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The f/ratio Myth and Digital Cameras

by Roger N. Clark

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There are often questions about smaller versus larger pixels and the corresponding camera size. Many of us would like smaller cameras that did just as good a job as larger ones. Is that possible? No, at least in terms of signal-to-noise that can be recorded. Here is why.

There is a common idea in photography that exposure doesn't change between different size cameras when working at the same f/ratio. For example, the sunny f/16 rule says a good exposure for a daylight scene is 1/ISO at f/16. Thus for ISO 100 film, you use a 1/100 second exposure on an 8x10 camera at f/16, a 4x5 camera at f/16, a 35mm camera at f/16, an APS-C digital camera at f/16, down to the smallest point and shoot camera at f/16 (assuming the small camera goes to f/16). The myth is that every camera will provide the same signal-to-noise images as long as the same exposure time and f/ratio lens is used, from the camera with the smallest sensor, to the camera with a large sensor.

The concept of constant exposure for a given f/ratio leads people to think cameras scale easily and still give the same image. But there is a fallacy in this idea, and that is the spatial resolution on the subject. The smaller camera, even at the same f/ratio, has a smaller lens which collects a smaller number of photons per unit time. The smaller camera gets the same exposure time because the UNIT AREA in the focal plane represents a larger angular size on the subject.

The rate of arrival of photons in the focal plane of a lens per unit area per unit time is proportional to the square of the f-ratio. Corollary: if you keep f/ratio constant, and change focal length then the photons per unit area in the focal plane is constant but spatial resolution changes.

So how does this apply to making smaller cameras?

The problem is that if you scale a camera down, say 2x, the aperture drops by 2x, the focal length drops by 2x (to give the same field of view), the sensor size drops by 2x, and the pixel size drops by 2x (to give the same spatial resolution on the subject). The aperture has collected only 1/4 the number of photons. If we kept the same sensor, then each pixel would collect the same number of photons because each pixel now sees a larger angular area. But we want the same resolution, so the pixels are 2 times smaller. The smaller pixels each collect 1/4 less photons since their area is divided by 4 to keep spatial resolution constant.

Another way to look at the problem is aperture collects light, the focal length spreads out the light, and the pixels are buckets that collect the light in the focal plane. BUT THE TOTAL NUMBER OF PHOTONS DELIVERED TO THE FOCAL PLANE IS ONLY DEPENDENT ON APERTURE (ignoring transmission losses of the optics). Thus, photons delivered to a pixel for a given resolution on the subject goes as the square of the aperture (and camera size)! Decreasing your camera by 2x means 4x less photons per pixel if you want to maintain field of view and megapixel count!

This is just what we observe with small cameras: their smaller sensors have smaller full well capacities, that get filled for a given exposure time with a smaller number of photons. That in turn means higher noise because there are fewer photons.

A good example is the Canon 20D with 6.4 micron pixels and a maximum signal at ISO 100 of 50,000 electrons, compared to the Canon S60 with 2.8 micron pixels with a maximum signal of about 11,000 electrons at ISO 100. The pixel size is (6.42 *6.42) / (2.82 * 2.82) = 5.2x scaling, similar to the 50000/11000 = 4.5 scaling of maximum recorded signal.

Then, for photon noise limited systems, the signal-to-noise ratio achievable in an image is the square root of the number of photons collected, so signal-to-noise ratio scales linearly with the camera pixel size.

If you tried to make a smaller camera that collects the same number of photons as a larger camera, you must keep the aperture constant. Given a camera, for example, with a 50 mm f/1.4 lens and then shrink the camera 2x, you would need a 25 mm f/0.7 lens that had double the resolution if you wanted to keep the same detail in the image. That means the smaller camera would not be much smaller, and might be more expensive due to the lens specifications.

Discussion: How Can You Predict the Noise Performance?

All decent digital cameras are photon noise limited for any signal above a few hundred of photons and read noise limited for less than a few tens of photons, for reasonable exposures of seconds and less. For long exposures into minutes, thermal noise adds to the equation. Every DSLR tested by myself and several others behaves this way, and so do the tested P&S cameras. Because the cameras have become so good (basically the last approximately 5 years), we can now understand the basic performance of systems and accurately predict what they can do.

Here is an explanation and test data for a DSLR that shows the Canon 1D Mark II DSLR, used in the example above, is photon noise limited: Procedures for Evaluating Digital Camera Sensor Noise, Dynamic Range, and Full Well Capacities; Canon 1D Mark II Analysis .

You can also estimate quantum efficiency and count photons with digital cameras: Digital Cameras: Counting Photons, Photometry, and Quantum Efficiency .

Table 1-3 at this page shows derived noise characteristics for photon-noise limited cameras: The Signal-to-Noise of Digital Camera images and Comparison to Film .

So, the bottom line is that photon noise limited cameras have predictable performance for normal photography (outdoor, even indoor lower light conditions). Photon noise limited performance is the best possible, and it is because cameras are performing at this impressive level that allows us to explore performance issues in ways never possible with film cameras.

Check out this web page for more info on this subject:

Another implication for the topics discussed on this page is the effects on depth-of-field. See: The Depth-of-Field Myth and Digital Cameras.

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First Published February 26, 2006
Last updated September 10, 2006.