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04-21-2010, 05:42 AM   #1
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Nine-Dimensional Photography

Preamble

I prepared the following article some time ago, not knowing if it would go on my blog or be published elsewhere. I tentatively submitted it to an online magazine but heard nothing back. I am sure it needs work, so treat it as a draft!

The audience is likely the photographer who is not quite a beginner but not so expert as to have figured all this out for themselves. I hope this structuralist approach can clarify some oft-misunderstood terms and concepts.


Nine-Dimensional Photography

How many dimensions does a photograph have? Seems like a pretty simple question. And the simple answer would be two: width and height. The paper it's printed on (if indeed it is printed... in this day and age that is not a given) also has a negligible depth, but the photo itself has two significant dimensions. This much is a commonplace.

The photograph is the end result of a photographic process. How many dimensions does this process have? By the end of this article we will be able to answer this question. And the answer is: Nine!

Have you ever wondered why photography is so challenging... enough to keep generations enthralled with the science and art of taking pictures? It's because the photographer faces the complex and subtle task of controlling a nine-dimensional process in order to render a two-dimensional result. Read on and I'll explain.


Framing An Image

What do you do when you take a picture? You point a camera at your subject and look through the viewfinder (or look at an LCD screen... same thing). The important thing is, you choose where to point. Changing your position or direction changes your perspective on the subject. Perspective is basically a vector, which we can represent as the distance from the photographer to the subject, z[1], and the angle of view, theta.

(Read notations like z[1] as z sub 1, that is, the "1" should be a subscript. Unfortunately I do not think that markup is supported in BBcode.)

Using a lens of a different focal length gives you a different field of view on your subject. (Twisting a zoom lens does the same thing.) Field of view is what angular slice of the subject we can see, so that's appropriately delta theta. This defines an arc from theta[1] to theta[2], centred on theta (assuming you are focusing in the centre of the frame).

Turn the camera ninety degrees and you have a portrait; keep it as it is and you have landscape orientation. You might also be able to change your aspect ratio, the ratio of width to height, either in camera or by choosing a different camera format in the first place. Ditto for the absolute number of pixels or inches or parsecs the image will have on a side. That's the image size.

Orientation, aspect ratio and image size all boil down to width and height. I'll call these the x and y dimensions, since those are the letters my physics professor always used.

These four dimensions mentioned so far define what falls within the frame of the picture. The choice of framing is one of the most important a photographer can make.

Note that so far we haven't discussed anything that's reliant on technology. If you form a rectangle with your hands and stare through it, the problem is the same.


Time and Spatial Parameters

The next four photographic parameters require a camera, though any camera will do. Two are time-based, two are spatial.

The first temporal parameter is the shutter actuation. When you press the shutter you start the time slice that defines the picture. This I'll represent by t[1]. The second temporal parameter is shutter speed. The faster the shutter speed, the more effectively we can freeze subject motion. This parameter, delta t, determines the end of the photo time slice. t[1] + delta t = t[2]. At t[2] the shutter closes and our photo is done.

Blink and you have the process in reverse. You decide when to start blinking (t[1]) and how long to keep your eyelids down (delta t). The length of time you cannot see is equivalent to the length of time the camera does see... but now we're entering the realm of metaphysics! To avoid this philosophical abyss, I recommend you do not blink while taking photos.

The next spatial parameter under consideration is the lens aperture. The larger the aperture the smaller the depth of field (DOF). What is DOF? It dictates how near objects must be to the focal distance to be in focus. Oh, it looks like we just sneaked in the other spatial parameter -- you did remember to focus the lens, didn't you?

But we have seen focal distance before -- it is the distance to the subject along the axis of the lens, that is, z. If you choose to focus on a slightly different point you've chosen a different subject, as far as we're concerned here. So it turns out we've discussed not four new photographic parameters in this section, but only three.

Those last two spatial dimensions are analogous to the temporal domain. The DOF dictates delta z, the distance around z that will be in focus, that is, the distance from z[1] to z[2]. This is not quite the same as the case of shutter actuation and shutter speed since z[1] is slightly in front of z and z[2] is behind it. But that's a technicality we can gloss over for now.

We'd say mathematically that z[1] < z < z[2] and delta z = z[2] - z[1].


Sensitivity

There's one last photographic variable and it's purely technological. To understand, it helps to remember that the faster the shutter speed (the smaller delta t), the less light into the camera. Likewise the smaller the aperture, the less light gets into the camera (the larger delta z). As a general principle, we want to capture as much light as possible. The more light that's dissipated or distorted on its way through the camera to the substrate that captures the picture, the less true our image.

Put in terms of information theory, the more light the less noise.

Since we rarely have an ideal amount of light, we must trade off image accuracy for noise. We do this using the last photographic parameter, which goes by the annoying name of ISO. Annoying, because the ISO is an international organisation that is responsible for setting thousands of standards, this being only one. (In fact it's ISO 5800.)

If the photographic substrate is film, ISO measures how sensitive the film is to light. Back in the old days we could refer to this dimension simply as film speed. If the photographic substrate is a bunch of light sensors on a computer chip, however, the term "film speed" makes little sense. I suppose "substrate sensitivity" is the most accurate term, but no-one says that, they say "ISO", as if it means something.

(And since everyone says it, it does mean something. Oh no, now we're into the realm of philosophy of language!)

I'll call this dimension s, because sensitivity is the word to emphasise.


Summary

Let me summarise the nine photographic dimensions in a list.
  • spatial dimensions:
    • frame:
      • width: x
      • height: y
    • perspective:
      • angle of view: theta
      • field of view: delta theta
    • depth:
      • focal distance: z[1]
      • DOF: delta z
  • temporal dimensions:
    • shutter actuation: t[1]
    • shutter speed: delta t
  • information dimension:
    • sensitivity: s

I find this structuralist approach to photography useful to clarify the complexities of the terminology. Before I worked this out for myself I was forever being confused by sloppy use of terms in the literature and on the net.


Postscript

Before anyone decides to mention dimensions not covered here, I will point out that the geometry I have covered is valid for SLR cameras. Those with front or rear plane controls add more dimensions, since the sensor/film plane is no longer required to be parallel to the lens focal plane. Maybe I will get to that in some other article!

04-21-2010, 07:40 AM   #2
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Interesting and insightful article Robin!

Mike
04-21-2010, 08:16 AM   #3
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That's a very cool way of looking at it, and nicely written up... As a geologist, I've spent years calling my field "the art of figuring out a four-dimensional jigsaw puzzle", it's good for getting puzzled looks but it explains the time aspect so well. Your essay does the same thing for things we don't usually think about in dimensional terms. Well done, Sir!!!

Jim
04-21-2010, 08:35 AM   #4
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Thank you both!

Would anyone out there with an expert grasp of graphical representations want to make some sort of a diagram for this? Or maybe I will try at some point. I do think a picture or two would help the exposition a great deal, especially for visual learners. And hey, we are visual people, we photographers!

04-25-2010, 02:38 AM   #5
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that an incredibly cool way to think about it. i love to see things broken down like that, so to me its almost like a visual cutaway of abstractness or something that can't be visually cutaway. er something like that.
well, i tried drawing the spacial bit out but x y and the feild of view seem to overlap. because no doubt when you change your fov you directly affect the x and y. and also because the x and y ratio is fixed, at least as far as i know. so i figure those should be coupled as one variable. so youd get a frame variable xy and a zoom variable dxdy. then rotation angle would just be completely separate along with the z and the dof dz. and i guess the rotation would be broken down into vertical and horizontal to get a full spherical grid going.
so the more i think about it the more and more variables i can throw in. like the trouble of having shutter speed only affect the time domain. because if you move your camera at all during an exposure you'll cover more unknown in the spacial domain. or the weight of a flash's affect on a photo with a long exposure giving that flash's time period alot more strength.
yeah im definately confusing myself more and more now as well. so yeah i like it
04-25-2010, 03:18 AM   #6
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kebabs, thanks for that comment though I am not sure I uderstand all of it!

It would be true that adding flash into the picture would complicate matters in the time domain. One would have the effect of the rising and falling curatins cutting acros the exposure time.

On contemplation I think you might be correct about angle of view conflicting with the frame dimensions X and Y. It might be the case that if angle of view is split into vertical and horizontal dimesnions, intead of just one, that x and y are redundant. I will think more on this. My brain does not seem to be as sharp today as when I was working on this article!
04-25-2010, 07:01 AM   #7
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a very simplistic illustration using paint

What would be good would be an animated interactive flash demo of the variables with a change in the angle of view as a result of pushing the lens face in and out etc - any takers?

04-25-2010, 08:18 AM   #8
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Arjay, thank you! That is an amazing addition. Just as my article is a "good start" to thinking about these things your diagram is a "good start" to visualising them. It really helps.

I can't help think this has been done before, in one way or another. I wonder if The Camera has this sort of a representation. Unfortunately I do not own a copy, but I should.
04-25-2010, 08:58 AM   #9
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Robin, very interesting and inspirational approach...

There is one additional thing I often try to pursue (or at least make an attempt) which is the "mood" of that moment/instant. I don't know you can count that as one additional dimension or not.
04-25-2010, 11:18 AM   #10
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QuoteOriginally posted by aleonx3 Quote
There is one additional thing I often try to pursue (or at least make an attempt) which is the "mood" of that moment/instant. I don't know you can count that as one additional dimension or not.
No, I don't think it would even be worthwhile trying to quantify mood. This article is only about all those non-important parts of a shot that one can define and put numbers on.

For subject, mood, atmosphere, message, theme etc. etc. I hope no-one will try to quantify anything. That would rather take any joy out of it!
04-26-2010, 01:47 AM   #11
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Between physicists ...

QuoteOriginally posted by rparmar Quote
  • spatial dimensions:
    • frame:
      • width: x
      • height: y
    • perspective:
      • angle of view: theta
      • field of view: delta theta
    • depth:
      • focal distance: z[1]
      • DOF: delta z
  • temporal dimensions:
    • shutter actuation: t[1]
    • shutter speed: delta t
  • information dimension:
    • sensitivity: s
Between physicists ...

I think this is somewhere in the middle between the photographic notion of what a photo defines and a physicist's one.

A truely physicist's definition would be one which exactly defines a taken photo:
  1. Camera position xc
  2. Camera position yc
  3. Camera position zc
  4. Camera position t
  5. Camera speed v (one average amplitude parameter only, representing shake or mode of stabilization (tripod etc.) and ignoring panning effects which would require many more parameters)
  6. Subject position xs (direction and focus distance)
  7. Subject position ys (direction and focus distance)
  8. Subject position zs (direction and focus distance)
  9. Framing orientation to gravity center alpha (0..90°)
  10. Framing field of view FoV (0..180°)
  11. Framing aspect ratio r
  12. Shutter speed s
  13. Lens aperture d
So, I don't agree to your nine degrees of freedom as 9 is arbitrary.

BTW, ISO is no such parameter as it doesn't effect the image. A camera is a photon counting device and doesn't require a sensitivity setting. It is merely a convenience parameter to guide the automatic exposure algorithms. You can always define the brightest pixel color channel to be one or record the photon count in the raw file. And the total number of photons is defined by d, s and FoV with no intervening ISO.

Last edited by falconeye; 04-26-2010 at 01:53 AM.
04-26-2010, 05:29 AM   #12
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While i reckon youve nailed alot of elements of a photo, the engie in me finds it hard to agree with the word `dimension` used. Fair enough you can represent most these things with a dimension and your not being litereal but it still bugs me. Not to nag but you cant really use an x-y-z coordinate system and say theta is another dimension as theta can be easily shown with the x-y-z coord's. According to your logic i could bake a cake, and it can be 9 dimension due to the temperature differences in the oven and the direction i held the spoon when mixing the mixture.

Apart from that, a nice read =p
04-26-2010, 05:38 AM   #13
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Hey Falk, I try hard to never call myself a physicist, and maybe this article demonstrates why!

But I am not sure of your list either.

First, let us ignore change over time, like speed of camera movement and so on. Assume for a moment (as I did, but did not state) that we don't care if the subject blurs or whatever.

Then, I think the main difference between us is that you are taking a fixed position outside the camera and I am attempting to define things relative to the camera. So, I don't care about the absolute (x, y, z) positioning of camera and subject. This brings our lists much closer together.

In fact, it seems each of us is correct with only one thing wrong apiece.

I was wrong in regarding sensitivity without aperture. And you are wrong in listing aspect ratio without indicating both dimensions (since an aspect ratio alone does not define the exact frame.

Make sense?
04-26-2010, 10:40 AM   #14
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@ifnot rparmar then skip post, pleasssse
QuoteOriginally posted by rparmar Quote
Then, I think the main difference between us is that you are taking a fixed position outside the camera and I am attempting to define things relative to the camera. So, I don't care about the absolute (x, y, z) positioning of camera and subject. This brings our lists much closer together.

In fact, it seems each of us is correct with only one thing wrong apiece.

I was wrong in regarding sensitivity without aperture. And you are wrong in listing aspect ratio without indicating both dimensions (since an aspect ratio alone does not define the exact frame.
We don't differ in any significant way, just playing a nice game of abstraction

But really, ... if you leave out x,y,z, then please leave out t as well. Which leaves you with eight dimensions.

It is eight, not seven after dropping ISO, because you have 4 framing parameters rather than 5 (plus focus distance).

I didn't miss the dimensions, but you missed the framing orientation angle which is only covered by the aspect ratio if it is restricted to 0° or 90°.

The dimensions are redundant. I give FoV, aspect ratio and distance. This already defines both frame dimensions in the subject space. I don't have to give frame dimensions in the image space (which would be the sensor size) or even a focal length as the captured image can be shown to be independent on these parameters.

If you replaced my lens diameter parameter d by an f-stop, then yes, I would need actually frame dimensions or the actual focal length as well. But you know what? This is why I picked lens diameter d over f-stop. Your DoF parameter does the same, you need no further parameter and d and DoF are interchangeable.
04-26-2010, 10:52 AM   #15
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All the Physics stuff seems so pointless. I also think that this seems to massively over complicating things, taking a photo should be simple and spontaneous.
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