JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Role of geodesics in Schrodinger's theory of color vision

Anil K. JainImage Processing Institute, Department of Electrical Engineering, University of Southern California,

Los Angeles, California 90007(Received 2 March 1973)

Relation between geodesics in color space and Schrodinger's theory of color vision is discussed. The FMC1formula is used to test this theory by examining intersection of geodesics on the constant-brightnesssurface. Validity of the FMC1 formula in numerical verification of Schr6dinger's concept is indicated.

Index Headings: Color; Colorimetry; Color vision.

In many situations, such as scene analysis, color imagerestoration, image evaluation, etc., where a visual modelfor color perception or color discrimination is utilized,the role of color-difference measure becomes veryimportant. The choice of a suitable color-differenceformula could be crucial to the success of such analysis.Indeed, a good color-difference formula can be used as avalidation measure for a visual model. Recently, someefforts have been madel"2 to compare the implicationsof various color-difference formulas on the basis of theirrelative agreements with the observed data. Results ofthese comparisons' suggest that the FMC1 formulamay be preferable in some respects to the other formulasrecommended by the CIE. In this paper I will discussthe use of this formula to test Schr6dinger's concept3 inthe theory of color vision. Success of this test could betaken as an indication of the suitability of the form ofthe FMC1 formula.

I. GEODESICS AND SCHRODINGER'SCRITERION

The FMC1 4 formula specifies the coefficients Ci. ofthe color-diff erence metric,

3 3

[ds'] = E Cijdxidxj, (1)il1 j=1

where xi, i= 1, 2, 3, are linearly related to the CIEX, Y, Z color tristimulus values; Cij are the metriccoefficients and depend on xi, i= 1, 2, 3. Using Eq. (1) asthe color-difference metric, the color difference betweenany two arbitrary colors (c, and C2) is given by 4

following Persels2 or Jain. 4 If (xyY) and (1,, Y)represent the coordinates of two colors cl and C2,

respectively, then according to Schr6dinger's theory,color c2 appears as bright as cl if, for fixed tg,F,.9, andP, Y is such that the color distance between cl and c2 isminimum. This means that, for the fixed color c, andfixed chromaticities (xy) of C2, the colors cl and c2 areequally bright if, among all possible geodesics thatoriginate from cl and terminate at 'the vertical linethrough (x,y) in the x, y, Y space, Y is such that thegeodesic between cl and c2 has minimum color distance.This implies that

(3)

where ly is the set of all possible luminance values inthe x, y, Y space. Figure 1 illustrates this meaning.The broken lines show the geodesics from cl to variouscolors on the constant-chromaticity line through (2,).The solid line is the geodesic satisfying Eq. (3). This iscalled the constant-brightness geodesic. For any fixedcl and any chromaticity (xy), there is a uniqueconstant-brightness geodesic. The loci of all theseconstant-brightness geodesics through x, g, Y is called

y

(2)s = min ds,Cl

i.e., the color distance between the colors cl and C2 isequal to the distance measured according to Eq. (1),along the curves of least distance between these colors.The curve of least distance xi(t) (i= 1, 2, 3, t=parameteralong the curve) is called the geodesic between cl and c2.There is a unique geodesic between any two points incolor space. Given the coordinates of cl and C2, thegeodesic between them can be numerically computed

I Ix, -1

II II - I

/ ',- -- ,Ia+ d a

___ - I CON(R , BRIE"I% ^AGEO1

I II I

(xy)

STANT -;HTNESSDES IC

(xy)

FIG. 1. Geodesics between a fixed color cl and a constant-chromaticity line. Broken lines show a geodesic with arbitraryvalue of Y on the constant-chromaticity line. Solid line satisfiesSchrddinger's criterion.

934

VOLUME 63, NUMBER 8 AUGUST 1973

'4)a%a.Eli

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GEODESICS IN SCHRODINGER'S COLOR THEORY

a constant-brightness surface, Based on Schr6dinger'stheory, described above, the following conclusions canbe made: (a) Any two colors on a constant-brightnessgeodesic are colors of equal brightness; (b) all constant-brightness geodesics are horizontal at white (or anygray).

The first conclusion can be justified by invokingBellman's principle of optimality.5 Suppose C3 is anypoint on the constant-brightness geodesic connectingcl and c2. Now, the point c2 is the point of least distance[according to Eq. (3)] from cl. Then according to theprinciple of optimality, c2 is also the point of leastdistance from C3 . Since c3 lies on the constant-brightnessgeodesic (or curve of least distance) from cl, and thedistance between cl and c2 is the sum of infinitesimaldistances along this curve, the point C3 must be thepoint of least distance from cl. This must hold true forany constant-brightness geodesic; since C3 is an arbitrarypoint on it, we can draw the conclusion that any twocolors on the surface of constant brightness are colorsof equal brightness (according to Schr6dinger'scriterion) and the geodesic connecting them is aconstant-brightness geodesic. We will use this to testSchrodinger's theory, by use of the FMC1 formula.

The second conclusion can be justified by applyingSchr6dinger's criterion to the color-difference metric.All of the color-difference formulas (including FMC1)recommended for test by the CIE indicate that allequi-color-distance infinitesimal ellipsoids with centerson white or any gray are symmetrical about constant- Y

TABLE I. Values of Y at the end pointsof the geodesics in Fig. 3.

Number ofgeodesic End point Y

1 (0.7, 0.25) 33.02 (0.6, 0.21) 35.03 (0.5, 0.15) 33.64 (0.33, 0.08) 32.45 (0.26, 0.05) 33.06 (0.19, 0.02) 23.57 400 nm 7.08 440 nm 15.59 447 nm 19.5

10 455 nm 25.011 473 nm 37.012 478 un 39.013 481 nm 39.514 485 nm 40.015 490 nm 40.016 496 nm 40.017 503 nm 39.018 540 nm 45.319 563 nm 48.720 572 nm 49.021 577 nm 48.622 583 nm 48.123 590 nm 46.424 600 nm 43.0

y

YA

I I

! I 'I. IX

CONSTANT-BRIGHTNESSGEODESICS

ACHROMATICPOINT

[.31+Ax,.31 | Ay)-W x,y

(.31, .316)

FIG. 2. Equi-color-difference ellipsoid at a gray.

planes i1n the x, y, Y space (i.e., the ellipoids are vertical,with no tilt). Because the point of least distance fromthe center on any vertical line in the x, y, Y space isthe point where the infinitesimal ellipsoid is tangent tothat line, the curves of least distance, or the infinitesimalconstant-brightness geodesics originating from a gray,are the infinitesimal straight lines joining the achro-matic point (gray) and the various intersection points.Clearly, these intersection points lie in the constant-Yplane through the achromatic center point; thereforethe constant-brightness geodesics and surface arehorizontal at any achromatic point (see Fig. 2). Theslope of these geodesics at any other point in the x, y, Yspace depends on the tilt of the equi-color-differenceellipsoid at the point. For any given color-differenceformula, the analytic conditions at the end point of aconstant-brightness geodesic can be derived. Theseconditions can be used to verify the validity of aconstant-brightness surface generated numerically orexperimentally.

II. CONSTANT-BRIGHTNESS SURFACE

Using the horizontal-slope-at-white (or gray) criterionof constant-brightness geodesics, Muth and Persels2

solved Eq. (3) by use of dynamic programming. Theygenerated several such geodesics from an achromaticpoint to various spectral colors and to various colors onthe line of purples. The values of Y (= Y in our notationof Sec. I) for YA 50, obtained by them for variouschromaticities are given in Table I. The accuracy ofthese values is +0.5 (or between 4 1-+ 7%, dependingon the value Y). I have generated geodesics betweenthe same achromatic point (YA = 50) and the same endpoints, by use of a gradient method described in anearlier work.4 Figure 3 shows these geodesics. Thesegeodesics are very similar to those of Muth and Persels.The geodesics in Fig. 3 were generated solely by use ofthe end-point coordinates (and the FMC1 formula, ofcourse); the horizontal-slope-at-gray criterion was notused. Within the accuracy of the Muth and Perselsdata, these geodesics were found to have horizontal

August 1973 935

I

ANIL K. JAIN

y

0.43

0.2- 1413

12

9 . 10 0.1 0.2 0.3 x 0.4

FIG. 3. Projection on the x,y plane ofsurface generated by the FMC1 formula.Y=45, points marked 0 have Y=40.

0.8x

0.5 0.6 0.7

the constant-brightnessPoints marked A have

slopes at the achromatic point YA = 50. Table II showsthis for typical x, y, Y coordinates (geodesic No. 1)of a geodesic.

TABLE II. The x, y, Y coordinates along thethe achromatic point YA=50 and the point

geodesic between(x=0.7, Y=0.25,

Y=.33.0). (Geodesic No. 1 in Fig. 3.)

x Y I

1 0.700 0.250 33.02 0.661 0.274 36.73 0.629 0.291 39.44 0.602 0.304 41.45 0.577 0.313 43.06 0.555 0.321 44.37 0.535 0.326 45.38 0.517 0.331 46.19 0.500 0.334 46.8

10 0.483 0.336 47.411 0.468 0.337 47.912 0.454 0.338 48.313 0.440 0.338 48.7

14 0.428 0.338 49.015 0.415 0.337 49.216 0.404 0.337 49.417 0.392 0.335 49.618 0.382 0.334 49.719 0.371 0.332 49.820 0.362 0.330 49.921 0.352 0.330 49.922 0.343 0.327 50.023 0.334 0.324 50.024 0.326 0.321 50.025 0.317 0.319 50.026 0.310 0.316 50.0

FIG. 4. Projection on the x,y plane of cross-spectrum geodesicsbetween the point (0.5, 0.15, 35.0) No. 3 and various spectralcolors on the constant-brightness surface.

Now, assuming that geodesics in Fig. 3 mapthe constant-brightness surface, then according toSchr6dinger's criterion, developed in Sec. I, a geodesicbetween any two points on this surface should also lieon this surface. This implies that, if cross-spectrumgeodesics are generated between noncomplementarypoints in the spectrum, they should intersect theconstant-brightness geodesics of Fig. 3. In other words,if a given constant-brightness geodesic intersects across-spectrum geodesic in the x, y plane, then at theintersection, their luminance (Y) values should beequal.

0 '0.2 0.4 X 0.6 0.8

FIG. 5. Projection of the x,y plane of cross-spectrum geodesicsbetween the point (0.26, 0.05, 33.0) No. 5 and various spectralcolors on the constant-brightness surface.

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GEODESICS IN SCHRODINGER'S COLOR THEORY

TABLE III. Points of intersection between various geodesics on the constant-brightness surface corresponding to Fig. 6.

Cross-spectrum geodesic Intersecting geodesic d= |xl-x2Number XI yl Yl Number X2 Y2 Y2 +|yl-y21 |E=IYu-Y 2 |

1 40 0.335 0.205 48.6 4 0.336 0.204 47.3 0.002 1.32 40 0.308 0.207 49.6 5 0.310 0.205 49.1 0.003 0.53 40 0.281 0.207 50.3 6 0.282 0.209 49.6 0.004 0.74 40 0.275 0.206 50.4 7 0.275 0.207 48.1 0.001 2.35 40 0.260 0.205 50.6 8 0.260 0.206 49.3 0.001 1.36 40 0.252 0.204 50.7 9 0.251 0.206 49.2 0.003 1.47 40 0.236 0.201 50.7 10 0.234 0.200 49.2 0.004 1.58 40 0.176 0.182 49.0 11 0.175 0.183 47.8 0.001 1.29 40 0.130 0.161 44.8 12 0.130 0.162 44.6 0.001 0.2

10 41 0.335 0.223 41.4 4 0.334 0.221 48.1 0.002 6.711 41 0.290 0.241 42.8 6 0.290 0.236 49.9 0.005 7.012 41 0.281 0.244 43.1 8 0.281 0.251 49.8 0.007 6.713 41 0.264 0.250 43.5 10 0.263 0.247 49.8 0.005 6.314 41 0.221 0.257 42.8 12 0.221 0.257 49.4 0.001 6.615 41 0.174 0.266 41.9 14 0.174 0.266 47.4 0.001 5.6

16 42 0.309 0.376 46.3 18 0.309 0.377 49.7 0.001 3.417 42 0.326 0.360 46.6 19 0.325 0.360 49.9 0.001 3.318 42 0.338 0.349 46.7 21 0.337 0.350 50.0 0.002 3.319 42 0.348 0.339 46.8 24 0.349 0.337 50.0 0.003 3.220 42 0.358 0.329 46.8 1 0.357 0.329 49.9 0.001 3.121 42 0.376 0.310 46.6 2 0.376 0.310 49.5 0.001 3.0

22 43 0.425 0.492 48.9 20 0.425 0.493 49.3 0.002 0.423 43 0.435 0.453 48.8 21 0.435 0.454 49.2 0.002 0.424 43 0.444 0.420 48.6 22 0.443 0.420 49.3 0.001 0.725 43 0.450 0.395 48.3 23 0.450 0.394 49.1 0.001 0.726 43 0.456 0.373 48.0 24 0.456 0.373 48.6 0.001 0.627 43 0.465 0.337 47.2 1 0.465 0.337 47.9 0.000 0.728 43 0.479 0.279 45.1 2 0.480 0.280 45.7 0.001 0.6

29 44 0.494 0.447 48.2 22 0.494 0.449 48.7 0.002 0.530 44 0.496 0.407 47.7 23 0.495 0.408 47.9 0.002 0.231 44 0.498 0.381 47.3 24 0.497 0.379 47.7 0.003 0.432 44 0.500 0.339 46.3 1 0.500 0.334 46.8 0.006 0.533 44 0.505 0.268 43.6 2 0.506 0.268 44.1 0.001 0.5

y y

0 0.2 0.4 X 0.6 0.8

FiG. 6. Intersection points of the cross-spectrum geodesics of FIG. 7. Intersection points of the cross-spectrum geodesics ofFig. 4 with the constant-brightness geodesics of Fig. 3 in the Fig. 5 with the constant-brightness geodesics of Fig. 3 in thex,y plane. xy plane.

August 1973 937

TAnLE IV. Points of intersection between various geodesics on the constant-brightness surface corresponding to Fig. 7.

Cross-spectrum geodesic Intersecting geodesic d= |XI-X21Number Xi Yl Yx Number X2 Y2 Y2 +1Y1-Y21 6=IY1-Y21

1 50 0.246 0.119 47.6 6 0.247 0.120 46.7 0.001 1.02 50 0.243 0.127 48.3 7 0.242 0.127 44.6 0.001 3.73 50 0.233 0.148 49.3 8 0.233 0.148 47.8 0.001 1.54 50 0.227 0.159 49.5 9 0.227 0.159 48.1 0.001 1.55 50 0.219 0.171 49.6 10 0.218 0.171 48.5 0.001 1.06 50 0.190 0.203 48.5 11 0.190 0.203 48.6 0.001 0.07 50 0.178 0.214 47.9 12 0.175 0.214 47.9 0.003 0.08 50 0.163 0.226 47.0 13 0.166 0.228 47.2 0.005 0.29 50 0.139 0.243 45.5 14 0.138 0.247 45.6 0.004 0.1

10 51 0.277 0.193 49.1 6 0.277 0.194 49.4 0.001 0.311 51 0.275 0.206 49.3 7 0.275 0.207 48.1 0.001 1.212 51 0.272 0.224 49.4 8 0.269 0.225 49.6 0.004 0.213 51 0.268 0.242 49.4 9 0.271 0.244 49.7 0.005 0.314 51 0.265 0.254 49.4 10 0.268 0.255 49.9 0.005 0.515 51 0.257 0.277 49.1 11 0.260 0.276 50.0 0.004 0.916 51 0.254 0.285 49.0 12 0.256 0.282 49.9 0.004 0.917 51 0.254 0.285 49.0 13 0.252 0.287 49.7 0.005 0.718 51 0.251 0.292 48.9 14 0.250 0.298 49.5 0.007 0.619 51 0.241 0.314 48.4 15 0.241 0.317 49.0 0.004 0.720 51 0.213 0.362 47.0 16 0.216 0.365 47.8 0.005 0.8

21 52 0.391 0.444 49.6 20 0.391 0.444 49.6 0.001 0.022 52 0.390 0.427 49.6 21 0.406 0.426 49.5 0.002 0.123 52 0.385 0.373 49.8 22 0.384 0.381 49.9 0.008 0.124 52 0.383 0.356 49.8 23 0.383 0.364 49.8 0.009 0.025 52 0.383 0.356 49.8 24 0.383 0.352 49.7 0.004 0.126 52 0.380 0.332 49.7 1 0.380 0.333 49.7 0.001 0.027 52 0.376 0.309 49.6 2 0.376 0.311 49.5 0.002 0.128 52 0.368 0.273 49.4 3 0.368 0.277 49.1 0.004 0.329 52 0.339 0.184 47.5 4 0.338 0.183 46.0 0.002 1.5

III. IMPLEMENTATION AND RESULTS

Several cross-spectrum geodesics between pairs ofpoints in Table I were generated and their intersectionpoints with the constant-brightness geodesics (Nos. 1-24) were examined. Figures 4 and 5 show projections ofsome of these geodesics on the x, y plane. Figures 6and 7 show intersections of these projections with theprojections of the constant-brightness geodesics ofFig. 3. All of these geodesics were determined digitally,by computing 51 points on each geodesic. The points ofintersection on the x, y plane were determined by findingthe pair of x, y coordinates on the two intersectinggeodesics that were nearest with respect to the city-block distance measure

d = I X1-x 21 +I yl-y 21, (4)

where xi, yi are the chromaticity coordinates of a cross-spectrum geodesic, and x2, Y2 are the chromaticitycoordinates of an intersecting constant-brightnessgeodesic of Fig. 3.

A more-refined estimate of the intersecting point wasmade by linear interpolation between the neighboringgeodesics surrounding the pair of points (x1,x1) and(x2,y2). The Y values of the intersecting geodesics werethen compared. Tables III and IV list the pair ofcoordinates nearest the intersection point on the two

geodesics, with their luminance difference

e-IY1-Y 21, (5)

where Y1 is the luminance value corresponding to xi, y,on a cross-spectrum geodesic, and Y2 is the luminancevalue corresponding to x2, Y2 on the intersectingconstant-brightness geodesic.

Tables III and IV show that, generally, the values ofe are quite small (i.e., Y1 - Y2), thereby indicating thatthe cross-spectrum geodesics do lie on the constant-brightness surface. These tables show intersectionpoints on only a few pairs of geodesics. Study of mostother intersection points leads to a similar conclusion.However, in a few cases (e.g., intersections of geodesicNo. 41 with constant-brightness geodesics 4, 5, 6, 7, . . .,etc.), the values of e are as high as 7.0 or 14%o. But,considering the possible errors (±0.5 or 1-7%) of thevalues of Y at the end points in Table I, the possibleerrors of the FMC1-formula optimization, and that thefrequency of occurrence of these errors is relativelysmall, these errors do not justify rejection ofSchrodinger's criterion.

IV. CONCLUSIONS

On the basis of results obtained in the study reported,I conclude that (a) Schr6dinger's criterion of constant-

938 ANIL K. JAIN Vol. 63

GEODESICS IN SCHRODINGER'S COLOR THEORY

brightness colors is probably valid, (b) the FMC1formula verifies this criterion nearly everywhere in thecolor solid.

The few intersection points that give rise to relativelyhigh values of E will be reexamined by generatingconstant-brightness surfaces with higher accuracy ofthe Y values at the end points. It may be possible todetermine analytically the end conditions for theconstant-brightness geodesics; if so, these conditionsshould be used in solving Eq. (3) for more-accurateresults. This aspect is being studied.

ACKNOWLEDGMENTS

The author is grateful to Dr. David L. MacAdam forsuggesting this work and also for his valuable discussions

during the course of its development. The author is alsothankful to Jim Pepin of Electronic Sciences Labora-tory, USC, for his help in producing computer plots ofvarious data reported here.

REFERENCES

'D. L. MacAdam, in Color Metrics, edited by J. J. Vos, L. F.C. Friele, and P. L. Walravon (AIC/Holland, c/o Institute ofPerception TNO, Soesterberg, Netherlands, 1972), p. 160.

2E. J. Muth and C. G. Persels, J. Opt. Soc. Am. 61, 1152(1971).

3E. Schr6dinger, in Sources of Color Science, edited by D. L.MacAdam (MIT Press, Cambridge, Mass., 1970), pp.155-182.

4A. K. Jain, J. Opt. Soc. Am. 62, 1287 (1972).5 R. E. Bellman, Adaptive Control Processes (Princeton U. P.,

Princeton, N. J., 1966), p. 56.

Technical Council

CHARLES J. KOESTER (Chairman), Research Labora-tory, American Optical Corporation, P. 0. Box 187,Framingham Centre, Mass. 01701

ROBERT V. POLE (Vice Chairman), IBM Corporation,Monterey & Cottle Roads, San Jose, California95114

The chairmen of the Technical Groups comprise(ex officio) the Technical Council.

Aeronautics and Space Optics-RAYMOND H. McFEEMcDonnell Douglas Astronautics Co., 5301 Bolsa

Ave., Huntington Beach, Calif. 92647Atmospheric Optics-FREEMAN F. HALL, JR.

NOAA, Wave Propagation Laboratory, Environ-mental Research Labs., Boulder, Colo. 80302

Color-C. J. BARTLESONKollmorgen Corporation, Box 950, Newburgh,

N. Y. 12550Information Processing, Holography, & Coherence-

JOSEPH W. GOODMANStanford Electronics Laboratories, Stanford, Calif.

94305Lasers and Electro-Optics-ANTHONY J. DEMARIA

United Aircraft Research Laboratories, East Hart-ford, Conn. 06108

Lens Design-DAVID S. GREYDavid Grey Associates, 60 Hickory Drive,

Waltham, Mass. 02154Optical Fabrication and Testing-FRANK COOKE

66 Summer Street, North Brookfield, Mass. 01535Optical Materials-BENNETT SHERMAN

General Telephone and Electronics Laboratories,208-20 Willets Point Blvd., Flushing, N. Y. 11360

Radiometry and Photometry-HENRY J. KOSTKOWSKINational Bureau of Standards, Washington, D. C.

20234Raman-ELLIS R. LIPPINCOTT

Dept. of Chemistry, University of MarylandCollege Park, Md. 20740

Spectroscopy-Ross A. McFARLANESchool of Electrical Engineering, Cornell Uni-

versity, Ithaca, N. Y. 14850Thin Films and Interferometry-

PHILIP W. BAUMEISTERInstitute of Optics, University of Rochester,

Rochester, N. Y. 14627Vision-LORRIN A. RIGGS

Hunter Laboratory, Brown University, Providence,R. I. 02912

939August 1973