Dynamic Range Is Not Exposure Range! Part II

By Ctein

Part I is here

The windmill I am currently tilting at is trying to get people to stop misusing the term "dynamic range" when they are talking about the exposure range of their camera. Last week I explained how real exposure ranges can be somewhat less than the dynamic range of the sensor. This week I'll explain how the exposure range can be significantly greater than the dynamic range.

Fig. 1. These four squares illustrate how a sensor responds to low levels of light. In an imaginary world photons rain down uniformly, so any exposure level less than one photon per pixel comes out pure black, because you can't have fractional amounts of a photon. This also means you can't record any intermediate gray levels between zero and one photons or between one and two photons per pixel.

Most Internet musings on the subject of sensor behavior and image characteristics don't have anything to do with the way the real, physical world works. Most notably, almost all the discussions of exposure range, number of gray levels recorded, and so on, imagine a uniform, noiseless world that is far removed from reality. In their mythical universe, figure 1 shows what happens when you expose a sensor to very low light levels, starting with perfect blackness (zero photons per pixel) on the left, up to one photon per pixel on the right. Axiomatically, you can't subdivide photons, so everything remains pitch black until you reach a level of one photon per pixel, and then suddenly you see a signal. Similarly, although not illustrated here, the next jump in response occurs when you reach a level of two photons per pixel; 1.5 photons per pixel just doesn't mean anything.

Right?

That's a rhetorical question. You know I'm going to tell you it's wrong. And here's why.

Counting statistics. In the real world, when the exposure works out to an overall flux of one photon per pixel, we don't actually get one photon per pixel. Light doesn't come in a uniform rain of photons. One photon is only the average. What actually happens is the likelihood of any one pixel seeing some number of photons follows something called a Poisson distribution. (Wikipedia will tell you more than you probably want to know about this). In reality, equal numbers of pixels receive either one or zero photons, totalling about 75% of all pixels. About 1/5 of the pixels will see two photons. About a sixteenth will see three, and lesser fractions four or more photons.

Fig. 2. In the real world, photons arrive randomly, obeying Poisson statistics. This figure illustrates what a 10x10 array of pixels might see when exposed to different levels of light—0, 0.25, 0.5 and 1 photon/pixel, average.

I've illustrated this in figures 2 and 3. Figure 2 shows 10x10 cells of pixels, highly magnified, so you can see the way Poisson distributions look. For figure 3 I tiled bunches of these together so you'd see something at 100% scale, like in a real photograph, instead of much-magnified. (These illustrations will be easier to see if you click on them to open the larger versions.) The tiling's why you see fine repeating patterns in figure 3. In real life those wouldn't be there, but I wasn't about to hand draw 100,000 pixels!

Fig. 3. The same light distributions as in figure 2, but now you're looking at 160 x 160 pixel zones, to better simulate the visual appearance of a real photograph. Ignore any repetitive fine textures; that's caused by me tiling the zones in figure 2 to fill the large areas.

Overall, seen from a distance, the 0-photon and one-photon-per-pixel zones in figures 1 and 3 look very similar. No surprise, there. Figure 3 looks grainy, because there is a mix of pixels with different numbers of photons. This is one of those places where having more pixels unequivocally wins out over having fewer; the grain gets finer.

The import of this statistical noise is that you can distinguish several levels of gray between integral-photon illuminances (see last week's column). In particular it means you can record light levels below the single-photon-average threshold of the detector. Suppose you set the exposure so that there is an average of only 1/2 photon per pixel. You don't get solid black, as the armchair "experts" imagine. A plurality of pixels do indeed see no photons, but a minority see one photon, and a small fraction see two. The resulting photograph will be grainy, but on average it will appear to be a darker tone than the one-photon picture and it will be usably lighter than solid black.

Even a 1/4 photon-per-pixel exposure produces a visibly-lighter-than-black tone, although it's getting marginal. Our original "10 stop dynamic range" sensor clearly has a real, useful exposure range greater than 11 stops, as you can see my illustrations.

Even mixing in the intrinsic system noise that I discussed last time doesn't necessarily wipe this out, as figure 4 illustrates. It gets harder to see the distinct zones, but they're still visible. Again, the more pixels you have to work with, the clearer the distinctions between tones.

Fig. 4. Toss in a moderate amount of system noise and you've got an approximation of real photography. The tones are grainy but distinct from pure black, even at the 0.25 photons per pixel level.

So, now you know that dynamic range may happen to match exposure range. Or, it's going to be better. Or worse. And sometimes the difference will be very significant.

As I said last week, my gripe derives from the fact that we had a useful and accurate term that was widely understood, and a bunch of semi-informed neologists decided to replace it with a more techie-sounding but incorrect and inaccurate term.

There's a very simple fix to this. Just stop saying "dynamic range." You're never actually talking about that; what you are talking about and interested in is "exposure range." Just start saying that, OK? I try to always do this, and so far no one has ever written to me saying, "I don't understand what you mean by exposure range." Trust me, no one will misunderstand you.

Next time I'm going to explain to you how Raw data isn't really all that "raw."

Ctein

Columnist Ctein illumines the darkness every Wednesday on TOP.

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Comments

[Peter]

Ctein,

I remember a different article you wrote where you said something to the effect that you had no problem with the idea of living in a non-deterministic world that is defined by randomness and statistics at the finest (quantum) levels of reality.

That comment seems to be apropos here.

[Camilo]

Great series. Looking forward to your next column. The "cooked/uncooked raw" argument is also a pet peeve of mine.

[Sasha Vasko]

Great point and explanation. The technique is actually well used (or used to be well used) in computer graphics to extend tonal range of inferior monitors/printers by introducing semi-random noise - dithering.

The pity is, and I suspect its the contributing factor in common hate for noise, that most, if not all raw processing software does not properly account for this while adjusting exposure/curves/etc., thus magnifying difference between nearby pixels, and turning nice smooth noise into a polka dot abomination.

[Mel]

Nice windmill, interesting science, good to promote accuracy and consistency. How will this help my photography? Whatever range we're going to call it is set when I buy the camera, right? If I want to expand it, what choices do I have?

[Stephen Connor]

Could we also go back to "photograph", instead of "capture" and its variants?

[Chris]

Ctein,
you describe some interesting aspects about sensor noise but i fail to see how your examples demonstrate an exposure range greater than the dynamic range. In Fig 4 you show a (hypothetical) sensor with very low intrinsic noise (lower than 1/4 photoelectron per pixel) but if i am not mistaken this is both leading to a high dynamic range and a large exposure range - reducing the intrinsic noise by half doubles the dynamic range, doesn't it?

[Sarge]

Ctein,

May I humbly propose that you explore further your "staircase" analogy which you referenced in last Wednesday's comment thread to Part 1.

"Bruce Fraser and I ... decided that the best way to explain this to people was to think of it thusly: the range (whether dynamic, exposure, input, output, whatever) is the height of the staircase. The bit-depth is how many stairs there are in the staircase." Ctein (reply to Ben S. in Part 1 comment thread)

"Height of the staircase" is lacking in precision. Is "range" (a) the total run (horizontal distance of a span or "flight" of stairs); or, (b) its total rise (height from floor-to-floor or floor-to-intermediate landing); or, (c) the ratio of total rise to total run? That is, its slope or pitch. (Locally, the slope is just the ratio of riser to tread, r/t. Globally, for the entire span of the stairs or a section thereof, the slope is ΔR/ΔT, where R is r(n+1), T is t(n), and n is the number of treads. The slope arrived at either way should be identical.)

Or does it really matter which? I submit that it does. The slope of a staircase is the better analogue.

There's a carpentry rule of thumb regarding the optimal (ergonomic) dimensions of stairs which goes:

(1) Riser + tread = 17" or 17.5", e.g.: 7½" r + 10" t = 17½", or

(2) Riser x tread = 70" or 75", e.g. from (1): 7.5" r x 10" t = 75"

Any significant deviation from the optimal would result in a staircase that is either too slow or too steep, such that climbing it will be tiring, if not unsafe. This also applies to ladders and spiral or winding stairs. A spiral staircase is more of a ladder than stairs. (This "rule of thumb" has been codified in the Universal Building Code and adopted by the American Insurance Association.)

I don't know if there is an imaging equivalent of this (17-17½) "golden mean." If it does, I suppose it will have something do with a more basic physical "constant" or constraint (the amplitude of a photon's probability wave?) which has imaging and other (e.g., audio) applications.

Impossible Staircase by Lionel and Roger Penrose     (Photo credit: wikimedia.org )

By inspection, the Penrose Staircase has a 4" r and a 13½ t, more or less. It is also pleasing to look at (for reasons explained below).

"...When noise enters the picture, you can usefully record a different number of stairs [risers and/or treads] than bit-depth would predict -- sometimes many less, sometimes many more." Ctein, ibid. 

I propose that the more spot-on analogy is an open staircase. That is, a staircase with no risers with only the stringers of the stairs holding the treads in place. An open staircase allows both light and air to circulate. (Here's an example of an open grand staircase.) What then are the digital imaging equivalent to "light" and/or "air" passing through an open staircase of a certain span (range). Is it signal, noise, or S/N ratio?

According to UBC a staircase can have any slope* between 7° through 50°. Above 50 degrees are step-ladders, with a 90° (caged) ladder being the limiting case. Below 7 degrees are ramps. Think of a ramp as a staircase whose risers and treads have converged almost to points. (Not all ramps are made of solid concrete. Ramps can be made of expanded or grated metal such as those used in low-rise, prefab steel car parks or the loading ramps of a trailer or truck bed.)

Any open staircase is more permeable to air than to light. With a light source directly overhead, the overlapping treads of a staircase (tread "nosing") will prevent light from illuminating the space underneath the staircase. However, the solid shadow cast by the stairs will not be pitch black, because of the ingress of "fill light" through the open risers. In the case of step-ladders, its rungs will cast only a faint, banded (if any) shadow. As for ramps, there might be no shadow at all as its gratings (if spaced wider than the girth of the grating members) will only diffuse the light rather than block it (akin to the effect of a cross-hatched or a neutral density lens filter). Note also that span-wise, ladders are short, ramps are long, and staircases are in-between.**

If, however, the light source is angled near- or perpendicular to the slope of the staircase, light will pass through depending only on how close to perpendicular the slope of the stairs is relative to the angle of the light source. When perpendicular, light reaching beneath the stairs will be maximized such that the stairs is more likely to diffuse rather than block the light passing through the open risers. That is, the stairs will cast no solid shadow on the floor beneath it or on the wall below its landing.

From the foregoing, I'll hazard that in the open staircase analogy, light is "signal," shadow is "noise" whether shot or read (or more precisely, a symptom of noise since it implies that the light signal has been attenuated), span or flight is "range," and the slope of the staircase is the analogue for sensor/lens/camera design subject to the laws of physics (the "golden mean"). The slope analogue is a broad one including but not limited to bit depth, sensor size (or crop), pixel size or density, amplification***(ISO), and so on. What I'm suggesting here is that the staircase analogy is richer and more sophisticated than it seems.

To be more apposite to the title of your column. I'll go out on a limb and propose that total rise, R (the height from floor-to-floor in a single-flight stairs), is analogous to dynamic range. While total run, T (horizontal distance from the front edge of the first tread to the front edge of the landing), is analogous to exposure range. A staircase with a 45° slope will have its R=T (i.e., a slope m equal to unity or 1). Therefore, for slopes higher than 45° DR > ER; below 45° ER > DR. More graphically, ladders have high DR, ramps have long ER, while stairs occupy the "Goldilocks" in-between. (This "hypothesis" breaks down at the limits due to infinities. For example a 90° ladder will have 0 ER and ∞ DR!)

My takeaway from Part II above, is that modern sensors has the capability to capture shadow detail to a an extent not possible in film or in low resolution sensors (such as those found in point-and-shoots). That the extension in the capabilities of digital cameras in terms of exposure range is found mostly at the "dark side" of the range, i.e., the pure blacks and the deeper shades of gray. In practical terms, if I want to shoot star trails or fireflies in available (star) light, I'll get good results if my camera has an extended exposure range. Going back to the staircase analogy, this means we're talking ramps instead of ladders when we talk of exposure range.

As a novice amateur photographer I don't really care that much for the exposure- or dynamic range (provided I know which is which), or ISO capabilities of my camera, so long as it enables me to make good photos. I'll make a further claim that staircase "ergonomics" (comfort and safety) is the building analogue for image quality. Extended ER, DR, and ISO capabilities ought to be evaluated not in isolation as ends unto themselves, but as being instrumental towards better IQ.

The rest of my comment has to do with "aesthetics."

There's also a building tradition dating back to classical times which holds that odd numbered stairs are more comfortable and safer than even numbered ones. In the Philippines (via Spain), this has been refined further since not all odd numbers are considered desirable (nor all even numbers undesirable).

A Filipino carpenter would count sequentially, thus: oro, plata, mata; oro, plata, mata... and so forth. It's okay if your foot falls on the the landing at the count of oro (gold) or plata (silver), but not if it falls on mata (death). Surprisingly, the 13th footfall is oro. There is always one more riser than the number of treads (n+1) in any staircase to account for the riser from the last tread to the landing. It's the risers that you count when doing an "oro, plata, mata." And you repeat this for each flight.

(One can check this out with the Penrose Staircase above where the corner "treads" are the landings. The Penrose staircase, although impossible, is well-proportioned and visually pleasing being compliant with both the "golden mean" and "oro, plata, mata" rules-of-thumb.)

This latter, I suggest, has more to do with aesthetics rather than ergonomics. I'm wondering if what the imaging equivalent of this "aesthetic" analogue is, if any. And whether it is applicable to, say, HDR?

(Disclaimer: I have forgotten most of my high school physics. I do read the science section of online broadsheets and news magazines and accessible books on physics and astronomy written by the likes of Steven Weinberg and Carl Sagan. I'm an economist by training, an amateur cabinet-maker, and a latecomer to photography. If I may make a shameless plug, check out my blog post: Carpentry vs. Photography: A Gear Acquisition Analogy.)

Notes:
*Graphically, a ramp or ladder differs from a staircase (or step-ladder) in that its y-intercept originates at zero. Whereas the equation of a straight line drawn tangent to the nosings of a flight of stairs has a positive intercept equal to the height of its risers. The slope of a ramp or ladder is its first derivative (m) at any point along it, provided it is uniformly inclined.
**Ergonomically speaking, stairs are designed for upright walkers or "bipeds", ladders are designed for climbing on all fours (hands and feet), ramps are designed for human- or self-powered wheeled conveyances (e.g., wheelchairs, carts, trolleys, luggage, or buggies). My favorite two-word definition of homo sapiens (to distinguish us from hominids or androids), is: "rational biped."
***Rotating the slope of a staircase is akin to ISO amplification to the extent that it allows more light to pass through. Note also that the slope of a staircase, ladder, or ramp already embodies information about tread and riser proportions given the "golden mean." Much like how prices contain information about the demand for and supply of products, given competitive markets.

[Earl Dunbar]

"Poisson statistics" sound very fishy to me ...

[DC Wells]

Those Poisson statistics sound fishy to me.

[Paul Amyes]

Mel the trick is not to expand the exposure range of your camera, but rather ensure that the subject brightness range fits within it. This is what Ansel Adams was doing with the Zone System and others have done using graduated neutral density filters or even HDR.

[ctein]

Dear Chris,

If you go back and reread Part 1 to review what the technical definition of dynamic range is, you'll see that it starts at a floor of 1 photoelectron. That should clear up your confusion.

Actually, the noise level I injected into figure 4 is closer to one photon per pixel, average, but that doesn't really matter. It's just a mathematically-generated illustration, not a real camera system. Part 1 and the previous article on noise referred to therein describe at more length the effects of noise upon tonal separation and discrimination.

~~~~~~

Dear Sarge,

You have way, way too much time on your hands. I am very glad to see you're putting it to good use [grin].

pax / Ctein

[toto]

Isn't "aperture" also often misused to mean f-stop?

[Chris]

Dear Ctein,
i see - if you define dynamic range this way you are right but since the 'smallest detectable signal' is no more a property of the sensor then it would no more be a very useful quantity to characterize its performance (the intrinsic noise level would no more have an influence on its dynamic range).

You could also say a sensor with very low intrinsic noise could detect a signal of less than 1 electron per pixel and therefore has a larger dynamic range - we are just not capable to provide such a signal.

[Hugh Crawford]

So the perfect monochrome sensor would have a dynamic range of one bit , where each pixel could only count one photon, but have infinite resolution, thus having infinite exposure range.

Assuming that there is no electrical noise, cosmic rays or thermal noise ( it's perfect right?)  for every photon there is a full receptor or "white" pixel boy no "white" pixels for any other reasons.

So for this physicly impossable perfect sensor, with only one bit of dynamic range but infinite resolution ( perfect for photographing my herd of spherical cows ) would also have an infinite exposure range where every photon was counted.

Question:
Would the image be characterized as very noisy or noiseless ?

In the real world if you crop a image* in half is the resulting image twice as noisy?

*not a hologram, cut a hologram in half and you get resolution divided by the square root of two if I remember correctly.

[ctein]

Dear Chris,

No, that's wrong. Dynamic range is a specific technical term, and the smallest detectable signal is one photoelectron. There are no such things as fractional electrons (or photons) in physical reality. The ability to detect non-integral signals is an emergent property of the statistics of taking multiple samples (in either space or time). It is not a sensor characteristic.

This is not about how I want to define dynamic range, this is what is.

If you want to argue this further, you should send me a private e-mail, as the comment section is not a place for lengthy discussions or debates.

~~~~~~

Dear Hugh,

Counting statistics noise is unavoidable when you're photographing with incoherent, “un-squeezed” light, which is always the case outside of the laboratory. Poisson noise cannot be eliminated by redesigning the sensor; it's inherent in the incoherent photon rain.

The general rule (there are exceptions) is that noise varies as the square root of the number of samples, and that holds well enough for Poisson statistics. So, if you cut the picture in half, the signal-to-noise ratio degrades by the square root of two.

pax \ Ctein
[ Please excuse any word-salad. MacSpeech in training! ]
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-- Ctein's Online Gallery http://ctein.com 
-- Digital Restorations http://photo-repair.com 
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[ctein]

Dear Stephen and Toto,

Combining your two questions into one answer because they're kind of connected.

Let's start off by remembering that language is the great consensus democracy and that synonyms exist so that people can use different words to refer to the same thing, with different implications and connotations.

That said, I have no quarrel with the folks who want to say capture instead of photograph, or, for that matter, image. The language has always had overloaded terms: we can talk variously about taking, making, capturing, and shooting a photograph, and the denotation is the same for all of them. Al Blaker wrote some years ago that the implications and connotations are important and he argued for the “making” and I agree with him. As I like to say, paparazzi TAKE photographs, I MAKE photographs. But it's a matter of taste. No one is forcing you to use one word or the other and there is no fundamental misunderstanding.

Similarly, I prefer to use “image” at the intermediate points between the “making” and the “viewing”, and I know that drives some people nuts. For me it feels natural; when I talked about darkroom methods, I didn't say “photograph” all the time and probably neither did you, as in:

“After I made the photograph with my camera, I developed the photograph and then put the photograph in the photograph carrier of the enlarger. After making the enlargement exposure, I put the photograph in the developer…”

Naaah, you had different words (film, negative, print, etc.) you used for the intermediate stages. With the fungibility of the electronic data and its non-corporeal nature, it's a little less clear-cut, but I'm still a lot more comfortable referring to those bit in the computer as an “image” than I am as a photograph. Once it's printed out and in my hands, then it's a photograph again.

That's just me. The great thing about language is that your mileage can differ.

It's a slightly different matter when the word has a very specific technical meaning. The consensus was still ultimately decide, but then there's a good reason for arguing.

Aperture is kind of an interesting gray case. Technically, it does refer to the size of the physical light-gathering opening, and F-number is the ratio of aperture to focal length. but it also simply means any kind of an opening and there's been a long-established custom of saying things like, “I exposed the scene at one quarter second with an aperture of f/2.8. Yes, technically, it should be "aperture-ratio" but the consensus shorthanded the form, and I think that ship sailed a century ago.

Now, because it is a technical misuse, you DO occasionally see confusion on this topic. Since the inter-web has given every noob a soapbox to stand on, you can find that confusion periodically popping up in discussion threads. Particularly when you see ill-informed pronouncements to the effect that absolute aperture is what really matters, not f/#. Almost never true, in ordinary photography.

Truth is, this was occasionally confusing to newcomers back in the film days. It's just that back then the noobs didn't have the microphone. Getting this straight was something you learned to understand as you learn more about photography, a fluke of the language, where words legitimately have more than one meaning.

pax \ Ctein
[ Please excuse any word-salad. MacSpeech in training! ]
======================================
-- Ctein's Online Gallery http://ctein.com 
-- Digital Restorations http://photo-repair.com 
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[Peter]

Ctein,

Is it possible that many people simply don't take the time to really understand the underlying math/science? Shortcuts taken in explaining the concepts of exposure has always been a bugbear of mine. It seems that more and more far out and imprecise analogies are used to explain things. Whereas f/stop is a pretty elegant concept owing to powers of 2 and how in the physical world you see a lot of things where the individual quantities vary, but the ratios between things stay constant.