What is Resolution?

In photography Resolution refers to the ability of an imaging system to capture fine detail from the scene, making it a key determinant of Image Quality. For instance, with high resolution equipment we might be able to count the number of tiny leaves on a distant tree, while we might not with a lower-res one. Or the leaves might look sharp with the former and unacceptably mushy with the latter.

We quantify resolution by measuring detail contrast after it has been inevitably smeared by the imaging process. As detail becomes smaller and closer together in the image, the blurred darker and lighter parts start mixing together until the relative contrast decreases to the point that it disappears, a limit referred to as diffraction extinction, beyond which all detail is lost and no additional spatial information can be captured from the scene.

Sinusoidal target of increasing frequency to diffraction limit extinction — Increasingly small detail smeared by the imaging process, highly magnified.

The units of resolution are spatial frequencies, the inverse of the size and distance of the detail in question. Of course at diffraction extinction no visual information is captured, therefore in most cases the criteria for usability are set by larger detail than that – or equivalently at lower frequencies. Thresholds tend to be application specific and arbitrary.

The type of resolution being measured must also be specified since the term can be applied to different physical quantities: sensor, spatial, temporal, spectral, type of light, medium etc. In photography we are normally interested in Spatial Resolution from incoherent light traveling in air so that will be the focus here.

Spatial Frequency at a Threshold for a Given Target

The threshold is application and target specific. For instance photographic equipment capabilities are often compared by capturing a slanted edge and measuring the spatial frequency at which detail contrast is halved, a metric sometimes referred to as MTF50. Or a perfect lens is said to just resolve 1600 line pairs divided by its f-number. Or a person with 20/20 vision is assumed to be able to resolve thirty line pairs per degree on the retina. There is Rayleigh’s limit for stars, Abbe’s limit for microscopy, Dawes’, Sparrow’s and many more. All of these criteria were chosen empirically in precise settings that must be referenced.

Therefore when referring to the resolution or resolving power of an imaging system as a spatial frequency, it is necessary to specify the target, threshold and conditions it refers to, though in many cases they are implied for a given field of work.

The ‘imaging system’ in question can stop at photographic hardware, whose job is to record visual information from the subject; or it can reach all the way to the eye, directly or through a display medium. In the former case resolution can be a useful indicator of the performance of the equipment – the focus of this article; in the latter it can be the basis of psychovisual qualities of IQ like perceived sharpness and Depth of Field.

Spatial Resolution = Telling Things Apart

The word resolution comes from the verb ‘to resolve’, which in imaging means to discern or to tell apart.

Resolve or discern what, you ask? Objects in space. Once the image is captured in the raw data and magnified, can you tell whether there are two separate objects, like two stars, or could it just be a single oval one?

Airy pattern Rayleigh 1.22 Unresolved 0.61 — Figure 1. Two object or one? Top: Just resolved (Rayleigh criterion). Bottom: Unresolved. Highly magnified, same peak intensity.

Or in a different context, at a given distance, can you tell what the letters on a Snellen Acuity chart are?

Figure 2. Snellen Acuity Chart, courtesy of Jeff Dhal, under licence.

The ‘objects’ in Snellen charts are mainly the alternating horizontal and vertical black and white lines the letters are made of.

Yet again, can you tell whether those tiny gray specs in the satellite photo are big rocks, buses or enemy tanks? These are all applications that require a certain resolving power: resolution at an appropriate threshold.

Mind the Gap

We know from previous articles that the equipment used to capture the image of a scene degrades it. In other words the geometric projection of the scene onto the sensing plane is blurred by the imaging system’s (impulse) response, aka its Point Spread Function, which in turn depends on the hardware: the lens, the sensor stack, the sensor itself, etc. Mathematically, the geometric image is convolved with the system’s PSF.

The more removed the hardware is from perfection, the blurrier the image, until two stars side by side may look like one; or a letter or tank is no longer intelligible. What makes the difference is the gap between two blurred objects side by side: if we are able to record and/or measure the gap appropriately, they can be resolved; if we are not, they are not.

One way to quantify the gap is to measure its intensity as it transitions from one object to the other. For instance, it is a well known fact that the smallest objects that can be imaged by a perfect lens with a circular aperture (distant points or stars) will show as Airy patterns in the image. Below are two increasingly close, incoherent, monochromatic point sources against a black background captured by a noiseless, diffraction limited imaging system, highly magnified. The intensity of the central slice through the images is shown below them:

Limit Resolution Thresholds, Rayleigh Criterion — Figure 3: Two Airy Patterns closer and closer together. Left to right 3a: Peak intensities are an Airy disk diameter apart; 3b: Rayleigh’s limit = one Airy disk radius apart; 3c: Sparrow’s limit = 0.94 lambdaN apart; 3d: unresolved. lambda is the wavelength of monochromatic light, N is the imaging system’s effective f-number. Peak intensity was normalized to one. Click on the image to see it better.

The plots show the central variation in intensity, or contrast, as the two objects are imaged closer and closer together. Clearly if there is maximum intensity swing in the gap (Figure 3a) the details are likely to be considered resolved. If there is none (Figure 3d) they are likely not. Somewhere in between there is a threshold contrast where the detail may be considered resolved for a specific application. For instance Figure 3b shows the Rayleigh criterion, which is often used by astronomers as the limit to determine whether they are looking at two stars or one.

Contrast, Michelson

A useful measure of the peak-to-valley swing is Michelson Contrast, equal to half the drop in the gap divided by its average intensity (see Figure 4 below):^[*]

(1) $\begin{equation*} M = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \end{equation*}$

Looking at the definition, it is clear that Michelson contrast is a relative measure independent of absolute intensity ( $I$ ); and its range is zero to one, since the gap can disappear (when the numerator is zero) and intensity is always non-negative (the largest $M$ occurs with the smallest $I_{min}$ possible, equal to zero).

For instance in Figure 3a the two objects are separated by a distance equal to the Airy disk diameter and the intensity in the gap reaches down to zero, so $I_{min}$ is zero and the contrast is one. In Figures 3b and 4 the two objects are closer together, a separation of $1.22\lambda N$ the so-called Rayleigh limit, they overlap to the point that the gap at its minimum intensity is about 26.5% lower than the peaks, so the contrast is 15.3% (.265/1.735).^[1] In 3c (separation of 0.94 $\lambda N$ , Sparrow’s limit) and in 3d they are so close that there is no gap in the central slice, so $I_{max}$ is equal to $I_{min}$ and contrast is zero.

Figure 4. Central slice of the intensity of two Airy Patterns at the Rayleigh limit.

Incidentally, $\lambda$ here refers to the wavelength of light and $N$ to the effective f-number of the lens as setup. $\lambda N$ takes the units of $\lambda$ , typically expressed in microns or millimeters. If for example $\lambda$ is 0.5 $\mu m$ and N is f/4, 1 $\lambda N$ represents a distance of 2 $\mu m$ , 2 $\lambda N$ a distance of 4 $\mu m$ etc.

Alternatively we could also project the two dimensional images to 1D perpendicularly to their distance, obtaining a Line Spread Function, and measure the swing in that. Results would be a little more meaningful and in fact that’s what we typically do when measuring Michelson Contrast via the slanted edge method in photography. It does not really matter as long as we are consistent since as we will see further down the choice of a threshold is arbitrary and application specific.

Line Pairs

The objects to be resolved are not necessarily points, they could be lines for instance, which carry more total intensity and are therefore easier to see. Many targets in the past tended to be line based.

The letters in Snellen Acuity charts are made of alternating white and black lines of equal width. Similarly the white lines on a black background in the 1951 USAF Test Target:

Figure 5. Left, 5a: Letter E from a Snellen acuity chart, entirely made up of black and white lines (or blocks) of equal width. Right 5b: USAF 1961 Resolving Power test target, made up of black and white lines of equal width, under license, cropped.

A white and a black line side by side make up a line pair, or an up and down intensity cycle. Since every line is of the same width independently of intensity, a line pair (lp) corresponds to two line widths (lw), these terms and symbols are often found in the units of resolution measurement.^[*]

Lines also become blurred by the imaging system just as points were, by convolution with its PSF.

Cycles

If we extend the concept to a set of repeating such line pairs, we obtain a pattern of black and white lines of equal width alternating at a given spatial frequency. The period is the extent of the line pair (a cycle), measured as the center-to-center spacing of lines of the same intensity

(2) $\begin{equation*} 1\: period (p) = 2\: line widths (lw) = 1\: line pair (lp) \end{equation*}$

and one over the period is the spatial frequency:

(3) $\begin{equation*} frequency\: (f) = \frac{1}{p} \end{equation*}$

Frequency is expressed in cycles (or sometimes line pairs) per the units of the period. For example if we measure the line width in millimeters, say 0.1mm, there are 0.2 mm per line pair and that’s the period. One over the period is frequency, so that would be 5 lp/mm. See the article on the units of spatial resolution for how to convert between their many variants.

What would happen if such a pattern were captured by a diffraction limited imaging system with black and white line pairs spaced so that they repeat every 1.22 $\lambda N$ mm, at the Rayleigh criterion for resolution?

Repeating Line Pairs

We can simulate how that would look like by convolving the image top left in Figure 6 with one of the Airy Patterns in figure 3, as shown below at bottom.

Figure 6. Top left: A target made up of alternating white and black lines of equal width. Top right: central slice showing the intensity across the target. Bottom left: the target after having gone through a diffraction limited imaging system. Bottom right: central slice showing the intensity across the target

Since the spatial frequency of the line pairs is one over the period per Equation (3), their frequency in this case is $\frac{1}{1.22\lambda N}$ lp/mm.

Sine Qua Non

The intensity swings shown in the resulting central slice above are not made up of just the specified frequency but also of many harmonics introduced by the sharp edges of the lines, complicating the interpretation of the measurement. To make the interpretating math simpler we can use a sinusoidal target at the same frequency as the repeating lines. That ensures that only the original frequency will be present in the resulting intensity slice, no matter the PSF:

Figure 7. Top left: A sinusoidal target with the same period as in Figure 6.. Top right: central slice showing the intensity across the target. Bottom left: the target after having gone through a diffraction limited imaging system. Bottom right: central slice showing the intensity across the target

Such a target could represent an illuminated distant grate made of cylindrical rods against a dark background, it is visually very similar to the one made of lines in Figure 6. Putting it through the same diffraction limited imaging system as above reduces the amplitude of the signal – but this time the single original frequency is preserved (bottom right), an effect called modulation. This is the reason that an ensemble of such measurements is often called the Modulation Transfer Function of the imaging system, as we will see further down.

Michelson Contrast is clearly 1.000 in the unblurred target at the top of Figure 7, while it is 0.091 after having gone through the diffraction limited imaging system (bottom).^[1] If for the given application anything blurrier than this would be considered unsuitable, we could say that the resolving power or limit for this equipment is a spatial frequency of $\frac{1}{1.22\lambda N}$ lp/mm.

Acceptable Resolution…

Different fields have different thresholds of acceptability for their details to be considered resolved so the limiting resolution is arbitrary and application specific. For instance the criteria are different in astronomy, microscopy, machine vision, defense, landscape vs portrait photography, reading – that’s why one finds so many definitions of resolution when investigating this subject.

There are different resolution requirements even within the same field. Going back to the tank metaphor, a given resolving power may be acceptable for telling whether there are one or more distant dark masses in a picture; but more resolution would be needed to tell whether the masses are big rocks or vehicles; more again to tell whether one of them is a tank; more yet to tell whether it’s friend or foe; etc.

Even though thus far we have assumed a target of only points, line pairs or sines, acceptable resolution varies also with the actual detail’s original shape, contrast, brightness and wavelength. It does not really matter as long as one mentions the target used and is consistent, since the thresholds are arbitrary.

… or Resolutions

In other words we ideally would like to know the resolving capabilities of our hardware (a camera, a telescope, a microscope, the eye) with object detail (lines, points, sines) separated by various distances so that in the future we could easily estimate its resolving power in various situations.

Therefore we measure the contrast in the captured image with increasingly small detail at increasingly close spacing, recording it in a plot so that when the time comes we will know whether it is suitable for a given application. For our monochrome, noiseless, diffraction limited system capturing the sinusoidal target this is what we would get (click on it to view it properly):

Figure 8. A sinusoidal target at shorter and shorter periods as seen by a diffraction limited imaging system, Note the loss of contrast as frequency increases, easily measured by reading off the central slice. Click on it to see the animation without the artifacts introduced by WordPress’ downsizing.

We can then express the spacing as a number of cycles or line pairs per unit distance and chart the measured Michelson Contrast as a function of spatial frequency in a plot usually referred to as the Spatial Frequency Response (SFR) or Modulation Transfer Function (MTF) of the equipment under test:

Figure 9. The contrast read off the sinusoidal target of increasing frequency in Figure 8. The theoretical orange line is the Modulation Transfer Function of diffraction caused by a perfect lens with a circular aperture.

You may recognize the orange curve as the theoretical Modulation Transfer Function of diffraction caused by a perfect lens with a circular aperture, as discussed in an earlier article.

If we know the threshold required by our application we can read the expected resolution of the equipment off such a plot as explored in the next section.

Vice versa, for a given detail fineness (spatial frequency) we can determine the expected contrast loss. This last application is often seen in lens manufacturer MTF charts where curves at, say, 30 lp/mm show contrast as a function of distance from the optical axis.

Which reminds us that in practice any such contrast readings vary quite a bit depending on where one looks in the image (closer to the optical axis is usually best) and with the direction of the captured detail (sagittal vs tangential). Therefore to characterize an imaging system we need to perform this exercise several times throughout the field of view. Fortunately for photographers this can be accomplished by taking a very careful single capture, thank you MTF Mapper.^[2]

Thresholds for Resolution Measurement

Photography hardware review sites tend to use the resolution at which contrast is halved as an indicator of the perceived sharpness capabilities of the equipment. This threshold is sometimes referred to as MTF50(%), see this article for just how good of an indicator it may be in practice.

We can read the spatial frequency at which MTF is 50% from the plot in Figure 9: about 0.4 cycles per $\lambda N$ . If the wavelength of light is 0.535 microns and effective lens f-number is 4, that would mean that MTF50 occurs at a resolution of about 187 lp/mm, which is correct for a diffraction limited system. Photographers would love to own a lens with such performance.

Alas, true diffraction limited systems are hard to come by so here is what the same plot looks like from a real camera and lens (blue dots):

Figure 10. Spatial Frequency Response of Phase One P45+ Digital Back with 180mm/4 lens at f/8, ISO 50 obtained through the slanted edge method. Data from Image courtesy Erik Kaffehr.

Here MTF50 was measured at about 0.4 cycles per pixel pitch ( $c/p$ ). Since the Phase One P45+ Back sports a sensor with pitch of about 6.8 microns that corresponds to about 60 lp/mm.

A newer digital camera with an equivalent set up might achieve around 90 lp/mm. We could then say that for this purpose it achieves one and a half times the resolution of the P45+ as setup. In other words, with the same 50% contrast loss we should be able to capture detail 1.5x smaller with the newer camera.

On the other hand some photographers ascribe great importance to a quality in captured images referred to as ‘microcontrast’. Jim Kasson suggests that contrast at a spatial frequency of 0.25 cycles per pixel is a good indicator of that quality, the higher the better the microcontrast.^[3]

For other applications an 80% or 90% contrast loss might be acceptable, corresponding to MTF20 or MTF10. Below that the resolution is close to being limited by noise or by diffraction extinction at $\frac{1}{\lambda N}$ lp/mm, therefore less useful in photographic practice.

Yet again, depending on how the final photograph is displayed and viewed, MTF80 may be more appropriate. Or even better, obtain an estimate of perceived sharpness using a weighted average of various contrast readings, as described in the referenced article.

Conclusion

So it becomes clear that Resolution as used in photography is a measure of the capabilities of an imaging system to capture fine detail in the image, expressed as the spatial frequency that results in a certain contrast loss with a given target and setup.

Therefore when referencing resolution or resolving power, both the target (point sources, line pairs, sines, slanted edge) and the threshold at which it occurs should be specified (20/20 vision, 1 arc minute, MTF50, MTF30, MTF20, MTF10, Rayleigh, Sparrow, extinction, etc.).

If you are interested in how the various hardware components used in digital photography contribute to the resulting system resolution (SFR, MTF) you can take a look at the series of articles starting here, covering

Also, how to measure spatial resolution for your equipment.

Notes and References

_{1. The Rayleigh criterion suggests a spatial frequency of 1/(1.22lambdaN) lp/mm, which corresponds to a Modulation Transfer of 9.1% with a diffraction limited lens with a circular aperture. This is only true with a perfect sinusoidal target at that frequency.

2. Frans van den Berg’s excellent open source MTF Mapper can determines the resolution, SFR, MTF of your camera and lens by capturing slanted edges at home.

3. Jim Kasson’s excellent blog can be found here.

4. Simulated images in this page should render accurately assuming an sRGB (or sGray) display color space. It is amazing how few do of the many found online.}_{5. The Matlab script used to generate the simulated images in this page can be downloaded from here.}

5 thoughts on “What is Resolution?”

Chris Dainty says:

May 8, 2024 at 9:27 am

As always, a beautiful tutorial! I look forward to future ones.

1. Jack says:
  
  May 8, 2024 at 9:32 am
  
  Thank you Chris, much appreciated.
  
  Jack
  
Bruce says:

June 21, 2024 at 12:39 am

Hi Jack!There is an empirical rule, that is when the resolution of a lens reaches half of a sensor’s Nyquist frequency, it basically meets the imaging needs of the sensor, is this empirical law reasonable?Thank you vary much!

1. Jack says:
  
  June 21, 2024 at 4:52 pm
  
  Hello Bruce, I don’t know, I prefer to look at the system MTF curve and its components, since sensor and lens interact resulting in contrast loss which varies with the chosen threshold frequency.
  
  Ideally you want to sample at least at twice the highest spatial frequency of the image on the sensor (3 or more times is better). So say that the highest frequency is a grate at 37 lp/mm on a sensor with 6.8um pitch (the P45+ of Figure 10 above). The period of the detail is 1/37 = 27um, which means that 27/6.8 = 4 pixels cover one cycle. 4 pixels per cycle corresponds to a spatial frequency of 1/4 = 0.25 c/p, half of monochrome Nyquist per your request.
  
  Looking at the resolution chart in Figure 10 you see that at 0.25 c/p system MTF is about 0.6, which should be a good contrast for pixel peeping, so you are achieving your objectives even though you can see that most of the contrast loss is due to the (good) lens at that spatial frequency.
  
  In fact if the threshold for your application were relaxed to MTF50, you should be able to properly capture detail at greater than 50 lp/mm while still sampling it at almost three times per period.
  
  Jack
  
  1. Bruce says:
    
    June 22, 2024 at 6:17 am
    
    Thank you vary much!

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