about the same as the excellent 35mm camera lens. I haven't yet added enlarging lenses to the MTFcurve simulation, so I use a little shortcut. I observe that  f50 for a 4000 dpi scanner alone (without sharpening) is around 60 lp/mm (the thin blue dashed line in the unsharpened 4000 dpi scanner resolution plot). From this we deduce that conventional darkroom prints are about as sharp as prints made from unsharpened 4000 dpi scans— at least for large prints where the sharpness isn't dominated by printer resolution. Sharpness for sharpened 2400 dpi scans is similar. I confirmed this when I compared scans of a  sharp negative in my report on the Canon CanoScan 4000 dpi 45mm/APS film scanner. From all this I conclude...

Prints from sharpened 4000 dpi scans are significantly sharper than conventional darkroom prints.

I've done a lot of conventional prints using rather fanatic technique, and I can report that a 13x19 inch print of the Aspens on the Pitkin Creek trail near Vail, Colorado, is sharper than any print I ever made conventionally. Very satisfying.

Summary

Modern high quality inkjet printers, such as the Epson 1270/1280/1290 and the 2100/2200, are capable of making prints about as fine as the human eye can resolve at distances of 10 inches (25 cm) or more. We discuss printers and prints in parts 3 and 4. Unlike film, lenses, scanners and sharpening, they can't be modeled with neat equations; they have to be measured. A few important points: A printer's dots per inch (dpi) rating is not its true resolution because it takes several dots to render a single pixel. At small magnifications image sharpness is limited by the printer. I'm extremely pleased with the intrinsic resolution of my Epson 1270. Prints up to letter size (8 1/2x11 inches)  made from sharp 35mm negatives are strikingly clear; larger prints are still satisfying, but they tend to be limited by the image rather than the printer. 13x19 prints are pleasingly sharp, but you know they weren't made from 4x5 originals.

The following images are the piece de resistance of the MTFcurve program: an illustration of resolution at each step of the imaging process. The entire plot is normalized for a tonal range from 0 to 1; there is no clipping of sharpening overshoots. Since both the film and sharpen function oversharpen somewhat (a good thing if not overdone), contrast in other bands has been reduced. This makes the bands without sharpening (original, lens only) look somewhat flat. If I had normalized for each band individually, the sharpened bands would appear to have less contrast. For better or worse, this is just one of several ways to display the results.
 

This is the image sequence for Fujichrome Velvia, the excellent 35mm lens, a 2400 dpi scan, and sharpening with ksharp = 0.44. The lens is very clearly better than the film.

The use of Fujichrome Velvia in these images is not an endorsement. Velvia is perceived by many photographers to be one of the sharpest slide films  because of its high contrast and adjacency effect boost. It indeed looks sharper in slides. But in a properly sharpened scanned image, Provia 100F (or Ektachrome E100VS, which has an identical MTF curve) can be as sharp or sharper— resolution can be boosted with less danger of oversharpening. The adjacency effect advantage is lost. Provia 100F has the additional advantage that it can capture a brightness range at least one f-stop larger than Velvia. Jonathan Sachs wrote an excellent article comparing the two. Michael Reichmann of Luminous-Landscape used Provia almost exclusively before he went all digital. See his comparison, Provia 100F and 400F.

This image is for a 4000 dpi scan of Velvia with the excellent 35mm lens with sharpening constant ksharp = 0.54. (Only the lower part is shown; the upper part is identical to the previous image.) Sharpening makes more of a visual improvement at this density than it does at 2400 dpi. Without sharpening the 4000 dpi scan isn't much sharper than the sharpened 2400 dpi scan.

The importance of sharpening is clearly apparent from these illustrations. The sharpening function is one of the great advantages of digital imaging.
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Sharpening is a vital part of digital image making. It is critical for bringing out your image's full potential quality.

Additional observations

Writing this has done more to satisfy my curiosity than to change my photographic technique, though it gave me the ammunition I needed to purchase a 4000 dpi 35mm scanner. Years ago I had a friend who met Albert Einstein shortly before he died. Einstein told him, "Never lose your holy curiosity." (The meeting was reported in Life Magazine, May 2, 1955. I finally located him with help from the Web.) I was so obsessed with this article during the month I wrote it (a lot of sleep lost) that I'm not sure if curiosity is a blessing or a curse. I just have to accept it.. 
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Download the MTFcurve program
The MTF curve program can be downloaded by clicking (or shift-clicking) on MTFcurve2.m. To run it you must have Matlab installed on your computer. A $99 student version is available at campus bookstores. To view instructions on running it, type
help mtfcurve2
Examples of running MTFcurve2 from the Matlab window have been shown previously. I'd appreciate hearing from you if you download and use MTFcurve2. There's no obligation of course, but there aren't many of us.
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Back: Part 1: Introduction | Part 1A: Film and lenses | Next: Part 3: Printers and prints
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Equations
Varying frequency

The equation for a pure sine wave tone h(x) (cosine differs from sine only by a phase of 90 degrees) is,

h(x) = cos(2 π  f x)
where f is the frequency. This waveform repeats itself every 1/f. It would seem logical and intuitive to use the following equation to vary the frequency for the figures in this report.
h(x) = cos(2π f (x) x)   THIS IS INTUITIVE BUT WRONG!!!
The true definition of frequency found in communications textbooks is the rate of change of phase of the function h(x).
For h(x) = cos(G(x)), frequency is defined as  f (x) = 1/(2π) dG(x)/dx, where d/dx is differentiation. Therefore,

G(x) = 2π ∫ f (x) dx.  

The result is quite different from the intuitive but wrong equation above. The pattern of black and white tones is created by applying the equation,

round((1+ cos(G(x)))/2)
where round is the Matlab function that rounds to the nearest integer, 0 and 1 in this case.

Sharpening

The Fourier transform F for a function f (x) shifted by v is,

F( f (x+v)) = F( f (x)) exp(-2 π j  v)
For the sharpening equation,
F((Dsharp(x)) = F((D(x) - ksharp/2*(D(x-v) + D(x+v))/(1-ksharp))
       = F(D(x))*(1-ksharp /2 (exp(-2 π j v) + exp(2 π j f v)))/(1-ksharp)
where j = sqrt(-1) and cos(x) = (exp(j x) + exp(-j x))/2. Therefore,
MTFsharp( f ) = F((Dsharp(x))/F((D(x)) = (1 - ksharp  cos(2 π f v))/(1-ksharp)

Zeiss statement on MTF

I've taken the liberty of reproducing the following statement from the archives of Camera Lens News, issue no. 7 (summer 1999), which I found buried deep within the Carl Zeiss Camera / Cine Lenses web page. They discuss how most photographers misunderstand MTF— my motivation for writing these pages. The final paragraphs are a slam at Schneider Optics, who publish computer simulated MTF plots (better than nothing). Zeiss's MTF curves can be found in the Technical information (PDF files) for individual lenses.
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Carl Zeiss MTF curves are measured – not just calculated
The Modulation Transfer Function (MTF) is nowadays widely used throughout the optical industry as the objective way to clearly represent and evaluate the performance of optical systems, like camera lenses for example. 

Trying to trace the usage of MTF in the optical industry we find first roots dating back as far as the year 1940. At that time an early version of MTF was first applied at Carl Zeiss in the Jena factory. Ever since Carl Zeiss pioneered the usage of this method, being convinced that Zeiss optics compared well by any means, MTF included. Other companies in the in-dustry followed decades later. Many refused to publish MTF data of their lenses arguing that it is not as easy as it appears to correctly interpret MTF curves. Today, with the majority of lens manufacturers and many photo magazines publishing MTF curves or similar representations to demonstrate the performance level of camera lenses, we often encounter cases where the curves are misinterpreted by enthusiasts quite grossly. Excellent lenses which earned high reputation amongst professional users are often underestimated by people who have no personal experience with the lens, but rely on their own interpretation of curves only. The opposite can also be seen: Mediocre optics sell impressive numbers, after stunning MTF data was published, albeit users with personal experience are rather disappointed. 

All this leads many seasoned photographers and practitioners to the question: is MTF really trustworthy? Optics designers will be quick to answer "Yes, it is!". They are the ones who can really interpret MTF curves. They have the professional background and additional information helping them to draw the right conclusions from these curves. 

They do know, for instance, that MTF curves can be influenced significantly, if the spectral energy distribution of the light used for measuring or assumed in calculation changes a bit. So MTF curves of the same lens coming from different sources may vary, even when "white light" is specified with all these curves. Even "white light" is not necessarily the exact same thing unless all the weighing factors for the different sections of the spectrum are made sure to be the same. 

Lens designers also know that there can be a huge difference between a lens performance designed on a computer and the actual performance of the real lens once in series production. This difference is not simply a fixed amount of loss, something like: "You design for 95% MTF and you lose 10%, thus getting 85% out of the production. So aim a little higher on the design computer and you will get a little more." Experienced optics manufacturers know that the opposite is usually true: If you aim for very high MTF values on the computer you create an optical system that is extremely sensitive in manufacturing. It will most likely force you to lower the MTF specification for final acceptance significantly to escape excessive scrapping costs. Thus you may have aimed for 98% MTF and get only 70% on average out of production. At least these lenses show very impressive MTF curves in the catalogues…

Carl Zeiss lens designers may, in a similar case, tend to rather aim for 93% instead of 98 % in order to actually get 90% MTF out of production – with very narrow tolerance bands. The MTF curves that Carl Zeiss publishes, are all measured curves of the first units actually built – not just calculated plots without information about how well they may be executed by the manufacturer.

As far as we can see today Carl Zeiss is still the only manufacturer to print MTF curves which are measured, which describe lenses that can actually be purchased, not just calculated curves of highly ambitious designs which may exceed the manufacturer's capability of turning them into reality.

Camera Lens News No. 7, summer 1999

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Back: Part 1: Introduction | Part 1A: Film and lenses | Next: Part 3: Printers and prints


Images and text copyright © 2000-2014 by Norman Koren.
Norman Koren lives in Boulder, Colorado, founded Imatest LLC in 2004, previously worked on magnetic recording technology. He has been involved with photography since 1964.


Images and text copyright © 2000-2014 by Norman Koren.
Norman Koren lives in Boulder, Colorado, founded Imatest LLC in 2004, previously worked on magnetic recording technology. He has been involved with photography since 1964.