Vector based modelling of colour difference

1 Vector Based Modelling of Colour Difference A Pilot Study of the DE2000 Colour Difference Model David P. Oulton and Stephen Westland 18 / 3 / 2016 Coloration Technology Vol. 133 issue 1 (Feb 2017) pp. 5 – 25 Publ: by John Wiley. Abstract A novel approach to colour difference modelling is presented whereby for any given CMC (1:1) or CIE DE2000 E C H and L colour difference, the equivalent CIE XYZ, L*a*b*, L*C*h0 coordinate changes are derived by optimizing the input RGB stimuli from which they are all calculated. Single dimension L or C or H difference loci expressed in DE2000 difference units are thus generated and the additive equivalence of Tristimulus values is likewise projected forward onto each locus and also onto a set of CIE DE2000 3-unit ellipse boundaries. Using the datasets thus generated, it is then shown firstly that the derived ellipses have well defined semi axes which explain the detailed orientation of the MacAdam ellipses in x,y,Y space. Unit CIE DE2000 difference is confirmed as a successful quantifying constant of visual difference over a wide range of Chroma Hue and Lightness differences. As a constant, CIE DE2000 unit difference is shown to have significantly variable value at high and low Chroma: Evidence is established for systematic changes in both Chroma and Hue difference sensitivity. A hitherto unresolved nonlinearity is revealed in the C* dimension of L*C*h0 space that is not replicated in the CIE DE2000 model. The derived difference loci appear to specify physically reproducible experimental stimuli that could be used in the estimation of visual difference magnitude. Overall, the data derived by the new approach and presented in this paper increase the probability that a true vector model of the visual difference response may eventually be derived. Key Words Colour difference, DE2000, vector modelling, visual difference loci, grey scale tracking. 2 Introdu0ction The aim of this pilot study is to explore the sequence of standardised and published data transformations by which CIE XYZ tristimulus values are calculated from stimulus power values; onward into CIEL*a*b* co-ordinate values; and finally into one or more definitions of unit colour difference as discussed by Berns [1]. The objectives are to analyse the numeric structure of these three conventional transformations; to examine their success and limitations as models of colour difference; and then to demonstrate a potentially superior alternative to the lo al opti izatio data fit approach to colour difference modelling that they exemplify. Two key innovative models enable the presented analysis. The first is denoted hereafter as Model 1 and the other uses a particular type of visual-difference locus denoted in this paper as a proto vector . The principles of visual additive equivalence in three-dimensions that underlie the success of the CIE System [2] are also invoked; and greyscale tracking [3] is used to project this property forward onto the detailed characteristics of Hue Chroma and Lightness difference. Model 1 takes as input two spectrally quantified 31-element stimulus power definitions and for each stimulus-pair the above noted colour space co-ordinates are derived sequentially, by optimizing the input RGB values from which they are all calculated. The co-ordinates are then output, together with appropriate scalar difference measures E, L, C, H and h0 etc. for each of the descriptive spaces. Model 1 can be used to quantify compare and equate the CIE XYZ, L*a*b* or L*C*h0 colour co-ordinates for any given pair of stimuli within the set of all possible spectrally defined input stimulus combinations. Thus for example, many alternative RGB spectral triplets (stimuli with power only at three distinct wavelengths) may be derived that have Tristimulus values X = Y = Z = 100 according to the CIE 1964 D65 Supplementary Standard Observer (SO) model. In the current context Model 1 will also be shown to generate stimulus pairs with well-specified complex differences. Stimulus pairs may thus be found that are different for example, by H = 0, L = 0, C = 3 units according to the definitive implementation of the CIE DE2000 colour difference model [4]. The presented methodology using Model 1 appears to be an example of output-only State Space estimation or Sub-space Identification discussed later, in The P ese ted A alysis . The visual-difference loci denoted as proto vectors and used in the analysis have well-defined strictly constant, symmetrical, transitive and proportionate additivity over a well-specified field [5]; however, the visual difference space in which they exist only has a nominally constant scalar basis. Such loci are thus not well-defined as vectors because they do not have a demonstrably constant scaling and are therefore denoted only as proto vectors . The point is that the presented analysis concerns the identification and refinement of candidate scalar metrics of visual difference and a demonstrably constant scaling may eventually be derived using them. 3 The principles of additive visual equivalence and greyscale tracking are invoked because they enable the derivation of Hue-specific proto vectors with a constant scalar basis over all wavelengths. This particular application of greyscale tracking was first described by Oulton [3] and is based on the fact that spectral Hues have mutually cancelling Chroma at each neutral axis equivalence point as discussed by Pridmore [6]. In the CIE Standard Observer model the proportionate visual additivity of stimulus power is expressed in terms of CIE x, y, Y co-ordinates such that the green sensation generated by monochromatic stimulus power at 530 nm exactly complements the magenta sensation generated by a pair of stimulus power values denoted as (530c); and the linear additivity of these two stimuli is represented on the x,y Chromaticity plane by a straight line that passes through the visual neutrality point. It will be shown in the following analysis that the ratios that quantify this linear additivity of physical stimuli may be held constant (by greyscale tracking), while the proto vectors of Chroma and Hue difference are derived for each spectral dimension. The primary stimulus wavelengths used in the analysis are 450, 530 and 650 nm. These were chosen firstly because they are close to the experimental primaries used by Wright [7]; secondly e ause they a e ep ese tati e of Tho to s PC observer [8]; and thirdly following the suggestion of Pridmore [9] that they allow the logical synthesis of non-spectral Magenta stimuli using mixtures of 450 and 650 nm stimulus power. Methods The Presented Analysis In the first part of the analytical process a comprehensive set of proto vectors and synthetic colourdifference ellipses are generated, all of which are specified in terms of output CIE DE2000 C, L and H 3-unit differences. The derivations invert the conventional modelling process, in effect exemplifying the Sub-Space Identification methodology described by Overshee et al. [10, 11] where output-only analysis systems are used as in the current context. Thus, instead of using CIE XYZ coordinates as the scalar basis for quantifying colour difference, CIE DE2000 L, C, and H. 3-unit CIE DE2000 differences are used as a candidate scalar basis and colour differences are in effect back-related in Model 1 to the equivalent XYZ co-ordinates etc. via constrained optimization of input stimulus power values. Model 1, which will be described in detail later, thus quantifies both the forward calculation of colour difference value from colour space co-ordinates and the reverse mapping of difference value onto CIE XYZ, x,y,Y and CIE L*a*b* values. The profound consequences of this inverse mapping are illustrated later in Figure 2 where the conventionally specified set of MacAdam ellipses [12] are compared with a similar set of ellipses generated by the inverse route. 4 The Additivity of Spectral Stimuli The scalar basis for the Standard Observer spectral sum that quantifies XYZ Tristimulus values is quantified physically by reference to stimulus power values. The spectral level elements of the sum therefore share a strictly constant common scaling. The property of visual stimulus additivity is then quantified proportionately over this stimulus power scaling by reference to its relative luminosity over wavelength (initially quantified by the CIE Colorimetry Committee using V) [13]. The CIE established the Standard Observer model of Tristimulus addition in 1931 by reference to Illuminant SE, which has equal stimulus power at all wavelengths; by adopting Illuminant SE as the matching stimulus that quantifies the additive equivalence of spectral stimuli; by normalizing the CMF values at this axis; and by declaring this axis to be the reference definition for visual neutrality. Clearly under these constraints the Standard Observer model is itself constrained to track the SE greyscale and it follows that for all possible Y the SE axis is represented by X=Y=Z; and all colours are specified proportionately by their Chromaticity co-ordinates relative to this axis such that x = X/X+Y+Z etc. Mathematically, vector space modelling likewise requires that: The property of multi-dimensional additivity is a strict constant that must be quantified at a demonstrably constant scaling and that this scaling constant must itself be quantified in each distinct dimension while holding constant the additivity ratios. These complementary constraints are also key procedural and analytical requirements of the modelling process that is enabled by greyscale tracking [3]. The intent in the presented analysis is therefore: 1. To treat the Standard Observer Tristimulus model of visual additivity as a strict constant of the mathematically distinct scalar model of the visual difference response. 2. To analyse the scaling of the spectral level visual-difference response, by means of a set of spectral proto-vectors whose visual scaling is also equated by reference to the Illuminant SE visual neutrality axis. Under the presented analysis, the Standard Observer visual neutrality definition will in effect be projected wavelength by wavelength onto the nominally uniform scaling quantified by CIE DE2000 colour difference; and critically in the current context the greyscale tracking constraint is shown to enable this projection. A series of derivations by Model 1a is reported and analysed where the scaling of the visual response in the Chroma difference dimension is quantified at constant Hue and Lightness and is at the same time back-related to the scalar metrics of colour space as quantified by RGB stimulus power triplets, CIE XYZ and x,y,Y and CIE L*a*b* co-ordinates. The derivation of 3-unit CIE DE2000 difference ellipses using Model 1b is also reported and analysed 5 Model 1 is built on a set of Excel spreadsheets and the input-output section of Model 1b (which differs only in its input-output section from Model 1a) is illustrated in Figure 1. Figure 1. As illustrated, the Stimulus 1 column specifies an ellipse centroid with one stimulus at a nonprimary wavelength (490nm) and Stimulus 2 has been optimized using the Solver function to be exactly 3 DE2000 units different from S1. Note also the specified direction of difference, +2.12 units C and 2.12 units H and that in the output section at the top, the co-ordinates for both stimuli are output for three different colour spaces. All of the calculations are by published and verified data transformations [1] and the iterative refinement of input stimulus power values is both constrained and resolved by means of the Excel Solver function. The optimization may then be in a specific subset of parameters and directions of difference in any of five spatial quantifications of colour. For example, Model 1a is dedicated to the generation of constrained single dimension Chroma Hue and Lightness proto-vectors each of which is scaled in terms of CIE DE2000 unit C, H or L colour difference. Model 1b is dedicated to the generation of CIE DE2000 3-unit colour difference ellipses, whose characteristics may then be analysed as quantified by co-ordinates in L*a*b*, XYZ, or RGB space and by CMC (1:1) and CIE DE2000 difference units. In Figure 1, the specific RGB values of stimulus S1 (the ellipse centroid) were chosen to lie at a known point on the 490 nm proto vector of Chroma change which was derived at constant Hue and 6 Lightness. The significance of this choice will become evident later. The RGB values of S2 that generate the ellipse point are adjusted iteratively until the required difference specification (as entered in Figure 1 under the table-heading DE 2000 ) is reached. Please also note that the lefthand-most column in Figure 1 specifies an RGB Primary triplet that matches Illuminant SE. It is emphasized that in the presented analysis all the proto vector derivations in Model 1 are elati e to a s ala asis ua tified at second ha d y efe e e to CIE DE2000 diffe e e u its rather than to direct visual observation data. It follows that the optimized outputs of Model 1 have strictly indicative and analytical value a d they poi t to a d athe tha ge e ate a defi iti e scalar metric. The generation of true candidate scalar models of visual difference must necessarily be quantified directly using experimental visual difference data; and it will be shown later that the adopted sub-space identification strategy suggests a new approach to the design of such experiments. Results and Analysis The first results to be presented and analysed concern the nature of the MacAdam ellipses. Figure 2 compares a set of 18 ellipses derived using Model 1 with the MacAdam set. For all the derived ellipses the individual CIE DE2000 difference values are constant and are shown below in Table 1. dL 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 dC 3.00 2.12 0.00 dH 0.00 2.12 3.00 -2.12 -3.00 -2.12 0.00 2.12 0.00 -2.12 -3.00 -2.12 2.12 dE 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 Table 1: The C and H values from the ellipse centroid to each boundary point. Consider the scalar basis of the two diagrams (candidate model left and Ma Ada s experimental image right [14]) in Figure 2. Both the MacAdam and the derived ellipses are scaled by reference to stimulus power and the colour differences in the MacAdam set are quantified only as stimulus differences. The actual scalar visual differences as distinct from additive stimulus ratios are an undefined constant in the MacAdam set. This is because in both the defining CMF match-equations of the CIE Tristimulus model and of the MacAdam ellipse definitions [16] the true visual difference scaling cancels out across all of the match equations that quantify the relevant visual-equality null points. 7 VISUAL DIFFERENCES PLOTTED ACCORDING to CIE x,y CHROMATICITY CO-ORDINATES Figure 2. At first glance these two diagrams differ only slightly as a quantification of colour difference. However, note particularly the differences in the DC and DH semi axes. The red semi axes in the left hand diagram now indicate derived ellipse orientations in DE2000 colour difference space, while those on the right represent discrete local sub-spaces. By contrast, the scalar basis of the derived ellipses is fully specified by CIE DE2000 value inputs, which quantify a nominally constant and uniform scalar model of visual difference value. The derived ellipse dimensions in effect relate the scaling of L, C, and H difference (according to the CIE DE2000 model) to the equivalent CIE x,y co-ordinates. This is likewise true of all possible alternative colour spaces derived from CIE Tristimulus space and all possible models and dimensions of colour difference. The relationships illustrated in Figure 2 are further considered late u de the headi g Ellipse O ie tatio . The scalar and additive visual response to spectral stimuli It is first emphasized that the CIE Standard Observer system quantifies the additive equivalence of spectral stimuli under the constant scalar basis quantified by stimulus power: That the visual scaling in the SO model is only equated three-dimensionally by balancing the additive proportions of the defining primaries both visually and numerically at the SE axis; and that the resultant visual scaling at each wavelength is an un-quantified constant of the Tristimulus sum. Clearly the true scalar value of visual difference at each wavelength must be determined in some way by independent experiment. It will next be shown that the data from visual experiments based on both complementary colour determination and Maxwell Method visual matching at the SE axis may be used to determine the 8 scalar characteristics of this constant in each spectral dimension. The point is firstly that the products of such mixtures collectively represent unique null points on the grey scale axis. Secondly, in such experiments the RGB scalar values may be precisely balanced to visual neutrality for many RGB triplet combinations in the same way as the defining primaries of the Standard Observer Tristimulus values XYZ are balanced numerically at the SE axis. It is emphasized that the additive equivalence ratio is a constant of all visual experiments at the SE axis. The significance of spectral stimuli that are complementary and thereby add to visual neutrality is well reported by Pridmore [9, 17]; and the significance of the SE axis is emphasized both above and by Oulton in [18]. Recall now the SE quantifying triplet specified in Fig.1. Col.1. It quantifies the unit scalar-value constant of the reference Primaries to which each of spectral stimuli illustrated in Table 2 will be equated. In Model 1 an RGB match to this quantifying triplet is first established using two of the constant primary wavelengths and one variable wavelength where the latter is selected according to the three right-hand-most columns in Fig 1. Once the match has been optimized by varying the R,G,B values of S2, the full numeric co-ordinates of this candidate stimulus wavelength are revealed simply by zeroing the input stimulus power values for the balancing primary pair. It follows that this property of scalar equivalence may thus be determined wavelength by wavelength as illustrated Table 2. Then, under the further constraint that Hue and Lightness differences are also held constant, distinct proto-vectors of Chroma change may be developed from each of these monochromatic or maximum Chroma balance points. In practice as shown in Table 3, the 530 nm proto vector is initiated by inserting the RGB values 0 , 938 , 0 from Table 2 into both S1 and S2 and then iterating the S2 values holding DH and DL at zero, until S2 differs from S1 by -3 DC units (see also Figure 1). The output co-ordinates are then recorded and the S2 values are copied into S1 creating a new zero-difference pair one step nearer the neutral point. Once this sequence has been repeated a sufficient number of times the locus of Chroma difference at constant Hue and Lightness may be made to terminate exactly at the Illuminant SE axis. Table 3 fully tabulates the proto vector quantified by the 530 nm Chroma change locus. In total 32 such well specified loci including a comprehensive sub-set of magenta Hue loci have been derived. 9 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 R G B X Y Z L a b L C h0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1049 835 721 692 738 873 1144 1687 2697 4696 0 0 0 0 0 0 0 0 0 2105 1570 1230 1028 938 907 941 1030 0 0 0 0 0 0 0 0 0 0 11304 2779 1168 734 585 574 655 874 1463 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21.54 21.07 20.41 19.70 18.98 18.11 16.89 14.59 10.50 3.55 0.95 4.27 10.58 19.12 29.38 42.95 62.31 78.81 72.59 68.79 66.43 64.93 64.00 63.45 63.05 62.76 62.63 2.25 2.18 2.17 2.46 3.12 4.43 7.24 13.93 32.15 61.88 62.20 64.07 66.81 70.33 74.50 79.97 87.77 85.63 62.24 47.94 39.05 33.41 29.88 27.83 26.33 25.24 24.75 97.35 97.35 97.36 97.39 97.42 97.49 97.62 97.93 98.72 76.93 30.27 12.28 5.44 2.51 1.09 0.34 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 16.77 16.41 16.37 17.74 20.51 25.04 32.34 44.13 63.46 82.85 83.02 84.00 85.41 87.16 89.16 91.67 95.07 94.15 83.04 74.79 68.79 64.49 61.56 59.74 58.35 57.31 56.83 158.47 157.85 154.84 145.50 129.98 106.00 68.00 4.02 -106.64 -261.76 -321.06 -256.30 -200.62 -156.58 -120.85 -86.88 -51.67 -12.96 22.47 50.05 70.80 86.03 96.59 103.19 108.27 112.09 113.86 -141.72 -142.34 -142.41 -140.07 -135.32 -127.55 -115.05 -94.94 -62.14 -12.82 36.45 73.01 99.06 119.30 136.99 152.71 163.53 162.33 143.17 128.95 118.61 111.19 106.13 103.00 100.60 98.81 97.99 16.77 16.41 16.37 17.74 20.51 25.04 32.34 44.13 63.46 82.85 83.02 84.00 85.41 87.16 89.16 91.67 95.07 94.15 83.04 74.79 68.79 64.49 61.56 59.74 58.35 57.31 56.83 212.60 212.55 210.37 201.96 187.64 165.85 133.64 95.02 123.43 262.07 323.12 266.50 223.75 196.85 182.68 175.70 171.50 162.85 144.92 138.32 138.13 140.58 143.50 145.80 147.80 149.43 150.22 318.19 317.96 317.40 316.09 313.85 309.73 300.58 272.43 210.23 182.80 173.52 164.10 153.72 142.70 131.42 119.64 107.53 94.56 81.08 68.78 59.17 52.27 47.69 44.95 42.90 41.40 40.72 Table 2. This quantifies a full set of scalar spectral stimulus values, such that by adding complementary primary stimulus inputs, they cancel out to visual neutrality at the Illuminant SE axis according to the Standard Observer model. Note particularly the visually equivalent Y values that are characteristic of each power value and wavelength and the required power minima at 450 530 and 600 nm. The Magenta stimuli are derived as binary mixtures of the 450 nm and 650 nm Primaries and the variable stimulus is an appropriate complementary Green denoted as 500c etc. Only the stimulus power of the green complementary is zeroed in this case, and it is the R:B ratio of the balanced triplet that quantifies the co-ordinates of the Chroma limit stimulus. R G B 0 204 400 584 755 911 1054 1182 1298 1401 1492 1573 1643 1701 1751 1795 1833 1866 1896 1897 938 911 884 858 834 811 790 771 754 738 724 711 700 691 683 676 670 665 660 660 0 17 40 66 96 127 158 189 219 247 273 297 319 338 354 369 381 393 403 404 X 19.12 23.86 28.58 33.19 37.59 41.75 45.62 49.19 52.46 55.42 58.10 60.49 62.56 64.33 65.85 67.18 68.35 69.38 70.29 70.33 Y 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 70.33 Z 2.51 5.35 9.12 13.59 18.53 23.72 28.97 34.15 39.15 43.90 48.34 52.42 56.05 59.21 61.97 64.42 66.60 68.54 70.26 70.33 L a b L C h0 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 -156.6 -134.5 -115.3 -98.5 -83.8 -71.0 -59.7 -50.0 -41.4 -33.9 -27.4 -21.8 -17.0 -13.0 -9.6 -6.7 -4.2 -2.0 -0.1 0.0 119.3 102.5 87.8 75.0 63.8 54.1 45.5 38.1 31.5 25.9 20.9 16.6 13.0 9.9 7.3 5.1 3.2 1.5 0.1 0.0 87.16 196.85 142.70 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 87.16 169.15 144.95 123.81 105.34 89.21 75.12 62.81 52.05 42.67 34.48 27.39 21.39 16.37 12.12 8.46 5.29 2.52 0.10 0.00 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.70 142.69 N/A Table 3. A specimen Chroma change locus is tabulated. The chosen locus quantifies the 530 nm proto vector that is also visible in Figure 3. Note particularly the linear scale of DE2000 0 Chroma difference in the right hand most column, and the constancy of both L and h . diff scale 54.12 51.12 48.12 45.12 42.12 39.12 36.12 33.12 30.12 27.12 24.12 21.12 18.12 15.12 12.12 9.12 6.12 3.12 0.12 0.00 10 Figure 3 illustrates a full set of such proto-vector loci. Note particularly both the systematic curvature of these constant Hue loci in the CIE x,y Chromaticity diagram; and also that the steps of equal Chroma difference are systematically nonlinear relative to the stimulus power metric of the CIE x,y,Y co-ordinate system. Figure 3. The clear non linearity of DE2000 DC difference steps over the metric of the x,y plane is evident and the curvature of the constant Hue loci is clearly similar to that of the Munsell Hue loci. The graphed set includes four non spectral Magenta loci. Ellipse Orientation The apparently complex nature of MacAdam ellipse orientation as quantified by CIE Standard Observer Chromaticity co-ordinates x,y is next analysed using quantified colour differences in CIE x,y,Y and L*a*b* co-ordinate space. The spatial properties of the conventional colour difference ellipses are specified by nominally orthogonal LCH semi axes. The directions in L*a*b* colour space thus denoted have piecewise scaling that is strictly local to a given ellipse; and it is the relationship between these locally significant L, C and H parameters that conventional colour-difference models attempt to quantify. 11 Recall Figure 1 and consider Figure 4. The widely changing orientation, and indeed in some cases the apparent curvature of the derived ellipses, is a direct result of the interlinked mapping between the different co-ordinate metrics revealed by Model 1. The fact that in general the ellipses lie exactly on the Chroma-change loci is clearly trivial because they were generated to do so. However as expressed by x,y stimulus value ratios, both the semi axis orientation and magnitude of these semi axes are now directly related numerically to the chosen continuous linear CIE DE2000 metric of visual difference E. Figure 4. By combining the data for Figures 1 and 3, the effect of the Chroma and Hue change Loci on ellipse orientation is emphasised. The black line semi axes in each ellipse indicate both orientation and magnitude of DE2000 DH differences. Several necessarily cautious inferences may be drawn from Figure 4 and from the derived datasets which the ellipses represent. It must be emphasized that the following observations are as yet unconfirmed by visual experiment, but they make good sense both logically and theoretically. Each of the derived ellipses may be interpreted visually as a true circle with nominally orthogonal Hue and Chroma axes, that is simply twisted out of the x,y plane. Fa s o th s o li ea t a sfo atio of the CIE 1931 x,y MacAdam ellipse co-ordinates to render them into circles [15] comes to mind in this context. At least to a first approximation the MacAdam ellipses have a similar orientation to the derived ellipses. In other words without any prior knowledge of the visual scalar difference metric that is an 12 undefined constant in CIE x,y,Y space (or indeed of the actual existence of this metric), MacAdam came remarkably close to illustrating it graphically. Finally, in the derived ellipses the variation in the relative magnitude of the C and H semi axes and their direction of action directly specify ellipse orientation. In Figure 4, the assumption that Chroma change axis is always longer than the Hue change axis and that the orientation of this axis is intrinsically radial is seen to break down. This is because in the Blue / Purple region of the Chromaticity diagram the Hue change axis is now shown to be logically oriented and the longer of the two. On one hand this correlates well with the intrinsically Red / Blue orientation of the MacAdam ellipses in the non-spectral region; and on the other hand with the need for a fourth ellipse-rotation parameter in the CIE DE2000 ellipse quantification calculations [4]. Cylindrical / Polar Difference versus Cartesian Difference scaling The derived ellipses in the CIELAB a*b* plane are next presented as a Cylindrical / Polar image in Figure 5 and as a Cartesian representation on the C / h0 plane as in Figure 6. Figure 5.Under this Cylindrical / Polar representation, the ellipses are oriented purely by Hue angle, Hue difference value appears to vary as a function of Chroma; and H may only be calculated indirectly by subtracting the C and L component differences from E. The more logical Cartesian representation of CIE DE2000 colour differences with nominally orthogonal C and H axes is adopted in Figure 6. Clearly in such a cartesian co-ordinate 13 representation the C / h0 Plane now directly visuallizes the nominally orthogonal Chroma and Hue dimensions of difference and the L dimension is perpendicular to this plane. Figure 6. When expressed as nominally orthogonal dimensions the C and H semi axes now correctly represent ellipse shape and magnitude and are much easier to interpret. The product of this co-ordinate change still expresses CIE DE2000 3-unit differences by linear projection of CIE L*a*b* values, but the re-distribution and re-shaping of the ellipses is certainly striking and clearly open to detailed interpretation. Firstly, do not be misled by the sudden appearance of approximately uniform and nearly circular 3unit colour difference boundaries in Figure 6. This is simply confirmation that the metric of Hue angle and Chroma change in Figure 6 now replicates the innately equivalent scaling of CIE DE2000 DC and DH difference in the presented inverse modelling approach. There are however some interesting by-products of this transformation. It suggests that the CIE DE2000 metric is close to success in establishing an invariant model of unit visual difference as a function of L*a*b* coordinates across all wavelengths and Chroma values in the C* > 20 < 70 region. Of course, a moment or two of thought might also suggest that this is logical, since the data set against which this metric was optimized probably lies almost entirely in this region. Next, consider the relationship between the denoted 3-unit difference-boundaries and their scalar representation outside this region. The tentative interpretation is that they indicate differences of visual sensitivity to change and that they are not currently modelled in the CIE DE2000 metric. 14 Under this proposition the 1640 Hue angle (510 nm) scaling of Chroma change relative to unit DC in Figure 6 may be considered to be nonlinearly compressed relative to unit DC with an apparent steady loss of sensitivity to Chroma change with increasing Chroma. Likewise the evident broadening of the Hue difference dimension as a function of Chroma also has a logical interpretation. The inference is that it charts a systematic increase in Hue-difference sensitivity with increasing Chroma. In the context of such potential variations in Hue difference sensitivity, the three lowest ellipses are also potentially significant as prima facia evidence that the apparent variation of Hue difference sensitivity over Chroma change may be independent of wavelength. Generating a Continuous Metric of Colour Difference Thus far in the current analysis, a set of Chroma Hue and Lightness difference proto-vectors have been related via Model 1 to unit CIE DE2000 difference as a candidate uniform vector space model of the human visual difference response; and the unit differences have been generated by the CIE DE2000 formula itself. Clearly the modelling method is thus intrinsically circular and it is therefore a potentially trivial exercise unless on one hand it reveals novel relationships that deviate from the current reference metric as in Fig 6 or on the other hand, that the intrinsic circularity may be demonstrated to be iteratively convergent via stepwise refinement of the candidate model. The point about the adopted inverted approach to subspace identification is that in some classic industrial applications Peeters et. al. [11] it has been shown to enable the transformation of partial and often discontinuous scalar models into equivalent uniform linear sub-space models that are the product of a set of smooth continuous functions. Clearly the ultimate goal should if possible be a linear metric of Hue Chroma and Lightness difference that is demonstrably constant across all spectral dimensions and as near as possible replicates experimentally determined visual differences. In this Pilot Study only the evident non-linearity of the Chroma proto vector loci relative to the linear CIE DE2000 DC loci has been fully investigated at the spectral level and the following further analysis will continue to be limited to these loci. Under the principles of vector space modelling Krantz [5, 19], each distinct C locus must be assumed to have a characteristically distinct nonlinearity unless and until a given scalar model is shown to have true vector additive properties under strictly constant scaling. To address this imperative and also avoid the inevitable circularity of unit definition in this pilot study, it is first necessary to change the scalar basis of the C loci from the CIE DE2000 metric, to an alternative that is independently calculated from C*. 15 Clearly, the proto vector set-members generated as Model 1 outputs are readily available for quantitative analysis. An appropriate set of linearization functions that linearize these relationships were therefore generated using the Pade approximant function see references [20, 21]. The resulting functional relationships are illustrated below in Figure 7. Figure 7. For a wide range of Hue angles a constant function appears to be sufficient to quantify the observed residual nonlinearity not described by the DE2000 scalar metric. Compared with conventional chroma difference modelling over wavelength the apparent uniformity of the graphed non linarites seems almost too good to be true and they are presented here only as a pilot study of what might eventually emerge. Under this clear disclaimer of their validity as significant scalar functions, it is however interesting to apply the derived single function to the data set graphed in Figure 6 and Figure 8 shows the result. Do not be misled by the pleasing circularity of the DE boundaries, it is once again the trivial product of the circular mapping. The most significant result is that all of the illustrated chroma difference intervals are now being calculated directly from CIE C* values rather than by the discontinuous CIE DE2000 model and that they are generated using a single continuous scalar function that is common to all of the wavelength-specific proto vectors. It is emphasized that the functional mapping is directly from L*C h0* space onto CIE DE2000 unit differences and it not only accurately replicates the output of the CIE DE2000 calculations, it also corrects their apparent failure at high and low Chroma. 16 C the Linearized version of C* is Expressed in DE2000 0 Units and Plotted Versus h Figure 8. The residual Hue difference broadening remains un-corrected, but by virtue of this additional non-linear correction of the DE2000 model, the resulting Chroma difference metric appears to be improved at both higher and lower Chroma values. The apparent systematic uniformity of this set of mapping functions is sufficiently encouraging to suggest a novel experimental strategy for visual difference quantification. Claus Witt suggests in the conclusions of his 1999 paper [22] that visual difference experiments that are dimension-specific where found useful in his colour difference modelling paper. The strategy in the current case is to use the CIE XYZ specifications of the specific Hue Lightness and Chroma proto vectors already generated in this pilot study as a quantitative basis for generating physically realized experimental stimuli in visual difference magnitude-estimation experiments. The proposition is that starting from the difference steps of the CIE DE2000 proto vectors, it should be possible to iteratively refine the component visual steps by direct experiment and if necessary introduce additional correcting parameters to represent for example, the apparent changes in Hue difference sensitivity over Chroma change. Lightness Difference In the presented study, several descriptors of Lightness difference have also been derived relative to the CIE Standard Observer model. However the scaling in this dimension has not been unified by greyscale tracking with that in the H and C dimensions. The results are again interesting as shown in Figure 9. 17 Lightness Difference Measures Plotted versus Y and its Derivatives Figure 9. The two graphs for the CMC model suggest it is potentially unreliable as a basis for quantifying low Lightness differences. The two graphs for the DE2000 model linearization are clearly better in this respect. Several tentative inferences may be drawn from Figure 9. Firstly, the CIE DE2000 model appears to significantly better than the CMC model as a linearized model of lightness difference. Secondly, that the CIE DE2000 DL versus Y relationship in Figure 9 looks remarkably similar to the Chroma difference relationships in Figure 7. Importantly, however, the Figure 7 nonlinearities are clearly additional to the L*a*b* lightness difference linearization which precedes the CIE DE2000 model. However, since the Chroma differences were generated at constant Lightness this similarity is in effect cross-dimensional and may eventually turn out to be significant. Conclusions It is particularly important to emphasize that this pilot study is strictly analytical and indicative in nature and it introduces no new experimental evidence about the colour difference property of the visual response. Three analytical innovations are however introduced that appear to reveal some key features of this response as currently quantified by the CMC and CIE DE2000 colour difference models. The first innovation is the optimizable numeric calculation system Model 1. This links the set of all possible three dimensional CMC and CIE DE2000 colour differences back quantitatively to their precursor RGB, XYZ, L*a*b* and L*C*h0 co-ordinate definitions. 18 The second innovation is the use of spectrally specified difference loci denoted as proto vectors. These have equated scalar value by reference to illuminant SE at the monochromatic Chroma limit and quantify individual loci of Chroma difference at constant Hue and Lightness and so forth. In principle a wavelength-by-wavelength analysis of the chromatic visual response is thus enabled and the Lightness and Hue dimension proto vectors can likewise be quantified as distinct parameters. The third innovation is to use CIE DE2000 colour difference units throughout the analysis as a scalar basis and reference space. The analysis is thereby related to a nominally continuous and uniform three-dimensional candidate model of visual difference, which may or may not be fully representative of the actual visual response. The point is that in principle the candidate model can be progressively refined by visual magnitude estimation experiments. The use of Excel Solver to optimize the input RGB stimuli in Model 1 and constrain the directions of difference between them was particularly successful. Several thousand pairs of RGB tripletwavelength stimuli were optimized and a five figure set of quantified colour differences in five distinct colour spaces where thereby made available for analysis. Using Model 1, it was possible to generate a set of 30+ spectral and non-spectral proto vectors that in principle range over all possible visual variations of spectral Hue and Chroma difference; and all such proto vectors are shown to quantify strictly single dimension L or C or H loci in each of the colour spaces. The principles of grey scale tracking are shown to enable the predictive success of the CIE Standard Observer; to enable the property of Tristimulus value additive equivalence thus established to be projected forward onto a model of spectral difference; and to further enable the generation of additively equivalent spectral proto vectors. A significant part of the reported analysis concerns the derivation of CIE DE2000 3-unit-difference semi axes and ellipsoid boundaries and their graphic properties when plotted in terms of CIE x,y,Y , L*a*b*, or LCh0 co-ordinates by associative calculation via Model 1. By contrast, these scalar differences are shown to be an unquantified three-dimensionally defined constant of the Standard Observer XYZ Tristimulus Value model which is the adopted scalar basis of conventional colour difference modelling. Several important inferences are reported. Firstly it is shown that the constant-Hue loci of Chroma change are systematically curved on the x,y plane; secondly that the new meaningfully quantified ellipse semi-axes appear to explain the hitherto problematic orientation of the MacAdam ellipses; and thirdly that they correlate with the effect of the ellipse rotation term in the CIE DE2000 colour difference calculation. 19 The next section of the analysis concerns the cylindrical / polar graphical representation of the ellipse boundaries in the a*b* plane and what happens when they are instead plotted as orthogonal differences in the C* h0 plane. Significantly misleading artefacts are shown to be present when colour differences are modelled on the CIELAB a*b*plane; and the noted distortions of the CIE DE2000 difference scaling in the a*b*plane are in effect removed when the ellipses are plotted using Cartesian co-ordinates. The final section of the analysis provides evidence toward the possibility that a smooth continuous scalar model of the visual difference response may eventually be derived. This pilot study is presented as a sequential analysis and indicative development toward novel colour difference modelling methods; and in this context the CIE X,Y,Z co-ordinate definitions of the proto vectors already derived in the pilot study, specify stimuli that can probably be physically reproduced as an initial starting point in visual-difference magnitude estimation experiments. The specified proto-vectors, could then perhaps be turned into true vector descriptions of the visual difference response by iterative refinement referenced to such visual experiments. Overall, the presented techniques of proto-vector modelling and grey scale tracking appear to enable a potentially significant vector based alternative approach to colour difference modelling. In principle this alternative may then achieve a long sought goal, the derivation of a statistically verifiable smooth and continuous vector space that is a true scalar model of the visual difference response. 20 References 1. RS Berns 2000, Standard Observer systems, i Bill eye a d Saltz a s P i iples of Colo Technology (2000), 3rd Edition, R.S. Berns (Ed.), John Wiley & Sons, New York. 2. B Rigg, Colorimetry and the CIE system, in Colour physics for industry (1997) R. McDonald (1997), pp 63-96, Soc. Dyers & Colourists, Bradford England. 3. DP Oulton, Selected papers on Colorimetric Theory and Colour Modeling, PhD Thesis (2010), Manchester University. 4. G Sharma, W Wu and EN Dalal, CRA, 30 (2005), 21. 5. DH Krantz, Journal of Mathematical Psychology, 12 (1975), 283. 6. RW Pridmore, CRA, 35 (1975), 122. 7. WD Wright, Trans. Opt. Soc., 31 (1930), 201. 8. WA Thornton, CRA, 17 (1992), 79. 9. RW Pridmore, CRA, 34 (2009), 233. 10. P Van Overshee and B De Moor, Subspace Identification for Linear Systems: Theory, Implementation, Applications (1996), Kluwer Academic Dordrecht. 11. B Peeters and G Roeck, Mechanical Systems and Signal Processing, 13 (1999), 855. 12. DL MacAdam, JOSA, 33 (1943), 675. 13. G Wyszecki and WS Stiles, Color Science Concepts and Methods (2000), 2nd Ed. Page 256. 14. Ibid. fig 2(54.4.1) Page 308 15. Ibid. fig 4(5.4.1.) Page 311. 16. G Wyszecki and GH Fielder, JOSA, 61 (1971), 1135. 17. RW Pridmore, CRA, 35 (2011), 394. 18. DP Oulton, CRA, 34 (2009), 163. 19. DH Krantz, Journal of Mathematical Psychology, 12 (1975), 304. 20. The Pade Approximant, Journal of mathematical Analysis and Applications, 2 (1961), 21. 21. The Pade Approximant in Theoretical Physics Vol 71 (1971) Elsevier 22. K Witt, CRA, 24 (1999), 78.