Ground Sampling Distance (GSD), Ground Resolution and Spatial Resolution are terms used when specifying remote sensing system resolution. They are all used to define the smallest feature a remote sensing system can resolve on the ground. If your system is not looking at the ground, resolution defines the smallest feature on whatever the system is imaging. Either way, we’ll refer to the thing your imaging system is looking at as “ground”.
The three most important factors that affect resolution in a system are:
Let’s consider them in order.
The simplest definition of resolution is the size of one camera pixel on the ground. This is dependent on the altitude of the system and focal length of the lens being used. The following figure helps define the terms.
By similar triangles, we see that
\( \frac{GSD}{H} = \frac{w}{F} \)
where GSD is the width of a pixel on the ground, H is the altitude, w is the width of a camera pixel and F is the focal length of the lens. By assuming a perfect lens with no diffraction effects, we can use simple geometry to define the system resolution. The system can perform no better than this.
The ground sampling distance of a remote sensing system depends strongly on its altitude. Below is a chart showing the altitude for commercial drones, stratospheric imaging systems and low earth orbit:
Altitude for Remote Sensing Systems
| Commercial Drones | Stratospheric Imaging | Low Earth Orbit(LEO) Imaging |
|---|---|---|
| 0-5 km (0-120m for consumer in US) | 15-40 km | 160-2000km |
The units of measure change to accommodate the GSD of different applications:
| Commercial Drones | Stratospheric Imaging | Low Earth Orbit(LEO) Imaging |
|---|---|---|
| cm/pixel | cm/pixel | m/pixel |
In general, small pixels and/or a long focal length lens give the best possible resolution. However, physics may impose a lower resolution limit.
Just because the camera has small pixels doesn’t mean that the system can resolve those pixels. One factor that can affect the spatial resolution limit is the lens aperture size, commonly defined using F/#.
The F/# of a lens is the focal length (F) of the lens divided by the aperture diameter (D).
\( F/\# = \frac{F}{D} \)
By definition, the aperture limits how much light can enter the lens. (If you are unfamiliar with F/#, see our page on Numerical Aperture and F/# here).
The limiting resolution of a lens system can be measured a few different ways. One measure of the resolution limit of an optical system is the Rayleigh Criterion.
The Rayleigh Criterion states that two points are “just” resolvable if they are separated by one Airy disc radius. An Airy disc is produced when a perfect lens images a star-like object (a point source of light). Instead of the image of the star being exactly a spot, diffraction from the lens aperture causes the image to resemble a bright central region surrounded by concentric rings of decreasing intensity. The Airy disc radius is defined as the distance from the central peak of the Airy disc to the first minimum (see Airy disc cross-section below).
The Airy disc radius can be calculated by:
\( R_A = 1.22 \, \lambda \, (F/\#) \)
Where RA is the radius of the Airy disc, λ is the wavelength of light in microns (0.54µm for green light), and F/# is the F-number of the lens. Below is a table showing the Airy disc radius for different F/#.
| F/# | Airy Disc Radius (µm) |
|---|---|
| 10 | 6.59 |
| 4 | 2.64 |
| 3 | 1.98 |
| 2 | 1.32 |
It is a good idea to ensure that the F/# of the lens is low enough to produce an Airy disc radius smaller than the pixel size of the camera. In the case of the GPixel GMAX3265 sensor with 3.2µm pixels, a recommended F/# would be F/4.9 or faster (lower F/# is “faster”). The F/# resulting in an Airy disc radius equal to the pixel size can be considered a hard limit on the “slowest” a lens can be and still resolve the pixel. Cell phone manufacturers have a strong inclination to use slow lenses and lots of computation to give acceptable images.
At high altitudes, like the stratosphere or space, another diffraction limit comes into play. Just as a lens must be “fast” enough (have a low enough F/#) to utilize camera pixels, it must have a large enough aperture to resolve detail on the ground. The same “one Airy disc radius” criterion applies, but now the F/# is the altitude divided by the lens aperture.
For a lens with an aperture of 100mm on a balloon at 20km (66,000 feet), this F/# is given by
\( F/\# = \frac{20{,}000 \, \mathrm{m}}{0.1 \, \mathrm{m}} = F/200{,}000 \)
That’s a really slow F/#! The resulting resolution (assuming a wavelength of 0.55µm) is
\( R_A = 1.22 \, \lambda \, (F/\#) = 1.22 \times 0.55 \times 10^{-6} \, \mathrm{m} \times 2 \times 10^{5} = 0.13 \, \mathrm{m} = 13 \, \mathrm{cm} \)
While this would be horrible resolution for a drone, it is reasonably good for a camera in the stratosphere. A satellite at 200km would require a 1m diameter aperture to attain the same resolution. An interesting historical note is that the Space Shuttle cargo bay was sized to fit a satellite with a 4m mirror, which could yield 3.3 cm resolution.
If you want to work backwards from the required resolution to the lens aperture, the formula is
\( \Phi = \frac{1.22 \lambda A}{R} \)
where Φ is the aperture diameter, A is the altitude and R is the required resolution. Just be careful with units.
| Lens Diameter (mm) | Ground Resolution (cm) |
|---|---|
| 67 | 20 |
| 100 | 13 |
| 134 | 10 |
| 270 | 5 |
Want to calculate GSD for your system? We know this can be a complicated process which is why we created a calculator to serve as a guide. To get started, fill in the green fields with your desired parameters to calculate Focal Length, Swath, GSD and the recommended F/# for your drone lens.
If you’d like to run the calculator offline or explore the source code, we’ve provided multiple formats:
GitHub Repository (Python source code for the calculator)
A lens can have a fast enough F/# and still not be able to resolve 1 pixel. This is due to optical aberrations (see our optical aberrations page here). So, how can we know that a lens has small enough aberrations to meet the system resolution requirements?
One way to answer this question is to observe that having the Airy disc diameter smaller than a pixel means that essentially all of the energy from a point on the ground should be focused onto a single pixel. A point on the ground is equivalent to the star mentioned above. The generalized form of the Airy disc is a Point Spread Function (PSF). PSF is defined as the image of a point source formed by a lens. If the lens has aberrations, they will spread out the PSF and reduce its peak amplitude. To make this more concrete, here is the PSF for a perfect F/4 lens:
Note that the peak is 1.0 and there is a faint ring around the main peak. The diameter of the trough between the main peak and the ring is the Airy disc diameter.
For comparison, here is an image of a PSF with significant aberration:
In this example, which is one of many possible forms for the PSF, the peak intensity is reduced to 0.416 and there is a secondary peak. Clearly, a lens with aberrations like this will not put the same amount of energy on a pixel as the above diffraction limited lens.
Although PSF helps visualize how lens aberrations inhibit sharp imaging, they don’t help quantify the deterioration of the image. To do that, we switch from the PSF to the MTF. The two are mathematically convertible but show different aspects of lens performance in the presence of aberrations.
MTF is the gold standard in the optics industry for evaluating lens performance (see our page on MTF here for more background information). MTF describes how the contrast varies with resolution, as seen in the chart below.
The chart has “Modulation” on the y-axis which is essentially equivalent to contrast. High modulation means that the image of a black object looks black and white looks white. As modulation decreases, blacks begin to look dark gray and whites like light gray. When modulation drops to zero, it is impossible to distinguish black and white; both are mid-gray.
The x-axis is “Spatial Frequency”, which is a measure of resolution. In loose terms, low spatial frequencies correspond to low resolution, and both increase proportionally. The units of spatial frequency are cycles per mm. A cycle consists of a pair of lines, one black and one white. For example, if we had 1 black line and 1 white line fit into 1mm we would have 1 cycle per 1 mm. This means that the desired resolution in millimeters can be converted to cycles per mm using the following formula:
If the resolution we want is 1 pixel (3.2 µm = 0.0032 mm) we get a spatial frequency of 156 cy/mm. This is called pixel frequency or Nyquist frequency.
Using MTF, we can compare the performance of different lenses with similar specifications.
Below is an MTF plot for a 60mm F/4 lens. Note that the top (black) curve is the diffraction limit.
MTF of 60mm F/4 Poorly Corrected Lens
The minimum acceptable MTF is 20% (0.2 on the plots) for the Nyquist frequency of your image sensor. If you were to use the GMAX3265 image sensor, Nyquist is 156 cy/mm. On the chart above, a few curves fall below 20% at < 156 cy/mm, so this lens would not be acceptable.
Below is another MTF plot for a 60mm F/4 lens, but this one is designed for higher resolution.
MTF of 60mm F/4 Well Corrected Lens
All fields are better than 20% at 156 cy/mm, so this lens is well designed for the GMAX3265 image sensor. It is not at the diffraction limit, but it is a good compromise between cost and performance.
So, pixel size, lens F/# and aberrations all affect the ultimate resolution of a system for remote sensing. We hope you find this synopsis helpful as you design your remote sensing system.
Feel free to Contact Us for more information or if you have any questions on your project.