At MTF basis for reckoning depth of field

In the matter of depth of field we ask this question:

For a given aperture and focus distance, over what range of object distances will the object be imaged with not over some degree of blurring from misfocus (which degree of blurring we choose based on our needs)?

We usually use as the metric of "degree of blurring" the diameter of the circle of confusion for an object at the distance of interest: the circular blur figure created from a point on the object in the case of misfocus.

To adopt a limit on the degree of blurring, we state a "circle of confusion diameter limit" (COCDL); the diameter of the circle of confusion at the chosen degree of blurring.

But how do we select a COCDL? Ideally, we would choose it taking into account how the image will be viewed (at what magnification compared to the image on the focal plane, and from what distance), together with what the viewer's expectations of "un-blurred-ness" will be.

But to provide a universal answer for "general use", especially in the film era, COCDL values were recommended based on:

• Some arbitrary (if perhaps reasonable) assumptions as to the size of the viewed image compared to the image on the focal plane, and the distance from which It will be viewed.

• The concept that the "acceptable" misfocus is that which can be noticed by the human eye, based on its resolution.

In modern times, there has been a trend to move away from this concept - which has a potentially great disconnect from the actual viewing situation - in favor of adopting as the "maximum acceptable blurring" that blurring which would noticeably degrade the resolution of the system compared to what it would be in a situation of perfect focus.

But defining this is problematical as well, in part because the "resolution" of a camera system does not even have a precise, objective definition.

Because of that conundrum, when we feel we need a quantifiable, objective metric for "resolution", we tend to base it on the MTF of the system. We may arbitrarily (there's that word again) choose the spatial frequency at which the MTF is 50%. This is arbitrary but fully reproducible.

I have earlier suggested that, in order to quantify the impact of diffraction on the resolution of the system, we might wish to choose as our criterion for "resolution noticeably degraded by diffraction" the situation in which the MTF reflecting the effect of diffraction is 50% at spatial frequency where the basic MTF is 50%.

Now we note that for any degree of blurring, the effect of the blurring can be reflected by an MTF (just as is true for the effect of diffraction).

So perhaps we should follow the same philosophy as for diffraction with regard to choosing a COCDL to be used to reckon the depth of field of a certain camera setup: we would choose as the limit the COCD for which the MTF reflecting the blurring has the value 50% at the frequency we consider as representing the resolution of the system (for perfect focus).

Typically that resolution is on the order of 75% of the Nyquist frequency (sn) dictated by the sensor sensel pitch (sn = 1/2p, where p is the sensel pitch).

If we go through all the math involved, we find that the "blur" MTF has the value of 50% at this "system resolution" frequency when the diameter of the circle of confusion is approximately twice the sensel pitch.

That is, we would choose the COCDL as 2p.

Best regards,

Doug

Esteemed colleagues,

My apologies for the above essay not being as clear or well-organized as it should have been.

After I had gathered all the information and my thoughts and started to write, Carla reminded me that in just a little while we were due to leave for an organization meeting. Rather than just putting the work on hold and finishing it when we got home, I just pressed on.

In any case, here is summary of the punch line:

• We often find that the "resolution" of a system (for perfect focus, with negligible effect of diffraction, and a "pretty good" lens) is about 75% of the Nyquist frequency determined by the sensel pitch.

• If we have, for objects at some distance, misfocus such that the diameter of the circle of confusion is twice the sensel pitch, then at the resolution frequency mentioned above (75% of the Nyquist frequency), the effect of the blurring will be roughly to cut the overall MTF to half its value with no blurring.

• If we find that amount of degradation of the system MTF an attractive, albeit wholly arbitrary, definition of a "significant effect of misfocus blurring", we can embrace it in our depth of field calculations by adopting a COCDL (circle of confusion diameter limit) of twice the sensel pitch.

Best regards,

Doug

By way of comparison, if we consider misfocus such that the diameter of the circle of confusion is one times the sensel pitch, then at a frequency of 75% of the Nyquist frequency (which we often find to be about the resolution frequency in the absence of diffraction and misfocus blur), the misfocus blur MTF has a value of about 83%.

Thus, in that situation we can reasonably say that the impact of that degree of misfocus blur on the system "resolution" is very slight.

Best regards,

Doug

Here we will look a little more closely at the nature of the MTF curve reflecting the effect on system response of blur from misfocus.

For my own convenience, I will use (with adaptation) an excerpt from very nice figure taken from David Jacobson's excellent "Lens Tutorial":

http://photo.net/learn/optics/lensTutorial#part5

Context

The figure is predicated on a certain scenario. Whole this may not be the one of most interest to us, I stay with it so as not to have to reconstruct the figures! We will see that the observations from the figure can be adapted to scenarios better matching our recent interests.

The predicates of the scenario are:

• The misfocus is that which gives a diameter of the blur figure (circle of confusion) of 0.030 mm.


• The diffraction is that for which the "diameter" of the Airy disk is 0.030 mm.


With the usual assumption of a wavelength of 555 nm, that diffraction would result from an aperture of about f/22, and that is the basis of the diffraction MTF curve in the figure to follow.

The figure

Here is the misfocus blur "MTF" curve:


As expected, the x-axis is in terms of spatial frequency, in cy/mm. Note that the y-axis is labeled "OTF" (Optical Transfer Function"), not MTF.

About the Optical Transfer Function

The actual factor that relates, at any spatial frequency, the "output" of an optical system to its "input" is in fact the Optical Transfer Function (OTF). This is a phasor quantity (often but inaccurately called a vector quantity). That means that its value has both an amplitude and a phase aspect.

The amplitude aspect tells the ratio between the amplitude of the output modulation to the amplitude of the input modulation. The phase aspect tells us if there is any spatial phase shift between the input modulation and the output modulation - if the two variations do not "align".

Now we cannot plot on a single axis such a phasor quantity.

But note that for a sinusoidal variation (and that is of course what any single frequency component is, by definition), a phase shift of 180° gives exactly the same result as a "polarity reversal." And of course a polarity reversal is the same thing as a change in the amplitude from a positive value to a negative value (with no phase shift).

Thus, if the only phase values our OTF takes on are 0° and 180°, we can consider the OTF to always have a phase value of 0° but amplitude values that are both positive and negative.

And of course we can plot that on a single axis on our chart.

Now the Modulation Transfer Function (MTF) is defined as the magnitude aspect of the OPF (that is, it ignores the phase aspect). And by formal definition, that is everywhere positive.

Now, getting a little ahead of the story, we note that the OTF curve for the misfocus blur (green) is in some places negative. That means that in this range of spatial frequency, the "polarity" of the output modulation is reversed compared to the input. That is, if we are considering a sinusoidal bar pattern at such a frequency, where the "black peak" should fall, we will see a "white peak", and vice-versa.

If we plotted the actual MTF curve (rather than the OTF curve), it would be incidental except that in the region where the OTF is negative, the MTF would be positive (that portion of the curve just being flipped vertically about the y-axis).

The range of the curves

We may be tempted to say, "surely there is no meaning to the curve above the Nyquist frequency, or if we think in terms of film, above its limiting resolution". But at this stage, there is no sensor, no Nyquist frequency. no limiting resultion of some presumed film. These curves only relate to the image on an abstract focal plane, for these two assumptions. The theoretical OTF extends to infinite spatial frequency.

• An aperture of f/22

• Misfocus such that the diameter of the blur figure (circle of confusion) is 0.030 mm.

Now let's get real

Now, let's indeed imagine that this relates to a hypothetical digital camera. Working backward from the COCD of 0.030 mm (chosen by the original author of the figure), let me suppose that this is in fact twice the sensel pitch, which must then be 0.015 mm.

If that is so, then the Nyquist frequency for this sensor must be 33.33 cy/mm. I have in fact drawn a marker line on the graph at that frequency.

Often we find that, in the absence of any significant impact of diffraction or misfocus blur, the resolution of the camera will turn out to be about 75% of the Nyquist frequency. Let's assume that this is the case for our hypothetical camera, a resolution of 25 cy/mm (which we would often call, in this context, "25 line pairs per mm"). I have also drawn a marker line on the graph at that frequency.

We see from the green curve that the OTF (the MTF will be the same) reflecting the effect of misfocus blur at that frequency (25 cy/mm) is about 44%. We would no doubt consider that to cause a substantial degradation of system resolution. Still, that might be a good benchmark for
"allowable" blur from misfocus - a COCDL of 0.030 mm might be appropriate.


Now, suppose that the sensel pitch was such that our arbitrary diameter of the circle of confusion was equal to one pixel pitch. That is, the pixel pitch would be 0.060, the Nyquist frequency would be 16.67 cy/mm, and thus the likely resolution of the camera would be 12.5 cy/mm. I have also drawn a marker line on the graph at that frequency.

Again looking at the green curve, we see that at this frequency, the OTF (and thus the MTF) would be about 83%. We would likely consider this "not consequential".

Summary

From all this we can see, from theoretical considerations, a choice of COCDL in the area of 2.0 times the pixel pitch might be desirable for doing depth of field considerations based on the "blur that does not consequentially degrade system resultion" philosophy.

Best regards,

Doug