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6.6. MTF - Modulation transfer function

Modulation transfer function (MTF) is commonly used to describe the outcome of the interaction of two or more PSF. More specifically, the combined intensity distribution resulting from point-image patterns close enough to affect each other. The MTF is a part of a complex function describing this phenomenon, called Optical transfer function, or OTF.

The OTF has two components: (1) MTF, also called contrast transfer function, and (2) phase-transfer function. The latter is present in asymmetrical aberrations, like coma and astigmatism, describing radial linear shift in the intensity pattern. Contrast transfer function alone shows the efficiency of contrast transfer from the object to its image for a single orientation in the aberrated image, normally that along the axis of aberration. FIG. 62 illustrates characteristic form of the MTF, showing contrast drop as a function of spatial frequency n for brightly illuminated object with high inherent contrast.

FIGURE 62: LEFT: Standardized MTF object form are parallel, equally spaced bright contrasty lines on dark background, with irradiance modulation defined as infinite sinusoidal distribution. Line width, equaling that of the bright/light pair, is inversely proportional to the spatial frequency n, normalized to 1 at the cutoff frequency D/λ cycles per radian (the inverse of the angular limit to resolution λ/D in radians), where image contrast drops to zero (linear cutoff frequency is the inverse of the linear limit to resolution λF, or 1/λF in lines per mm, for λ in mm). This form of intensity distribution can be represented as periodic sinusoidal wave, with the contrast determined by the relative intensity of its maxima vs. minima. Image of this pattern has the relative intensity and contrast dampened due to the effect of diffraction, whether without or with aberrations present, At any given frequency, MTF is a ratio of the output (image) to input (object) modulation amplitude. Mathematically, it is a Fourier transform of the aperture's PSF (more specifically, it is an integrated sum of the PSFs for every point over its intensity distribution profile, i.e. convolution of object's Gaussian image and aperture's PSF). At the limit of resolution for aberration-free aperture, the line width is nearly equal to the aperture's FWHM.
RIGHT: Image contrast c is expressed in units of object contrast. Image contrast transfer P in a perfect aperture (yellow overlap area) is the ultimate limit to the overall contrast and resolution set by diffraction for this type of object. Normalized to 1, it equals the relative overlapping area of two circles of unit diameter, at a center separation n, the normalized spatial frequency (top right). Image contrast level A in the aberrated aperture (1/4 wave P-V of spherical aberration) is lower than P over the entire range of frequencies. Relative contrast RC shows better contrast differential between the two, in units of the perfect aperture contrast level c* normalized to 1 over the frequency range. The low-contrast thresholds for bright (LCB) and dim (LCD) details, is an informal, but useful addition by Rutten and Venrooij. It indicates the approximate cutoff levels for typical low-contrast details, requiring higher minimum contrast at the resolution limit than the standard bright, contrasty MTF pattern (original plots are constructed by starting with 0.1 contrast at zero frequency, decreasing toward zero at regular cutoff frequency, with limit to resolution determined by the minimum contrast required by the eye; the LCB and LCD lines above are based on these plots, only with the contrast at zero frequency normalized to 1). As shown, the approximate resolution limit is 1/2 and 1/7 of the standard MTF cutoff frequency for bright and dim low-contrast objects, respectively. Note that different patterns of intensity distribution have different contrast transfer (more general term, since object's intensity pattern does not have to follow any form of modulation); for instance, periodic square wave (i.e. pattern of clearly defined dark and bright lines) will have contrast transfer higher at all frequencies either in aberration-free aperture (solid blue line), or for any given level of aberration.

Note that the above graph represents monochromatic MTF (as indicated by the cutoff frequency 1/LF). Polychromatic MTF cutoff frequency - and to a slight extent, the overall shape - is determined by the intensity distribution within the wavelength range.

For aberration-free clear aperture, the contrast transfer ct can be expressed simply as:

with the angle α in degrees found from cos(α)=ν. Graphically, the contrast transfer ct equals the relative overlapping area of two identical circles, in units of the circle area (FIG. 62, top right), with the circle diameter normalized to 1, and the center separation s=ν varying from 0 when the circles are coinciding (ν=0), to 1 when only touching (the cutoff frequency ν=1).

CONTRAST TRANSFER IN A PERFECT UNOBSTRUCTED APERTURE
MTF frequency	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Contrast transfer	1.0000	0.8681	0.7483	0.6310	0.5138	0.4008	0.2983	0.1957	0.1069	0.0414	0.0000

Similarly, normalized contrast transfer ct for reduced aperture (still of unit diameter, with v=0 when the two circles are touching, smaller inside the larger), equals the overlapping area with the smaller circle appropriately reduced in diameter, with overlapping area being in units of the smaller circle area. The actual range of resolvable frequencies of a smaller aperture is in proportion to the aperture reduction factor.

It is important to understand that the MTF graph, such as the one above, does not set absolute values for the contrast drop, or limit to resolution. Both are strictly applicable only to the particular MTF object form used for its calculation: a pattern of bright lines on dark background, λF/2ν wide linearly, F being the focal ratio ƒ/D (i.e. linear width of the bright line at resolution limit is λF/2, or nearly one fifth of Airy disc diameter). Actual contrast drop-off and limiting resolution will vary with the specific properties of details observed, background, and peculiarities of eye perception, or detector properties.

One example is the resolution threshold for low-contrast MTF-like planetary details which is, according to the LC threshold level in FIG. 62, approximately half of that for brightly illuminated contrasty object. Another is a dark line on light background, which can be detected at an angular width several times smaller than the angular diffraction resolution limit for point-images of ~λ/D radians, because it is the Edge Spread Function, not the PSF, that is applicable in determining its diffraction intensity distribution.

An actual object that comes very close to the standardized MTF scenario is a pair of nearly equally bright stars at the optimum brightness level. Resolution-wise, the MTF is nearly identical to the empirical Dawes' limit in double star observing. However, for pairs farther from the optimum brightness level, or, especially, pairs with significant difference in brightness, the resolution limit is lower, or much lower.

Still, despite the MTF being standardized to a single object form sample and brightness level, it is considered to be a reliable general indicator of the effect of wavefront aberrations - or any other factor affecting wave interference in the focal zone - on image quality. As mentioned, given relatively low RMS wavefront level of any aberration will result in near-identical overall contrast loss, but the specifics will vary somewhat. FIG. 63 illustrates variations in the aberrated PSF (left) and MTF (right) for common wavefront aberrations of 1/13.4 and 1/6.7 wave RMS (graphs generated by Aberrator freeware, Cor Berrevoets).

FIGURE 63: PSF and MTF plots for aberration-free aperture (top pattern), and for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), as well as for this error doubled (~0.15 wave RMS).

● the effect of 1/4 and 1/2 wave P-V wavefront error of defocus on the PSF intensity distribution (left) and image contrast (right). Doubling the error nearly halves the peak diffraction intensity, but the average contrast loss nearly triples (evident from the peak PSF intensity).

● 1/4 and 1/2 wave P-V of lower-order spherical aberration. While the peak PSF intensity change is nearly identical to that of defocus, wider energy spread away from the disc results in more of an effect at mid- to high-frequency range. Central disc at 1/2 wave P-V becomes larger, and less well defined. The 1/2 wave curve indicates ~20% lower actual cutoff frequency in field conditions.

● 0.42 and 0.84 wave P-V wavefront error of coma. Both, intensity distribution (PSF) and contrast transfer change with the orientation angle, due to the asymmetric character of aberration. The worst effect is along the axis of aberration (red), or length-wise with respect to the blur (0 and π orientation angle), and the least is in the orientation perpendicular to it (green).

● 0.37 and 0.74 wave P-V of astigmatism. Due to the tighter energy spread, there is less of a contrast loss with larger, but more with small details, compared to previous wavefront errors. Contrast is best along the axis of aberration (red), falling to the minimum (green) along 45° (π/4) axis. Astigmatism expands the central maxima more than other aberrations. Thus, for given magnitude, it has more of a potential to reduce limiting resolution than spherical aberration or coma.

◄ 6.5. Strehl ratio ▐ 6.6. MTF 2 ►

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