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On the two meanings of "modulation transfer function" (MTF)

Doug Kerr

Well-known member
In mathematics, when the value of one variable depends on the value(s) of one or more other variables, the first variable is said to be a function of the other variable(s). Hold that thought.

In what follows, imagine a "black and white" scene and "black and white" photography.

We can think of the scene as an array of varying luminance. If the luminance were constant across the scene, the scene would be "flat gray", and (except for some scientific work) of no interest to photograph.

We look to the camera lens to transfer this array of luminance into a corresponding array of illuminance in the image that we let fall on film or a digital sensor, thus creating a photograph.

We can think of the variation of luminance in the scene (or illuminance in the image) as modulation of the luminance (illuminance), a term borrowed (mostly) from electrical engineering. The term in fact basically merely means variation.

If we consider the luminance values of two (probably nearby) points in the scene, we can quantify the difference in terms of the metric depth of modulation. This is obtained by taking the difference between the two luminance values and dividing it by the average of the two luminance values.

If we consider the modulation of illuminance between the corresponding points in the image created by the lens, we can quantify the depth of modulation there in the same way.

Our wish would be that the lens would faithfully transfer the modulation of luminance that defines the scene into a precisely-corresponding modulation of luminance in the image. But, for many reasons is does not.

Thus we become interested in a certain metric: the ratio of the depth of modulation of illuminance (say, between two closely spaced points) in the image to the depth of modulation of luminance between those two points in the scene that was its source. Ideally, that ratio would always be 1, but in practice it is not.

I will for the moment (and at some later times) refer to this ratio as the modulation transfer ratio, a term that to me clearly describes what it is, and does not suggest anything else.

But in fact, in the practice I discuss here, the modulation transfer ratio is usually called the modulation transfer function (MTF). The term "ratio" is just lost, and the term "function" is tacked on to remind us that the MTF is a function of several other variables.

If we consider a zoom lens, those other variables are (principally):

• The spatial frequency of the modulation of image luminance (a measure of how rapidly the luminance changes; looking at it another way, a measure of the "fineness of the detail" described by that variation of luminance).

• The focal length (for a zoom lens)

• The aperture.

• The distance from the center of the image.

• The direction of the "track" along which the variation of luminance is noted.

Now, we may wish to plot the value of the MTF vs. one of those other variables, with the remaining variables being given fixed values (maybe several fixed values, resulting in several curves). Two such presntations we commonly encounter are:

A. Plotting MTF vs. spatial frequency. (Perhaps there will be several curves for, for example, different values of the aperture.)

B. Plotting MTF vs. distance from the center of the image. (Perhaps there will be several curves for, for example, different spatial frequencies (often only two, sometimes three).)

Presentation ""B" is the one most often published by lens manufacturers.

Presentation "A" leads us to the determination of the "resolution" of the lens (at least in one of the common definitions of that). Form B does not. It lets us see, for the spatial frequencies represented on its different curves, what the MTF of the lens is for different distances from the center of the image.

Suppose I am interested in the widely-used criterion that the "resolution" of a lens will be stated as the spatial frequency at which the MTF drops to 50% of its value for low frequencies (perhaps at the center of the image). Can I tell that from MTF presentation "B". Not usually.

A second meaning of "MTF"

I described earlier that the term modulation transfer function is essentially a synonym for what is more clearly called modulation transfer ratio. It is the quantity that corresponds to the vertical axis of both "presentation A' and "presentation B". So that axis is usually labeled "MTF".

But of course, in making either of these two "plots", we are dealing not just with our interest in a particular value of the MTF (perhaps at a specific spatial frequency) but rather with how it varies with (for example) spatial frequency, or maybe distance from the center of the image. In other words we are interested in the function that describes the variation of the MTF. It is what the plot shows. What would we call that?

Well, we could call it the modulation transfer function function. (MTFF?). But we don't. We call it the modulation transfer function (MTF).

But i thought that term meant a value of the modulation transfer ratio, not how it varies.

Yes, i means that too.

So what we suffer from here is the inclination of mathematicians to label a value determined by a function with the name of the function.

For example If we know that the temperature varies with the time of day, and plot the function that describes that relationship (we decide to call relationship that the "temperature-time function"), with temperature on the vertical axis and time of day on the horizontal axis, the mathematician would not label the vertical axis "temperature" but rather "temperature-time function".

So we in effect have a situation that the modulation transfer function (MTF) of a lens varies according to several factors, the description of that variation as we observe it being called the modulation transfer function (MTF).

Silly as that sounds, we have gotten used to it.

But be cautious when you hear "MTF" that you know which of two things is meant.

Fortunately, most often, the curves are spoken of as "MTF curves". That is really quite appropriate, so long that we recognize that this means that the curves show the variation of MTF, and not that the curves "show the MTF".

Best regards,

Doug
 

Jerome Marot

Well-known member
The modulation transfer function is a function which, for a given lens, associates:
as input: a vector comprising
  • the spatial frequency of the source (as determined by the Fourier transform of the object)
  • the orientation of the spatial frequency of the source
  • distance from the optical center (an only that for a lens which is symmetric around the optical axis, i.e. not an anamorphic lens)
  • wavelength
  • aperture
  • focal length (for a zoom lens)
  • focussing distance

to an output which is a real number. So it is a function from ℝ^7 -> ℝ.

Plots of this function in a plane are, mathematically speaking, projections of the curve, which is defined in ℝ^8 space, onto an ℝ^2 space.

It is always a function, functions are mathematical objects which can have more than one parameter as input.

Obviously, if you are presented in a curve plotted in ℝ^2, you can only link it to its physical meaning if you have an idea of what parameters are used for the curve. In that sense, it is good that you remind us that not all curves have the same meaning, even when they are all labelled "MTF".

The modulation transfer function is itself a projection of the lens transfer function, which entirely characterises an optical system.
 

Doug Kerr

Well-known member
Hi, Jerome,

The modulation transfer function is a function which, for a given lens, associates:
as input: a vector comprising
  • the spatial frequency of the source (as determined by the Fourier transform of the object)
  • the orientation of the spatial frequency of the source
  • distance from the optical center (an only that for a lens which is symmetric around the optical axis, i.e. not an anamorphic lens)
  • wavelength
  • aperture
  • focal length (for a zoom lens)
  • focussing distance

to an output which is a real number. So it is a function from ℝ^7 -> ℝ.

And that output, like the function itself, is called the modulation transfer function (MTF).

Plots of this function in a plane are, mathematically speaking, projections of the curve, which is defined in ℝ^8 space, onto an ℝ^2 space.

It is always a function, functions are mathematical objects which can have more than one parameter as input.

Obviously, if you are presented in a curve plotted in ℝ^2, you can only link it to its physical meaning if you have an idea of what parameters are used for the curve. In that sense, it is good that you remind us that not all curves have the same meaning, even when they are all labelled "MTF".

The modulation transfer function is itself a projection of the lens transfer function, which entirely characterises an optical system.

Thanks for that nice and useful discussion.

Best regards,

Doug
 
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