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Modern microminiature digital photo and video cameras are built according to the principles typical for the facet eye of insects. The desire to increase the number and size of the pixels used by the photomatrix obviously has technological limitations that determine the information capacity of the system. At the same time, there are theoretical limitations of information capacity. First of all, these are diffraction limitations that determine the limit of spatial resolution for micro-optical systems. In addition, these are energy limitations related to the signal-to-noise ratio during exposure. In the article these factors limiting the information capacity of the system are considered and formulas are obtained that allow to determine the rational combination of parameters of the facet optoelectronic system and calculate its information capacity taking into account the noted limitations. The parameters of known foreign developments are close to those calculated by the above formulas.
DOI: 10.22184/1993-7296.FRos.2019.13.1.66.73
DOI: 10.22184/1993-7296.FRos.2019.13.1.66.73
Теги: diffraction facet system information capacity microlens photomatrix дифракция информационная емкость микролинза фасеточная система фотоматрица
INTRODUCTION
In facet optoelectronic systems, to a certain extent, the principles of constructing the eye of insects, crustaceans and some other invertebrates are implemented. The optical model of the facet eye can be represented as a set of conjugated microlenses and photoreceptors that perceive the radiation flux of a certain direction (Fig. 1). Such small «eyes», called ommatidia, are directed in all directions and can cover a huge angular field up to the full sphere. The number of ommatidia can be tens of thousands, e. g., a dragonfly has 30,000 of them. Such advantages of the facet eye as a large angular field, miniature size, large depth of the imaged space, invariably attract the developers of optoelectronic systems of circular viewing of space, video surveillance, control, technical vision of robots, pattern recognition, photo and video. However, at small values of the focal length of microlenses, the spatial resolution, determined by the angle Δα, is obviously low.
Optoelectronic systems for various purposes do not copy the arrangement of the facet eye, but implement certain principles inherent in its structure. Thus, all-around vision system can contain multiple channels located around the circumference, providing the most coverage of a 360-degree field. A review of facet optoelectronic systems of various types is given in [1, 2].
According to the principle of construction, the systems that use microlenses that create an image on a photomatrix, micro-optical facet systems, are the closest to the facet eye. Such systems provide minimization of dimensions and weight of the device.
The information capacity of micro-optical facet systems has certain limits related not only to the number of elements of a photomatrix, but also to fundamental physical limitations: diffraction and the signal-to-noise ratio (energy limitations).
The purpose of the article is to find the relationship between the parameters of the optical system, the parameters of the photomatrix and the conditions of illumination, allowing to calculate the information capacity of the facet optoelectronic system and to determine the rationality of the combination of its basic elements.
1. DIFFRACTION LIMITATIONS
In recent years, attempts to technically implement facet vision have been made in a number of research laboratories in the USA, Japan, Switzerland, and Germany [3–5]. They are aimed at solving an important problem of increasing spatial resolution due to the use of not a single element radiation detector (photoreceptor), but a photomatrix, and the choice of mutual placement of microlenses and photomatrix. Various technical and technological solutions are proposed, but from the point of view of determining information capacity, the facet vision system can be represented as a set of units (facets) containing microlenses, each of which is associated with a photomatrix. The image created by the microlens covers the area of the photomatrix with a certain number of elements (pixels). Facets can be located on the dome surface, due to which a wide angular field of the system is achieved [6]. The technological process developed at the Fraunhofer Institute of Optics and Precision Engineering in Jena (Germany) made it possible to manufacture a flat chamber with a thickness of only 2 mm. A conceptual prototype of such a device is known as facetVISION [7].
Let’s assume that the microlens has a light diameter D (Fig. 2). The minimum image area required to overlap the field of view of one microlens with the location of microlenses in rows and columns at equal intervals is equal to D2. Taking into account the overlap of the angular fields of microlenses required for stitching an image, the required image area should be larger, however, the overlapped areas are not included in the information capacity of the resulting («crosslinked») image. Under diffraction limitations, the diameter of the scattering circle (Erie circle) is equal to
,
where λ is the radiation wavelength, f ‘ is focal length of the microlens.
We assume that the area of a pixel determined by diffraction is equal to the area of a square with side d, i. e.,
.
Then the number of pixels in the angular field of the microlens will be equal to
,
where is the relative aperture of the microlens.
It is known that the image can be restored without distortion, if the number of samples per pixel is at least two [8], i. e., the number of photomatrix elements in the angular field of a microlens should be 2 N1, and if the facet system contains n microlenses, then the total number of photomatrix elements should be make up
. (1)
2. ENERGY LIMITATIONS
If it is possible to select m gradations of brightness on the photomatrix element, and the brightness values are equally probable, then, using the Hartley formula, the information capacity of the brightness field is determined through statistical entropy as
,
or
. (2)
The number of gradations of brightness, depending on the signal-to-noise ratio µ, is equal to
, (3)
where k is a coefficient numerically equal to the minimum acceptable signal-to-noise ratio. The signal-to-noise ratio is recorded using either a linear or a logarithmic scale.
The signal-to-noise ratio (S / N), measured in decibels, is defined as
,
or
,
where uc is the signal level, is the r. m. s. value of the noise. In accordance with the expert assessment of image quality in television, recommended by the International Advisory Committee on Radio Engineering, satisfactory image quality can be obtained with a signal-to-noise ratio S / N of at least 30 dB (µ = 32).
The signal and noise levels and the signal-to-noise ratio can be expressed in terms of the corresponding numbers of signal nc and noise nш electrons:
,
where nвн ш is the number of internal noise electrons, nфш is the number of external noise electrons. External noises (photon noise) are the result of a discrete nature and follow the Poisson law (statistics). The number of noise electrons also follows this statistics, and according to it, the photon noise is equal to the square root of the number of signal electrons, i. e. . Thus, with significant signal-to-noise ratios, when n2фш >> n2вн ш, the signal-to-noise ratio will be equal to the square root of the number of signal photons:
. (4)
The number of signal electrons per accumulation cycle on the photomatrix element at wavelength λ, is defined as
, (5)
where Eλ is the spectral irradiance of the photomatrix element (hereinafter – the energy values taken in the narrow spectral range Δλ are used, while Eλ = Eeλ Δλ, Eeλ is the spectral density of energy irradiance, Anu is the pixel area, tн is accumulation time, is the photon energy at the wavelength λ, ηλ is the quantum yield, h = 6,626 · 10–34 J · s is Planck’s constant.
It is easy to show that the spectral irradiance of the photomatrix element is determined by the relation:
, (6)
where Eобλ is the spectral irradiance of the object, rλ is the spectral reflectance ratio of the object, τоλ is the spectral transmittance of the optical system, τaλ is the spectral transmittance of the atmosphere, is the relative aperture of the objective (microlens). From relations (4), (5) and (6), taking into account the fact that the area of the photomatrix element Anu should be half as much as d2, we obtain:
µ = 1,22 λ . (7)
Then from (2), (3) and (7) we obtain the formula determining the information capacity of the facet optoelectronic system, in the following form:
. (8).
3. ANALYSIS AND CONCLUSIONS
The analysis of the relations obtained shows that increasing the number of photomatrix elements to enhance the information capacity of the system is advisable to a certain limit. A rational number of elements, as it follows from formula (1), is determined by the parameters of the microlens and the number of microlenses, which in turn depends on technological factors. On the other hand, for a given number of photomatrix elements, it is possible to calculate the required number of microlenses in the facet system. So, e. g., when using a 10-megapixel matrix, with relative aperture of the microlens , with the wavelength λ = 0.5 µm, and with the diameter of the microlens D = 0.5 mm, 118 microlenses are enough. As D decreases, the required number of microlenses increases dramatically.
The information capacity of the system significantly depends on the energy relations in accordance with (8). Knowing the object irradiance Eобλ and other parameters included in (7) and (8), one can calculate the signal-to-noise ratio µ and the contribution of the logarithmic multiplier to the information capacity of the facet optoelectronic system. On the other hand, by determining the required value µ by the criteria of image quality, one can find the necessary level of illumination of the subject. If in the formula (5) we put the number of electrons nλ equal to the noise number nλш, then the corresponding irradiance of the pixel will be equal to the threshold value
.
Assuming Eобλ = Eпλ, we obtain, taking into account relation (6), the value of the threshold irradiance of the object, at which the pixel signal is equal to noise:
.
Then the maximum spectral irradiance of the object in the range of linearity of the energy characteristic is equal to:
.
For typical values: signal-to-noise ratio S / N 40 dB (µ = 100), λ = 0.5 µm, Anu = 25 · 10–12 (photomatrix element size 5 µm),tH = 0,02 с, ηλ =0,4, rλ = 0,6, τoλ = 0,8, τaλ = 1, = 0,5 we obtain Enλ = 1,98 W / m2, = 66 · 10–4 W / m2 which in terms of light units corresponds to object illumination 270 lux.
Thus, the goal set in the article seems to have been achieved. The obtained formulas determine the relationship between the parameters of the optical system, the parameters of the photomatrix and the illumination conditions, allowing us to calculate the information capacity of the facet optoelectronic system and determine the rationality of the combination of its main elements: the number of photomatrix elements, its format, element size, microlens number, diameter and relative aperture microlenses. The calculation using the given formulas shows that some of the parameters sparingly published by the developers of the facet systems, primarily, the number of microlenses and the parameters of photomatrices, correspond to the calculated values.
In facet optoelectronic systems, to a certain extent, the principles of constructing the eye of insects, crustaceans and some other invertebrates are implemented. The optical model of the facet eye can be represented as a set of conjugated microlenses and photoreceptors that perceive the radiation flux of a certain direction (Fig. 1). Such small «eyes», called ommatidia, are directed in all directions and can cover a huge angular field up to the full sphere. The number of ommatidia can be tens of thousands, e. g., a dragonfly has 30,000 of them. Such advantages of the facet eye as a large angular field, miniature size, large depth of the imaged space, invariably attract the developers of optoelectronic systems of circular viewing of space, video surveillance, control, technical vision of robots, pattern recognition, photo and video. However, at small values of the focal length of microlenses, the spatial resolution, determined by the angle Δα, is obviously low.
Optoelectronic systems for various purposes do not copy the arrangement of the facet eye, but implement certain principles inherent in its structure. Thus, all-around vision system can contain multiple channels located around the circumference, providing the most coverage of a 360-degree field. A review of facet optoelectronic systems of various types is given in [1, 2].
According to the principle of construction, the systems that use microlenses that create an image on a photomatrix, micro-optical facet systems, are the closest to the facet eye. Such systems provide minimization of dimensions and weight of the device.
The information capacity of micro-optical facet systems has certain limits related not only to the number of elements of a photomatrix, but also to fundamental physical limitations: diffraction and the signal-to-noise ratio (energy limitations).
The purpose of the article is to find the relationship between the parameters of the optical system, the parameters of the photomatrix and the conditions of illumination, allowing to calculate the information capacity of the facet optoelectronic system and to determine the rationality of the combination of its basic elements.
1. DIFFRACTION LIMITATIONS
In recent years, attempts to technically implement facet vision have been made in a number of research laboratories in the USA, Japan, Switzerland, and Germany [3–5]. They are aimed at solving an important problem of increasing spatial resolution due to the use of not a single element radiation detector (photoreceptor), but a photomatrix, and the choice of mutual placement of microlenses and photomatrix. Various technical and technological solutions are proposed, but from the point of view of determining information capacity, the facet vision system can be represented as a set of units (facets) containing microlenses, each of which is associated with a photomatrix. The image created by the microlens covers the area of the photomatrix with a certain number of elements (pixels). Facets can be located on the dome surface, due to which a wide angular field of the system is achieved [6]. The technological process developed at the Fraunhofer Institute of Optics and Precision Engineering in Jena (Germany) made it possible to manufacture a flat chamber with a thickness of only 2 mm. A conceptual prototype of such a device is known as facetVISION [7].
Let’s assume that the microlens has a light diameter D (Fig. 2). The minimum image area required to overlap the field of view of one microlens with the location of microlenses in rows and columns at equal intervals is equal to D2. Taking into account the overlap of the angular fields of microlenses required for stitching an image, the required image area should be larger, however, the overlapped areas are not included in the information capacity of the resulting («crosslinked») image. Under diffraction limitations, the diameter of the scattering circle (Erie circle) is equal to
,
where λ is the radiation wavelength, f ‘ is focal length of the microlens.
We assume that the area of a pixel determined by diffraction is equal to the area of a square with side d, i. e.,
.
Then the number of pixels in the angular field of the microlens will be equal to
,
where is the relative aperture of the microlens.
It is known that the image can be restored without distortion, if the number of samples per pixel is at least two [8], i. e., the number of photomatrix elements in the angular field of a microlens should be 2 N1, and if the facet system contains n microlenses, then the total number of photomatrix elements should be make up
. (1)
2. ENERGY LIMITATIONS
If it is possible to select m gradations of brightness on the photomatrix element, and the brightness values are equally probable, then, using the Hartley formula, the information capacity of the brightness field is determined through statistical entropy as
,
or
. (2)
The number of gradations of brightness, depending on the signal-to-noise ratio µ, is equal to
, (3)
where k is a coefficient numerically equal to the minimum acceptable signal-to-noise ratio. The signal-to-noise ratio is recorded using either a linear or a logarithmic scale.
The signal-to-noise ratio (S / N), measured in decibels, is defined as
,
or
,
where uc is the signal level, is the r. m. s. value of the noise. In accordance with the expert assessment of image quality in television, recommended by the International Advisory Committee on Radio Engineering, satisfactory image quality can be obtained with a signal-to-noise ratio S / N of at least 30 dB (µ = 32).
The signal and noise levels and the signal-to-noise ratio can be expressed in terms of the corresponding numbers of signal nc and noise nш electrons:
,
where nвн ш is the number of internal noise electrons, nфш is the number of external noise electrons. External noises (photon noise) are the result of a discrete nature and follow the Poisson law (statistics). The number of noise electrons also follows this statistics, and according to it, the photon noise is equal to the square root of the number of signal electrons, i. e. . Thus, with significant signal-to-noise ratios, when n2фш >> n2вн ш, the signal-to-noise ratio will be equal to the square root of the number of signal photons:
. (4)
The number of signal electrons per accumulation cycle on the photomatrix element at wavelength λ, is defined as
, (5)
where Eλ is the spectral irradiance of the photomatrix element (hereinafter – the energy values taken in the narrow spectral range Δλ are used, while Eλ = Eeλ Δλ, Eeλ is the spectral density of energy irradiance, Anu is the pixel area, tн is accumulation time, is the photon energy at the wavelength λ, ηλ is the quantum yield, h = 6,626 · 10–34 J · s is Planck’s constant.
It is easy to show that the spectral irradiance of the photomatrix element is determined by the relation:
, (6)
where Eобλ is the spectral irradiance of the object, rλ is the spectral reflectance ratio of the object, τоλ is the spectral transmittance of the optical system, τaλ is the spectral transmittance of the atmosphere, is the relative aperture of the objective (microlens). From relations (4), (5) and (6), taking into account the fact that the area of the photomatrix element Anu should be half as much as d2, we obtain:
µ = 1,22 λ . (7)
Then from (2), (3) and (7) we obtain the formula determining the information capacity of the facet optoelectronic system, in the following form:
. (8).
3. ANALYSIS AND CONCLUSIONS
The analysis of the relations obtained shows that increasing the number of photomatrix elements to enhance the information capacity of the system is advisable to a certain limit. A rational number of elements, as it follows from formula (1), is determined by the parameters of the microlens and the number of microlenses, which in turn depends on technological factors. On the other hand, for a given number of photomatrix elements, it is possible to calculate the required number of microlenses in the facet system. So, e. g., when using a 10-megapixel matrix, with relative aperture of the microlens , with the wavelength λ = 0.5 µm, and with the diameter of the microlens D = 0.5 mm, 118 microlenses are enough. As D decreases, the required number of microlenses increases dramatically.
The information capacity of the system significantly depends on the energy relations in accordance with (8). Knowing the object irradiance Eобλ and other parameters included in (7) and (8), one can calculate the signal-to-noise ratio µ and the contribution of the logarithmic multiplier to the information capacity of the facet optoelectronic system. On the other hand, by determining the required value µ by the criteria of image quality, one can find the necessary level of illumination of the subject. If in the formula (5) we put the number of electrons nλ equal to the noise number nλш, then the corresponding irradiance of the pixel will be equal to the threshold value
.
Assuming Eобλ = Eпλ, we obtain, taking into account relation (6), the value of the threshold irradiance of the object, at which the pixel signal is equal to noise:
.
Then the maximum spectral irradiance of the object in the range of linearity of the energy characteristic is equal to:
.
For typical values: signal-to-noise ratio S / N 40 dB (µ = 100), λ = 0.5 µm, Anu = 25 · 10–12 (photomatrix element size 5 µm),tH = 0,02 с, ηλ =0,4, rλ = 0,6, τoλ = 0,8, τaλ = 1, = 0,5 we obtain Enλ = 1,98 W / m2, = 66 · 10–4 W / m2 which in terms of light units corresponds to object illumination 270 lux.
Thus, the goal set in the article seems to have been achieved. The obtained formulas determine the relationship between the parameters of the optical system, the parameters of the photomatrix and the illumination conditions, allowing us to calculate the information capacity of the facet optoelectronic system and determine the rationality of the combination of its main elements: the number of photomatrix elements, its format, element size, microlens number, diameter and relative aperture microlenses. The calculation using the given formulas shows that some of the parameters sparingly published by the developers of the facet systems, primarily, the number of microlenses and the parameters of photomatrices, correspond to the calculated values.
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