Sampling Theorem

Appropriate sampling Undersampling
The image on the left is sampled appropriately, the image on the right however is undersampled.
The loss of image detail is proportional to the undersampling, and as such the quality of the subsequent image analysis.

To sample a microscopic image with a CCD camera, you should adher to the Nyquist sampling theorem and the Whittaker-Shannon Sampling theorem. The sampling theorem states that in order to reconstruct a function after discrete sampling, the samples should be taken at intervals equal to 1/2 of the upper cutoff (Nyquist) frequency of the original function. The Nyquist sampling theorem states that, when converting from an analog signal (sound or a microscope image) to digital, the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version. If the sampling frequency is less than this limit, then frequencies in the original signal that are above half the sampling rate will be aliased and will appear in the resulting signal as lower frequencies (seen as the blocks in the undersampled image shown above).
The actual sampling rate required to reconstruct the original signal will be somewhat higher than the Nyquist frequency, because of quantization errors introduced by the sampling process.

Separating adjacent features requires the presence of at least one intervening pixel of disparate intensity value. For this reason, the best spatial resolution that can be achieved occurs by matching the diffraction-limited resolution of the optical system to two pixels on the CCD in each linear dimension. This is called the Nyquist limit.
Nyquist frequency: The highest frequency that can be reproduced accurately when a signal is digitally encoded (e.g. CCD camera) at a given sample rate.
Theoretically, the Nyquist frequency is half of the sampling rate.
For example, when a digital sound recording uses a sampling rate of 44.1kHz, the Nyquist frequency is 22.050kHz. If a signal being sampled contains frequency components that are above the Nyquist limit, aliasing will be introduced in the digital representation of the signal unless those frequencies are filtered out prior to digital encoding.
Expressing this mathematically for a CCD camera mounted on a microscope and light with wavelength lambda we get:

(0.61 x lambda / N.A.) x Magnification = 2.0 x (pixel size)

Let's use this result to work through some practical examples.

Example 1: Given a CCD camera with a pixel size of 6.8 µm, visible light (lambda = 0.5 µm), and a 1.3 N.A. microscope objective, we can compute the magnification that will yield maximum spatial resolution.
 

M = (2 x 6.8) / (0.61 x 0.5 / 1.3) = 58

 

Thus, a 60x, 1.3 N.A. microscope objective provides a diffraction-limited image for this CCD camera without any additional magnification. Keep in mind, however, that this assumes that the condensing system also operates at an N.A. of 1.3. This high N.A. means the condenser must be operated in an oil-immersion mode, as well as the objective.
 
 
Example 2: Given a CCD camera with a pixel size of 15.0 µm, visible light (lambda = 0.5 µm), and a 100x microscope objective with an N.A. of 1.3, we can compute the magnification that will yield maximum spatial resolution.
 

M = (2 x 15.0) / (0.61 x 0.5 / 1.3) = 128

 

Since the microscope objective is designed to operate at 100x, we would need to use an additional projection optic of approximately 1.25x in order to provide the optimum magnification.
 

The table below gives you the minimum magnification, necessary to detect all the spatial details a microscope can resolve, with a single CCD B/W camera placed on the microscope. The same principle will hold for most 3CCD color cameras, but not for a single-CCD color camera with a Bayer-grid as it has a reduced and unequal spatial sampling rate for each color.
Choose the appropriate Numerical Aperture (N.A) and the pixel size (for square pixels, width) of the CCD-array of the camera, to calculate the apropriate magnification. The values given here are for an optical (widefield) microscope, a single CCD B/W camera and green light with a wavelength of 520 nm.

  CameraPixel (micron) 6 7 8 9 10 11 12 13 14 15 16 17 18
N.A. Resolution (micron)                          
0.1 2.65 4 4 5 6 6 7 7 8 9 9 10 11 11
0.15 1.77 6 6 7 8 9 10 11 12 13 14 15 16 17
0.2 1.33 7 9 10 11 12 14 15 16 17 19 20 21 22
0.25 1.06 9 11 12 14 15 17 19 20 22 23 25 26 28
0.3 0.88 11 13 15 17 19 20 22 24 26 28 30 32 33
0.35 0.76 13 15 17 19 22 24 26 28 30 32 35 37 39
0.4 0.66 15 17 20 22 25 27 30 32 35 37 40 42 45
0.45 0.59 17 19 22 25 28 31 33 36 39 42 45 47 50
0.5 0.53 19 22 25 28 31 34 37 40 43 46 49 53 56
0.55 0.48 20 24 27 31 34 37 41 44 48 51 54 58 61
0.6 0.44 22 26 30 33 37 41 45 48 52 56 59 63 67
0.65 0.41 24 28 32 36 40 44 48 52 56 60 64 68 72
0.7 0.38 26 30 35 39 43 48 52 56 61 65 69 74 78
0.75 0.35 28 32 37 42 46 51 56 60 65 70 74 79 84
0.8 0.33 30 35 40 45 49 54 59 64 69 74 79 84 89
0.85 0.31 32 37 42 47 53 58 63 68 74 79 84 89 95
0.9 0.29 33 39 45 50 56 61 67 72 78 84 89 95 100
0.95 0.28 35 41 47 53 59 65 71 76 82 88 94 100 106
1 0.27 37 43 49 56 62 68 74 80 87 93 99 105 111
1.05 0.25 39 45 52 58 65 71 78 84 91 97 104 110 117
1.1 0.24 41 48 54 61 68 75 82 88 95 102 109 116 122
1.15 0.23 43 50 57 64 71 78 85 92 100 107 114 121 128
1.2 0.22 45 52 59 67 74 82 89 97 104 111 119 126 134
1.25 0.21 46 54 62 70 77 85 93 101 108 116 124 131 139
1.3 0.20 48 56 64 72 80 88 97 105 113 121 129 137 145
1.35 0.20 50 58 67 75 84 92 100 109 117 125 134 142 150
1.4 0.19 52 61 69 78 87 95 104 113 121 130 139 147 156

Undersampling (magnification too low) will result in loss of detail in the digital image and will have a negative influence on the quality of the image analysis.
Oversampling (magnification too high) will not add more to the spatial detail of the digital image for analysis.

See also

References

*Nyquist, Harry
Certain topics in telegraph transmission theory
AIEE Trans., vol. 47, pp. 617644, Jan. 1928.
*Shannon, Claude E.
Communications in the presence of noise,
Proc. IRE, vol. 37, pp. 1021, Jan. 1949.

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The author of this webpage is Peter Van Osta and my resume can be found here.