What is the Frequency?

February 23, 2010

Last week’s post, comparing excessive bits in music recording with megapixel overkill in cameras, drew more comments than usual.

So today, I will recklessly continue my strained analogy between audio and optics. I hope you’ll bear with me.

Anyone who has researched stereo gear or sound-recording equipment will have come across graphs of frequency response, like this one:

Shure SM-58 Microphone Response Curve

Frequency Response Curve for a Microphone

The horizontal axis shows the pitch of the sound, increasing from low notes on the left to high ones on the right. The wavy curve shows how the output signal rises and falls with these different sound frequencies.

In a mic like this, air pressure needs to to physically shove a diaphragm and a coil of wire back and forth. When you get to the right end of the graph, with air vibrating 20,000 times per second, mechanical motion has trouble keeping up; the response curve nosedives.

Now, other brands of microphones exist where the frequency response is almost ruler-flat across the spectrum. So is this just a poor one?

In fact, you have to look at this response curve in the context of human hearing ability. As it happens, our ears don’t have a very flat frequency response either:

Equal Loudness Curves

Our Ears Have a Sensitivity Peak

Note that the orientation of this graph is “upside down” compared to the first one. These curves show that our hearing is most sensitive to pitches around 3 or 4 kHz; tones of other frequencies must be cranked up higher to have subjectively equal loudness.

(Fletcher & Munson from Bell Labs published the first comprehensive measurements of this effect in 1933. So you’ll still hear mention of “Fletcher-Munson curves.” But the measurements have been redone several times since. Click the graph for a PDF of a recent, and hopefully definitive study.)

The upshot is, the rolled-off ends of this microphone’s frequency response may not be that obvious to our ears. And actually, the SM58 has become the standard stage mic for vocalists. Its ability to tolerate nightly abuse in a bar room or on a music tour offers a real-world advantage outweighing any lack of extended treble.

If you’ve ever seen live music, you’ve undoubtedly heard vocals through an SM58. Did the roll-off above 10 kHz change how you felt about the singer’s performance? I doubt it.

Shure SM58 Side

You Recognize This

Last week I made a post about lens sharpness, referring to MTF graphs. While the little test target I posted on this blog has solid black & white bars (which was easiest for me to create), formal MTF testing presents a lens with stripes varying in brightness according to a sine function—just like audio test tones:

MTF Sine Target

Increasing Spatial Frequencies

The brightness difference between the white and black stripes at a very low frequency defines “100% contrast.” Then, as you photograph more and more closely-spaced lines, you measure how the contrast ratio drops off, due to diffraction and aberrations.

Ultimately, at some frequency, the contrast percentage plummets into the single digits. At that point details become effectively undetectable.

In fact you can draw a “spatial frequency response” curve for a lens—very much like our audio frequency response graph for the microphone:

Spatial Frequency Response Graphs

Lens Spatial Frequency Response Curves

(This chart comes from a deeply techie PDF from Carl Zeiss, authored by Dr. Hubert Nasse. It discusses understanding and interpreting MTF in great detail, if you’re curious.)

Unlike the earlier MTF graph I showed, here the horizontal axis doesn’t show distance from the center of the frame. Instead, it graphs increasing spatial frequencies (that is, stripes in the target getting more closely spaced) at a single location in the image.

The lower dotted curve shows how lens aberrations at f/2 cause contrast to drop off rapidly for fine details. The heavy solid line shows how contrast at f/16 would be limited even for a flawless lens, simply due to diffraction.

The lens performance at f/5.6 is much better. It approaches, but does not quite reach, the diffraction limit for that aperture. Results like this are representative of many real lenses.

Now, our natural assumption is that greater lens sharpness is always useful. But as I mentioned in my earlier post, our eyes have flawed lenses too (squishy, organic ones), limited by diffraction and aberrations. They have their own MTF curve, which also falls off as details become more closely spaced.

Might we say that our visual acuity has its own “Fletcher-Munson curves”?

To answer that I’m going to ask you to open a new tab, for this link at Imatest (a company who writes optical-testing software). This is their discussion of the human “Contrast Sensitivity Function.”

The top graph shows the approximate spatial frequency response for human vision.  It shows that subjectively, we are most sensitized to details at a spacing of about 6–8 cycles per degree of our visual field.

Even more startling is the gray test image below. Notice how the darker stripes seem to fill a “U” shape?

In fact, at any given vertical height across that figure, every stripe has the same contrast. You are seeing your own spatial sensitivity function at work. It’s the middle spacings that we perceive most clearly.

You can have a bit of fun moving your head closer and further from your computer screen (changing the cycles/degree in your vision). Notice how your greatest sensitivity to the contrast moves from the left to the center of the image? (Ignore the right edge, where the target starts to alias against our computer-screen pixel spacing.)

It might surprise you that there is a low-frequency roll-off to our vision’s contrast detection. After all, imagine filling your field of vision with ONE giant white stripe, and one black one—surely you’d notice that?

But, MTF measurements don’t use stripes with hard edges. Instead MTF uses smoothly-varying sinusoidal brightness changes, as I showed above.

Our eyes are great at finding edges. But smooth gradients over a large fraction of your visual field are much less obvious. Our retinas locally adjust their sensitivity—which is why we get afterimages from camera flashes or from staring at one thing for too long—and tend to subtract out those large-scale patterns.

A Contrast Sensitivity Function

Matching a Sensitivity Curve to Test-Subject Data (Red Dots)

Defining the whole human contrast sensitivity function precisely can get extremely complex.  However we can make a few points:

The ultimate limit for human acuity might be at 45 or 50 cycles per degree (sources differ). But that spacing is right at the edge of perceivability.

It’s only under bright light that we achieve our greatest acuity; but we generally view photographs under dimmer indoor illumination. There, even spacings of 20 cycles per degree might show seriously degraded contrast.

In a second PDF, Dr. Nasse discusses a “Subjective Quality Factor” for lenses (see page 8). This is an empirical figure of merit, which integrates lens MTF over the spatial frequencies that our eyes find most significant. The convention is to use 3 to 12 cycles per degree—near our vision’s peak sensitivity.

My prior post on MTF mentioned that 10 and 30 lp/mm (on a full 24 x 36mm frame) were lens-test spacings chosen for their relevance to human vision. Actually that oversimplified slightly. We didn’t specify how the photographs would be viewed.

In fact, those criteria correspond to “closely examining” an image. If you put your eye very near to a print (a distance half the image diagonal) the photo practically fills your view. It’s about the limit before viewing becomes uncomfortable. In those conditions, the 10 and 30 lp/mm of lens tests translate to 4 and 12 cycles per degree of your vision.

The conditions of “regular” viewing (at a distance maybe twice the image diagonal) relax the detail requirements significantly.

So how much detail do our photos really, truly need?

Ultimately, that’s a personal question, which photographers must answer for themselves. It depends how large you print, and how obsessively you examine.

But the eye’s own limited “frequency response” suggests we may not always need to worry about it.


One Response to “What is the Frequency?”

  1. […] remember from my earlier discussion that detectable detail doesn’t equate very well with subjectively significant […]

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