# What is the Nyquist Limit and what is its significance to photographers?

The Nyquist Limit is frequently mentioned in the context of lens and sensor resolution.
What is it and what is its significance to photographers?

Here is an example of it being used by DPReview.com in their resolution testing. Please note that the following is a simplification of how things actually work

Background:

In digital photography, a light pattern is focused by the lens onto the image sensor. The image sensor is made up of millions of tiny light-sensitive sensors whose measurements are combined to form a 2-dimential array of pixels. Each tiny sensor produces a single light intensity measurement. For simplicity, I will look at the 1-dimensional case. (Think of this as a slice that looks at only a single row of pixels).

Sampling:

Our row of tiny sensors, each of which is measuring a single point of light, is performing sampling of a continuous signal (the light coming through the lens) to produce a discrete signal (light intensity values at each evenly spaced pixel).

Sampling Theorem:

The minimum sampling rate (i.e., the number of sensors per inch) that produces a signal that still contains all of the original signal’s information is known as the Nyquist rate, which is twice the maximum frequency in the original signal. The top plot in the figure below shows a 1Hz sine wave sampled at the Nyquist rate, which for this sine wave is 2Hz. The resulting discrete signal, shown in red, contains the same information as the discrete signal plotted beneath it, which was sampled at a frequency of 10Hz. While a slight over simplification, it is essentially true that no information is lost when the original sample rate is known, and the highest frequency in the original signal is less than half the sample rate.  Effects of under sampling:

If the sample frequency were less than 2 times the maximum frequency of the signal, then the signal is said to be under sampled. In that case, it is not possible to reconstruct the original continuous signal from the discrete one. An illustration of why this is the case can be found in the figure below. There, two sine waves of different frequencies sampled at the same rate produce the same set of discrete points. These two sine waves are called aliases of each other. All discrete and digital signals have an infinite number of aliases, which correspond to all the sine waves that could produce the discrete signals. While the existence of these aliases may seem to present a problem when reconstructing the original signal, the solution is to ignore all signal content above the maximum frequency of the original signal. This is equivalent to assuming that the sampled points were taken from the lowest possible frequency sinusoid. Trouble arises when aliases overlap, which can happen when a signal is under sampled.

But Photographs Don't Look Like Sinusoidal Waves. How is all this Relevant?

The reason all of this matters for images is that through application of the Fourier Series, any signal of finite length can be represented as a sum of sinusoids. This means that even if a picture has no discernable wave pattern, it can still be represented as a sequence of sinusoids of different frequencies. The highest frequency that can be represented in the image is half the Nyquist rate (sampling frequency).

Meanings of Similar Terms:

Nyquist rate - The lowest possible sampling frequency that can be used while still guaranteeing the possibility of perfect reconstruction of the original continuous signal.

Nyquist frequency - The highest frequency continuous signal that can be represented by a discreet signal (for a given sampling frequency).

These two terms are two sides of the same coin. The first gives you a bound on sampling rate as a function of max frequency. The second gives you the max possible frequency as a function of sampling rate. See Wikipedia: Nyquist frequency for further reading.

Nyquist Limit is another name for Nyquist frequency. See wolfram.com: Nyquist Frequency

• Superb answer! The part about under sampling is particularly useful. Apr 10 '11 at 19:12
• Thanks. I adapted it from a paper I wrote a few years ago for one of my electrical engineering classes.
– Sean
Apr 10 '11 at 19:16
• So, here's a question I have. The photosites aren't actually theoretical point samples; they cover an actual area. (Or, in the one-dimensional case, a short length — but not a point.) Does this have any practical impact on application of the theory to reality? Apr 10 '11 at 23:35
• @mattdm - That's a very interesting question. In the context where I studied sampling (time changing electrical signals), the duration over which each sample was taken was never large relative to the sample rate, so it was never an issue. As far as I am willing to speculate, the effect might be similar to applying a low-pass filter that had a cutoff frequency very near to the sampling frequency. Such a filter would attenuate (but not completely remove) the very high frequency content of the image.
– Sean
Apr 11 '11 at 1:08
• This video might help you visualize aliasing: youtube.com/watch?v=yIkyPFLkNCQ -- The "frequency" keeps increasing until it hits the Nyquist frequency (at about 0:37), after which the wave appears to reverse direction and decrease in "frequency" back down to 0. May 2 '11 at 7:44

The Nyquist Limit is mostly used in digital sound recording, but it also applies to digital photography.

In digital sound recording, the highest frequency sound that you can possibly record is half of the sampling frequency. A sound recording av 44100 kHz can not record any sound frequencies above 22050 Hz.

In photography it means that you can't possibly capture a wave pattern where the waves are closer together than two pixels.

In sound recording, everything is frequencies, so the Nyquist Limit is always relevant. In photography you don't often have wave patterns that are affected, so it's mostly used as a theoretical limit of the resolution of the sensor.

You can see the effect of this limit in a few situations where there is a horisontal or vertical wave patterns in a photo, like for example taking a picture where there is a window at a distance with the blinds pulled. If the blades in the blind are closer than two pixels, you can't distinguish the separate blades. However, you are more likely to see a wave pattern that is not exactly horizontal of vertical; it is in that case you will instead see the effect of jagged edges or moiré patterns which occur before the Nyquist Limit.

• Everything in photography is also frequencies. Digital cameras take a sample of an analog signal. At that point, it doesn't really matter if the signal is sound or light. This answer seems to imply that the limit only applies to certain patterns in a scene, which isn't right. Apr 10 '11 at 12:22
• It doesn't matter. The image is still an analog signal. The point is that all photographs have a pattern that span an area of pixels. In fact, every photograph is such a pattern, spanning all of the pixels. In some cases (as the ones you are talking about) you may see artifacts caused by the sampling. But in all cases, the resolution is limited. (A more interesting objection is that photosites are not points but actually cover an area; I have no idea how that factors in.) Apr 10 '11 at 16:41
• @Guffa, @mattdm, the light falling on the sensor is a wave pattern. The Nyquist limit applies because each photo site is a sample of the incident wave form. The Nyquist Limit says that we can only reproduce a sampled waveform if the sampling frequency is >= 1/2 the incident frequency. The number of photo sites determines the sampling frequency and therefore the Nyquist Limit. Apr 10 '11 at 17:17
• @Guffa, a digital image is a 2D wave pattern (really three, one for each color channel), not in terms of the frequency of light waves but in terms of alternating light and dark pixels that make up the image. The fact that light is itself a wave is not directly relevant to the use of the Nyquist–Shannon theorem for measuring sensor resolution.
– Sean
Apr 10 '11 at 17:38
• @Guffa: The analog image projected by a lens is indeed a wave pattern, and the full extent of wave theory can be applied to photographic images. When we talk about waves in terms of images, were not talking discrete light waves, but the wave nature of lighter and darker elements of a 2D image. In most simplistic terms, a maximally bright pixel is the peak of a wave, where as a minimally dark pixel is the trough of the wave, when only factoring in luminosity. The problem becomes more complex when you account for R, G, and B colors, but the concept remains the same. Apr 10 '11 at 18:38

Just to add to the previous answers... if you have a pattern beyond the Nyquist limit, you may experience aliasing — i.e. it may show as a lower frequency pattern in the image. This used to be very apparent on things like checked jackets on TV. Therefore, you do need a low pass anti-aliasing filter before sampling so that this artifact is not a problem.