Sampling and sampling density
Definition
The word "sample" does not refer, in this image processing context, to the object you are imaging, but to a VoXel!!! "Sampling is the process of converting a signal (e.g., a function of continuous time or space) into a numeric sequence (a function of discrete time or space). The process is also called analog-to-digital conversion, or simply digitizing". (Wikipedia).
The sampling density or sampling rate is the number of recorded samples (PiXels or VoXels) per unit distance when converting from an analog signal to digital. Thus, the larger the Sample Size (or Sampling Distance, i.e. the size of one VoXel), the smaller the sampling density. It is like beans in a vase: the larger the size of the beans, the less of them fit in the vase, so the density of beans decreases and vice versa. (Not the physical density of the beans, but the number of beans in a given volume!!!)
The sampling density is a Microscopic Parameter that describes the conditions of your image acquisition, it is not something you can tune later on. It is determined by the way you configure your microscope (usually by the zoom factor in Confocal Microscopes, or by the magnification in combination with the detector size -maybe a CCD cell- in Wide Field Microscopes, or by the axial and lateral steppers). It establishes a direct connection between one VoXel or PiXel in the image and a real volume in the physical space. See Parameters in the software below.
Example
An image with 256 x 256 pixels that covers a physical area of 100 x 100 µm2 has a sampling density of 256 / 100 = 2.56 samples per micron both along X and Y. Equivalently, the Sample Size along any of these directions is 100 / 256 ~ 0.391 µm = 391 nm. Whether this is enough for the current conditions is established by the ideal sampling rate (see below).
Changing the zoom factor of your scanning microscope to scan a smaller physical area, say 70 x 70 µm2, while keeping the same number of pixels in the recorded image, 256 x 256 pixels, allows you to gain resolution by reducing the sample size: 70 / 256 ~ 0.273 µm = 273 nm. You acquire more samples along a given physical distance, allowing you to distinguish more details. This is limited though, by the diffraction on the lenses, and sampling beyond the ideal rate does not improve things.
The microscopic zoom can be contrasted with 'digital magnification': cropping and interpolation of the acquired data. Digital (post-acquisition) zoom does not provide more physical information or better resolution, structures are not better resolved. It does consume storage and memory resources. Two distinct features that look overlapped due to the resolution limitations will also look overlapped after a digital zoom.
Find another example here.
Ideal sampling
The ideal sampling density depends on the optics. (Ideal means necessary to capture all the information). As the Point Spread Function (PSF) is the basic "brick" of which the images are "built", one should record details at least on the scale of the PSF to gather all the available information. Failing in that may spoil any attempt of Doing Deconvolution, because deconvolution works on the PSF scale. If your digital sampling occurs at a physical scale much larger than that of the PSF, deconvolution simply can't happen!!! You would be recording many PSF's inside each image pixel, and it wouldn't be possible to disentangle them anymore.
See Quality Vs Sampling and Ideal Sampling for details.
Parameters in the software
It is recommended that image acquisition is done accordingly to the Nyquist Rate as much as possible. If this was not achieved, the actual sampling distances must be used in any case as parameters when Doing Deconvolution. Imaging with one sampling density but using different values for deconvolution will produce wrong Image Restorations. Don't fake the Microscopic Parameters to avoid warning messages in the software!!!
In the Huygens Software the sampling you enter to describe your image is compared to the Nyquist Ideal Sampling. If it is large or too large you get orange or red colors in the field entries, warning you about UnderSampling. This is merely informative, nothing is done to your data: the image will be deconvolved anyway using the actual parameters you entered (which is the correct thing to do). But the further you are from the ideal sampling the less deconvolution will improve the image. Therefore, the coloring and the warnings are simply to let you know that your original data won't improve too much, and that you'd better reacquire your images again with a more correct sampling...
To have the good parameters it is also important being aware of any Pixel Binning active in the microscope.