Good question, Vladimir.

The modulation transfer function (MTF) is frequently used by scientists in both academia and industry to assess the  contrast between features within images formed by an optical system. Typically, image-forming systems with higher contrast will make better images, though I think there is often some confusion as to what "better" means in this context and how exactly the MTF is related to contrast. So, to try to help you think about what this means, I'll give you a couple points to consider.

The first thing to consider when trying to understand the connection between contrast and MTF is that contrast is usually defined in optics text books as a measure of the difference between the irradiance values at two different points in the image. The MTF, however, is the magnitude of the Fourier transform of the point-spread function of the system. For a system forming two-dimensional images on a flat screen, this means that the MTF is a function of two variables: spatial frequency in the x- and y-directions like Michael mentioned above.

From the start, it's confusing to claim that a two point measurement like contrast is determined from a multivariate function of spatial frequency such as the MTF. It would be better, in my opinion, to say that the MTF is something of a global, or average measure of how well small features are contrasted against larger features in an image. Thus, the MTF is a predictor of what the two-point measure of contrast will be between features of different sizes in an image.

In other words the MTF will tell us how well these features will stand out against the background and is therefore useful as a descriptor of an optical system. Now, if no noise were present in the system and you could measure images with unlimited dynamic range, the level of contrast would not affect your ability to resolve two closely-spaced small objects. This is not the case in the real world, though, so contrast and the MTF effectively tell you whether you're going to be able to resolve two closely-spaced features in an image or not. A higher contrast means you have a better ability to resolve two objects.

The other point I want to stress is that the MTF is really just a convenient mathematical tool that reduces the amount of information represented in the optical transfer function (OTF), just as Parviz said above. The OTF is the Fourier transform of the point spread function (PSF). So, if you know exactly what the PSF of your system is, then you in principle know the MTF; you just have to extract it from the PSF by doing a few mathematical operations.

I'm not too familiar with the effect of turbulence on MTF, though I would imagine that it reduces the effective MTF of your imaging system. You could consider your effective imaging system as the turbulence + the optics, with the turbulence acting as an independent component. I'm not sure if this is usually how people in the field consider turbulence, though.

Finally, remember that MTF is a function, so you can't quite strictly say that the value of the MTF decreases when aberrations are present, because the MTF has multiple values depending on the spatial frequency. Instead, you can say that the value of the MTF decreases for any given spatial frequency in the presence of aberrations. It decreases especially fast for high spatial frequencies, which ultimately reduces your ability to resolve small features because their contrast is poor.

I hope this helps!

Kyle

MTF is a mathematical tool to interprete the physical meaning. Turbulence, aerosol, fog , volcanic ash etc. can be  modeled with various matrics based on absorption and  Rayliegh, Mie and multiple-scattering effects. There are a lot of relevant  papers in periodicals pertaining atmospheric modeling. 

Good question, Vladimir.

The modulation transfer function (MTF) is frequently used by scientists in both academia and industry to assess the  contrast between features within images formed by an optical system. Typically, image-forming systems with higher contrast will make better images, though I think there is often some confusion as to what "better" means in this context and how exactly the MTF is related to contrast. So, to try to help you think about what this means, I'll give you a couple points to consider.

The first thing to consider when trying to understand the connection between contrast and MTF is that contrast is usually defined in optics text books as a measure of the difference between the irradiance values at two different points in the image. The MTF, however, is the magnitude of the Fourier transform of the point-spread function of the system. For a system forming two-dimensional images on a flat screen, this means that the MTF is a function of two variables: spatial frequency in the x- and y-directions like Michael mentioned above.

From the start, it's confusing to claim that a two point measurement like contrast is determined from a multivariate function of spatial frequency such as the MTF. It would be better, in my opinion, to say that the MTF is something of a global, or average measure of how well small features are contrasted against larger features in an image. Thus, the MTF is a predictor of what the two-point measure of contrast will be between features of different sizes in an image.

In other words the MTF will tell us how well these features will stand out against the background and is therefore useful as a descriptor of an optical system. Now, if no noise were present in the system and you could measure images with unlimited dynamic range, the level of contrast would not affect your ability to resolve two closely-spaced small objects. This is not the case in the real world, though, so contrast and the MTF effectively tell you whether you're going to be able to resolve two closely-spaced features in an image or not. A higher contrast means you have a better ability to resolve two objects.

The other point I want to stress is that the MTF is really just a convenient mathematical tool that reduces the amount of information represented in the optical transfer function (OTF), just as Parviz said above. The OTF is the Fourier transform of the point spread function (PSF). So, if you know exactly what the PSF of your system is, then you in principle know the MTF; you just have to extract it from the PSF by doing a few mathematical operations.

I'm not too familiar with the effect of turbulence on MTF, though I would imagine that it reduces the effective MTF of your imaging system. You could consider your effective imaging system as the turbulence + the optics, with the turbulence acting as an independent component. I'm not sure if this is usually how people in the field consider turbulence, though.

Finally, remember that MTF is a function, so you can't quite strictly say that the value of the MTF decreases when aberrations are present, because the MTF has multiple values depending on the spatial frequency. Instead, you can say that the value of the MTF decreases for any given spatial frequency in the presence of aberrations. It decreases especially fast for high spatial frequencies, which ultimately reduces your ability to resolve small features because their contrast is poor.

I hope this helps!

Kyle

One way to appreciate the MTF is to close one eye and look at a hair comb a comfortable distance away, you will see alternating light and dark based on the spaces between the comb's teeth. As you move the comb further away it becomes a more uniform gray pattern.  Moving the comb back is comparable to having more "line pairs" or increasing the spatial frequency. They greying comes from the change in the visibility that the lens of your eye can transfer onto your retina. Not a perfect example, but it is a simple way to grasp a physical meaning. Cheers!

In my opinion, when stars are observed, the OTF is equivalent to the Airy disk. The latter corresponds to the first diffraction maximum of the telescope aperture. When the lens forms an image of a nearby object on a screen, its aperture forms a diffraction image on the same screen.

MTF is a generalization of OTF -- valid for frequencies other than the optical range -- in the same way as an effective aperture is associated to a horn or reflector antenna.

I suppose that your second question draws on an analogy of light with sounds. In that case a small paper disk is used to measure the direction of a sound field. In that analogy turbulence, as well as aberration cause distortion.

The modulation transfer function (MTF) is the modulus of the optical transfer function (OTF).  The OTF is generally a complex-valued function and if sum the squares of the real and imaginary parts and take the square root of that sum you have the MTF. (Note that the MTF is NOT a generalisation of the OTF as suggested by Vesely above). 

So what is the OTF?  The OTF can be described  as a function of spatial freqency, s, that describes how truly the pattern in the image plane of an optical system represents the pattern in the object field being imaged.  To understand this more precisely we need to have a basic understanding of Fourier representations.  Without being mathematically rigorous we may take it as true that a scene can be represented as the weighted sum of sinusoidal functions of light and dark having frequencies that are integral multiples of a fundamental frequency - for a 2-D image we must take functions in the horizontal and vertical directions.  The optical system is considered to be a passive device that simply scales each freqency by he optical magnification and then reproduces each of those sinusoids with some lack of perfection.  Two types of loss are captured in the transfer function representation - a relative reduction in amplitude and some shift in the image from the geometrically perfect positioning.  The OTF captures both of these through its two components - the modulation transfer function that gives the ratio of amplitudes of each of the sinusoids, and the phase transfer function that describes the lateral shifts in terms of fractions of a period by which each of the sinusoids is shifted.  Imagimg defects such as astigmatism or coma cause both modulation loss as well as phase shifting, while spherical aberration causes only modulation loss while (pure) distortion causes only phase shifting.

Atmospheric effects will cause both amplitude and phase degradations and these will be non-stationary with time.  Inhomogeneities in the atmosphere may produce characteristic structures in te phase transfer function such as a linear dependence of phase on spatial frequency (achieved also with a tilted lens in an imaging system).

In my previous contribution I tried to avoid suggesting that the mathematical tools of the Fourier transform are mandatory to deal with Fraunhofer's diffraction. I just explained in a few words the physical meaning I would attach to the MTF, connecting it with the much older concept of Airy disc. The “wavy” effect of the aperture of the telescope on the images of stars, which I called OTF, was noticed well before it was mathematically interpreted. On the plane of the image of a diffraction limited system one always sees that effect of the aperture, which is not its image. According to this physical interpretation the blur of stars, and of the images of incoherently lit objects, is due to the “Fraunhofer diffraction image” of the aperture. Therefore, the MTF can be conceived as a generalization of the OTP, that takes into account how that train of thought evolved from the Airy disk to the analysis of optical lenses, and finally toward the standard conventions utilized in electrical engineering about amplitude and phase.

I agree that, for Ronchi rules in nearly monochromatic light the OTF is due to them. In fact, in the applications of the linear filter theory to electro-optics systems, where the Fourier transform is applied, contrast variation depends also on  the constituent Fourier components of the imaged object.