Resolution

Resolution and recognition are concepts related to processing and use of a noisy and/or smoothed data. Often these concepts are applied to the analysis of bi-dimensional pictures, images; but applied also to uni–dimensional arrays, for example, registered spectra at the physical experiments.

Roughly, resolution refers to the ability to distinguish single object from a pair of close objects at the picture.

Recognition refers to the ability to identify an object at the image, to guess, that it belongs to the certain set.

In vicinity of the limit of the imaging system to proceed the sufficient information, the terms resolution and recognition become related.

Introduction
Any measurement has limits, related with the noise and smoothing of data. Usually, the devices measuring the length cannot measure it with error smaller than the size of atoms, id est, 0.1 nanometer. The devices measuring time usually cannot measure it with an error smaller than a femtosecond. Such estimates are far form the fundamental Planck's units of time and lengths ( \(5.39\!\times\!10^{-44}\)second and \(1.6\!\times\!10^{-35}\)meter ) but the most of measurements have even worse precision.

The important question is how to characterize the precision and resolution at the measurements; this refers both to the measurement of a single value and to analysis of images obtained with some optical systems (spectra, registrograms, traces, photographies, etc.).

The question of qualification of some details a diagrams in terms "clearly seen" (or "resolved") to "barely seen" (or "barely resolved") and "not seen" ("not resolved") is considered in this article.

Resolution
Term "resolution" may have different meanings.

At the digital representation of image (for example, specifying the image at the computer's display), term "resolution" may mean the number of pixels represented in each of two directions .

At the registration of images with optical devices, the term resolution or minimum resolvable distance is the minimum distance between distinguishable objects in an image, although the term is loosely used by many users of microscopes and telescopes to describe resolving power. .

In such a way, term "resolution" is very ambiguous. In this article, resolution refers to the ability to distinguish objects, and, therefore, is closely related to the recognition.

History
The the first attempts to provide the formal criteria of "resolution" are attributed to Joseph Louis Lagrange, Hermann von Helmholtz, Ernst_Abbe and Lord Rayleigh (John William Strutt, 3rd Baron Rayleigh). These criteria refer to the observation of image of a pair of dots, obtained by an idealized ideal lens (cylindrical or spherical) of a given size and focal length. The dots are declared to be resolved, if the diffraction pattern of one of them is centered at the first node of the diffraction pattern due to another one. Roughly this leads to the estimate of the minimal distance
 * \( d=\lambda/(2~ \mathrm {NA})\)

where \(\lambda\) is wavelength of waves used for the probing, and NA is the numerical aperture of the idealized imaging system.

At the scientific slang, the distance \(d\) is called "limit of resolution" or simply "resolution". Such a slang may cause a confusion, because the term "increase the resolution" may have meaning of reduction of \(d\); and "High resolution optics" refers to small values of \(d\).

Various ways to improve the resolution are related with use of confocal optics and/or the interferometric microscopy ; any attempts to see the details smaller than \(d\) can be qualified as "super–resolution". In particular the 4Pi Microscope is claimed to provide \(d\) for 2 orders of magnitude smaller, than in the criterion above (and much smaller than wavelength). . Even more options appear with the use of the Stimulated Emission Depletion Microscope and/or other methods involving the nonlinear response of the optical materials. However, the experimental verification (or refutation) of the claims about the "super-resolution" needs certain justification of term "resolution".

Super-resolution and recognition
In the idealized case, if the interference pattern due to each of dots of the sample is clearly seen, the pair of dots can be distinguished from a single dot even if distance between them is much smaller than \(d\) in the criterion above; this is determined by the precision of the registration of the image. In such a way, the "resolution" defined with a common sense and the criterion above gives only qualitative specification of an optical system, it indicates the correct order of magnitude of the the size of objects that are seen. In particular, the ability to resolve the objects depend on the contrast: the dark–gray spots at the light–gray background are less likely to be "resolved" than the brilliant dots at the black background.

In the simplest case, one deals with just dots. Then, the ability to resolve these dots refers to the ability to distinguish one single dot from the pair of dots. The correct question about such a dots can be placed as follows:  At which significance level the registered pattern is different from that due to a single dot?.

For example, of the image of an idealized dot is circular with many decimal digits, and the deviation from the circular pattern is registered in the second digit, does this mean that the sample is recognized, "resolved" as a pair of dots?

In such a way, the ability to "resolve" the details is strongly related to the ability to recognize, identify them. The a priori knowledge of some properties of the sample greatly improves the resolution; for example the knowledge that a single dot has circular symmetry allows to distinguish it from a pair of very close dots.

There exist various ways to approach the "maximal" resolution , and correct formulation of the concept of resolution should be elaborated. The concept of recognition seems to be the appropriate base for the correct definition of resolution.

Resolution as recognition
Assume there are several classes of objects. The difference between these objects may be in their size, in their shape, in their orientation, in their colors, in the topological structure of black and white regions, etc. Assume that at some "ideal resolution", the problem of the identification, recognition is already solved. This makes the difference between the problem of resolution and the classical problem of recognition by M.Bongard . This means that there exist a formal algorithm that sort the samples by the sets specified. Then, the resolution of an image means that with good efficiency (say, 99%, or \(1\!-\!10^{-9}\), etc.) the algorithm may perform the same sorting using such images.

In such a way, the term "resolution" cannot be applied to a single image; it should be applied to some class of images of objects from different (at least two) sets.

If the algorithm can perform the sorting, by the images, then these images are qualified as "resolved", and if the algorithm cannot, the images are qualified as "not resolved".

Future work
The practical verification and comparison of various methods for microscopy (as well as for any imaging systems) should be based on the ability to classify the objects from different sets. The list of the Bongadr's problems could be used as the test images for the preliminary verification of the algorithms of imaging. In such a sense, the advanced imaging system cannot be separated from the system of recognition.

The more advanced classification may interpret the images as pictures of biological sells, slices of neurons, as bacterias, as viruses, as molecules, as atoms – as far as possible while the precision of measurement allows to jump under the classical "limit of resolution" specified in the beginning.

The first stage of work may deal with artificial, simulated objects, similar to the simplest Bongard problems from the first raw in. Then, the samples of the cross-sections of the realistic structures should be considered, for example, the slices of the brains of the insects labeled with specific antibodies . The advanced treatments of images may include the recognition of the 3-dimensional structures from the set of images of the bi–dimensional slices. At the marginal case, the resolution of such 2-dimensional structures should be better than bi-dimensional: the additional information from neighbor slices improves the ability of the recognition.

The digital treatment of data seems to be unavoidable in the interferometric microscopy ; however, the processing should be extended to the plots of reconstructed images. In the case of successful recognition of the object (for example, parametrization of location, shape, color of each neuron recognized by the set of its slices), the resulting plot can have "infinite" resolution, id est, no imaging defects are supposed to be seen at the final diagram.

Conclusion
The general specification of "resolution" of any optical system may have only qualitative sense.

The correct attribution of any specific "resolution" to an imaging system should be related to some certain sets of objects that the system allows to distinguish, to classify, to identify, to recognize. The optimized treat of the images should lead to the diagrams without any "noisy" defects typical for the primary data.

The specification of the class of objects suitable for the analysis and automatic recognition are essential for the methodologically-correct comparison of various imaging systems.

The recognition-oriented imaging should have applications in many areas and, in particular, in the interpretation of the neurobiological structures in terms of the circuit modeling . Only in such applications the term "resolution of an optical system" can be interpreted as a physical quantity.

Copyleft 2011 by Dmitrii Kouznetsov.