# What Is Modulation Transfer Function?

In an earlier column, we spoke of the maximum resolution capability of a film stock, expressed in line pairs per millimeter. Although this is a valuable parameter with which to establish a frame of reference, we said at the time that things are really more complex than a single number, and we mentioned the concept of modulation transfer function (MTF). What is MTF, and what may we determine from it?

Film's pixels, or light-sensitive grains, are randomly arranged, giving it the same spatial resolution in all directions. Using resolution in the horizontal direction as an illustration, if we photographed a test chart containing a sequence of parallel vertical lines that are alternately white and black, the resolution number indicates the maximum number of such pairs that may occupy a horizontal distance of one millimeter, and still be resolved by a viewer. A much more informative parameter is MTF.

In order to understand MTF, we must first define modulation within this context. Our set of white and black vertical bars, as it appears on the test chart, may be considered to be a square wave grating: It consists of alternating white (fully reflective) and black (fully absorptive) bars with an instantaneous step transition between each bar and the next. Their contrast ratio, the ratio of maximum light intensity to minimum light intensity, will ideally approach infinity. As the light coming from the test chart, and the modulated medium that stores or transmits the light information (film, in this case) degrade the original square-wave signals, the square-wave transitions degenerate into gentler sine-wave transitions (the MTF declines). The net result of this decline is that the contrast ratio between light and dark bars will be reduced to some degree. When the test chart is illuminated, it is obvious that the maximum light intensity will be reflected from the white bars and the minimum light intensity will come from the black bars. The light's modulation is defined by the equation:

Modulation = (Lmax - Lmin)/(Lmax + Lmin)

Where Lmax and Lmin are the maximum and minimum light intensities respectively.

In the ideal condition, Lmin is 0, and in that case, the modulation equation becomes Lmax/Lmax, or 1. In the worst case, Lmin equals Lmax, and in that case, modulation equals 0. The light reflecting off the chart or object and entering the lens is called the stimulus. When the stimulus passes through a lens, an image is projected onto a sensor or film. The lens will degrade the contrast ratio of the stimulus to some degree, and the image modulation will therefore be a lower number than that of the stimulus. The modulation transfer function is the expression of how well the stimulus modulation is transferred to the image. It is the ratio of the modulation of the image, Mi, and the modulation of the stimulus, Ms:

MTF = Mi/Ms

We can see that MTF will always be less than 1 for a real-world example.

SPATIAL RESOLUTION

Spatial resolution is a measure of how well spatial detail is preserved. We have seen that the resolving power is a number that tells us how many line pairs per millimeter may be resolved. This number tells us something, but it certainly does not tell us everything. From it we may calculate the highest frequency that can be reproduced with some selected contrast ratio, but many other questions present themselves. We know that the contrast ratio of the image will be less than that of the stimulus, but how much less? What is the degree of contrast degradation at the highest frequency that may be resolved? What is the degree of contrast degradation at half the highest frequency, or at any other frequency below the maximum? Does the contrast ratio degrade slowly or abruptly as frequency increases? MTF curves answer these questions.

An MTF curve is obtained by plotting Mi/Ms, or the extent of preservation of the contrast ratio, on the vertical axis, versus frequency on the horizontal axis. The alert reader will recognize that this is something akin to an audio frequency response plot. An MTF plot might look like the hypothetical curve shown in Fig. 1. The Y-axis is often labeled "contrast" or "response", and it is usually plotted either as a ratio from 0 to 1, or in percent. The X-axis is often labeled "spatial frequency" or "frequency". The hypothetical example in Fig. 1 indicates that response falls off quickly and almost linearly as frequency increases. This would not be a desirable MTF curve. Fig. 2 shows three hypothetical curves, from each of which we may deduce several things. Curve A indicates a combination of relatively good contrast (the curve is high on the vertical axis) and relatively good resolving power (the curve stays high for a good horizontal distance and falls off gently). Curve B indicates good contrast at low frequencies, but poor overall resolving power, as the curve falls precipitously starting at a relatively low frequency. Curve C indicates good resolving power, as the curve stays above zero for a long horizontal distance, but poor overall contrast, as it falls to a low level quickly and stays there.

(click thumbnail)Fig. 1
(click thumbnail)Fig. 2
(click thumbnail)Fig. 3
An example of a film stock's MTF is shown in Fig. 3, which is the published set of MTF curves for Kodak Vision 200T film stock, also known as 5274 (35 mm) and 7274 (16 mm). Vision 200T is a tungsten-balanced color negative stock with a published speed of ASA 200. The first thing we notice is that there are three curves. Color film has red, green, and blue dye layers, and each dye layer has its own response characteristic and its own MTF curve. We also note that these curves are plotted on a log-log graph in order to clearly show the full areas of interest. A typical single maximum resolution number is often selected to correspond to the point where the green dye layer's response is 50 percent. This compromise is chosen because the eye is most sensitive to colors in the green region, and a response of 50 percent is sufficiently high to yield a very good contrast ratio. Using these criteria the single maximum resolution figure for Vision 200T would be about 75 line pairs per millimeter. Some of the curves go over 100 percent because they are normalized so that 100 percent corresponds to a somewhat low frequency.

WHAT THE CHART SAYS

Examining this MTF chart can tell us a lot about this film. We see that the blue layer has the best response, while the red layer has the poorest response. These are typical characteristics of color film stocks, owing to the materials used to make them. The blue layer has response of at least 70 percent out to 50 line pairs per millimeter, and falls off gently thereafter. The green layer's curve is the same shape as the blue layer's, but a little lower. The red layer's response falls off more quickly than the other two, its 50 percent response point being about 40 cycles/mm.

It is apparent that this MTF chart tells us a lot more about what we might expect from 5274/7274 film stock than a single maximum resolution figure. It essentially shows us contrast ratio vertically and resolution horizontally for each dye layer. In a broad sense, when we see the MTF curves staying high and horizontal for a good distance out on the resolution axis, and falling off gently rather than abruptly crashing, we know that this stock will give us sharply detailed pictures, provided that we light it correctly and focus our lens well.