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Modulation Transfer Function

By far the most sophisticated resolution criterion for the objective evaluation of an optical system is the polychromatic Modulation Transfer Function (MTF).
The MTF is a quantitative measure of the effectiveness of an optical system (lens or lens system) to recreate the contrast detail in a scene. Mathematically, the MTF is the magnitude of the complex Optical Transfer Function. Before the development of MTF techniques (in the 1950s), the subjective evaluation of an optical system by viewing a resolution chart (bar patterns of varying pitch) had proven unreliable and lacking consistency. Between different observers, variations of 20 - 30% were not unusual. Further, the use of single - value performance measures (resolving power, classical Strehl ratio, ...) to predict image quality was imprecise.
Understand that MTF and image quality are not one and the same.
Image quality is dependent on the information transfer (to recreate the contrast detail) from object to image, the effect of noise on the perceptual process and psychophysical factors, (inherent characteristics of the human visual system). The optical system is just one component of an 'imaging chain' that includes human vision.
Today, most optical designs can benefit from automatic optimisation programs to control the aberration balance, the manufacturing tolerances and thereby maximise the performance. Modern lenses are engineered to offer high performance (and build quality) using aspherical elements and glasses that reduce chromatic effects.
Nearly all optics manufacturers publish abridged MTF data (though optical and mechanical tolerances can be specified, the computed MTF is for a typical lens)
for their range of commercial products. Moreover, the MTF data are wavelength dependent, the spectral quality of the light must be characterised. MTF performance that is limited by diffraction alone is referred to as diffraction - limited and cannot be exceeded. Bear in mind, diffraction is a fundamental property of wave propagation (that is always present), the deleterious effects are unavoidable.
Think of diffraction - limited performance as the benchmark for performance excellence.
Real lenses are non - diffraction - limited, residual (primary and
higher - order) aberrations (distortion of the wavefront from an ideal spherical shape that is predicted by geometrical optics) and defocus degrade the MTF performance.

The MTF of a photographic lens may be presented in various guises, ultimately this 'quality criterion' must be related to the actual behaviour of the lens. A typical MTF representation is the modulation transfer (modulationimage/modulationobject) as a function of the spatial frequency, at various field angles, for example, 0 (on - axis), 10, 40, ... . Modulation is essentially contrast (the relative difference between light and dark features) that is defined in a particular way. A sinusoidal object produces a sinusoidal image with reduced contrast. The contrast (C) for a sine wave pattern is illustrated. Other patterns (bar patterns (square wave response)
and real scenes) may be decomposed into a set of harmonically related sine waves using the techniques of Fourier analysis (after J Fourier, 1768 - 1830).

C = (Imax − Imin) /(Imax + Imin)

The spatial frequency, the rate of change of brightness from dark to light (that corresponds to one bar and one space of a resolution chart) is measured in line pairs per millimetre (lp/mm), low lp/mm relate to coarse detail, high lp/mm relate to fine detail. The MTF characterises the transfer of contrast (from object to image) as a function of the spatial frequency. That is to say, the spatial frequency of the object (scene) structure. Lower spatial frequencies are transferred at a higher contrast, on a resolution chart the black and white bars are separated, higher spatial frequencies are transferred at a lower contrast, on a resolution chart the black and white bars are merged. Notice the inverse relationship between contrast and spatial frequency. Fine detail is degraded more than coarse detail. By convention, the MTF is normalised to unity (1 is 100% modulation transfer) at zero spatial frequency.
Off - axis, the sagittal (or radial) MTF and tangential (or meridional) MTF are recorded independently. To gain an appreciation, consider the object (subject) to be a spoked wheel that is placed perpendicular to the optical axis, the sagittal MTFs relate to lines parallel to the radii (the spokes), the tangential MTFs relate to lines perpendicular to the radii (the rim).
On paper, all of the aberrations are correctable and the MTF can approach the diffraction limit from below. A real world lens has an MTF close to unity (100%) at low spatial frequencies, gradually falling with increasing spatial frequency. As the field angle increases there is a marked deterioration of the MTF.

To demonstrate trends, the MTF is plotted as a function of the normalised spatial frequency f/fc, where f is the absolute spatial frequency.                                      

Several points are noteworthy: 

  • the blue curve shows the diffraction - limited performance, notice that for lower spatial frequencies the curve is almost a straight line.
  • the green curves show two lenses with the same diffraction cutoff frequency, the dashed curve is for a lens that has good contrast transfer at low frequencies,
    the solid curve is for a lens that has good contrast transfer at high frequencies and superior detail rendition. Bear in mind, the effect of contrast on the subjective impression of image quality is dependent on the noise in the image. 
  • manual focusing (particularly at close focus distances) must be accurate, focus error can induce a significant modulation reduction and a loss of image quality. For a linear defocus z, the diameter of the corresponding blur circle is z/f/#. The red curve shows the MTF for a defocused lens, reproduced from MATHCAD worksheets. The defocus is described by a dimensionless parameter Δ, where
    Δ = 1/2(f/#)2 z/λ
    . Notice how the MTF falls and goes negative (below zero), the effect is a contrast (black and white) reversal, termed spurious resolution.

You can see that a lens functions as a low pass filter of the scene detail. The spatial spectrum is attenuated selectively, slightly blurring the image (without aberrations and diffraction there would be no blurring, the MTF would be unity at all spatial frequencies). The relevance of low - pass filtering can be easily demonstrated by a familiar example, consider a uniform scene, rolling hills that are crossed by the cables of electricity pylons, low - pass filtering would attenuate the high frequency components, the cables, and leave the low frequency components, the rolling hills, relatively unchanged. 
The diffraction cutoff frequency (
fc) is given by 1/λf/#, where the MTF approaches zero. At spatial frequencies above the diffraction cutoff frequency, the scene detail is removed. The table lists the dependence on f - number (f/#), for light at the centre wavelength (λ = 0.555 μm) of the visible spectrum.

     f/#   Cut - Off (lp/mm)
     2.8                644
     5.6                322
       11                164
      22                  82

These are very high frequencies compared to the cutoff frequency of the human eye.
For a photopic pupil diameter of about 2 mm, the performance of the human eye
is near diffraction - limited. Of course, this does not take account of any retinal limitations. If you have 20/20 vision (the human eye can resolve about one minute
of arc), at the distance of most distinct vision, the resolving power corresponds to a spatial frequency of about 7 lp/mm. Projected onto the image plane, about 60 lp/mm for the 35 mm format. If you have 20/10 vision (under ideal conditions the human eye can resolve about thirty seconds of arc), at the distance of most distinct vision, the resolving power corresponds to a spatial frequency of about 14 lp/mm. Over all age groups, the average visual acuity is somewhere between. Further, for dark lines on a light background and for light lines on a dark background, the perception of detail is different. On a resolution chart, one line pair is one (black) bar and one (white) space.
Even though all camera components (lens, film/image sensor, electronic processor, ...) effect the ability to resolve detail, the human eye is the limiting element.

An alternative MTF representation is the modulation transfer
as a function of the image diagonal (measured from the optical axis) at different spatial frequencies,
for example, 10, 20, 40, ... lp/mm. The top to bottom curves relate to the lowest to highest spatial frequencies. Recall that the MTF decreases as the spatial frequency (lp/mm) increases.
Most MTF data provided by optics manufacturers are charted like so, from the centre to the edge of the image.
For the 35 mm (film) and full frame (digital) format the semi - diagonal dimension is 21.65 mm, for the sub full frame (digital) formats the semi - diagonal dimension varies, (APS - H) 17.25 mm, (Nikon DX) 14.20 mm, (APS - C) 13.35 mm, (Four Thirds) 10.80 mm, ... .
Lenses that are designed exclusively for digital cameras have image circles that
are matched to the image sensor format to guarantee centre to edge image quality.
To demonstrate trends, the MTF (for a 35 mm format lens) is plotted as a function of the image diagonal.

Several points are noteworthy: 

  • at f/11, there are no specific aberrations (closing the lens reduces the residual aberrations), across the image circle the contrast transfer of low spatial frequency detail (10 lp/mm) is constant and close to unity (100%).
  • at f/2.8 there are uncorrected aberrations, across the image circle the contrast transfer of low spatial frequency detail (10 lp/mm) though relatively high is variable, at the edge of the image the MTF degrades rapidly. 
  • at f/11, there are no specific aberrations (closing the lens reduces the residual aberrations) but the effects of diffraction are more evident, across the image circle the contrast transfer of higher spatial frequency detail (40 lp/mm) is constant (but depressed from the 10 lp/mm curve), at the edge of the image
    the MTF degrades rapidly. Mounted on a sub full frame DSLR body, the lens is capable of excellent sharpness from the centre to the edge of the image. 

In practice, the MTF is not measured at discrete spatial frequencies but automatically over a continuum of spatial frequencies, at various field angles, using an instrument that can present an edge or line intensity distribution to the lens under test (there are many techniques based on the ISO 12233 and 15529 standards). The captured image (edge or line spread function) is sampled and processed by software that uses a mathematical procedure called Fourier transformation to compute the sine wave spatial frequency response.
One final point, imaging a bar pattern produces the square wave spatial frequency response or Contrast Transfer Function (CTF). By Fourier analysis, the MTF can be derived from the CTF (MTF = π/4[Σ CTF]). 

To assess the global performance of a lens, you must inspect a family of MTF curves that are (computed or measured) at different aperture and focus distance settings. Some manufacturers provide only computed (nominal) MTF curves at infinity and
close focus distances. Even with meticulous quality control procedures, the computed MTF data may not characterise the true performance of the production lens, due to manufacturing and (assembly) defects.
What to look for on an MTF chart - 

  • The closer to unity (at full aperture, above 80%) the 10 lp/mm curve on the chart, the higher the contrast quality in the image. Curves should be even,
    from the centre to the edge of the image. 
  • The closer to unity (at full aperture, above 40%) the 40 lp/mm curve on the chart, the higher the resolution quality and perceived sharpness in the image. Curves should be even, from the centre to the edge of the image.  
  • Throughout the aperture range (lens closed and open), from the centre to the edge of the image, the MTF curves should not be widely separated. Most 35 mm lenses provide their best performance (sharpness) at about f/8 - f/11.
    Further, the sagittal and tangential MTF curves should more or less coincide, divergent MTF curves may indicate uncorrected off - axis aberrations (astigmatism, coma, ...) and transverse chromatic aberration. The former can be reduced by closing the lens, provided that diffraction blur is not evident. 
    In general, for the higher priced lenses (chromatically corrected), the MTF curves are less separated. 

Learn how to interpret MTF charts, the concepts are not difficult to grasp.

To compare different manufacturers products, always check the scaling of the MTF chart and the particular test conditions. Compare products side - by - side (the same aperture and focal length) and relate the MTF data to actual conditions, your proposed use of the lens, for example, general purpose or specialised photography (action, landscape, wildlife, ...). The MTF chart can help you to evaluate lens performance (to identify limitations) and to form an opinion on value for money
(you are probably purchasing on a budget). For a balanced appraisal, the MTF charts should be read with accompanying lens performance data (distortion, flare, ...). Above all, look for performance excellence at apertures and focal lengths that are best suited to your photographic technique and range of subject matter. Finally, read photographic magazines and browse manufacturers websites, their product reviews (test results) may provide useful advice.

The application of MTF techniques to evaluate optical systems is analogous to the impulse response techniques (that uses a mathematical procedure called Laplace transformation) to evaluate electronic systems. In fact, each camera component has an associated MTF, lens, film/image sensor, electronic processor and integral colour LCD viewing screen, where the camera MTF is the product of the component MTFs. MTF analysis is part of the rigorous treatment that camera and optics manufacturers use to predict product performance. The application of the MTF methodology to photographic systems is reviewed in a forthcoming book.
In conclusion, the Modulation Transfer Function (MTF) is an objective measure of performance that is reproducible (the MTF can be computed and measured) and correlates with our subjective image quality evaluation.

All images and text imajtrek