Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision

American Journal of Physics

Volumn 67, Issue 11, 1999, Pages 946-953

  Soffer, Bernard H ^a   Lynch, David K ^b  

^a Aerospace Corporation ^*  (United States)

^b AEROSPACE CORPORATION   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033446361     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19170     Document Type: Article

References (38)

1. Keneth R. Lang, Sun Earth and Sky (Springer-Verlag, Berlin, 1995), p. 210.
2. J. D. Mollon, "The Uses and Origins of Primate Colour Vision," J. Exp. Biol. 146, 21-38 (1989).
3. James T. McIlwain, An Introduction to the Biology of Vision (Cambridge UP, Cambridge, 1996), pp. 5-6.
4. Evan Thompson, Colour Vision (Routlidge, London and New York, 1995), pp. 51, 169.
5. Albert Rose, Vision Human and Electronic (Plenum, New York and London, 1973), pp. 49-50.
6. Robert M. Boynton, Human Color Vision (Holt, Rienhart and Winston, New York, 1979), p. 51 This misleading statement is repeated in Peter K. Kaiser and Robert M. Boynton, Human Color Vision, 2nd ed. (Optical Society of America, Washington, DC, 1996), p. 6.
7. Robert M. Boynton, Human Color Vision (Holt, Rienhart and Winston, New York, 1979), p. 51 This misleading statement is repeated in Peter K. Kaiser and Robert M. Boynton, Human Color Vision, 2nd ed. (Optical Society of America, Washington, DC, 1996), p. 6.
8. Robert M. Boynton, "Human color perception," in Science of Vision, edited by K. N. Leiboveic (Springer-Verlag, New York, 1990), p. 218.
9. David K. Lynch and William Livingston, Color and Light in Nature (Cambridge UP, Cambridge, 1995), p. 222.
10. Laurie White, Infrared Photography Handbook (Amherst Media, Amherst, New York, 1996), pp. 28-32.
11. Robert Sekuler and Randolph Blake, Perception, 3rd ed. (McGraw-Hill, New York, 1994), p. 201.
12. Several confusingly similar sounding but distinct spectral radiometric quantities: spectral radiant power; spectral radiant emittance; spectral irradiance; spectral radiant intensity; and spectral radiance are all employed to describe spectral density distributions. They differ in whether or not they are intensities per unit area, per unit solid angle, both together or neither one, and whether the radiation is emanating from an emitter or impinging on a surface. Many of the papers we quote from, and refer the reader to, do use these different quantities for their purposes, and make the errors we describe using them. As these quantities are all are spectral densities, i.e., quantities per unit bandwidth interval, e.g., per unit wavelength interval or per unit frequency interval, they all illustrate the argument of this paper equally well. They all suffer exactly the same peak shifts and distortions that we describe. For the thesis of this paper we may normalize these quantities by setting all areas and solid angles equal to one and use all of these terms interchangeably without fear of spoiling our argument. Furthermore, for the purposes of this paper we make all powers relative and normalize them as well. Separate from the thesis of this paper, however, it should be pointed out that in spite of the efforts of many international committees to standardize the polyglot nomenclature of radiometry and photometry, not many seem to abide strictly by the naming standards. The confusion in the set of quantities that are intensities per unit area is even more than a matter of naming them, as some authors use projected area, including the cosine obliquity factor, and some others, using the same nomenclature, do not. All this is yet another reason to exclaim CAVEAT LECTOR.
13. Gail P. Anderson et al., "MODTRAN2: Evolution and applications," Proc. SPIE 222, 790-799 (1994).
14. Stanley Thomas Henderson, Daylight and its Spectrum (American Elsevier, New York, 1970).
15. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry, 3rd ed. (Wiley, New York, 1975), p. 71. We reluctantly introduce the photometric quantity relative "luminous efficiency" at this point only because the vision literature to which we refer the reader, almost exclusively, uses it to describe the eye's sensitivity. It is the perceptual sensitivity of human photopic, or cone vision to quasimonochromatic light, relative to its maximum sensitivity, usually as a function of wavelength, represented by the fictitious eye of a standardized observer. It is a good practical model for the bright-light adapted eye and it is proportional to the spectral power sensitivity over a large gamut of power. It is not a density distribution function. We do not really need to consider, for the argument of this paper, the sophisticated subtleties of photometry, how many lumens per watt there may be, or even what a lumen might be. Older synonymous terms for luminous efficiency, still to be found, are "visibility factor" and "luminosity factor."
16. n multiples) of a given note's frequency. If we pick, for example, middle C = 256 Hz, the frequency of the note one octave higher, by definition, is 512 Hz. We hear all those n higher octave notes, ignoring intensity-dependent effects, still as a kind of replica of C, although higher in pitch. In representing musical spectral power or noise density distributions, the very same representational issues that we have been discussing for optical power naturally arise. But here, if the intention is to represent human perceptual hearing, then the logarithmic representation is clearly to be preferred and thus, by settling on it, many of the representational pitfalls and paradoxes can be avoided. Audio engineers have introduced several spectral logarithmic measures and quantities. For example, the logarithmic equivalent of equally distributed, or so-called "white," noise is called appropriately "pink" noise. It preferentially weights the lower frequencies logarithmically, thereby putting equal noise power into each octave. To the ear, pink noise sounds uniformly distributed.
17. M = 0.3666. Note that the optimum wavelength, for a given temperature, does not coincide with the peak of the Planck distribution. Benford was the same person whose name has come to be associated what is variously referred to as the "Dirty [first pages of the] Logarithm Table Phenomenon," "the Anomalous Distribution of First Digits," or more usually "Benford's Law," though this empirical law may have been first noticed at least a century ago by Simon Newcomb [Am. J. Math. 4, 39-40 (1881)]. See Frank Benford, "The Law of Anomalous Numbers," Proc. Am. Philos. Soc. 78, 551-572 (1938). The first digits of naturally occurring numbers, such as of the physical constants, expressed in scientific or floating point notation, are not distributed uniformly randomly as naively might be expected, but rather they have the ubiquitous reciprocal, or inverse probability density distribution. This is all the more pronouncedly so, when many disparate kinds of quantities are examined all together. The integrated cumulative probability is thus logarithmic, and, by subtraction, the probability of a first digit N occurring is log(N+1) - log N, favoring small Ns. Many theoretical and empirical studies of this law have been published, but one simple and direct way of proving and appreciating it is to realize that the inverse probability density distribution, with its strong fractal invariance properties, satisfies the requirement of being independent of the arbitrary choice of scale or units, and the base or modulus, that the numbers are expressed in. Benford's Law is thus seen to be a property of our number system. Those with a taste for paradox will savor this result and many may be tempted to make wagers with the unwary.
18. M = 0.3666. Note that the optimum wavelength, for a given temperature, does not coincide with the peak of the Planck distribution. Benford was the same person whose name has come to be associated what is variously referred to as the "Dirty [first pages of the] Logarithm Table Phenomenon," "the Anomalous Distribution of First Digits," or more usually "Benford's Law," though this empirical law may have been first noticed at least a century ago by Simon Newcomb [Am. J. Math. 4, 39-40 (1881)]. See Frank Benford, "The Law of Anomalous Numbers," Proc. Am. Philos. Soc. 78, 551-572 (1938). The first digits of naturally occurring numbers, such as of the physical constants, expressed in scientific or floating point notation, are not distributed uniformly randomly as naively might be expected, but rather they have the ubiquitous reciprocal, or inverse probability density distribution. This is all the more pronouncedly so, when many disparate kinds of quantities are examined all together. The integrated cumulative probability is thus logarithmic, and, by subtraction, the probability of a first digit N occurring is log(N+1) - log N, favoring small Ns. Many theoretical and empirical studies of this law have been published, but one simple and direct way of proving and appreciating it is to realize that the inverse probability density distribution, with its strong fractal invariance properties, satisfies the requirement of being independent of the arbitrary choice of scale or units, and the base or modulus, that the numbers are expressed in. Benford's Law is thus seen to be a property of our number system. Those with a taste for paradox will savor this result and many may be tempted to make wagers with the unwary.
19. M = 0.3666. Note that the optimum wavelength, for a given temperature, does not coincide with the peak of the Planck distribution. Benford was the same person whose name has come to be associated what is variously referred to as the "Dirty [first pages of the] Logarithm Table Phenomenon," "the Anomalous Distribution of First Digits," or more usually "Benford's Law," though this empirical law may have been first noticed at least a century ago by Simon Newcomb [Am. J. Math. 4, 39-40 (1881)]. See Frank Benford, "The Law of Anomalous Numbers," Proc. Am. Philos. Soc. 78, 551-572 (1938). The first digits of naturally occurring numbers, such as of the physical constants, expressed in scientific or floating point notation, are not distributed uniformly randomly as naively might be expected, but rather they have the ubiquitous reciprocal, or inverse probability density distribution. This is all the more pronouncedly so, when many disparate kinds of quantities are examined all together. The integrated cumulative probability is thus logarithmic, and, by subtraction, the probability of a first digit N occurring is log(N+1) - log N, favoring small Ns. Many theoretical and empirical studies of this law have been published, but one simple and direct way of proving and appreciating it is to realize that the inverse probability density distribution, with its strong fractal invariance properties, satisfies the requirement of being independent of the arbitrary choice of scale or units, and the base or modulus, that the numbers are expressed in. Benford's Law is thus seen to be a property of our number system. Those with a taste for paradox will savor this result and many may be tempted to make wagers with the unwary.
20. Veronika A. Szalai and Gary W. Brudviig, "How plants produce dioxygen," Am. Sci. 86, 542-551 (1998).
21. Project Cyclops CR 114445, revised ed., NASA/Ames Research Center, Moffett Field, California, 1973, p. 41.
22. Boynton, Ref. 6.
23. Albert Rose, Ref. 5, p. 50.
24. V. I. Krasovsky, N. N. Shefov, and V. I. Yarin, "Atlas of the airglow spectrum 3000-12400 Å.," Planet. Space Sci. 9, 883-915 (1962).
25. M. H. Pirenne, "The thermal radiation inside the eye and the red end of the spectral sensitivity curve," J. Physiol. (London) 106, 25 (1947); or Proc. Physiol. Soc. 106, 25 (April, 1947). Pirenne's old numerical estimates differ markedly from Rose's analysis.
26. M. H. Pirenne, "The thermal radiation inside the eye and the red end of the spectral sensitivity curve," J. Physiol. (London) 106, 25 (1947); or Proc. Physiol. Soc. 106, 25 (April, 1947). Pirenne's old numerical estimates differ markedly from Rose's analysis.
27. S. Duke-Elder, The Eye in Evolution (C. V. Mosby Co., St. Louis, 1958), p. 619.
28. Ricardo C. Pastor, Hiroshe Kimura, and Bernard H. Soffer, "Thermal stability of polymethine Q-Switch solutions," J. Appl. Phys. 42, 3844-3847 (1971).
29. Ricardo C. Pastor, Bernard H. Soffer, and Hiroshe Kimura, "Photostability of polymethine saturably absorbing dye solutions," J. Appl. Phys. 43, 3530-3533 (1972).
30. M. A. Ali, "Temperature and vision," Rev. Can. Biol. 34, 131-186 (1975).
31. See, e.g., A. Knowles and H. J. A. Dartnall, in The Eye, edited by Hugh Davson, 2nd ed. (Academic, London, 1977), Chap. 8.; and Boynton, in Ref. 6, pp. 104, 107.
32. See, e.g., A. Knowles and H. J. A. Dartnall, in The Eye, edited by Hugh Davson, 2nd ed. (Academic, London, 1977), Chap. 8.; and Boynton, in Ref. 6, pp. 104, 107.
33. Frederick J. G. M. van Kuijk, "Effects of ultraviolet light on the eye: Role of protective glasses," Environ. Health Perspect. 96, 177-184 (1991).
34. James K. Bowmaker, "The evolution of vertebrate visual pigments and photoreceptors," in Vision and Visual Dysfunction, edited by John R. Cronly-Dillon and Richard L. Gregory (CRC, Boca Raton, 1991), pp. 63-81.
35. V. I. Govardovskii and L. V. Zueva, "Spectral sensitivity of the frog the in the ultraviolet and visible region," Vision Res. 14, 1317-1321 (1974).
36. Bernard H. Soffer and David K. Lynch, "Has evolution optimized vision for sunlight?," Annual Meeting of the OSA, Long Beach CA, 12-17 October 1997, SUE4, p. 70; David K. Lynch and Bernard H. Soffer, "On the solar spectrum and the color sensitivity of the eye," Optics and Photonics News, March, 1999; Bernard H. Soffer and David K. Lynch, "The spectral optimization of human vision: Some paradoxes, errors and resolutions," Trends in Optics and Photonics, edited by Toshimitsu Asakura, International Commission for Optics Book 4 (Springer-Verlag, Heidelberg and New York, 1999).
37. Bernard H. Soffer and David K. Lynch, "Has evolution optimized vision for sunlight?," Annual Meeting of the OSA, Long Beach CA, 12-17 October 1997, SUE4, p. 70; David K. Lynch and Bernard H. Soffer, "On the solar spectrum and the color sensitivity of the eye," Optics and Photonics News, March, 1999; Bernard H. Soffer and David K. Lynch, "The spectral optimization of human vision: Some paradoxes, errors and resolutions," Trends in Optics and Photonics, edited by Toshimitsu Asakura, International Commission for Optics Book 4 (Springer-Verlag, Heidelberg and New York, 1999).
38. Bernard H. Soffer and David K. Lynch, "Has evolution optimized vision for sunlight?," Annual Meeting of the OSA, Long Beach CA, 12-17 October 1997, SUE4, p. 70; David K. Lynch and Bernard H. Soffer, "On the solar spectrum and the color sensitivity of the eye," Optics and Photonics News, March, 1999; Bernard H. Soffer and David K. Lynch, "The spectral optimization of human vision: Some paradoxes, errors and resolutions," Trends in Optics and Photonics, edited by Toshimitsu Asakura, International Commission for Optics Book 4 (Springer-Verlag, Heidelberg and New York, 1999).