Better Photography Metrics

With the rise of digital cameras, we have seen incredible growth in the variety of camera sensor sizes. With film cameras, there were not only fewer cameras but the 35mm film format was ubiquitous. So-called "full-frame" camera sensors are now a minority. Values like focal length and f-stop work well when most cameras have the same sensor format, but are less useful otherwise. I propose new metrics that give better intuition about the relative performance of photography equipment, without relying on the 35mm sensor format as a frame of reference.

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Focal Length

Focal length seems to be the most troublesome optical system parameter. It is the number most often converted to 35mm equivalents. Most compact consumer-grade cameras will omit actual focal length from advertising material, since the actual numbers give very little useful information. The main parameter implied from the 35mm equivalent focal length is field-of-view. It makes more sense to skip converting to a 35mm equivalent and simply advertising a lens' field of view directly. Not only would it preclude calculating equivalent focal lengths for comparison, but it would actually give you more information than focal length does, since you could immediately start to visualize how wide or narrow your view would be.

f-stop

Second, f-stop numbers are inherently confusing and unintuitive. Even though f-stop is independent of sensor size and gives a very good estimate of the light gathering power of a lens, it is a poor metric for two reasons: A higher f-stop number indicates less light gathering power, and different f-stop numbers cannot be compared to each other directly with a simple linear relationship. That is, f-stop is inversely proportional to aperture diameter, but the light gathering power is proportional to the square of aperture diameter. This means that f-stop numbers must be squared to be compared with a linear relationship. I recommend writing f-stop numbers as a squared square-root. That is, f/1.4 becomes f/√2, f/2.0 becomes f/√4, f/2.8 becomes f/√8, etc.

This change allows for linear comparison between numbers under the radical sign. However it does not solve the problem of having an inverse relationship when we probably want an arguably more-intuitive direct relationship. The only solution I can think of for this problem is reformulating f-stop from a divisor of focal length to a multiple of focal length. For example: f/1.4 becomes (1/2)f√2, f/2.0 becomes (1/4)f√4, f/2.8 becomes (1/8)f√8. While this nomenclature may seem more cumbersome, a photographer would only need concern themselves with the multiplier in front of the f, omitting the radical expression. This allows for direct comparison of f-stop numbers with a linear relationship and gives a direct relationship between light-gathering power and this number. It's immediately apparent that f/1.4 gives you half the light of f/1.0 when you write these numbers instead as (1/2)f and (1)f, respectively (with the omitted √2 and √1 being implicit).

If this implicit multiplier is too confusing, and "backwards-compatibility" with current f-stop numbers is desired, another possibility is simply re-factoring in a way that keeps 'old' numbers, with these 'new' squared factors as divisors. Examples: f/1.4 becomes f1.4/2, f/2 becomes f2/4, f/2.8 becomes f2.8/8, f/4 becomes f4/16, etc. The divisor tells you light gathering power relative to f/1, while the numerator tells you an equivalence to 'old' values, i.e. the same fraction written as 1/f-number. It would be nice to replace the old system completely, but this might be a nice compromise.

Defocus Blur

Another problem that has arisen with many different sized camera sensors is that there aren't good numbers to indicate how much control a given optical system gives a photographer over defocus blur (limiting depth of field). Usually, the focal length of a non-35mm sensor system is converted into an equivalent field of view for a 35mm system, using the crop factor. The f-stop number is then scaled by this same crop factor. A photographer can then compare these numbers to lenses within the 35mm system, which they are presumably more familiar with. This is fine for an experienced photographer, but those less experienced will probably find this procedure confusing and labor-intensive. Here the relationship with f-stop is linear and simple, but the relationship with focal length is complex. Doubling focal length gives you about one-fourth the depth of field. Even given focal length and f-stop in 35mm equivalents, it's still difficult to understand how changing these numbers changes the depth of field. For example, it's not immediately obvious whether a 50mm f/1.8 lens can give you a shallower depth of field than a 75mm f/4 lens.

As a solution, I suggest adding a new sensor-independent number to camera system and lens parameters. This number would simply be the aperture diameter multiplied by the focal length (which is focal length squared divided by f-number). This number is proportional to hyperfocal distance, and is easily calculated with commonly used lens parameters. By listing this number with lens parameters, photographers can more easily understand the depth-of-field capabilities of different lenses in different sensor-format systems. Compared to using 35mm equivalents, this number would allow for a much more straight-forward comparisons. A higher lens number means a higher control over limiting depth of field. Also, differences in these depth-of-field units translate directly into proportional differences of depth of field.

The one drawback of this approach is that I have not factored in sensor size. If you view images from different sensor formats at the same size, then this needs to be factored in. The image from a smaller sensor needs to be enlarged more to be viewed at the same size as an image from a larger sensor. What is acceptably sharp on the enlargement from a larger sensor may not be acceptably sharp at that same size when using a smaller sensor. Basically, we need to factor in the "circle of confusion" (CoC) of a given sensor, which is a function of various parameters. However, if we only seek to compare sensor and lens combinations to one another, we only need to know the relative size of circles of confusion to one another, controlling for the required viewing size and resolution of the image. These ratios come back to the crop factor, the ratio of sensor diagonals. Factoring in the CoC, we would simply quote Hyperfocal distance for a a lens/sensor combination at its widest aperture, using some standard CoC number for each sensor size. I'm really not sure why things like the wikipedia topic on this issue seem to be so obfuscating. Comparing hyperfocal distances at maximum aperture seems to simplify things considerably.

Sensor Size and Resolution

Full-frame, or medium-format cameras have much more light-gathering ability than smaller camera sensors because they are larger in area. Consequently individual photo-sensing pixels are usually larger as well, although usually not by the same ratio. For examples, the 4/3rds sensor format is about 1/4th the area of a full-frame sensor. This gives the sensor about 1/4th the light gathering power, as a whole. Most current 4/3rds sensors are either 12 or 16 megapixels. If these sensors were scaled up to full-frame with the same pixel density, they would be 48 and 64 megapixels, respectively. The full-frame Nikon D800 has a 36 megapixel sensor. This means that a 12 megapixel 4/3rds sensor has pixels with surface area almost as large as those of this particular Nikon. If a 4/3rds sensor had around 8 megapixels, then we might expect the light gathering performance per-pixel of this sensor to be on-par with the full-frame camera's performance.

Pixel size is important for the same reason sensor size is important, light gathering power, but also because it determines how finely detail can be resolved. Larger pixels mean more light gathering, but also, less spatial resolution. This can be thought of as a trade-off between temporal and spatial resolution. Bigger pixels allow for faster shutter speeds, but give you less detail. You can increase both temporal and spatial resolution if you make the sensors bigger, but if you constrain sensor size then you have to make a choice between the two. As I write this, camera makers still seem to be (slowly) increasing megapixel counts. This is a predictable consequence of primarily emphasizing the number of pixels on a sensor: pixel counts are optimized and unmeasured parameters are largely ignored. (This seems a universal principle: what you measure is where your effort goes, so choose your metrics wisely.)

As a solution, I suggest adding pixel area to the stats for camera sensors. This serves two purposes. First, it allows simple comparisons to be made between cameras with different sensor sizes. It gives a good indication of the per-pixel temporal resolution performance of a sensor. Second, measuring this number provides slight push-back on the race for more and more megapixels. Given another metric to measure performance, camera companies would presumably seek to optimize both pixel counts and pixel areas, balancing temporal and spatial performance. Most photographers do not require the level of spatial resolution offered by current camera sensors. However, temporal resolution, low-light performance, etc. could still be improved almost without exception. Having a specific metric which correlates strongly with this performance will push camera companies to improve in this direction, and help keep them from pushing for more impressive megapixel numbers. If we have more megapixels than most people can use, but many are still asking for better low-light performance, then clearly the current pixel size that most manufacturers choose is not ideal, and there is still room for optimization. Personally, I would gladly trade better low-light performance for lower resolution images.

Examples
advertised specsproposed specs
Canon s10024-120mm (35mm equiv)74°-17° FOV (2-17m HF Distance)
F/2 - F/5.9F2/4 - F5.9/35
12 megapixels12 megapixels (3.6 pm² effective pixel area)
   
Olympus PEN E-P314-42mm kit lens (28-84mm full-frame equivalent)66°-24° FOV (4-21m HF Distance)
F/3.5 - F/5.6F3.5/12 - F5.6/31
12 megapixels12 megapixels (19 pm² effective pixel area)
   
Canon EOS 5D Mark III24-105mm kit lens74°-20° FOV (5-92m HF Distance)
F/4F4/16
22 megapixels22 megapixels (39 pm² effective pixel area)
   
Olympus 75mm F/1.8 μ4/375mm (150mm full-frame equivalent)14° FOV (208m HF Distance)
F/1.8F1.8/3.2

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