Part 2- DOF and Distance

If you haven't read my photography disclaimer, I recommend you read it now.

Photography with SLRs essentially involves navigating the multi-dimensional space created by many variables - aperture, shutter speed, ISO, exposure (ev), DOF, noise, focal length, focusing distance etc. In part, your effectiveness as a photographer depends on how well you understand this space and how well you can navigate it.

In this writeup I look at DOF. Before you read this article, I expect you already know what DOF is, and that you have some experience trying to control it. Browse around and a bit and read about DOF before you look at this article. Know what the terms mean and such. Here are two little articles that you introduce some terminology for you.

http://www.dofmaster.com/dofjs.html
http://www.dpreview.com/learn/?/key=depth_of_field

Here are some slightly more explanatory articles:

http://www.luminous-landscape.com/tutorials/understanding-series/dof.shtml 
http://www.cambridgeincolour.com/tutorials/depth-of-field.htm

Most books, like Bryan Peterson's popular "Understanding Exposure", give you a some perspective into DOF, how aperture affects DOF and how to use it creatively. I have always been a bit confused by depth of field because it never seemed to work quiet the way I expected. I always seemed to me that DOF relationships are not linear in the space of the variables that affect it. What I mean by that is that when I change a variable that affects DOF, say the aperture, it does not seem to change DOF at the same rate.

So what variables affect DOF? As best as I know they are not just aperture, but include aperture, focal length, focusing distance and camera type. Look at the URLs above, they have  DOF calculators there that accept values for these variables and give you various DOF values back. What we are going to do here is to plot some graphs between these to better visualize how one variable affects another. My graphs are based of the equations that run the DOF calculator on dpreview and are hence only as accurate as those equations are. What value you get out of this depends on how closely you look at the graphs and try to visualize what the curves mean in terms of actual photography.

This time we are going to look at the most frequently expressed relationship, that of DOF and Aperture.

The graph below shows DOF plotted against aperture for a focal length of 50mm and focusing distance of 1m (ie the object that you are focusing on is 1m away). This plot is for the Digital Rebel XT (the EOS 350d) and should hold for any 1.6x senor DSLR such as the Rebel, Rebel XTi and such.

The aperture has been varied from f/1 to f/32. Assuming we (ever) have an affordable lens at f/1. There are two lines in the picture. The lower one corresponds to the nearest point that will be in focus and the upper one corresponds to the farthest point that will be in focus. So at f/4, the region of focus starts at roughly 2.5cm in front of the point of focus and extends to about 2.5cm behind it. The total DOF at f/4 is about 5cm.

Since you are at the same position with respect to the object you are photographing and that focal length is 50mm throughout, changing the aperture will affect the depth of field and exposure time (the shutter speed) needed to maintain the same exposure value.

Notice how the graph tends to curve a bit? Lets see how the DOF curve looks when plotted at different distances. Here are the curves for some smaller distances all in the same graph.

For each focusing distance there are two lines, the lower one being the nearest point in focus and the farther one being the farthest point in focus. Some interesting things to note: 1) The DOF does not increase very much if you are focusing on close objects. 2) For short distances, of under a meter, the DOF seems to grow linearly.

Lets look at some greater distances.

Here the non linearities become very apparent. This graphs plots DOF curves at distances of 1 to 5 meters. Look at the y-axis, note that each unit corresponds to 2 meters. The pair of green lines representing the two meter focusing distance seems to increase almost linearly. At 3 meters, aperture values of 22 and above have rather large farthest points of focus.

The general trend is this: at greater focusing distances the father point of focus increases exponentially on increasing the aperture. In fact its worse than just exponential, the values hit infinity relatively quickly. So at focusing distance of 5m at f/22, the farthest point of focus has hit infinity.

Lets look at some greater distances.

At larger distances, smaller apertures lead to infinite DOF values. At 50m, f/4 already gives you infinite DOF! Even at 10m, there is a region of focus that is about 4-5m at f/4. Another way to look at this is that DOF tricks make sense only if the object in question is relatively close to you - i.e. within 2 or 3 meters. Well, that's not entirely true - we haven't heard what focal length has to say about it. But it does seem true at 50mm.

Here are some greater distances that show degenerate version of the above curves.

Now lets go back to the first graph where we were at 1m and 50mm focal length. Lets look at some graphs that show the effect of changing the focal length. Along the x-axis we still vary the aperture value and on the y-axis we plot the DOF. We are just going to draw multiple graphs based on changing the focal length (while keeping the distance fixed).

So at wider apertures, the DOF increases. The effect of going from 50mm to 60mm, visually will also be that the object occupies a larger part of your frame. This is roughly what you get were you to stay at 50mm and step closer to the object. From this can we conclude that if the object stays roughly the same size in your screen, the DOF stays the same? I don't know yet. But the graphs seem to indicate so.

Let look at some wider focal lengths. The widest non-fisheye lens one can get for the Rebel series is the 10-22mm lens. (Its an awesome lens btw.) So lets start plotting at 10.

At 10mm f/5.6, we have already hit infinite DOF even when the object is just 1m away. At 18mm we hit infinity at about f/15. At 18mm and f/4 we have a decently small area of DOF, however remember how easily this degenerates based on your distance graphs? So if you want to take advantage of DOF when you are at the wide end, you must be pretty close to the object and have a wide open aperture (the smaller the numeric value of aperture, the larger is the actual aperture i.e. more light is allowed onto the sensor).

Lets look at some longer ranges.

Notice that the units of the y-axis have changed. At longer ranges (70mm to 120mm) the DOF stays small and concise, good enough to bring out that beautiful bokeh. Greater zooms have narrower DOFs - I don't bother to plot those graphs here. Its easy enough to imagine how they'd look.

So what do we have to take away. DOF does increase with aperture, but usually only for short distances. For greater distances there is non-linear increase in DOF often hitting infinity. Similarly at wider focal lengths you get a larger DOF for the same aperture. On increasing the focal length DOF does get narrower until some point where the DOF-aperture relationship starts to look linear again.

Distance from the object and the zoom (focal length) both DOF inversely. Increasing distance increases DOF and increasing zoom decreases it. In other words, if you were to step back but zoom in tighter to compensate, the increased distances and the increased zoom might compensate for each other giving you roughly the same DOF. (I would be able to assert this with some certainty if I have equations for the size of the image on the sensor.)

That said, if you want to take a picture whose beauty depends on DOF and you are willing to vary the object size in the final image (maybe you can compensate by cropping?), understanding the relationship between distance/zoom and DOF will come in handy. In the later parts of this write-up I hope to go into that. Let me know what you think.

Part 2 - DOF and Distance