An Oddity of the Nyquist Limit in N Dimensions

photo of Harry Nyquistphoto of Claude ShannonA non-rigorous description of the Nyquist frequency or the Nyquist limit (named after the engineer Harry Nyquist) is simply that it is half the sampling rate of a "signal" (UV-visible light spectrum, audio file, image, whatever) that is discretely sampled. This comes out of the field of information theory which was developed by the engineer/mathematician Claude Shannon and is a concept that should be familiar to everyone who uses digital images.

In the cryoEM community, the term Nyquist limit is sometimes replaced by "Shannon sampling limit" and is rather casually referred to as the resolution limit of an image that is imposed by the pixel size of the digitization process (e.g., the film scanner or CCD camera). As noted above, the Nyquist frequency is half the sampling rate, which translates into the reciprocal of twice the pixel size of a digitized image. It is always important to note that this is often not the useful resolution limit of the image (which might be limited by factors such as staining, defocus or the environment of the microscope), but rather reflects the highest theoretical resolution that can be attained given the discrete sampling of the image.

low mag image of waffle gridIn practical terms, when examining the discrete Fourier transform of a digital image, the horizontal and vertical edges of the transform directly reflect the Nyquist frequency. For example, the image shown to the right is a 512 x 512 pixel image from a larger image recorded with the Gatan UltraScan 4000 CCD camera on our JEOL JEM 3200FS. The diffraction grating should have a spacing of 463 nm. The sampling in this image was calculated to be 5.6 nm/pixel (based on the distance between lines in the replica diffraction grating or the frequency of the diffraction maxima from the replica grating which are seen in the Fourier transform below). The Nyquist frequency (limit) of this image is the reciprocal of twice the pixel size or 1/11.2 nm (0.0893 nm-1). Fourier transform of image to right

The arrows here indicate the Nyquist frequency and point to the edges of the Fourier transform that are above and to the left of the origin of the Fourier transform. To be more correct, these arrows should point to the one pixel along the edge of the Fourier transform that is exactly above the origin and to the one pixel along the edge of the Fourier transform that is exactly to the left of the origin (or to the edge pixels below and to the right of the origin that are related to these according to Friedel's law).

There is clearly very little information outside the central region of this Fourier transform. This is a consequence of the low resolution nature of this image, and is a property of the "contents" of the image (i.e., what is shown in the image) but not of the digital nature (discrete sampling) of the image. For example, if the replica diffraction grating had been better defined (more regular, sharper edges), the diffraction from it would have extended further towards the edge of the transform (towards higher resolution).

However, it is possible to create a series of perfect digital images that show strong information at high resolution. The following series of checkerboard patterns are 384 x 384 pixel images that were created using a computer and that are "perfect" - the pixel values are binary (0 or 1, black or white) and the number of black or white "tiles" along each edge of the image is an even integer. In such an image, the edges of the tiles are infinitely sharp and the pattern of black/white tiles repeats perfectly in both dimensions.

The corresponding Fourier transforms (the scaled power spectra from the Fourier transforms are shown here) are also 384 x 384 pixel images and reflect the perfect nature of the original images: each diffraction maxima is exactly 1 pixel wide/tall (roll the mouse over the first Fourier transform to see a magnified image of this region) and there are maxima at every reciprocal lattice position in the Fourier transform. The power spectra are also exactly what one expects from the Fourier transform of a checker-board pattern: the Fourier transform of a square wave is a series of evenly spaced delta functions, and the lattice of points shown in these Fourier transforms is the two-dimensional analogy of that.

Note: The power spectra shown here have been scaled to show the diffraction maxima present at every possible position. Although only the first power spectrum appears to show variation in the strength of the maxima across the field of view, this actually happens in all the power spectra shown (and in the one-dimensional case, the values of these maxima follow a simple sinc curve). Anyone who wants to recreate these examples and examine the raw data should use images that are 512 x 512 to eliminate certain artifacts that are not visible in these images.

As the size of the tile in the checkerboard grows smaller, the spacing between the diffraction maxima grows larger. As this spacing grows larger and larger, there appears to be less and less information in the Fourier transforms (fewer and fewer maxima), but that information starts at higher and higher resolution (the diffraction maxima begin further and further from the origin). The final image likely appears to have a single diffraction maxima at the origin (which would correspond to an image with a constant value in every pixel). However, there is actually an additional diffraction maxima in the lower left corner of the transform which is faintly visible against the color frame that surrounds this Fourier transform (and a magnified view of this corner appears when the mouse is rolled over it).

Note2: Although the Friedel-related maxima should occur in the upper right corner of this image, it is missing due to the display algorith used to generate this figure.

64 x 64 pixel tiles:checkerboard with 64 x 64 pixel tiles Fourier transform of image at left:FFT of image at left
32 x 32 pixel tiles:checkerboard with 32 x 32 pixel tiles Fourier transform of image at left:FFT of image at left
16 x 16 pixel tiles:checkerboard with 16 x 16 pixel tiles Fourier transform of image at left:FFT of image at left
8 x 8 pixel tiles:checkerboard with 8 x 8 pixel tiles Fourier transform of image at left:FFT of image at left
4 x 4 pixel tiles:checkerboard with 4 x 4 pixel tiles Fourier transform of image at left:FFT of image at left
2 x 2 pixel tiles:checkerboard with 2 x 2 pixel tiles Fourier transform of image at left:FFT of image at left
1 x 1 pixel tiles:checkerboard with 1 x 1 pixel tiles Fourier transform of image at left:

The frequency (or resolution) measured in Fourier transforms is a function of the distance from the origin of the Fourier transform (located in the center of all the transforms shown here) and longer distances correspond to higher frequencies (higher resolutions, smaller spacings in the original image). In the last image/Fourier transform pair, the single diffraction maximum in the Fourier transform occurs at a radial position that is sqrt(2) times the distance from the origin to the edge of the Fourier transform in either the horizontal or vertical directions (i.e., the distance that defines the Nyquist frequency or limit). This long distance (and the corresponding high resolution) is clearly larger than the Nyquist limit (and thus higher resolution than the Nyquist frequency allows) but also clearly corresponds to a real feature of the Fourier transform: if the maxima at this distance are removed from the Fourier transform and a back-transform is calculated, the resulting image is featureless. As a matter of fact, every Fourier transform based on the checkerboard images above contains some diffraction maxima that are further from the origin than the distance measuring the nominal x/y Nyquist limit.

How is this seeming contradiction explained? There are at least two different ways to think about it:

However one choses to think about this, bear in mind that

  1. Shannon sampling is the correct way to think about and treat discretely sampled data;
  2. the Nyquist limit is an absolutely true property related to Shannon sampling;
  3. every point in a Fourier transform is within the Nyquist limit based on the sampling of the original image.

One implication of this is that if the data in a Fourier transform are filtered using any sort of mask with a radial limit, real information is (potentially) being discarded. This happens when (for example) a three-dimensional cryoEM reconstruction is blindly filtered "to the Nyquist limit" (i.e., to the radial limit defined by 1/(2 * step size)). In most cases, there is little to no real information at these resolutions. However, one should always keep in mind that there is the possibility of information in these locations, and one should at least understand one's data as well as possible (and these images recorded on the 3200FS using either our Gatan UltraScan 4000 CCD camera or a demo Direct Electron DE-12 camera show that there can be real information that would be lost if an image were to be blindly filtered to the Nyquist limit).