Spectral locus interpolation with splines in optical instruments

The article deals with the problem of representation of the spectral locus curve on the CIE xy chromaticity diagram. The issue is that spectral locus is specified at discrete set of points corresponding to given set of wavelengths of monochromatic colors. Thus, the problem appears which way to calculate chromaticity coordinates for any other wavelength. Three different methods how to interpolate the spectral locus were considered. The method we proposed to build an interpolant curve is based on using of Bezier curves. This approach allows to calculate chromaticity coordinates for any arbitrary wavelength directly on the CIE xy chromaticity diagram. Obtained analytical expressions for the spectral locus curve will increase the evaluation accuracy of the luminous radiation dominant wavelength value which defines the hue and the saturation value of the emitting color and other things.


Introduction
In 1931 the International Commission on Illumination (CIE) established [1]  with a step ∆λ forms a point specified contour on the XY-plane, these points are nodes for interpolation procedure [2,3] that enables to obtain the spectral locus continuous curve for the monochromatic colors which defines the color locus area. The straight line between two endpoints on the XY-plane corresponding the wavelengths 380 nm and 780 nm is called the line of purple colors (LPC).
An interpolation of a set of points {x[i],y[i]} must provide [4,5] a possibility to obtain the interim values of the (x,y)-chromaticity coordinates of monochromatic colors with an arbitrary wavelength step which is not equal to 5 nm according to the CIE standards. Combining those values a continuous and «smooth» spectral locus graph can be obtained. A task to interpolate the tabulated spectral locus is

The solution of the interpolation problem for the spectral locus
To solve the interpolation problem using the first and the second approaches mentioned above the standard method of spline interpolation can be used, cubic spline interpolation technique in patricular [8]. Main disadvantage of the cubic spline technique is that resulting interpolant curve usually has undesirable «oscillations». There is an alternative and original approach to interpolate the spectral locus curve represented as a trajectory which is drawn on the XY-plane by chromaticity coordinates point when it moves according to wavelength change λ[i] from 380 nm to 780 nm. To interpolate spectral locus directly on XY-plane, Bezier interpolation [9,10] technique seems to be useful tool. Locally specified Bezier curves may be calculated and then jointed with each other in the interpolation nodes to create a joint Bezier spline for spectral locus curve. For that purpose quadratic and cubic Bezier curves are used.
An interpolating function which is used to build locus curve between wavelengths from λ[i] to where and P2[i] on the XY-plane which define the interpolating function properties in each interval; An interpolating function which is used to build locus curve between wavelengths from λ[i] to λ[i+1] on the basis of the cubic Bezier curve must be represented as: where the function angle(x,y) computes the angle(in radians) of the directing vector which starts in the XY-plane coordinate origin and terminates in the point defined with the coordinates (x,y), In the i-th interpolation interval the starting tangent with the inclination angle α[i] to the X axis must be made through the point (x[i],y[i]) of the interpolation interval beginning and the terminating tangent with the inclination angle α[i+1] to the X axis must be made through the point The selection method of the Bezier curve type and its construction method are shown in figure 2 using as an example two parts of a spectral locus graph.
For the spectral locus ending nodal points in the expressions (3) and (4) When the control points P1[i] and P2[i] are found on the different sides from the supporting line which is drawn through the nodal points (x[i],y[i]) and (x[i+1],y[i+1]) the preliminary control points must be finally set in the positions (5), (6) and in this interval a cubic Bezier curve must be drawn. In other case, when preliminarly calculated control points P1[i] and P2[i] are found on the same side (both in the upper part, in the lower part, in the left part or in the right part) from the supporting line, then calculating of a cubic Bezier curve become pointless. In this case one control point Q[i] must be assigned instead of two preliminarily calculated. This new point is the point where the starting and terminating tangents are crossed. As the result, a quadratic Bezier curve must be build.
The control point coordinates Q[i] which are used to interpolate spectral locus by a quadratic curves (2) must be calculated with the following formulae: The control points position (5), (6) and (7), (8)  combine the Bezier curves in the nodal points to ensure the continuity of the interpolating function first derivative. The decision function n[i] which enables a preferred selection of the quadratic (2) or the cubic (3) interpolated curve in the i-th interpolation interval may be represented as: If n[i]>0 a quadratic Bezier curve must be selected and if n[i]≤0a cubic Bezier curve must be selected. As a result an interpolating function system which defines the spectral locus on the XY-plane must be represented as:  In figure 2 the nodal points are connected with straight solid lines and that connection method gives a possibility to evaluate visually the proposed interpolation method efficiency in comparison with the method of linear interpolation on XY-plane. The spectral locus interpolating curves obtained with three different methods are shown in figure 3 at a larger scale in order to reveal some features of each method of interpolation. а) b) Figure 3. The spectral locus interpolation results: a) in the band 380 nm -420 nm, b) in the band 510 nm -525 nm (the dotted curve is obtained with the method of CIE XYZ tristimulus values interpolation by cubic splines, the pointed curvethe chromaticity coordinates data interpolation by cubic splines, the solid curvethe proposed interpolation method on XY-plane using Bezier splines).