## Tuesday, 23 October 2012

### The problem

If you have ever played around with macro photography at a magnification of 1x or more, you will have encountered the curse of shallow Depth of Field (DOF). It is often desirable in portrait photography to isolate the subject by having only the subject in focus, with the background nicely out of focus, i.e., you want relatively shallow DOF.

Unfortunately, there is such a thing as too little DOF, where it becomes difficult to keep the entire subject in focus, or at least the parts you would like to keep in focus. This problem pops up in macro photography all the time. Consider this example shot at 2.8x magnification (and then cropped a bit):
 An ant at 2.8x magnification, lens aperture f/4
Depending on your preference, you may decide that the DOF in this shot is just fine. Personally, I would have liked to have more of the hair running down the centre line of the ant's head, especially between the mandibles and the antennae, to be in focus.

Normally, you can just stop down the lens to increase DOF. Unfortunately, there is the small matter of diffraction that gets in your way. To fully appreciate the problem, first consider the way in which magnification affects the effective aperture (also called the "working aperture"). The aperture of the lens has to be scaled according to magnification, so that
Ne = N * (1 + m)
where Ne denotes the effective aperture, N denotes the aperture as set on the lens, and m is the magnification ratio. At 1:1 magnification, which is usually considered the start of the "macro" range, the value of m is equal to 1.0. This means that the image projected onto the sensor is physically the same size as the object being photographed. (Please note that this equation is an approximation that assumes that the pupil ratio is equal to 1.0; many real lenses have pupil ratios that differ significantly from 1.0, and a different form of the equation is required to include the pupil ratio in such cases, but the overall behaviour of the equation is unchanged. For simplicity, I assume a pupil ratio of 1.0 in this article.)

For non-macro photography, the value of m is usually small, around 0.1 or less, which implies that the effective aperture Ne is approximately equal to N, the aperture selected on the lens. Under these conditions, you can just stop down the lens to increase DOF, at least to around f/11 or f/16 on modern sensors with roughly 4.8 micron photosite pitch values. Going beyond f/11 on these sensors will increase DOF, but diffraction softening will start to become visible.

In the macro domain, though, it is an entirely different kettle of fish. The shot of the ant above serves as an example: at 2.8x magnification, with the lens set to f/4, we obtain an effective aperture of f/15.2. If we stop down the lens just a little bit, say to f/5, we are already at f/19. The minimum aperture of the lens I used is f/22, which gives us a mind-bogglingly small effective aperture of f/83.6. Keep in mind that we were running into visible diffraction softening even at a lens aperture of f/4.

How bad is this diffraction softening? Well, assuming an aberration-free lens that is in its diffraction-limited range, the f/4 lens on a D7000 body would give us a maximum resolution of 39.6 line pairs per mm. A 12x8 inch print would thus render at 3 lp/mm (divide the 39.6 by the print magnification factor of ~13), which is barely acceptable at approximately 152 DPI.

Bumping up the f-number to f/22 on the lens gives us only 8.7 lp/mm of resolution on the sensor, or 0.675 lp/mm (34 DPI) in print. To reach roughly 152 DPI we can print at a maximum size of 2.6x1.75 inch; anything larger will look visibly softer than the f/4 example in the preceding paragraph.

So how much DOF do we have at 2.8x magnification with the lens set to f/4? Only about 76 micron, or less than 1/10th of a millimetre. Stopping down the lens to f/22 increases DOF to 418 micron, or 0.418 mm. I would consider a DOF of 0.3 mm to be workable for many insects, depending on their pose relative to the focus plane.

To summarise: At magnifications of greater than 1x, we can increase DOF by stopping down the lens, but the effective aperture quickly becomes so small that diffraction softening destroys much of the detail we were hoping to capture.

Can we work around this problem?

### Defining "depth of field"

Compact cameras usually have significantly more DOF than SLR cameras. The explanation of this is actually quite straightforward, if rarely heard. But first we must define what we mean by depth of field. (You can safely skip ahead to the next section if you are confident that you know what DOF is.)

Shallow DOF is not the same thing as a blurred-out background; shallow DOF just means that the region of the image which will be acceptably sharp is shallow. Nothing is said about the character of the part of the image which is not in focus. The common misconception that long focal lengths produce shallow DOF is based on the confusion of these concepts: lenses with longer focal lengths produce more blurry backgrounds, but their DOF is actually very similar to lenses with shorter focal lengths after you have moved back to compensate for the smaller field of view.

DOF is only meaningful in the context of specific viewing conditions. First up, DOF only pertains to the region of the image which is acceptably sharp under the conditions in which the final image is viewed. This is usually taken to be a given print size viewed at a specified distance. Working back from what constitutes an acceptably sharp print, we arrive at the definition of the circle of confusion (CoC). In other words, we take a single, sharp point in the print, and "project" this back onto the sensor. This projected point forms a little disc on the sensor, with a diameter equal to the CoC.

Reasoning backwards, we can see that at the sensor, anything image feature smaller than the disc formed by the CoC will be compressed to a single point in the final print. Any feature larger than the CoC will be printed across more than one dot in print, and will thus be visible to the viewer. This ties the CoC to our definition of the region of acceptable sharpness: point light sources that are in front of (or behind) the exact plane of focus (in the scene) will project onto the sensor as small discs. If these discs are smaller than the CoC, then they will appear in focus. If the point source is moved away from the exact plane of focus, it will reach the distance where the disc that it forms on the sensor first matches, and then exceeds, the CoC, at which point it will start to appear blurry.

The DOF is thus the range of distances between which image features are rendered as points in the print, i.e., appear acceptably sharp. At this point it should be clear that the size of the print relative to the size of the sensor has a direct impact on the perceived DOF in print, since a small-sensor image will have to be magnified more to reach the desired print size, compared to the image formed on a large sensor (with the same field of view). This implies that the CoC of a small sensor will be proportionally smaller, i.e., the CoC for a 35 mm sensor size ("full-frame") is usually taken as 0.03 mm, while a compact camera with a 5.9 mm sensor width will have a CoC of 0.0049 mm.

### Why small sensors have more DOF

So why does a small-sensor compact camera appear to have a large depth of field? The physical size of the aperture is the key concept. An f/4 lens with a focal length of 105 mm will have a physical aperture diameter (actually, entrance pupil) of 26.25 mm. A 42 mm lens at f/4 will only have an aperture diameter of 10.5 mm.

If the physical diameter of the aperture is small, the DOF will be large; just think of how a pinhole camera works. The catch is of course that a pinhole camera will suffer from diffraction softening, but you can hide this softening if your film/sensor is large enough that you do not have to magnify the image in print (i.e., contact prints).

A small sensor requires a lens with a shorter focal length to achieve the same field of view as another lens fitted to a large sensor. For example, both the 105 mm lens and the 42 mm lens will produce the same field of view if we attach them to an APS-C sized sensor and a 5.9 mm width sensor, respectively. The 5.9 mm width sensor, though will have a larger DOF because it has a smaller physical aperture.

Note that you can substitute a change in position for the change in focal length. Say you use a 50 mm lens on a full-frame camera at f/2.8. To achieve the same subject size on an APS-C (crop) camera, you would have to move further backwards to match the field of view of the full-frame camera. You can keep the lens at f/2.8, but the full-frame image will have a shallower depth of field, even though we did not change the physical aperture size. The trick is to realize that DOF is also a function of subject distance (magnification), so that the APS-C camera will have increased depth of field because it is further from the subject.

It is instructive to play with VWDOF (available here) to observe these effects first-hand.

### A small-sensor dedicated macro camera at 1:1

Can we solve the problem of insufficient macro DOF by using a small sensor?
Will diffraction prevent us from obtaining sufficient detail with such small photosites?

The idea is simple: What would happen if you increased the linear pixel density of your DX camera by a factor of four (each pixel is replaced by a 4x4 group of pixels)? We would end up with extremely small photosites, but still within the limits of what is currently possible. Then, instead of increasing the magnification of our lens, which would decrease our effective aperture a lot, we just crop out the centre 25% of our higher-resolution image. This effectively gives us an additional 4x magnification. If we are going to crop out the centre 25% of every image, then we might just as well use a smaller sensor. So, we keep the same number of pixels, but we use a sensor that is only 1/4 the size of our initial sensor (DX, in this case). Once we have a smaller sensor, we can replace our lens with a shorter focal length lens, which will be smaller, lighter, hopefully cheaper, but also easier to build to achieve exceptional sharpness. To illustrate this idea, a practical example now follows.

Suppose we have a subject that is 28.3 mm wide (roughly the width you can achieve with a 105 mm lens at 1x magnification on a Nikon DX sensor). What we want to achieve is to capture the exact same subject, but with different sensor sizes. To achieve this, we will have to vary both the focal length and the lens magnification factor.

To illustrate this concept, I will define three (hypothetical) cameras:
1. DX: 4980 pixels over 23.6 mm sensor width,
focal length = 105 mm,
required magnification = 1x,
lens aperture = f/4,
effective aperture = f/8
2. FX: 4980 pixels over 36 mm sensor width,
focal length = 127 mm,
required magnification = 1.525x,
lens aperture = f/4,
effective aperture = f/10
3. 1/4DX: 4980 pixels over 5.9 mm sensor width,
focal length = 42 mm,
required magnification = 0.25x,
lens aperture = f/4,
effective aperture = f/5

With these specifications, all the cameras will capture the subject at a distance of 210 mm, with the same field-of-view (FOV) of roughly 7.71 degrees, and the same subject size relative to the image frame. The 1/4DX camera is just what the name implies: scale down the sensor size by a factor of four in each dimension, but keep the same number of pixels. You can calculate the photosite pitch by dividing the sensor width by the number of pixels; the 1/4DX sensor would have a pitch of 1.18 micron, which is close to what is currently used in state-of-the-art cellphone sensors.

We define DOF the same way for all the cameras, i.e., being able to produce the same relative circle of confusion when looking at a final print size of 12 inches wide. The actual circle of confusion for the 1/4DX camera will have to be much smaller (1/4 of the DX CoC) to compensate for the fact that we have to magnify the image 4x larger than the DX image to arrive at a 12 inch print.

Computing the actual DOF using VWDOF:
1. DX:      0.314 mm
2. FX:      0.261 mm
3. 1/4DX: 0.784 mm

This is looking promising. The 1/4DX sensor gives us roughly 2.5 times more DOF (compared to DX) in the final print. The FX sensor gives us less DOF, which is more or less what we expected.

Using the MTF Mapper tool mtf_generate_rectangle, we can model the MTF50 values after taking into account the softening effect of diffraction (MTF50 values are a quantitative metric that correlates well with perceived sharpness). This allows us to calculate how sharp the lens will have to be for the 1/4DX camera to work, as well as what our final resolution will be once we have scaled the images up to 12-inch prints.

The actual resolution of the final print, which arguably is the thing we want to maximise, turns out as follows:
1. DX:       4.31 line pairs / mm
2. FX:       4.6 lp/mm
3. 1/4DX:  2.5 lp/mm

What happened? Diffraction destroyed a lot of our resolution on the 1/4DX sensor. In short, we decreased our photosite pitch by a factor four, which means that diffraction softening will now already be visible at f/4 (probably by f/2.8, even). We expected our smaller pixels to be able to capture smaller details, but diffraction blurred out the details to the point where very little detail remained at the pixel level.
We can try to remove the AA filter on the 1/4DX sensor (which we arguably no longer need at f/4 with a 1.18 micron photosite pitch) to win back a little resolution, which will give us 2.7 lp/mm in print. Not a huge gain, but we might as well use it.

One tiny little detail is also quite important: the 1/4DX camera would require a really, really sharp lens: around 129 lp/mm. I think this is still possible, though, but we'll need a lens designer's opinion.

So going for a smaller sensor with 4 times higher (linear) pixel density does indeed give you more DOF in a single shot, provided that you keep the lens aperture the same. The price for this is that the sharpest part of the printed picture will be noticeably less sharp than the prints produced on the DX camera. DOF increased by a factor of 2.5, but overall resolution decreased by a factor of 1.724.

We could take our DX camera and stop it down to roughly f/9.1 on the lens, producing an effective aperture of f/18.2. This would produce comparable resolution to the 1/4DX camera (around 2.7 lp/mm in a 12-inch print), and the DOF would be 0.713 mm, which is ever so slightly less than what the 1/4DX camera would produce.

### Detours at 1:1

Ok, so if we lose resolution but gain DOF by stopping down the aperture, what would happen if opened up the aperture on the 1/4DX camera a bit?

Well, at a lens aperture of f/1.6, the effective aperture would be f/2. This would produce a 12-inch print at a resolution of 5.5 lp/mm, which is higher than any of the other options above. But the DOF is exactly 0.314 mm, so we are no better off in that respect, except that we have increased our final print resolution slightly.

A slightly less extreme aperture of f/2.25 would give us an effective aperture of f/2.8, which will match the print resolution of the DX camera, and give us 0.441 mm of DOF. That is a decent 40% increase in DOF over the DX camera, and still gives us the same print resolution.

Proceeding along this path, we can choose a lens aperture of f/3.2, resulting in an effective aperture of f/4, yielding print resolution of 3.29 lp/mm. This is about 30% less resolution, but we have a DOF of 0.627 mm. The DX camera will reach the same print resolution (3.29 lp/mm) at an aperture of f/6.8 (effective aperture is f/13.5), which will yield a DOF of 0.533, so the 1/4DX camera is looking less attractive at this point.

### What happens at 2.8x magnification?

We can repeat the process at 2.8x magnification, which gives us the following cameras:
1. DX: 4980 pixels over 23.6 mm sensor width,
focal length = 105 mm,
required magnification = 2.8x,
lens aperture = f/4,
effective aperture = f/15.2
2. 1/4DX: 4980 pixels over 5.9 mm sensor width,
focal length = 58.5 mm,
required magnification = 0.7x,
lens aperture = f/2.6,
effective aperture = f/4.4
Note that the aperture for 1/4DX was chosen so that the print resolution of the 1/4DX camera matches that of the DX camera. With these settings, the DX camera has a DOF of 0.076 mm, and the 1/4DX camera has a DOF of 0.0884 mm, so no real improvement in DOF if we keep the print resolution the same.

### Conclusion

There does not appear to be a free lunch here. It is possible to increase the DOF of a 1/4DX camera relative to that of the DX camera, but it requires an extremely sharp lens. The lens will fortunately be quite small, so it may be feasible to construct such a lens.

The price we pay for the increased DOF is that we have much smaller photosites, which will have a negative impact on image quality in the form of noise. Specifically, photon shot noise is related to the full-well capacity of a photosite, which in turn is linked to the photosite size. So while the 1/4DX camera may be able to offer slightly more DOF at just the right settings (e.g., lens aperture set to f/2.25), the trade-offs in terms of noise are unlikely to make this approach attractive.

It would be interesting to explore the parameter space more methodically. For example, what if we try a 1/2DX camera instead? I am betting against it, but I probably should run the numbers and see ...