DigitUnderstanding image sharpness part 1:Introduction to
resolution and MTF curvesby Norman KorenSite map/guide to
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galleriesSearch WWWSearch www.normankoren.comTable ofcontentsfor
theimagesharpnessseriesPart 1: IntroductionIntroduction to
modulation transferfunction (MTF)Definition | Virtual chart | MTF
dataand other linksHuman visual acuityPart 1A: Film and LensesPart
2: Scanners and sharpening4000 vs. 8000 dpi scansPart 3: Printers
and printsPart 4: Epson 1270 resultsPart 5: Lens testingPart 6:
Depth of field anddiffractionDigital cameras vs. film, part 1 |part
2Part 8: Grain and sharpness:comparisonsImage sharpness and detailA
photograph's detail is an integralpart of its appeal.
Manyphotographers spend a great deal oftime, energy and money
acquiringequipment to make sharp images.Back in the film era, if
35mm didn'tsatisfy them, they invested in mediumformat, 4x5, 8x10,
or larger. (I knowUnderstanding resolution and MTF
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PMtwo who use 8x20 inch cameras.) Thedigital versus film debate is
nowmostly settled (2007), but there is stillsome debate over the
relationshipbetween the number of megapixelsand image quality. I
love sharpnessand detail, but I take my camera gearon long hikes,
so I prefer to carrylightweight equipment. I need toknow what it
can achieve, how to getthe most out of it and what I'm tradingoff
by not going to a larger format,apart from saving my back.
That'swhat 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 itrelates to image quality. A sample is shown on the
right. The top is a target composed of bands of increasingspatial
frequency, representing 2 to 200 line pairs per mm (lp/mm) on the
image plane. Below you can see thecumulative effects of the lens,
film, lens+film, scanner and sharpening algorithm, based on
accurate computermodels derived from published data. If this
interests you, read on. It gets a little technical, but I try hard
tokeep it readable.This page introduces MTF and relates it to
traditional resolution measurements.Part 1A illustrates its effect
on film and lenses.Part 2 continues with scanners (image sensors)
and sharpening algorithms.Part 3 discusses printers and prints, and
how to characterize their sharpness and resolution.Part 4 presents
detailed printer test results.Part 5 discusses lens testing using a
new downloadable target with continuously varying
spatialfrequency.Part 6 discusses depth of field (DOF), emphasizing
sharpness at the DOF scale limits.Part 7 compares digital cameras
with film, and addresses the question, "How many pixels does it
takefor a digital sensor to outperform 35mm film?"Part 8 compares
grain and sharpness for three scanners with a well-crafted enlarger
print, and we lookat grain aliasing and software solutions.The
companion website,Imatest.com, describes asoftware tool you can use
tomeasure MTF and otherfactors that contribute toimage quality in
digitalcameras and digitized filmimages.Green is for geeks. Do you
get excited by a good equation? Were you passionateabout your
college math classes? Then you're probably a math geek a member of
amaligned and misunderstood but highly elite fellowship. The text
in green is for you. Ifyou're normal or mathematically challenged,
you may skip these sections. You'll neverknow what you
missed.Understanding resolution and MTF
http://www.normankoren.com/Tutorials/MTF.html2 of 12 11/3/2010 1:05
PMIntroduction 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
(typicallythe 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 highlyinconsistent. Lines per mm would have been
more useful if it were measured at a well established
contrastlevel, but that was not so easy; it would have required
expensive instrumentation. The problem of specifyingresolution 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
confusesmany photographers. The goal of this series is to shed
light on the subject (literally as well as figuratively). Iinclude
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
ata given spatial frequency relative to low frequencies.Spatial
frequency is typically measured in cycles or line pairs per
millimeter (lp/mm), which isanalogous to cycles per second(Hertz)
in audio systems. Lp/mm is most appropriate for filmcameras, 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, whichhave a wide variety of
sensor sizes.High spatial frequencies correspond to fine image
detail. The more extended the response, thefiner 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
persecond, now called Hertz. Audio components amplifiers,
loudspeakers, etc. are characterized byfrequency response curves.
MTF is also a frequency response, except that it involves spatial
frequencycycles (line pairs) per distance (millimeters or inches)
instead of time. The mathematics is the same. The plotson these
pages have spatial frequencies that increase continuously from left
to right. High spatial frequenciescorrespond to fine image detail.
The response of photographic components (film, lenses, scanners,
etc.) tendsto roll off at high spatial frequencies. These
components can be thought of as lowpass filters filters that
passlow frequencies and attenuate high frequencies.Line pairs or
lines?All MTF charts and most resolution charts display spatial
frequency in cycles or line pairs perunit length (mm or inch). But
there are exceptions. An old standard for measuring TV
resolutionuses line widths instead of pairs, where there are two
line widths per pair, over the total heightof the display. When
dpreview.com recommends multiplying the chart values in its lens
tests by100 to get the total vertical lines in the image, they
refer to line widths, not pairs. Confusing, butI try to keep it
straight. Imatest SFR displays MTF in cycles (line pairs) per
pixel, line widths perpicture 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 apure tone (a sine wave). At frequencies wherethe MTF
of an imaging system or a component(film, lens, etc.) is 100%, the
pattern isunattenuated it retains full contrast. At thefrequency
where MTF is 50%, the contrastUnderstanding resolution and MTF
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PMhalf its original value, and so on. MTF isusually normalized to
100% at very lowfrequencies. But it can go above 100%
withinteresting results.Contrast levels from 100% to 2%
areillustrated on the right for a variable frequencysine pattern.
Contrast is moderately attenuatedfor MTF = 50% and severely
attenuated forMTF = 10%. The 2% pattern is visible onlybecause
viewing conditions are favorable: it issurrounded by neutral gray,
it is noiseless (grainless), and the display contrast for CRTs and
most LCDdisplays is relatively high. It could easily become
invisible under less favorable conditions.How is MTF related to
lines per millimeter resolution? The old resolution
measurementdistinguishable lp/mm corresponds roughly to spatial
frequencies where MTF is between 5% and 2% (0.05to 0.02). This
number varies with the observer, most of whom stretch it as far as
they can. An MTF of 9% isimplied 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) wherecontrast has dropped by
half.One important detail: MTF is not the same as grain. Grain
increases with film speed: MTF is less sensitive tofilm speed. MTF
corresponds to the bandwidth of a communications system; grain
corresponds to its noise. Graincan be characterized by a frequency
spectrum (higher frequencies correspond to finer grain patterns) as
wellas 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 measurecalled "Print Grain Index." Later in this series
I hypothesize that the Shannon information capacity of animaging
system a function of bandwidth and noise correlates with perceived
image quality. The MTF curve on the right is for Fuji's highly
regardedProvia 100F slide film. It's typical except for one detail:
MTFisn't 100% at low spatial frequencies. This is an errorperhaps
the work of an overly creative marketing department.The 50% MTF
frequency ( f50 ) is about 42 lp/mm. MTF isonly shown as far as 60
lp/mm. The resolution of this film israted as 60 lp/mm for 1.6:1
chart contrast and 140 lp/mm for1000:1 chart contrast. The latter
number may be of interest toastronomers, but it has little to do
with the perceived imagesharpness of any realistic scenes.The
figure below represents a sine pattern (pure frequencies)with
spatial frequencies from 2 to 200 cycles (line pairs) permm on a
0.5 mm strip of film. The top half of the sine patternhas uniform
contrast. The bottom half illustrates the effects ofProvia 100F on
the MTF. Pattern contrast drios ub half at
42cycles/mm.Understanding resolution and MTF
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PMA more precise definition of MTFbased on sine patterns: MTF is
the contrast at a given spatialfrequency ( f ) relative to contrast
at low frequencies. These equations are used in the page on
Lenstesting 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. (Thisseries uses charts of continuously varying
frequency.) Definitions:. VBThe 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.VminThe minimum luminance for a pattern near spatial
frequencyf(a "valley" or"negative peak").Vmax The maximum luminance
for a pattern near spatial frequencyf(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 frequencyf .
Normalizingcontrast 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 spreadfunction the response of an
imaging system to an infinitesimal point or line of light. This
definitionis technically accurate and equivalent to the sine
pattern contrast definition, but can't be visualized aseasily
unless you're an engineer or physicist. View image galleriesHow to
purchase prints...An excellent opportunity to collect high
qualityphotographic prints and support this website.Imaging
systemsSystems for reproducing information, images, or sound
typically consist of a chain of components. ForUnderstanding
resolution and MTF http://www.normankoren.com/Tutorials/MTF.html5
of 12 11/3/2010 1:05 PMexample, audio reproduction systems consist
of a microphone, mike preamp, digitizer or cutting stylus, CDplayer
or phono cartridge, amplifier, and loudspeaker.Film imaging systems
consist of a lens, film, developer, scanner, image editor, and
printer (for digital prints) orlens, film, developer, enlarging
lens, and paper (for traditional darkroom prints). Digital
camera-based imagingsystems consist of a lens, digital image
sensor, de-mosaicing program, image editor, and printer. Each of
thesecomponents has a characteristic frequency response; MTF is
merely its name in photography. The beauty ofworking 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/mmnumbers typical of the old resolution
measurements. It takes some getting used to if you grew up with the
oldmeasurements.The response of a component or system to a signal
in time or space can be calculated by the
followingprocedure.Convert the signal into frequency domain using a
mathematical operation known as the Fouriertransform, which is fast
and easy to perform on modern computers using the FFT ( Fast
FourierTransform) algorithm. The result of the transform is called
the frequency components or FFT of thesignal. Images differ from
time functions like sound in that they are two dimensional. Film
has the sameMTF in any direction, but not lenses.1.Multiply the
frequency components of the signal by the frequency response (or
MTF) of the componentor system.2.Inverse transform the signal back
into time or spatial domain. 3.Doing this in time or spatial domain
requires a cumbersome mathematical operation called convolution. If
youtry it, you'll know how the word "convoluted" originated. And
you'll know for sure why frequency domain iswidely
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 ncomponents, each
of which has an MTF curve similar to those shown below, can
beapproximated by the equation, 1/r = 1/r1 + 1/r2 + ... + 1/rn
(equivalently, r = 1/(1/r1 + 1/r2 + ...+ 1/rn )). This equation is
adequate as a first order estimate, but not as accurate as
multiplyingMTF's. [I verified it with a bit of mathematics,
assuming a second order MTF rolloff typical ofthe 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 MTFUnderstanding resolution and MTF
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PMTo visualize the effects ofMTF, we have created avirtual target
0.5 mm inlength, shown greatly enlargedon the right. The
targetconsists of a sine pattern and abar pattern, both of
whichstart at a low spatialfrequency, 2 line pairs permillimeter
(lp/mm) on the left,and increase logarithmically to200 lp/mm on the
right.The mathematicsfor generating thisfunction is rathertricky.
It isdiscussed at theend of part 2.The red curve below theimage
represents the tonaldensities (0 and 1) of the barpattern. The
vertical scale100 through 102is for theMTF curves to come, not
forthe tonal density plot.Understanding resolution and MTF
http://www.normankoren.com/Tutorials/MTF.html7 of 12 11/3/2010 1:05
PM The plot on the left illustratesthe response of the
virtualtarget to the combined effectsof an excellent lens
(asimulation of the highly-regarded Canon 28-70mmf/2.8L) and film
(a simulationof Velvia). Both the sine andbar patterns (original
andresponse) are shown. You'llfind these plots throughoutthis
series as we simulatelenses, film, scanners,sharpening, and
finally,digital cameras.The red curve is the spatialresponse of the
bar pattern tothe film + lens. The bluecurve is the combined
MTF,i.e., the spatial frequencyresponse of the film +
lens,expressed in percentage oflow frequency response,indicated on
the scale on theleft. (It goes over 100%(102).) The thin blue
dashedcurve is the MTF of the lensonly.The edges in the bar
patternhave been broadened, andthere are small peaks oneither side
of the edges. Theshape of the edge is inverselyrelated to the MTF
response:the more extended the MTFresponse, the sharper
(ornarrower) the edge. Themid-frequency boost of theMTF response is
related tothe small peaks on either sideof the edges.The leftmost
edge in the plot is a portion of the step response of the system
(film + lens). A muchlower spatial frequency is required to
represent it properly. The impulse response the responseof the
system to a narrow line (or impulse) is also of interest. The
impulse response is thederivative of the step response (d(step
response)/dx).The MTF curve is related to the impulse response by a
mathematical operation known as theUnderstanding resolution and MTF
http://www.normankoren.com/Tutorials/MTF.html8 of 12 11/3/2010 1:05
PMFourier 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 thetopic 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
thismagnification (260x), a full frame 35mm image (24x36mm) would
be 240 inches (6.2 meters) high and 360inches (9.2 meters) wide. A
bit excessive, but if you stand back from the screen you'll get an
feeling for theeffects of the lens, film, scanner (or digital
camera), and sharpening on real images.The companion
website,Imatest.com, describes asoftware tool you can use tomeasure
MTF and otherfactors that contribute toimage quality in
digitalcameras and digitized filmimages.Links to general articles
on MTFUnderstanding 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 graphsby Klaus
Schroiff. Another useful explanation.Photodo has several excellent
articles on MTF and image quality. Recommended.MTF Engineering
Notesfrom 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 presentsmuch 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
SystemsClass notes from ECE 102, Center for Electronic
ImagingSystems, University of Rochester. Taught by Dr. Michael
Kriss. Connected with the U of R Image ProcessingLab.RIT Center for
Imaging Science class material is a serious resource well worth
exploring. Basic Principlesof Imaging Science 1. Lectures 17 and 18
on MTF and imaging microstructure are particularly
interresting.Help support this site. Link to Adoramafor your
photographic purchases.Human visual acuityThe ability of the eye to
resolve detail is known as "visual acuity." The normal human eye
can distinguishpatterns of alternating black and white lines with a
feature size as small as one minute of an arc (1/60
degreeUnderstanding resolution and MTF
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PMor /(60*180) = 0.000291 radians). That, incidentally, is the
definition of 20-20 vision. A few exceptionaleyes may be able to
distinguish features half this size. But for most of us, a pattern
of higher spatial frequencywill 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 focallength
of 16.5 mm), thiscorresponds to objects oflength = (angle in
radians)*d= 0.000291*d. For example,for an object viewed at
adistance of 25 cm (about 10inches), the distance youmight use for
close scrutinyof an 8x10 inch photographicprint, this would
correspondto 0.0727 mm = 0.0029inches. Since a line paircorresponds
to two lines ofthis size, the correspondingspatial frequency is
6.88lp/mm or 175 lp/inch.Assume now that the imagewas printed from
a 35mmframe enlarged 8x. Thecorresponding 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 theMTF
of a digital camera above 2800 LW/PH (Line Widths per Picture
Height) measured by Imatest SFR, hasno effect on the appearance of
the print. That's why the highest spatial frequencies used in
manufacturer'sMTF charts is typically 40 lp/mm, which provides an
excellent indication of a lens's perceived sharpness in an8x10 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
thesmallest feature of importance, viewed at 25 cm, is 0.01 inches
3 times larger. It shouldn't be a surprise thatfocus 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 theright from the Handout #9: Human Visual Perception from
Stanford University course EE368B - Image andVideo 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 30CPD,
corresponding to a one minute of an arc feature size, MTF drops
from 0.4 for the 2 mm pupil to 0.16 forthe 5.8 mm pupil. (Now you
know your eye's f-stop range. It's similar to compact digital
cameras.) AnotherStanford page has Matlab computer models of the
eye's MTF.Understanding resolution and MTF
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1:05 PMThe human eye's MTF, which is limited athigh angular
frequencies by the eye'soptical system and cone density, does
nottell the whole story of the eye's response.Neuronal interactions
such as lateralinhibition limit the eye's response at lowangular
frequencies, i.e., the eye isinsensitive to very gradual changes
indensity. The eye's overall response is calledits contrast
sensitivity function (CSF).Various studies place the peak CSF
forbright light levels (typical of print viewingconditions) between
6 and 8 cycles perdegree. The graph on the left uses
anapproximation (equations below) thatpeaks just below 8
cycles/degree.CSF is used in a measure of perceptualimage sharpness
called Subjective QualityFactor (SQF), which includes MTF,
CSF,print size, and typical viewing distance.SQF has been used
since the 1970s insideKodak and Polaroid, but it was difficult
tocalculate, and hence remained obscure,until it was incorporated
into Imatest SFRin 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 onInformation Theory, pp.
525-535, Vol. 20, No 4, (1974), cited on this page of Kresimir
Matkovic's 1998 PhDthesis.CSF( f ) = 2.6 (0.0192 + 0.114 f )
exp(-0.114 f )1.1The 2.6 multiplier can be removed and the equation
can be simplified somewhat. The dc term (0.0192) can bedropped 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 testingresource center and Stanford
University. Page 3 from Stanford has a plot of the MTF of the human
eye. Ibelieve the x-axis units (CPD) are Cycles per Degree, where a
pair of 1/60 degree features corresponds to 30CPD.Next: Part 1A:
MTF in film and lenses | Part 2: Scanners and
sharpeningUnderstanding resolution and MTF
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1:05 PMImages and text copyright 2000-2010 by Norman Koren.Norman
Koren lives in Boulder, Colorado, founded Imatest LLC in
2004,previously worked on magnetic recording technology. He has
been involved withphotography since 1964.Understanding resolution
and MTF http://www.normankoren.com/Tutorials/MTF.html12 of 12
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