DigitUnderstanding image sharpness part 1:
Introduction to resolution and MTF curves
by Norman Koren

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Table of contents

for the
image sharpness
series

Part 1: Introduction
Introduction to modulation transfer function (MTF)
Definition | Virtual chart | MTF data and other links
Human visual acuity
Part 1A: Film and Lenses
Part 2: Scanners and sharpening
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

Image sharpness and detail

Cumulative effect of the film, lens, scanner and sharpening on the targetA photograph's detail is an integral part of its appeal. Many photographers spend a great deal of time, energy and money acquiring equipment to make sharp images. Back in the film era, if 35mm didn't satisfy them, they invested in medium format, 4x5, 8x10, or larger. (I know two who use 8x20 inch cameras.) The digital versus film debate is now mostly settled (2007), but there is still some debate over the relationship between the number of megapixels and image quality. I love sharpness and detail, but I take my camera gear on long hikes, so I prefer to carry lightweight equipment. I need to know what it can achieve, how to get the most out of it and what I'm trading off by not going to a larger format, apart from saving my back. That's what motivated this study.

The sharpness of a photographic imaging system or of a component of the system (lens, film, image sensor, scanner, enlarging lens, etc.) is characterized by a parameter called Modulation Transfer Function (MTF), also known as spatial frequency response. We present a unique visual explanation of MTF and how it relates to image quality. A sample is shown on the right. The top is a target composed of bands of increasing spatial frequency, representing 2 to 200 line pairs per mm (lp/mm) on the image plane. Below you can see the cumulative effects of the lens, film, lens+film, scanner and sharpening algorithm, based on accurate computer models derived from published data. If this interests you, read on. It gets a little technical, but I try hard to keep it readable.

The companion website, Imatest.com, describes a software tool you can use to measure MTF and other factors that contribute to image quality in digital cameras and digitized film images.
Imatest: affordable software for measuring sharpness and image quality

Green is for geeks. Do you get excited by a good 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.

Introduction to modulation transfer function (MTF)

Back in my youth, lens and film resolving power was measured in lines (or line pairs) per millimeter (lp/mm)— easy to understand, but poorly standardized. It was obtained by photographing a chart (typically the USAF 1951 lens test chart) and looking for the highest resolution pattern where detail was visible. Because perception and judgment were involved, measurements of the same film or lens were highly inconsistent. Lines per mm would have been more useful if it were measured at a well established contrast level, but that was not so easy; it would have required expensive instrumentation. The problem of specifying resolution and perceived sharpness was solved with the introduction of the Modulation transfer function (MTF), a precise measurement made in frequency domain. This made optical engineers happy, but confuses many photographers. The goal of this series is to shed light on the subject (literally as well as figuratively). I include software you can run yourself if you have Matlab, a popular program with engineers and scientists.
 
MTF is the spatial frequency response of an imaging system or a component; it is the contrast at a given spatial frequency relative to low frequencies.

Spatial frequency is typically measured in cycles or line pairs per millimeter (lp/mm), which is analogous to cycles per second  (Hertz) in audio systems. Lp/mm is most appropriate for film cameras, where formats are relatively fixed (i.e., 35mm full frame = 24x36mm), but cycles/pixel (c/p) or line widths per picture height (LW/PH) may be more appropriate for digital cameras, which have a wide variety of sensor sizes.

High spatial frequencies correspond to fine image detail. The more extended the response, the finer the detail the sharper the image.

Most of us are familiar with the frequency of sound, which is perceived as pitch and measured in cycles per second, now called Hertz. Audio components— amplifiers, loudspeakers, etc.— are characterized by frequency response curves. MTF is also a frequency response, except that it involves spatial frequency— cycles (line pairs) per distance (millimeters or inches) instead of time. The mathematics is the same. The plots on these pages have spatial frequencies that increase continuously from left to right. High spatial frequencies correspond to fine image detail. The response of photographic components (film, lenses, scanners, etc.) tends to roll off at high spatial frequencies. These components can be thought of as lowpass filters— filters that pass low frequencies and attenuate high frequencies.

Line pairs or lines?
All MTF charts and most resolution charts display spatial frequency in cycles or line pairs per unit length (mm or inch). But there are exceptions. An old standard for measuring TV resolution uses line widths instead of pairs, where there are two line widths per pair, over the total height of the display. When dpreview.com recommends multiplying the chart values in its lens tests by 100 to get the total vertical lines in the image, they refer to line widths, not pairs. Confusing, but I try to keep it straight. Imatest SFR displays MTF in cycles (line pairs) per pixel, line widths per picture height (LW/PH; derived from TV measurements), and line pairs per distance (mm or in).

The essential meaning of MTF is rather simple. Suppose you have a pattern consisting of a pure tone (a sine wave). At frequencies where the MTF of an imaging system or a component (film, lens, etc.) is 100%, the pattern is unattenuated— it retains full contrast. At the frequency where MTF is 50%, the contrast half its original value, and so on. MTF is usually normalized to 100% at very low frequencies. But it can go above 100% with interesting results.

Contrast levels from 100% to 2% are illustrated on the right for a variable frequency sine pattern. Contrast is moderately attenuated for MTF = 50% and severely attenuated for MTF = 10%. The 2% pattern is visible only because viewing conditions are favorable: it is surrounded by neutral gray, it is noiseless (grainless), and the display contrast for CRTs and most LCD displays is relatively high. It could easily become invisible under less favorable conditions.

How is MTF related to lines per millimeter resolution? The old resolution measurement distinguishable lp/mm— corresponds roughly to spatial frequencies where MTF is between 5% and 2% (0.05 to 0.02). This number varies with the observer, most of whom stretch it as far as they can. An MTF of 9% is implied in the definition of the Rayleigh diffraction limit.

Perceived image sharpness (as distinguished from traditional lp/mm resolution) is closely related to the spatial frequency where MTF is 50% (0.5)— where contrast has dropped by half.
One important detail: MTF is not the same as grain. Grain increases with film speed: MTF is less sensitive to film speed.

 MTF corresponds to the bandwidth of a communications system; grain corresponds to its noise.
Grain can be characterized by a frequency spectrum (higher frequencies correspond to finer grain patterns) as well as amplitude (intensity or contrast). Because there is no simple formula that determines how spectrum, amplitude and print magnification affect our perception of grain, Kodak has devised a subjective measure called "Print Grain Index." Later in this series I hypothesize that the Shannon information capacity of an imaging system— a function of bandwidth and noise— correlates with perceived image quality. 
 
The MTF curve on the right is for Fuji's highly regarded Provia 100F slide film. It's typical except for one detail: MTF isn't 100% at low spatial frequencies. This is an error— perhaps the work of an overly creative marketing department. The 50% MTF frequency ( f50 ) is about 42 lp/mm. MTF is only shown as far as 60 lp/mm. The resolution of this film is rated as 60 lp/mm for 1.6:1 chart contrast and 140 lp/mm for 1000:1 chart contrast. The latter number may be of interest to astronomers, but it has little to do with the perceived image sharpness of any realistic scenes.

The figure below represents a sine pattern (pure frequencies) with spatial frequencies from 2 to 200 cycles (line pairs) per mm on a 0.5 mm strip of film. The top half of the sine pattern has uniform contrast. The bottom half illustrates the effects of Provia 100F on the MTF. Pattern contrast drios ub half at 42 cycles/mm.

A more precise definition of MTF  based on sine patterns: MTF is the contrast at a given spatial frequency ( f ) relative to contrast at low frequencies. These equations are used in the page on Lens testing to calculate MTF from an image of a chart consisting of sine patterns of various frequencies, where the sine pattern contrast in the original chart is assumed to be constant with frequency. (This series uses charts of continuously varying frequency.) Definitions:
. 
VB The minimum luminance (or pixel value) for black areas at low spatial frequencies. The frequency should be low enough so that contrast doesn't change if it is reduced.
VW The maximum luminance for white areas at low spatial frequencies.
Vmin The minimum luminance for a pattern near spatial frequency  f  (a "valley" or "negative peak").
Vmax The maximum luminance for a pattern near spatial frequency  f  (a "peak").
C(0) = (VW-VB)/(VW+VB) is the low frequency (black-white) contrast.
C( f ) = (Vmax-Vmin)/(Vmax+Vmin) is the contrast at spatial frequency  f . Normalizing contrast in this way dividing by Vmax+Vmin (VW+VB at low spatial frequencies) minimizes errors due to gamma-related nonlinearities in acquiring the pattern.
MTF( f ) = 100%*C( f )/C(0)
.
MTF can also be defined as is the magnitude of the Fourier transform of the point or line spread function the response of an imaging system to an infinitesimal point or line of light. This definition is technically accurate and equivalent to the sine pattern contrast definition, but can't be visualized as easily unless you're an engineer or physicist.
 
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Imaging systems

Systems for reproducing information, images, or sound typically consist of a chain of components. For example, audio reproduction systems consist of a microphone, mike preamp, digitizer or cutting stylus, CD player or phono cartridge, amplifier, and loudspeaker.

Film imaging systems consist of a lens, film, developer, scanner, image editor, and printer (for digital prints) or lens, film, developer, enlarging lens, and paper (for traditional darkroom prints). Digital camera-based imaging systems consist of a lens, digital image sensor, de-mosaicing program, image editor, and printer. Each of these components has a characteristic frequency response; MTF is merely its name in photography. The beauty of working in frequency domain is that the response of the entire system (or group of components) can be calculated by multiplying the responses of each component.

Typical 50% MTF frequencies are in the vicinity of 40 to 80 lp/mm for individual components (lenses, film, scanners) and often as low as 30 lp/mm for entire imaging systems much lower than the 80-160 lines/mm numbers typical of the old resolution measurements. It takes some getting used to if you grew up with the old measurements.

The response of a component or system to a signal in time or space can be calculated by the following procedure.
  1. Convert the signal into frequency domain using a mathematical operation known as the Fourier transform, which is fast and easy to perform on modern computers using the FFT ( Fast Fourier Transform) algorithm. The result of the transform is called the frequency components or FFT of the signal. Images differ from time functions like sound in that they are two dimensional. Film has the same MTF in any direction, but not lenses.
  2. Multiply the frequency components of the signal by the frequency response (or MTF) of the component or system.
  3. Inverse transform the signal back into time or spatial domain.
Doing this in time or spatial domain requires a cumbersome mathematical operation called convolution. If you try it, you'll know how the word "convoluted" originated. And you'll know for sure why frequency domain is widely appreciated.

Resolution of an imaging system (old definition) Using the assumption that  resolution is a frequency where MTF is 10% or less, the resolution r of a system consisting of n components, each of which has an MTF curve similar to those shown below, can be approximated by the equation, 1/r = 1/r1 + 1/r2 + ... + 1/r (equivalently, r = 1/(1/r1 + 1/r2 + ... + 1/rn )). This equation is adequate as a first order estimate, but not as accurate as multiplying MTF's. [I verified it with a bit of mathematics, assuming a second order MTF rolloff typical of the curves below. It's not sensitive to the MTF percentage that defines r. The approximation, 1/r2 = 1/r12 + 1/r22 + ..., is not accurate.]

A virtual chart for visualizing MTF

To visualize the effects of MTF, we have created a virtual target 0.5 mm in length, shown greatly enlarged on the right. The target consists of a sine pattern and a bar pattern, both of which start at a low spatial frequency, 2 line pairs per millimeter (lp/mm) on the left, and increase logarithmically to 200 lp/mm on the right.
The mathematics for generating this function is rather tricky. It is discussed at the end of part 2.
The red curve below the image represents the tonal densities (0 and 1) of the bar pattern. The vertical scale— 100 through 102—  is for the MTF curves to come, not for the tonal density plot.
Input image for MTF visualization
MTF illustration for Fujichrome Velvia, excellent 35mm lens  The plot on the left illustrates the response of the virtual target to the combined effects of an excellent lens (a simulation of the highly-regarded Canon 28-70mm f/2.8L) and film (a simulation of Velvia). Both the sine and bar patterns (original and response) are shown. You'll find these plots throughout this series as we simulate lenses, film, scanners, sharpening, and finally, digital cameras.

The red curve is the spatial response of the bar pattern to the film + lens. The blue curve is the combined MTF, i.e., the spatial frequency response of the film + lens, expressed in percentage of low frequency response, indicated on the scale on the left. (It goes over 100% (102).) The thin blue dashed curve is the MTF of the lens only.

The edges in the bar pattern have been broadened, and there are small peaks on either side of the edges. The shape of the edge is inversely related to the MTF response: the more extended the MTF response, the sharper (or narrower) the edge. The mid-frequency boost of the MTF response is related to the small peaks on either side of the edges.


The leftmost edge in the plot is a portion of the step response of the system (film + lens). A much lower spatial frequency is required to represent it properly. The impulse response the response of the system to a narrow line (or impulse) is also of interest. The impulse response is the derivative of the step response (d(step response)/dx).

The MTF curve is related to the impulse response by a mathematical operation known as the Fourier transform ( F ), which is well-known to engineers and physicists.

        MTF response = F(impulse response)
        impulse response = F-1(MTF response)

F-1 is the inverse Fourier transform. We'll spare the gentle reader from further equations the topic is quite understandable without them.

The image above represents only 0.5 mm of film, but takes up around 5 inches (13 cm) on my monitor. At this magnification (260x), a full frame 35mm image (24x36mm) would be 240 inches (6.2 meters) high and 360 inches (9.2 meters) wide. A bit excessive, but if you stand back from the screen you'll get an feeling for the effects of the lens, film, scanner (or digital camera), and sharpening on real images.

The companion website, Imatest.com, describes a software tool you can use to measure MTF and other factors that contribute to image quality in digital cameras and digitized film images.
Imatest: affordable software for measuring sharpness and image quality

Links to general articles on MTF

Understanding MTF: The Modulation Transfer Function Explained by Michael Reichmann of Luminous-landscape.com.  Excellent introduction.
What is an MTF ...and Why Should You Care? by Don Williams of Eastman Kodak.
How to interpret MTF graphs  by Klaus Schroiff. Another useful explanation.
Photodo has several excellent articles on MTF and image quality. Recommended.
MTF Engineering Notes  from Sine Patterns LLC, a purveyor of lens test charts. Lots of equations.
Image Processing page from efg (Earl F. Glynn)  Serious links to (mostly) serious academic literature. Fascinating for geeks. Click here if the link doesn't work.
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.
An Evaluation of the Current State of Digital Photography by Charles Dickinson. RIT bachelor's thesis, 1999. Uses MTF analysis.
Introduction to Electronic Imaging Systems  Class notes from ECE 102, Center for Electronic Imaging Systems, University of Rochester. Taught by Dr. Michael Kriss. Connected with the U of R Image Processing Lab.
RIT Center for Imaging Science class material is a serious resource— well worth exploring. Basic Principles of Imaging Science 1. Lectures 17 and 18 on MTF and imaging microstructure are particularly interresting.

Human visual acuity

The ability of the eye to resolve detail is known as "visual acuity." The normal human eye can distinguish patterns of alternating black and white lines with a feature size as small as one minute of an arc (1/60 degree or π/(60*180) = 0.000291 radians). That, incidentally, is the definition of 20-20 vision. A few exceptional eyes may be able to distinguish features half this size. But for most of us, a pattern of higher spatial frequency will appear nearly pure gray. Low contrast patterns at the maximum spatial frequency will also appear gray.

At a distance d from the eye (which has a nominal focal length of 16.5 mm), this corresponds to objects of length = (angle in radians)*d = 0.000291*d. For example, for an object viewed at a distance of 25 cm (about 10 inches), the distance you might use for close scrutiny of an 8x10 inch photographic print, this would correspond to 0.0727 mm = 0.0029 inches. Since a line pair corresponds to two lines of this size, the corresponding spatial frequency is 6.88 lp/mm or 175 lp/inch. Assume now that the image was printed from a 35mm frame enlarged 8x. The corresponding spatial frequency on the film would be 55 lp/mm.

This means that for an 8x10 inch print, the MTF of a 35mm camera (lens + film, etc.) above 55 lp/mm, or the MTF of a digital camera above 2800 LW/PH (Line Widths per Picture Height) measured by Imatest SFR, has no effect on the appearance of the print. That's why the highest spatial frequencies used in manufacturer's MTF charts is typically 40 lp/mm, which provides an excellent indication of a lens's perceived sharpness in an 8x10 inch print enlarged 8x. Of course higher spatial frequencies are of interest for larger prints.

Standard Depth of Field (DOF) scales on lenses are based on the assumption, made in the 1930s, that the smallest feature of importance, viewed at 25 cm, is 0.01 inches— 3 times larger. It shouldn't be a surprise that focus isn't terribly sharp at the DOF limits. See the DOF page for more details.

The statement that the eye cannot distinguish features smaller than one minute of an arc is, of course, oversimplified. The eye has an MTF response, just like any other optical component. It is illustrated on the right from the Handout #9: Human Visual Perception from Stanford University course EE368B - Image and Video Compression by Professor Bernd Girod. The horizontal axis is angular frequency in cycles per degree (CPD). MTF is shown for pupil sizes from 2 mm (bright lighting; f/8), to 5.8 mm (dim lighting; f/2.8). At 30 CPD, corresponding to a one minute of an arc feature size, MTF drops from 0.4 for the 2 mm pupil to 0.16 for the 5.8 mm pupil. (Now you know your eye's f-stop range. It's similar to compact digital cameras.) Another Stanford page has Matlab computer models of the eye's MTF.

The eye's contrast sensitivity functionThe human eye's MTF, which is limited at high angular frequencies by the eye's optical system and cone density, does not tell the whole story of the eye's response. Neuronal interactions such as lateral inhibition limit the eye's response at low angular frequencies, i.e., the eye is insensitive to very gradual changes in density. The eye's overall response is called its contrast sensitivity function (CSF). Various studies place the peak CSF for bright light levels (typical of print viewing conditions) between 6 and 8 cycles per degree. The graph on the left uses an approximation (equations below) that peaks just below 8 cycles/degree.

CSF is used in a measure of perceptual image sharpness called Subjective Quality Factor (SQF), which includes MTF, CSF, print size, and typical viewing distance. SQF has been used since the 1970s inside Kodak and Polaroid, but it was difficult to calculate, and hence remained obscure, until it was incorporated into Imatest SFR in 2006.
The following formula for CSF is relatively simple, recent, and fits the data well. The source is J. L. Mannos, D. J. Sakrison, ``The Effects of a Visual Fidelity Criterion on the Encoding of Images'', IEEE Transactions on Information Theory, pp. 525-535, Vol. 20, No 4, (1974), cited on this page of Kresimir Matkovic's 1998 PhD thesis.
CSF( f ) = 2.6 (0.0192 + 0.114 f ) exp(-0.114 f )1.1
The 2.6 multiplier can be removed and the equation can be simplified somewhat. The dc term (0.0192) can be dropped with very little effect.
CSF( f ) = (0.0192 + 0.114 f ) exp(-0.1254 f )

Additional explanations of human visual acuity can be found on pages from the Nondestructive testing resource center and Stanford University. Page 3 from Stanford has a plot of the MTF of the human eye. I believe the x-axis units (CPD) are Cycles per Degree, where a pair of 1/60 degree features corresponds to 30 CPD.

Next: Part 1A: MTF in film and lenses | Part 2: Scanners and sharpening


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.