Telescope performance

The telescope performance describes the performance of a visually used long -range optical instrument such as a telescope or binoculars . Historically, telescope performance was initially defined as the quotient of both visual acuities (when observing with a telescope) and (when observing with the naked eye). Visual acuity is understood to mean the reciprocal values ​​of the object sizes that can just be recognized (expressed in arc minutes), which are determined in the course of a series of tests on visual test panels . Later, the concept of telescope performance was expanded to include other aspects of visual perception in order to also integrate viewing thresholds that are relevant for object recognition or target acquisition. Since the telescope performance is made up of a combination of technical parameters of the instrument, of physiological parameters such as the pupil size of the observer, as well as of perceptual psychological aspects such as pattern recognition and observation experience, their metrological determination is dependent on test series and statistically collected sample means . ${\displaystyle L=S_{F}/S_{A}}$ ${\displaystyle S_{F}}$ ${\displaystyle S_{A}}$

Empirical modeling of telescope sharpness performance

In the 1950s, H. Köhler and R. Leinhos ( Carl Zeiss AG ) developed an empirically determined universal formula for the telescope performance (related to visual acuity). The visual acuity was determined on test subjects by evaluating Landolt rings , with the result:

${\displaystyle L=m^{1-2x}\left({\frac {D}{p}}\right)^{2x}T^{x}}$.

Here stands for the magnification of the optical instrument, for its objective diameter, and for its degree of transmission , while the pupil width of the observer indicates, which depends on the ambient luminance . The value of the exponent is also a function of the ambient brightness, but can be applied approximately in three brightness ranges as follows: (in daylight), (in twilight), and (under residual light in the night). ${\displaystyle m}$ ${\displaystyle D}$ ${\displaystyle T}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle x=0}$ ${\displaystyle x=1/4}$ ${\displaystyle x=1/2}$

During the day, the telescope performance is therefore determined exclusively by the magnification of the instrument . Neglecting the individual pupil width and the degree of transmission, the binoculars performance is calculated at twilight , which corresponds to the well-known twilight factor first proposed by A. Kühl in 1929 and later established as DIN standard 58386 on the initiative of Zeiss . In advanced darkness, the telescope performance results in the relationship in which the objective diameter dominates, but on which the transmission of the device also has a considerable influence. ${\displaystyle L=m}$ ${\displaystyle L\sim {\sqrt {mD}}}$ ${\displaystyle L=D{\sqrt {T}}}$

Telescope power as contrast threshold useful power

An alternative way to telescope performance leads through the laws of perception to object recognition. Max Berek ( Ernst Leitz AG ) combined the existing literature data on the sighting thresholds of objects, which were usually circular targets of different diameters and gray values, into a universal law of perception. For this purpose it is assumed that the circular target object of the apparent extent (in arc minutes) has a surface luminance of and is placed in front of a (homogeneous) background of the ambient luminance, which results in a Weber contrast of . Furthermore, let the adaption luminance of the eye be the result of the contrast threshold, i.e. H. the minimal contrast that leads to the sighting of the object ${\displaystyle \sigma }$ ${\displaystyle L_{o}}$ ${\displaystyle L_{h}}$ ${\displaystyle K=(L_{o}-L_{h})/L_{h}}$ ${\displaystyle L_{a}}$

${\displaystyle {\sqrt {K}}={\frac {1}{\sigma }}{\sqrt {\frac {\phi (L_{a})}{L_{a}}}}+{\sqrt {\frac {b(L_{a})}{L_{a}}}}}$,

The two functions of the adaption luminance, (the characteristic luminous flux function ) and ( characteristic luminance function ), have been extracted from test series and tabulated. Their meanings become apparent in the two borderline cases: 1. Very small object diameter , which leads to the borderline case with the above equation . This is Ricco's theorem , which states that with a given adaption luminance, the luminous flux at the viewing threshold is a constant - which can therefore be tabulated as a function . 2. The limiting case of a large apparent extent of the object leads to , the Weber-Fechner's law , in the scope required for viewing minimum surface luminance of an object is independent of its size and can be written as the product of the adaptation luminance and contrast. ${\displaystyle \phi }$ ${\displaystyle b}$ ${\displaystyle \sigma }$ ${\displaystyle \phi (L_{a})\approx \sigma ^{2}L_{a}K}$ ${\displaystyle \sigma ^{2}L_{a}K}$ ${\displaystyle \phi (L_{a})}$ ${\displaystyle b(L_{a})\approx L_{a}K}$

The crucial step that leads to telescope performance is to apply a universal law of perception of this kind not only to the image that an observer perceives with the naked eye, but also to the virtual image presented to the eye by an optical instrument . Berek suggested that the quotient of the two contrast thresholds (with and without an instrument) should be called the contrast threshold useful power and that this should be understood as an alternative approach to the definition of the telescope performance. Berek receives for the contrast threshold when observing through the instrument:

${\displaystyle {\sqrt {{\frac {T}{\mu }}K}}={\frac {1}{\sigma m^{*}}}{\sqrt {\frac {\phi (\mu L_{a})}{\mu L_{a}}}}+{\sqrt {\frac {b(\mu L_{a})}{\mu L_{a}}}}}$.

Here stands for the effective transmittance of the instrument, which is composed of the sum of the transmittance of the useful light and the proportion of scattered or false light . is the objective diameter divided by the larger of the two values: diameter of the exit pupil and pupil width of the observer. By transforming this expression, it is also possible to determine threshold brightnesses and the threshold diameter or, equivalently, the threshold distance of a target object with a given contrast, as required. In the latter case, it is the maximum distance the target may be so that it can just barely be recognized by the instrument. ${\displaystyle \mu =T+\nu }$ ${\displaystyle T}$ ${\displaystyle \nu }$ ${\displaystyle m^{*}=D/d^{*}}$

One advantage of this approach to defining telescope performance is that it can be applied to other models of perception as well. For example, the law of perception developed by Matchko and Gerhart and derived from the extensive data set by Blackwell and McCready is able to integrate further parameters such as the duration of sighting and the apparent movement of target objects into the evaluation of the sighting thresholds.

Application example

Contrast threshold useful power of binoculars with different key figures (left: lens diameter, right: magnification), as a function of the ambient luminance (on a logarithmic scale).

The contrast threshold useful power is shown in the figure opposite for binoculars with different key figures as a function of the ambient luminance. For this purpose, the following assumptions were made: The adaptation luminance of the eye is identical to the ambient luminance, i.e. the observer's eye is adapted to the ambient brightness. This assumption is justified as long as the object only fills a small part of the image compared to the background. Watson's universal formula was used to calculate the respective pupil widths, the age of the observer being set to 30 years and the subjective viewing angle of the binoculars to 60 °. The target object has an apparent diameter of one arc minute (which corresponds approximately to the resolution limit of the eye in daylight). The binoculars have a transmittance of , stray light is not present. The characteristic functions, and , originally given in awkward tabular form, are now available as analytical approximation formulas. ${\displaystyle T=0.9}$ ${\displaystyle \phi (L_{h})}$ ${\displaystyle b(L_{h})}$

In the figure, the ambient luminance increases from left to right, with the lighting values ​​being divided into three areas (night, twilight, daylight). The individual curves show the lens diameter of the binoculars on the left and the corresponding magnifications on the right. The yellow curve, for example, describes the contrast threshold useful power of 7 × 42 binoculars as a function of the ambient brightness. It can be seen that the telescope performance defined in this way is clearly determined by the magnification of the instrument in daylight, while at night the objective diameter largely determines the performance of the instrument. This is in agreement with the empirical approximation formula for telescope sharpness performance by Koehler and Leinhos. However, there are significant deviations in the twilight phase: the twilight factor for 7 × 42 binoculars delivers a performance value of 17.1, for a 10 × 32, on the other hand, a higher value of 17.9. The comparison of the contrast threshold useful power of both devices (in yellow and magenta), however, shows a clear performance advantage of the 7 × 42 in twilight. The situation is similar with the pairing 8 × 50 (blue curve, twilight factor: 20) and 12 × 42 (red-yellow curve, twilight factor: 22.4). This example clearly shows that different approaches to the definition of the telescope performance, especially in twilight, i.e. the phase of mesopian vision , can lead to significantly different results.

Scope Limitations

The theoretical approaches to telescope performance are based on a number of implicit assumptions that relate to the test setup : Effects of color perception are neglected because the test series were carried out exclusively with test panels of different gray values. This limitation on brightness contrasts has no influence on performance in advanced (nautical) twilight or at night, but significant color effects are to be expected in daytime vision and in civil twilight (e.g. Purkinje effect ). Furthermore, the influence of hand restlessness on telescope performance, which scales with magnification, was neglected. The above relationships therefore only apply to fixed optics and must be corrected accordingly for moving target objects ( air defense ). In the practice of target acquisition, neither the exact direction nor the distance of the object is known - a wide field of view and a high depth of field have a positive effect on the success rate, with lower magnifications again having an advantage. The target objects should also be sufficiently close so that the effects of air turbulence ( seeing ) or atmospheric extinction can be neglected. Otherwise, extensions of the model must also integrate the influences of image flickering as a function of magnification and its contrast as a function of distance ( Lambert-Beer law ).

Criticism of the definitions of telescope performance

Since the two definitions of the telescope performance as visual acuity useful performance and contrast threshold useful performance, especially in the twilight phase, provide qualitatively different results, the introduction of the twilight factor (initially in 1929 by Kühl, then via the universal formula by Köhler & Leinhos) led to controversial discussions. In particular, the equality of both parameters, objective diameter and magnification, initially seemed to contradict the experience that a large exit pupil is preferable when observing in the twilight phase . König & Köhler write about this: This result [... that the magnification assumes a position on an equal footing with the objective diameter ...] should be pointed out once again; because it contradicts an opinion that has been widespread for decades, which was also supported by the binocular manufacturers and which said that in twilight vision all that mattered was binoculars with the largest possible diameter of the exit pupil or the largest possible value of the pupil square to use . To this, Berek replies: The question, which has often been raised recently, to what extent a smaller exit pupil of the telescope can be compensated for by a higher magnification is pointless; ... mind you for sighting, not for visual acuity . Köhler & Leinhos intensify in their comparative analysis of both models: Berek's theory differs from Kühl's [which leads to the twilight factor] fundamentally in that Berek only relies on experimental findings for free viewing, but purely the transition to telescope viewing theoretically and thus ignores all physiological influences. When evaluating the twilight performance of binoculars, you come to the conclusion: There is ... not the slightest agreement with Berek's theory.

On the other hand, it has meanwhile been proven that Berek's model delivers predictions that are compatible with the independently collected test series by Blackwell and therefore should not have any technical errors worth mentioning. Rather, Merlitz points out that the experimental set-ups that lead to the two approaches to binoculars address different mechanisms of visual perception: The telescope performance related to visual acuity, which leads to the twilight factor, requires the identification of Landolt rings, a detailed analysis and mastering imposes foveal vision on the observer . Under these conditions, over-enlargement, i.e. H. the use of an instrument whose exit pupil diameter is less than the diameter of the eye pupil is still conducive to detail recognition, although the contrast suffers. In contrast, Berek's approach is based on the determination of perception thresholds - a process in which extrafoveal (peripheral) vision is also significantly involved.

An example may clarify this difference: The ornithologist who still has to read the rings of a bird after sunset is dependent on foveal detail vision in his work, and is therefore well advised with the twilight number when choosing the appropriate instrument. In contrast, a soldier who has to grasp a well camouflaged target during twilight, or an amateur astronomer who wants to find a weakly glowing, diffuse comet after sunset, observes in the area of ​​the respective sighting thresholds. A sighting takes place in this phase of the mesopian vision z. Sometimes already extrafoveal, since the contrast thresholds of the retinal rods , also due to the intensifying effect of convergent receptive fields , are lower than those of the foveal cones . Ultimately, the telescope performance is therefore a context-bound variable, for the assessment of which there can be no single, solely valid access.

literature

• Albert König, Horst Köhler: The telescopes and range finders . Springer-Verlag Berlin, 1959, ISBN 9783642491245 .

• Holger Merlitz: Hand binoculars: function, performance, selection . Verlag Europa-Lehrmittel Haan-Gruiten, 2019, ISBN 978-3-8085-5775-4 .
• Paul R. Yoder, Jr., Daniel Vukobratovich: Field Guide to Binoculars and Scopes . SPIE PRESS, 2011, ISBN 978-0-8194-8649-3 .

Individual evidence

1. ↑ Albert König, Horst Köhler: The telescopes and range finders . Springer-Verlag, Berlin 1959, p. 100.
2. ↑ ^{a b} H. Koehler, R. Leinhos, Investigations on the Laws of Telescopic Vision, Optica Acta: International Journal of Optics, 4: 3, pp. 88-101 (1957).
3. ↑ Albert König, Horst Köhler: The telescopes and range finders . Springer-Verlag, Berlin 1959, p. 103.
4. ↑ A. Kühl, Centralztg. f. Opt. U. Mech. 50, s. 202 u. 218 (1929).
5. ↑ M. Berek, On the basic physiological law of the perception of light stimuli , Zeitschrift für Instrumentenkunde 63, p. 24 (1943).
6. ↑ ^{a b} M. Berek, The useful power of binocular earth telescopes , Z. Phys. A 125, p. 657 (1949).
7. ↑ RM Matchko, GR Gerhart, Parametric analysis of the Blackwell-McCready data , Opt. Eng. 37, p. 1937 (1998).
8. ↑ RM Matchko, GR Gerhart, ABCs of foveal vision , Opt. Eng. 40, p. 2735 (2001).
9. ^ HR Blackwell, DW McCready, Foveal Detection thresholds for various duration of single pulses , University of Michigan Engineering Research Institute rep. 2455-13-F (1958).
10. ↑ AB Watson, JI Yellott, A unified formula for light-angepasst pupil size , J. Vis. 12, p. 1-16 (2012).
11. ↑ ^{a b c} H. Merlitz, Berek's model of target detection , J. Opt. Soc. At the. A 32, p. 101 (2015).
12. ↑ D. Vukobratovich, Binocular performance and design , Proc. of SPIE 1186, Current Developments in Optical Engineering and Commercial Optics , ed. RE Fischer, HM Pallicove, WJ Smith (1989).
13. ↑ H. Merlitz: Hand binoculars: function, performance, selection , Verlag Europa-Lehrmittel, ISBN 978-3-8085-5775-4 , p. 144 (2019).
14. ↑ Albert König, Horst Köhler: The telescopes and range finders . Springer-Verlag, Berlin 1959, p. 105.
15. ^ HR Blackwell, Contrast thresholds of the human eye , J. Opt. Soc. At the. 36, p. 624-643 (1946).