Virron Res. Vol. 15, pp. 613-616.
Pergamon Press 1975. Printed in Great
Britam
RESEARCH
NOTE
A MATHEMATICAL APPROACH TO EXPLAIN SUBJECTIVE COLOR PERCEPTION LEONARDPOLIZZOTTO 103 Lexington
Street, Auburndale,
Massachusetts
02166, U.S.A.
and
A. F'EURA
ROBERT Worcester
Polytechnic (Received
Institute,
Worcester,
Massachusetts
19 May 1973; in revisedform
01609, U.S.A.
20 August 1974)
INTRODUCTION
Subjective color is a sensation evoked by intermittent visual stimuli, as opposed to real or physical color which is evoked by steady visual stimuli. Hargroves and Hargroves (1971) and Sheppard (1968) provide good summaries concerning the many investigations of subjective color. The most widely used source for producing a black-white flicker resulting in subjective color is the top invented by Benham (1894, 1895) or modified forms of it. This paper discusses the application of statistical communication theory and harmonic analysis techniques to the pulses produced by the rotation of a modified Benham’s disc, referred to as the disc (see Fig. 1) in order to explain human subjective color perception. The results suggest that phase angle modulation is the process used by the visual system in the perception of subjective color. ASSUMPTION
FOR ANALYSIS
With the disc rotating at approximately 6 Hz in the clockwise direction, the colors red, green and blue appear in the arcs indicated in Fig. 1. When the direction of rotation is reversed, blue and red exchange positions. This implies that the observed color is not dependent on the radial distance of the arcs but rather on their position with respect to the black half of the disc. Other colors can be synthesized on the disc by varying the arc length and position. The patterns that yield orange and yellow are also shown in Fig. 1. Since the rotating disc produces color sensations an analogy was made between the elements of the disc and the variables of physical color. The variables of physical color are the dominant wavelength or hue, the 613
Fig. 1. A modified
Benham’s disc. H = hue pulse and S = saturation pulse.
purity or saturation, and the luminance or brightness. As seen in Fig. 1 the disc consists of a complete black half and a half consisting of black arcs on a white background. The width of the arcs and the space between them were each equal to approximately 1cm. All white portions of the disc were considered as “on” and all black as “off”. An arbitrary 0” point was established and the on-off pulse trains resulting from the arcs of the rotating disc were determined (see Fig. 1). As was noted before a change in the arc length and position results in a change in color. For this reason, the pulses derived from the arcs were termed the hue color variable, the first element of the disc. Between each arc there is white, which produces a pulse that is on for 180” and off for 180”. When it is eliminated (by blacking out the areas between the arcs), no colors appear as the disc is rotated. The 180
LWNARDPOLIZZOTTO and ROHEKT A. PMJKA
613
white arcs between the black arcs were therefore termed the saturation color variable, the second element of the disc. This provides the mixture with white and determines the purity of the color. The final element of the disc, luminance, is considered to be the amount of light reflected from the disc. METHOD
OF ANALYSIS
The eye. in some yet unexplained manner. compares the hue and saturation waves, processes this information. and sends it to the central nervous system with the result that color is perceived. Since the visual system is a communication system. a mathematical means for comparing and analyzing signals in a communication system. namely crosscorrelation, will be applied to the hue,.fh(t), and saturation. /i(t) pulses. The crosscorrelation function. ~"S(T) provides a measure of similarity between,f,(t) and,fJt) as a function of a searching parameter. 5. r is a continuous time displacement in the range considered and is independent oft (Lee, 1960). C&(T) is defined as:
;
; .f;,(t) x .f,t + 7) dt. 7Z
(1)
Tis the period of the wave or the inverse of the disc frequency (Hz) in seconds. With the white portion of the saturation wave on only one side of an arc, or hue wave, the subjective color perceived is less vivid and occurs only at the edge of the arc bordering the saturation wave. Thus, each arc must be bordered on both sides by a white saturation wave and two sequential crosscorrelation calculations must be performed. The first crosscorrelation c/I~S(T),was calculated between the hue pulse and one of the bordering saturation pulses, .fJr). The second crosscorrelation, 4&t), a “double” crosscorrelation function. was calculated between the first crosscorrelation function c/I~,&~)and the other identical bordering saturation pulse .&t). The results of this analysis are in’ the time domain. The frequency domain representation of a time function is found by taking the Fourier transform. This transformation was applied to the double crosscorrelation functions which yielded the magnitude and phase spectra in the frequency domain.
Fig.
2. Double [4~,&)1.
crosscorrelation
functions
of
blue
green I+H,,s~~)I~ and red [4~,,~.d~)l.
primary colors in light) were determined and are shown in Fig. 2. This graph shows that each color has a distinct shift in time. Further observation reveals that the blue and red double crosscorrelations, 4r,,J,ss(~)and &&T), have identical shapes while the green is much broader. A prediction was made that green would result if a double crosscorrelation function was synthesized such that its shape was similar to 4H,ss(~) and c$~,~&) and shifted such that its midpoint was at T/4, the midpoint of the crosscorrelation function for the original green G, as seen in Fig. 2. The disc pattern and resulting hue pulse [He,(t)] are shown in Fig. 1 while the double crosscorrelations for G and GI are compared in Fig. 3. The prediction indeed proved correct! A new green resulted, however, our observers identified it as a much darker green. This indicated that two factors are necessary for color identification, the shape and time shift of the double crosscorrelation function. Yellow is an equal mixture of red and green. Based on this, we hypothesized that the crosscorrelation function for yellow 4H,sJ~) would be the normalized sum of c$~,~~T) and 4H,Ss(7). To verify this, the yellow c$~,,.~(z) was determined from a double crosscorrela-
FIh,s&,l(~) = 2j”“’’ 4m&) cos(nw, T) d7 (4 where CO, is the fundamental frequency and equals 2n/7: UJ = 110,. and II = the number of the harmonic (Lathi. 1965).
PREDICTIONS
AND
RESULTS
Twenty observers were requested to identify what they saw on the rotating disc under various lighting conditions (sunlight, incandescent and fluorescent light). The observers were approx 1 m from the disc rotating at 6 Hz. From these observations, the colors were identified as indicated in Fig. 1. There was virtually no difference in the subjective color perceived for the various lighting conditions. The double crosscorrelation functions for blue, green and red (the three
Fig. 3. Double
crosscorrelation
functions
of original
[4H,Sd7)l and the new green C~H,,,SS(~~~~
green
Mathematical approach to subjective color perception tion calculation compared to
615
using the hue pulse in Fig. 1 and then
As shown in Fig. 4, the predicted and actual double crosscorrelations for yellow are similar. It should be pointed out that a simple combination of the hue pulses of red and green )[Ha(t) + H,(t)) when compared to those of the original yellow do not show similarity. This further supports our contention that in order to predict and analyze the subjective color produced by the disc, the double crosscorrelation functions of the hue and saturation pulses must be examined. A similar analysis was used to predict the double crosscorrelation function for orange. An orange sensation generally results from a mixture of 3 parts red and 1 part green. Therefore, an orange double crosscorrelation function c$~,,~S(T)was synthesized by taking 3/4 $%,,SS(7)+ 1/4 ~H,S&). This was then compared to 4HqSs(r). Both the synthesized and actual were again similar as shown in Fig. 5. The above predictions confirmed the use of crosscorrelation techniques as a means to analyze the information from the rotating disc and predict the subjective colors that would result from any combination of pulses on the disc. Many investigators of subjective color (Troland, 1921; Fry, 1945; Roelofs and Zeeman, 1958), have postulated that subjective color might be explained by a temporal modulation theory. Modulation is a phenomenon of the frequency domain. All our analyses to this point have been in the time domain. In order to examine the extent of the modulation of the disc pulses a conversion from the time domain to the frequency domain was necessary. The Fourier transform of the double crosscorrelation function results in the magnitude and phase spectra of the disc pulses in the frequency domain. Each time shift of the double crosscorrelation function results in
Fig. 5. Double crosscorrelation functions of original orange [&JH,&T)] calculated from the hue and saturation disc
pulses and the synthesized orange [~H,,,&T)]. a distinct phase angle for each color. Applying equation 2 to the double crosscorrelation functions for each color, at the fundamental frequency, ml, yields an instantaneous phase deviation e(t), for each color. They are :
&l(t) =
2,
&(t) = ‘2
e,(t) =
and
2,
/3,(t) =
f&(r) = 2g.
22, (3)
The definition of angle modulation is f(t) = A sin [CO,t + e(t)] where A is the amplitude, wi is the fundamental frequency, and O(t) is the angle modulation in radians. Phase angle modulation results when e(t) is proportional to the modulating signal. Each color has a distinct instantaneous phase deviation, e(t) (equation 3) which suggests that subjective color information is transmitted via a phase angle modulated process. DISC MODIFICATIONS
Fig. 4. Double crosscorrelation functions of original yellow [~H,s~(T)] calculated from the hue and saturation disc pulses and the synthesized yellow [dH,&~)].
The effects of varying the amount of the solid black area on the disc were investigated. As the amount of solid black was decreased with the arcs remaining constant, it became increasingly difficult to distinguish between the colors. With the solid black sector eliminated, only black rings appeared. As the solid black decreases from a half circle the saturation pulse increases. Simultaneously, the hue pulses are altered by the addition of an “on” pulse at T/2. The duration of the new hue pulse depends on the amount of decrease of the solid black. These pulse variations cause the double crosscorrelation functions to spread out, lose their peaks and overlap one another. This causes each color to lose its unique shape and phase shift. Another investigation involved changing the arc lengths (for R, G, and B) such that they were equally
LEONARD POLIZZOTTO and ROBERTA. PEURA
616
spaced in the remaining white area after the solid black sector was altered. With the solid black only 120”, each arc now increases to 80”. For colors to appear, the disc must now be rotated at approx 8 Hz. Essentially, the hue and saturation pulses are “spread out”. This causes the double crosscorrelations to spread out if the disc was rotated at 6 Hz, the control frequency. However, if the disc speed is increased from 6 to 8 Hz, the new pulses and double crosscorrelations are the same as the original. At 6Hz for example, a 60” arc appears for l/36 set, at 8 Hz, an 80” arc also appears for l/36 sec. Similarly, when the black sector was increased to 240 and the arcs each only 40”, the speed of rotation for best color perception was 4 Hz. During the above described disc modifications the width and the space between the arcs remained constant so as not to introduce too many variables. Future investigations will be concerned with these modifications. These results further suggest that the shape and time shift of the double crosscorrelation functions, as determined by the arc duration are important for subjective color perception. SUMMARY
This paper draws an analogy between the elements ofa modified Benham’s disc and the variables of physical color. The hue variable of a particular color was identified as the on--off pulse resulting from an arc. The saturation variable was suggested to be the white bordering the two sides of an arc, while luminance was considered to be the amount of incident light reflecting from the disc. We have shown that statistical communication theory proves to be a useful analysis technique to examine subjective color perception. The hue pulse for a particular color was crosscorrelated with a saturation pulse. resulting in a single crosscorrelation. Since each hue is bordered on both sides by a saturation pulse, the single crosscorrelation was then crosscorrelated with the other bordering saturation pulse. This analysis indicated that each color had its own distinct shape and time shift. In order to analyze the disc pulses in the frequency domain the Fourier transform of the double crosscor-
relation function was taken. The results showed that each color has a distinct instantaneous phase deviation (see equation 3). This suggests that subjective color information is transmitted in the visual pathway via an angle modulated system. This is in agreement with Fry (1945) who has suggested a modulation theory for color vision where each color has a distinct modulation pattern or wave shape, which is superimposed on a basic frequency. Roelofs and Zeeman (1957) have also indicated that a form of angle modulation is used to transmit color information. They suggested that messages are sent to the brain by oscillating current through the optic nerve. An increase or decrease of the frequency of oscillation seems to be the factor that distinguishes subjective color. We have also shown that, using the double crosscorrelation waveforms for blue, green and red, the double crosscorrelation waveforms of other colors such as yellow and orange can be predicted. The double crosscorrelation waveforms for blue, green and red are combined in the same proportions as are the three primary colors in light to yield the desired color. We have shown that statistical communication techniques can be effectively used to suggest a possible explanation for the perception of subjective color. REFERENCES
Benham C. E. (1X94) The artificial
spectrum top. ~Vafurr. Land. 51, 200. Benham C. E. (1895) The artiticral spectrum top. Nature. Land. 51, 32 I. Fry G. A. (1945) A photo receptor mechanism for the modulation theory of color vision. J. opr. Sot. Am. 35, 114135. Hargroves J. A. and Hargroves R. A. (1971) Bibliography of work on Hashing lights. Ksiorl Rrs. (Suppl.) 2, I71 I 1969. Lathi B. P. (1965) Sipak Sq~srenzs md Communicutions. Wiley, New York. Lee Y. W. (1960) Statisticul Theory e/ Communicution. Wiley. New York. Roelofs C. 0. and Zeeman W. P. C. (19%) Benham’s Top and the color phenomena resulting from interaction with intermittent light stimuli. Acfa psychol. 13, 334356. Sheppard J. J. (1968) Humurl Color Prrwption. Elsevier. New York. Troland L. T. ( I91 I ) The enigma of color vision. rl~. .I. ~/IF.siol. Opt. 2, 23%4X.
Pergamon Press 1975. Printed in Great
Britam
RESEARCH
NOTE
A MATHEMATICAL APPROACH TO EXPLAIN SUBJECTIVE COLOR PERCEPTION LEONARDPOLIZZOTTO 103 Lexington
Street, Auburndale,
Massachusetts
02166, U.S.A.
and
A. F'EURA
ROBERT Worcester
Polytechnic (Received
Institute,
Worcester,
Massachusetts
19 May 1973; in revisedform
01609, U.S.A.
20 August 1974)
INTRODUCTION
Subjective color is a sensation evoked by intermittent visual stimuli, as opposed to real or physical color which is evoked by steady visual stimuli. Hargroves and Hargroves (1971) and Sheppard (1968) provide good summaries concerning the many investigations of subjective color. The most widely used source for producing a black-white flicker resulting in subjective color is the top invented by Benham (1894, 1895) or modified forms of it. This paper discusses the application of statistical communication theory and harmonic analysis techniques to the pulses produced by the rotation of a modified Benham’s disc, referred to as the disc (see Fig. 1) in order to explain human subjective color perception. The results suggest that phase angle modulation is the process used by the visual system in the perception of subjective color. ASSUMPTION
FOR ANALYSIS
With the disc rotating at approximately 6 Hz in the clockwise direction, the colors red, green and blue appear in the arcs indicated in Fig. 1. When the direction of rotation is reversed, blue and red exchange positions. This implies that the observed color is not dependent on the radial distance of the arcs but rather on their position with respect to the black half of the disc. Other colors can be synthesized on the disc by varying the arc length and position. The patterns that yield orange and yellow are also shown in Fig. 1. Since the rotating disc produces color sensations an analogy was made between the elements of the disc and the variables of physical color. The variables of physical color are the dominant wavelength or hue, the 613
Fig. 1. A modified
Benham’s disc. H = hue pulse and S = saturation pulse.
purity or saturation, and the luminance or brightness. As seen in Fig. 1 the disc consists of a complete black half and a half consisting of black arcs on a white background. The width of the arcs and the space between them were each equal to approximately 1cm. All white portions of the disc were considered as “on” and all black as “off”. An arbitrary 0” point was established and the on-off pulse trains resulting from the arcs of the rotating disc were determined (see Fig. 1). As was noted before a change in the arc length and position results in a change in color. For this reason, the pulses derived from the arcs were termed the hue color variable, the first element of the disc. Between each arc there is white, which produces a pulse that is on for 180” and off for 180”. When it is eliminated (by blacking out the areas between the arcs), no colors appear as the disc is rotated. The 180
LWNARDPOLIZZOTTO and ROHEKT A. PMJKA
613
white arcs between the black arcs were therefore termed the saturation color variable, the second element of the disc. This provides the mixture with white and determines the purity of the color. The final element of the disc, luminance, is considered to be the amount of light reflected from the disc. METHOD
OF ANALYSIS
The eye. in some yet unexplained manner. compares the hue and saturation waves, processes this information. and sends it to the central nervous system with the result that color is perceived. Since the visual system is a communication system. a mathematical means for comparing and analyzing signals in a communication system. namely crosscorrelation, will be applied to the hue,.fh(t), and saturation. /i(t) pulses. The crosscorrelation function. ~"S(T) provides a measure of similarity between,f,(t) and,fJt) as a function of a searching parameter. 5. r is a continuous time displacement in the range considered and is independent oft (Lee, 1960). C&(T) is defined as:
;
; .f;,(t) x .f,t + 7) dt. 7Z
(1)
Tis the period of the wave or the inverse of the disc frequency (Hz) in seconds. With the white portion of the saturation wave on only one side of an arc, or hue wave, the subjective color perceived is less vivid and occurs only at the edge of the arc bordering the saturation wave. Thus, each arc must be bordered on both sides by a white saturation wave and two sequential crosscorrelation calculations must be performed. The first crosscorrelation c/I~S(T),was calculated between the hue pulse and one of the bordering saturation pulses, .fJr). The second crosscorrelation, 4&t), a “double” crosscorrelation function. was calculated between the first crosscorrelation function c/I~,&~)and the other identical bordering saturation pulse .&t). The results of this analysis are in’ the time domain. The frequency domain representation of a time function is found by taking the Fourier transform. This transformation was applied to the double crosscorrelation functions which yielded the magnitude and phase spectra in the frequency domain.
Fig.
2. Double [4~,&)1.
crosscorrelation
functions
of
blue
green I+H,,s~~)I~ and red [4~,,~.d~)l.
primary colors in light) were determined and are shown in Fig. 2. This graph shows that each color has a distinct shift in time. Further observation reveals that the blue and red double crosscorrelations, 4r,,J,ss(~)and &&T), have identical shapes while the green is much broader. A prediction was made that green would result if a double crosscorrelation function was synthesized such that its shape was similar to 4H,ss(~) and c$~,~&) and shifted such that its midpoint was at T/4, the midpoint of the crosscorrelation function for the original green G, as seen in Fig. 2. The disc pattern and resulting hue pulse [He,(t)] are shown in Fig. 1 while the double crosscorrelations for G and GI are compared in Fig. 3. The prediction indeed proved correct! A new green resulted, however, our observers identified it as a much darker green. This indicated that two factors are necessary for color identification, the shape and time shift of the double crosscorrelation function. Yellow is an equal mixture of red and green. Based on this, we hypothesized that the crosscorrelation function for yellow 4H,sJ~) would be the normalized sum of c$~,~~T) and 4H,Ss(7). To verify this, the yellow c$~,,.~(z) was determined from a double crosscorrela-
FIh,s&,l(~) = 2j”“’’ 4m&) cos(nw, T) d7 (4 where CO, is the fundamental frequency and equals 2n/7: UJ = 110,. and II = the number of the harmonic (Lathi. 1965).
PREDICTIONS
AND
RESULTS
Twenty observers were requested to identify what they saw on the rotating disc under various lighting conditions (sunlight, incandescent and fluorescent light). The observers were approx 1 m from the disc rotating at 6 Hz. From these observations, the colors were identified as indicated in Fig. 1. There was virtually no difference in the subjective color perceived for the various lighting conditions. The double crosscorrelation functions for blue, green and red (the three
Fig. 3. Double
crosscorrelation
functions
of original
[4H,Sd7)l and the new green C~H,,,SS(~~~~
green
Mathematical approach to subjective color perception tion calculation compared to
615
using the hue pulse in Fig. 1 and then
As shown in Fig. 4, the predicted and actual double crosscorrelations for yellow are similar. It should be pointed out that a simple combination of the hue pulses of red and green )[Ha(t) + H,(t)) when compared to those of the original yellow do not show similarity. This further supports our contention that in order to predict and analyze the subjective color produced by the disc, the double crosscorrelation functions of the hue and saturation pulses must be examined. A similar analysis was used to predict the double crosscorrelation function for orange. An orange sensation generally results from a mixture of 3 parts red and 1 part green. Therefore, an orange double crosscorrelation function c$~,,~S(T)was synthesized by taking 3/4 $%,,SS(7)+ 1/4 ~H,S&). This was then compared to 4HqSs(r). Both the synthesized and actual were again similar as shown in Fig. 5. The above predictions confirmed the use of crosscorrelation techniques as a means to analyze the information from the rotating disc and predict the subjective colors that would result from any combination of pulses on the disc. Many investigators of subjective color (Troland, 1921; Fry, 1945; Roelofs and Zeeman, 1958), have postulated that subjective color might be explained by a temporal modulation theory. Modulation is a phenomenon of the frequency domain. All our analyses to this point have been in the time domain. In order to examine the extent of the modulation of the disc pulses a conversion from the time domain to the frequency domain was necessary. The Fourier transform of the double crosscorrelation function results in the magnitude and phase spectra of the disc pulses in the frequency domain. Each time shift of the double crosscorrelation function results in
Fig. 5. Double crosscorrelation functions of original orange [&JH,&T)] calculated from the hue and saturation disc
pulses and the synthesized orange [~H,,,&T)]. a distinct phase angle for each color. Applying equation 2 to the double crosscorrelation functions for each color, at the fundamental frequency, ml, yields an instantaneous phase deviation e(t), for each color. They are :
&l(t) =
2,
&(t) = ‘2
e,(t) =
and
2,
/3,(t) =
f&(r) = 2g.
22, (3)
The definition of angle modulation is f(t) = A sin [CO,t + e(t)] where A is the amplitude, wi is the fundamental frequency, and O(t) is the angle modulation in radians. Phase angle modulation results when e(t) is proportional to the modulating signal. Each color has a distinct instantaneous phase deviation, e(t) (equation 3) which suggests that subjective color information is transmitted via a phase angle modulated process. DISC MODIFICATIONS
Fig. 4. Double crosscorrelation functions of original yellow [~H,s~(T)] calculated from the hue and saturation disc pulses and the synthesized yellow [dH,&~)].
The effects of varying the amount of the solid black area on the disc were investigated. As the amount of solid black was decreased with the arcs remaining constant, it became increasingly difficult to distinguish between the colors. With the solid black sector eliminated, only black rings appeared. As the solid black decreases from a half circle the saturation pulse increases. Simultaneously, the hue pulses are altered by the addition of an “on” pulse at T/2. The duration of the new hue pulse depends on the amount of decrease of the solid black. These pulse variations cause the double crosscorrelation functions to spread out, lose their peaks and overlap one another. This causes each color to lose its unique shape and phase shift. Another investigation involved changing the arc lengths (for R, G, and B) such that they were equally
LEONARD POLIZZOTTO and ROBERTA. PEURA
616
spaced in the remaining white area after the solid black sector was altered. With the solid black only 120”, each arc now increases to 80”. For colors to appear, the disc must now be rotated at approx 8 Hz. Essentially, the hue and saturation pulses are “spread out”. This causes the double crosscorrelations to spread out if the disc was rotated at 6 Hz, the control frequency. However, if the disc speed is increased from 6 to 8 Hz, the new pulses and double crosscorrelations are the same as the original. At 6Hz for example, a 60” arc appears for l/36 set, at 8 Hz, an 80” arc also appears for l/36 sec. Similarly, when the black sector was increased to 240 and the arcs each only 40”, the speed of rotation for best color perception was 4 Hz. During the above described disc modifications the width and the space between the arcs remained constant so as not to introduce too many variables. Future investigations will be concerned with these modifications. These results further suggest that the shape and time shift of the double crosscorrelation functions, as determined by the arc duration are important for subjective color perception. SUMMARY
This paper draws an analogy between the elements ofa modified Benham’s disc and the variables of physical color. The hue variable of a particular color was identified as the on--off pulse resulting from an arc. The saturation variable was suggested to be the white bordering the two sides of an arc, while luminance was considered to be the amount of incident light reflecting from the disc. We have shown that statistical communication theory proves to be a useful analysis technique to examine subjective color perception. The hue pulse for a particular color was crosscorrelated with a saturation pulse. resulting in a single crosscorrelation. Since each hue is bordered on both sides by a saturation pulse, the single crosscorrelation was then crosscorrelated with the other bordering saturation pulse. This analysis indicated that each color had its own distinct shape and time shift. In order to analyze the disc pulses in the frequency domain the Fourier transform of the double crosscor-
relation function was taken. The results showed that each color has a distinct instantaneous phase deviation (see equation 3). This suggests that subjective color information is transmitted in the visual pathway via an angle modulated system. This is in agreement with Fry (1945) who has suggested a modulation theory for color vision where each color has a distinct modulation pattern or wave shape, which is superimposed on a basic frequency. Roelofs and Zeeman (1957) have also indicated that a form of angle modulation is used to transmit color information. They suggested that messages are sent to the brain by oscillating current through the optic nerve. An increase or decrease of the frequency of oscillation seems to be the factor that distinguishes subjective color. We have also shown that, using the double crosscorrelation waveforms for blue, green and red, the double crosscorrelation waveforms of other colors such as yellow and orange can be predicted. The double crosscorrelation waveforms for blue, green and red are combined in the same proportions as are the three primary colors in light to yield the desired color. We have shown that statistical communication techniques can be effectively used to suggest a possible explanation for the perception of subjective color. REFERENCES
Benham C. E. (1X94) The artificial
spectrum top. ~Vafurr. Land. 51, 200. Benham C. E. (1895) The artiticral spectrum top. Nature. Land. 51, 32 I. Fry G. A. (1945) A photo receptor mechanism for the modulation theory of color vision. J. opr. Sot. Am. 35, 114135. Hargroves J. A. and Hargroves R. A. (1971) Bibliography of work on Hashing lights. Ksiorl Rrs. (Suppl.) 2, I71 I 1969. Lathi B. P. (1965) Sipak Sq~srenzs md Communicutions. Wiley, New York. Lee Y. W. (1960) Statisticul Theory e/ Communicution. Wiley. New York. Roelofs C. 0. and Zeeman W. P. C. (19%) Benham’s Top and the color phenomena resulting from interaction with intermittent light stimuli. Acfa psychol. 13, 334356. Sheppard J. J. (1968) Humurl Color Prrwption. Elsevier. New York. Troland L. T. ( I91 I ) The enigma of color vision. rl~. .I. ~/IF.siol. Opt. 2, 23%4X.
