Light as particles – photon noise
My previous article considered diffraction blur, a phenomenon explained by the wave nature of light. In this article I will look at noise in digital image capture, which is best explained by the particle nature of light. This article introduces the subject; for more information and details of a method for measuring noise, including results of measurements on real cameras, see How to measure noise in digital camera raw images.
Light can be considered as particles, photons, and because these are of a non-zero size the noise is more significant in cameras with smaller sensors and is more noticeable at lower light levels (e.g. in the darker areas of an image). This is the dominant noise source in modern compact cameras as other sources of noise have been reduced by improvements in technology.
To understand why in low light a sensor with smaller pixels shows more noise it is useful to consider the analogy of “buckets in the rain”. Consider a bucket left out in a rain shower. The bucket represents the “photon well” of a single pixel on the sensor and each raindrop represents a photon. Clearly a bucket with twice the area will collect twice the number of raindrops in a given time.
In a digital sensor each photon generates an electron so a pixel with double the area will collect twice the number of electrons. During an exposure in low light we can expect to collect thousands rather than millions of photons. Poisson statistics tells us that the probable error (the spread in values, i.e. the noise) is the square root of the number counted.
Notice that four times the signal gives us only twice the noise. Doubling the signal to noise ratio (SNR). This 4x could be, for example, a 4x longer exposure of the same subject – or it could be a 4x increase in sensor area. The concept of SNR is important because we cannot amplify a signal without amplifying the noise by the same factor so that the SNR remains the same.
Effect of pixel dimensions
Consider two hypothetical cameras, both having sensors of 6000 x 4000 pixels (=24Mp). Suppose the first is a full-frame camera, 36 x 24 (mm) sensor size, and the second has a sensor exactly one quarter the area, 18 x 12 (mm). I will refer to this as “Crop-factor 2” Other things being equal, each pixel on the full-frame sensor will be 4x the area of a pixel on the smaller sensor and in the same light level will collect 4x the signal.
Suppose each of the cameras is used to take the same low light shot. Using the same numbers as above we could have:
The table shows that because the smaller sensor has pixels which are one quarter the area the same light level will produce a signal one quarter the size. When this is amplified to match the signal from the larger sensor, the noise is amplified to a level twice that of the full-frame sensor.
Another consideration is thermal noise. Any electronic amplifier whether it’s squeezed into your digital camera or sits on a shelf in your hi-fi cabinet contains components which generate noise – even in the absence of a signal. This is mainly thermal noise (also known as Johnson-Nyquist noise) and is generated by the thermal agitation of the electrons in the components. As the name implies it increases as things get hotter. Amplifiers for very special purposes are refrigerated to keep the noise acceptably low.
The fact that in general a smaller sensor will require a larger degree of amplification might mean a further contribution to the total noise. However, this is probably less significant than the photon noise and whereas photon noise is fundamental (in that it is amenable to simple analysis) thermal noise depends very much on the detail design of the circuits. With current technology, photon noise is likely to be more significant than thermal noise.
So what does this mean for compact cameras?
If we compare the area of a single pixel in, say, the Canon 5D Mk II with the Canon G15 compact (based only on pixel pitch) we have a ratio of 41.1/3.6 = 11.4, more than eleven times the area. I have no knowledge of other noise sources in these cameras and I have not quantified the photon noise theoretically, but this does explain why compact cameras cannot compete with larger formats in low light.
Having access to a 5D Mk II, a G9, and a G15, I took a few test shots to compare the noise in the dark areas. The results are shown below. Each camera was put on the tripod in turn and all shots were taken at 400 ISO, f/4, aperture priority. The images covered the same field of view and the red square in the image below shows the area sampled at 100%.
In every case the raw image was processed in RawTherapee by the amaze demosaicing algorithm with 2 steps of false colour correction. No noise smoothing was applied.
I could not resist plotting pixel values along a horizontal row using ImageJ. In each of the 1:1 samples I selected a single row of pixels as shown left. The graphs are shown below. The horizontal scale is pixel number, the vertical scale is the pixel value.
The above graphs show the pixel values along a single row selected from the three images in approximately the same place in the image.
When is a raw file not a raw file?
We can never know just how much pre-processing (including noise filtering) has been carried out on the raw data before it is saved as what we term a raw file. Compare the images from the G9 and the G15; The sensor dimensions and pixel count are the same but the later camera shows a lower noise level. The G9 has a CCD sensor and the G15 a CMOS sensor which probably accounts for much of the improvement. The processing is also different, the G9 using the DIGIC III processor and the G15 using the DIGIC 5 processor.
A final thought
Whilst much of what goes on inside digital cameras will always remain hidden, one aspect of the process of image capture is truly in the public domain and that is the simple process of photon counting which inevitably obeys Poisson statistics. As we saw with diffraction where smaller holes produce softer images, smaller pixels produce noisier images, both factors supporting the argument for physically larger sensors – not necessarily more pixels.
It could be argued that in the image produced by the 5D Mk II we are seeing not the noise generated in the camera but the noise inherent in the printing of the book cover. (This will, of course, be present in all the images but is swamped by camera generated noise in the G9 and G15). A better test would be a comparison of out-of-focus grey areas. The performance of the 5D Mk II may well be much better than this image suggests.
Since writing this article I have measured (yes, measured!) the noise in different cameras. For the results and a theoretical analysis of the results follow the link:
Roger N Clark, Digital Cameras: Does Pixel Size Matter?
Factors in Choosing a Digital Camera
(Does Sensor Size Matter?)
A very comprehensive article with lots of data