Color reproducing device for reproducing matched colors and an outputting device for outputting information for reproducing a color of a coated surface

United States Patent

United States Patent	5,590,251
Takagi	December 31, 1996

Abstract

By inputting a predetermined number of first data to a color reproducing device, a predetermined number of reproduced colors are respectively measured, and a plurality of relationships of correspondence between inputted first data and measured second data are determined. On the basis of the plurality of relationships of correspondence between the first data and the second data, relationships of interpolated correspondence, which express relationships between second data other than the measured second data and first data corresponding to the second data other than the measured second data, are estimated. After second data which is the same as or closest to a color to be reproduced is selected, first data corresponding to selected second data is selected on the basis of the relationships of correspondence and the relationships of interpolated correspondence. Selected first data is then inputted to the color reproducing device, and a color to be reproduced is reproduced.

Inventors:	Takagi; Atsushi (Toyota, JP)
Assignee:	Toyota Jidosha Kabushiki Kaisha (Toyota, JP)
Appl. No.:	08/286,350
Filed:	August 5, 1994

Foreign Application Priority Data


Aug 06, 1993 [JP]			5-196082
Nov 30, 1993 [JP]			5-300386

Current U.S. Class: 345/604

Field of Search: 395/130-131,129,127,126,9

References Cited

U.S. Patent Documents


4457718	July 1984	Lerner
4477833	October 1984	Clark et al.
4605687	August 1986	Panush
4615940	October 1986	Panush et al.
4959711	September 1990	Hung et al.
4971841	November 1990	Panush et al.
5033857	July 1991	Kubota et al.
5045933	September 1991	Tatsumi
5056234	November 1991	Hung et al.
5151751	September 1992	Nakajima et al.
5185661	February 1993	Ng
5231472	July 1993	Marcus et al.
5231504	July 1993	Magee
5255350	October 1993	Hermann et al.
5296923	March 1994	Hung
5315380	May 1994	Ingraham et al.
5317425	May 1994	Spence et al.
5333069	July 1994	Spence
5363318	November 1994	McCauley
5386496	January 1995	Arai et al.
5412491	May 1995	Bachar
5416890	May 1995	Beretta

Foreign Patent Documents


62-149760	Jul., 1987	JP
64-1151	Jan., 1989	JP
488584	Mar., 1992	JP
5322657	Dec., 1993	JP
5324850	Dec., 1993	JP
5322655	Dec., 1993	JP
WO9120047	Dec., 1991	WO

Other References

R A. Prosser et al., "Color Matching Spanning the Visible and Near-Infra: Use of the Cubic Spline Function in Interpolation", Technical Report Natick/TR-90/006, Nov., 1989. .
E. Allen, "Basic Equations Used in Computer Color Matching", J. Opt. Soc. Am., vol. 56, No. 9, pp. 1256-1259, 1966. .
E. Allen, "Basic Equations Used in Computer Color Matching, II. Tristimulus Match, Two-Constant Theory", J. Opt. Soc. Am., vol. 64, No. 7, pp. 991-993, 1974. .
"Image Processing for Color Reproduction", Imaging Part 1, Supplementary vol. Kodera, 1988 pp. 44-55 (Shashin Kogyo and compiled by the Electrophotography Society. .
A. Takagi et al, "Accurate Rendering Technique Based on Colorimetric Conception:", Computer Graphics, vol. 24 No. 4, 1990, pp. 263-272. .
T. Oshima et al, "System for Color Design of a Car", Eurographics '92, vol. 11 No. 3, 1992 pp. C381-C390. .
A. Tagaki et al, "Rendering Technique Based on Colorimetric Conception", Computer Graphics, vol. 24, No. 4, 1990 pp. 263-272. .
Foley, James D., et al, Computer Graphics: Principles and Practices, Addison-Wesley Publishing Co., 2nd ed. pp. 574-589 and 754-773, (1990). .
Haskell, Barry H., "Interpolative, Predictive and Pyramid Transform Coding of Color Images", ICASSP '88: Acoustics, Speech and Signal Processing Conference, 1988, pp. 785-787. .
Meyer, Gary W., et al, "Color-Defective Vision--Color-Defective Vision and Computer Graphic Displays", IEEE Computer Graphics and Applications Magazine, Sep. 1988, vol. 8, Iss. 5, pp. 28-40. .
Abe, Satoshi et al, "A Neural Network Approach for RGB to YMCK Color Conversion", Tecon '94-1994 IEEE Region 10 Conference on Frontiers of . . . , 1994, pp. 6-9. .
Kleiman, Larry, "XYZ Technology", Instruments and Measurement Technology Conference, 1992, pp. 84-89. .
Trussel, H. J., "DSP Solutions Run the Gamut for Color Systems", IEEE ASSP Magazine, Apr. 1993, vol. 10, Iss. 2, pp. 8-23. .
Lee, Hsien-Che, et al, "Modeling Light Reflection For Computer Color Vision", IEEE transactions on Pattern Analysis and Machine Intelligence, Apr. 1990, vol. 12, Iss. 4, pp. 402-409. .
Wolff, Lawrence B., "Diffuse Reflection", Computer Vision and Pattern Recognition (CVPR), 1992, pp. 472-478. .
Yaohua, Pang et al, "Development of Anti-Glare Coating for High Resolution Color Display Tubes", Electronic Manufacturing, 1991 11th IEEE/CHMT International Symposium, 1991, pp. 371-373. .
Onishi, Hiroshi et al, "Thermal Dissolution Ink Transfer for Full-Color Printing", IEEE Transactions on Electron Devices, Jan. 1993, pp. 69-74. .
Dias, Maria Lurdes, "Ray Tracing Interference Color", IEEE Computer Graphics and Applications Magazine, Mar. 1991, pp. 54-60. .
Sato, Yoichi et al, "Temporal-Color Space Analysis of Reflection", Computer Vision and Pattern Recognition, 1993, (CVPR), pp. 570-576. .
Ikeda, Hiroaki et al, "A Study on Colometric Errors Caused by Quantizing Color Information", IEEE Transactions on Instrumentation and Measurement, 1992, pp. 845-849. .
Healy, Glen et al, "A Color Metric for Computer Vision", Computer Vision and Pattern Recognition, (CVPR), 1988, pp. 10-17. .
Robertson, Philip K., "Perceptual Color Spaces", IEEE Computer Graphics and Applications Magazine, 1988, pp. 50-64..

Primary Examiner: Herndon; Heather R.
Assistant Examiner: Buchel; Rudolph
Attorney, Agent or Firm: Cushman, Darby & Cushman

Claims

What is claimed is:

1. A color reproducing device to which first data expressing a color expressed in a first colorimetric system is inputted, and which reproduces a color corresponding to the inputted first data and to be expressed in a second colorimetric system different than the first colorimetric system, comprising:

a measuring device measuring each of a predetermined number of colors reproduced by inputting a predetermined number of first data to said color reproducing device, and outputting a predetermined number of second data expressing a color expressed in the second colorimetric system;

estimating means for estimating relationships of interpolated correspondence expressing relationships between second data other than the measured second data and first data corresponding to the second data other than the measured second data, on the basis of a plurality of relationships of correspondence between the first data inputted to said color reproducing device and the measured second data; and

selecting means for, after second data which is the same as or closest to data of a color to be reproduced is selected on the basis of the relationships of correspondence and the relationships of interpolated correspondence, selecting first data corresponding to the selected second data on the basis of the relationships of correspondence and the relationships of interpolated correspondence,

wherein the selected first data is inputted to said color reproducing device, and said color reproducing device reproduces the color to be reproduced.

2. A color reproducing device according to claim 1, wherein the plurality of relationships of correspondence are nonlinear relationships.

3. A color reproducing device according to claim 1, wherein the first colorimetric system is an RGB colorimetric system, the second colorimetric system is an XYZ colorimetric system, the first data is data relating to at least one of a color material and a bright material, and the second data is data relating to tristimulus values.

4. An outputting device for outputting information for reproducing a color of a coated surface which is formed with one or a plurality of layers on an object to be coated and in which each of the layers is formed of at least one component material, comprising:

a color material mixing device to which characteristic values expressing respective quantities of all of the component materials forming the coated surface are inputted, and which generates a paint for forming the coated surface on the basis of the inputted characteristic values;

a measuring device measuring physical quantities expressing one of a spectral reflectance distribution and tristimulus values of each of a predetermined number of coated surfaces formed by coating on the object to be coated a paint which is generated by inputting to said color material mixing device a predetermined number of characteristic values in which at least one of the component material quantities is respectively different;

estimating means for estimating relationships of interpolated correspondence expressing relationships between physical quantities other than the measured physical quantities and characteristic values corresponding to the physical quantities other than the measured physical quantities, on the basis of a plurality of relationships of correspondence between the characteristic values inputted to said color material mixing device and the measured physical quantities;

selecting means for, after physical quantities relating to physical quantities of a paint color to be reproduced have been selected on the basis of the relationships of correspondence and the relationships of interpolated correspondence, selecting the characteristic values corresponding to the selected physical quantities on the basis of the relationships of correspondence and the relationships of interpolated correspondence; and

outputting means for outputting the selected characteristic values as information for reproducing the color of the coated surface.

5. An outputting device according to claim 4, wherein the physical quantities relating to physical quantities of the paint color to be reproduced are physical quantities which are the same as or closest to physical quantities of the paint color to be reproduced.

6. An outputting device according to claim 4, wherein said color material mixing device generates the paint on the basis of the information outputted from said outputting means.

7. An outputting device according to claim 4, further comprising:

display means for reproducing and displaying the color of the coated surface on the basis of the information outputted from said outputting means.

8. An outputting device according to claim 4, wherein said selecting means determines coordinate values on a coordinate expressing colors of a predetermined colorimetric system with respect to each of the tristimulus values in the relationships of correspondence and relationships of interpolated correspondence, and sets a plurality of coordinate values among the determined coordinate values as reference color coordinate values for expressing reference colors, and consecutively selects, as the characteristic values corresponding to the physical quantities, the characteristic values corresponding to the coordinate values between the tristimulus values of the paint color to be reproduced and at least one of the reference color coordinate values.

9. An outputting device according to claim 8, wherein said color material mixing device generates the paint on the basis of the information outputted from said outputting means.

10. An outputting device according to claim 8, further comprising:

display means for reproducing and displaying the color of the coated surface on the basis of the information outputted from said outputting means.

11. An outputting device according to claim 4, wherein said selecting means determines varied-angle characteristics of the coated surface expressing flip-flop relationships between a varied angle which is a light-receiving angle varied during reception of light reflected from the coated surface and brightness at the varied angle, on the basis of the spectral reflectance distributions in the relationships of correspondence and in the relationships of interpolated correspondence, and selects the characteristic values corresponding to the determined varied-angle characteristics as the characteristic values corresponding to the physical quantities.

12. An outputting device according to claim 11, wherein said color material mixing device generates the paint on the basis of the information outputted from said outputting means.

13. An outputting device according to claim 11, further comprising:

display means for reproducing and displaying the color of the coated surface on the basis of the information outputted from said outputting means.

14. An outputting device according to claim 4, wherein said selecting means determines a particle-size distribution of each of the component materials for each of the characteristic values in the relationships of correspondence and the relationships of interpolated correspondence, and determines depth indexes expressing the depth of paint colors on the basis of the determined particle-size distributions and the spectral reflectance distributions in the relationships of correspondence and in the relationships of interpolated correspondence, and selects the characteristic values corresponding to the determined depth indexes as characteristic values corresponding to the physical quantities.

15. An outputting device according to claim 14, wherein said color material mixing device generates the paint on the basis of information outputted from said outputting means.

16. An outputting device according to claim 14, further comprising:

display means for reproducing and displaying the color of the coated surface on the basis of the information outputted from said outputting means.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method of color reproduction, a method of reproducing a coating color, and a method of selecting a coating color. More particularly, the present invention concerns a method of color reproduction by determining in advance correlations between input values and an outputted color for reproducing a specific color in an apparatus for color printing, color display, or the like, as well as a method of reproducing a coating color and a method of selecting a coating color so as to reproduce a coating color of a coated surface intended by a designer or the like when obtaining a coated surface by coating the surface with a paint or the like or when displaying a coated surface on a color CRT.

2. Description of the Related Art

As is known, the color of an object surface, a color original image, or the like can be specified in a standardized manner by determining the position where chromaticity coordinates, which are expressed by using tristimulus values X, Y, and Z of the color in the CIE (International Commission on Illumination) standard XYZ colorimetric system, are located in a chromaticity diagram. Namely, if the spectral distribution I(.lambda.) of light reflected from or transmitted through the object or the like can be measured, the tristimulus values X, Y, and Z can be determined from the following Formula (1): ##EQU1## where k=100.multidot..intg.{I(.lambda.)y(.lambda.)d.lambda.},

x(.lambda.), y(.lambda.), z(.lambda.): CIE color matching functions,

.lambda.: wavelength

This value Y shows the brightness of the light having I(.lambda.), and the color can be specified by plotting points on a chromaticity diagram of an orthogonal coordinate system, in which values of x and y obtained from the following Formulae (2) are set as chromaticity coordinates, and x is plotted as the abscissa and y as the ordinate in a conventionally known manner (all the colors are included within a slanted bell shape):

Recently, there has been a need for reproduction of colors which can be specified in a standardized manner as described above. For instance, in the field of design, there has been a demand for color-reproducing techniques which make it possible to faithfully reproduce necessary colors for the purpose of evaluation of color design. As examples of apparatus which require color reproduction of color original images, there are display units for displaying color-reproduced images that are color-reproduced on the basis of color data on color original images, as well as color copying apparatus (color hard-copying apparatus) for copying color-reproduced images that are color-reproduced on the basis of color data on color original images.

As a method of color reproduction during color hard copying in the color hard-copy apparatus, a method of color reproduction is known for faithfully reproducing a necessary color by using image processing (Kodera: "Image Processing for Color Reproduction" in Supplementary Volume "Imaging Part 1" of the Shashin Kogyo (Photo Industry) published by Shashin Kogyo and compiled by the Electrophotography Society).

However, since the color can be specified by the mixing of pigments of YMC colors, RGB signals for the display, or the like, the RGB colorimetric system based on the three primary colors (reference stimuli) is in most cases used as the colorimetric system for specifying actual colors. Hence, conversion of color data in the RGB colorimetric system to and from color data in the XYZ colorimetric system is required.

By taking the aforementioned color copying apparatus as an example, color specification in this color copying apparatus is generally effected by mixing predetermined color materials (R material, G material, and B material) to form a reproduced image (a copy of the original) and output the same. Because respective colors of this reproduced image can be specified by data in the XYZ colorimetric system, i.e., color data using the tristimulus values X, Y, and Z, on the basis of reproducing conditions such as a mixing ratio of the color materials and values of measurement by a spectrophotometer or the like, each of these colors may be considered as a function f for computing the tristimulus values X, Y, and Z using data (r, g, and b) of the color materials as parameters, as shown in the following Formula (3):

Since the density based on these color materials can generally be changed in 256 gradations for r, g, and b, respectively, it is possible to reproduce a color original image in 256.sup.3 (=16,777,216) combinations.

Here, since the color data when the color original image is read can be converted to color data in the XYZ colorimetric system, as described above, it is possible to compute the tristimulus values X, Y, and Z of the colors to be reproduced as a reproduced image. Accordingly, if the data (r, g, and b) of the color material, in which the tristimulus values X, Y, and Z, i.e., output values of the function f, are equal to the tristimulus values X, Y, and Z of the color data on the color original image, are determined, color reproduction is possible with high accuracy. For this reason, it is possible to form a reproduced image with a high level of color reproducibility by determining an inverse function f.sup.-1 of Formula (3) in advance and then by using the data (r, g, and b) on the color material determined by this inverse function f.sup.-1.

However, in the aforementioned color copying apparatus, since color formation is generally based on the subtractive mixture of color stimuli, Formula (3) above becomes nonlinear. For this reason, it is difficult to determine the inverse function f.sup.-1. To overcome this problem, it is conceivable to determine and store in advance all of the tristimulus values X, Y, and Z and the data on the color materials with respect to the 256.sup.3 combinations mentioned above and to extract data on the color materials which are in a relationship of the inverse function f.sup.-1 during reproduction processing. However, the amount of computation for processing in advance is enormously large, and a storage area for storing the relationships determined also becomes enormously large, so that this scheme is not practically feasible.

In addition, an object surface, such as the body of a vehicle, is formed by a coated surface having a coating color obtained by applying a paint or the like. To obtain a coated surface of a desired coating color intended by a user, a designer, and the like, a paint or the like obtained by mixing a plurality of pigments and the like by using a color sample as a reference is applied to the object.

A method is conventionally known in which, with respect to an object surface having uniform optical properties, the color of the object is reproduced and displayed three-dimensionally and realistically with accuracy with the semblance of the actual object by computing coloring on the basis of a ray tracing method using the reflectance of the object surface, such as the spectral reflectance factor (A. Takagi et al. "Computer Graphics," Vol. 24, No. 4, 1990, and the like). In this method, color specification values (tristimulus values) of the CIE standard XYZ colorimetric system are first determined on the basis of a spectral reflectance factor and the like of the object surface. These tristimulus values are then converted to color specification values peculiar to the colorimetric system through a linear combination transformation, are subjected to .gamma. correction, and are converted to RGB gradients, thereby reproducing the object color and displaying an image. According to this method, if the reflectance of the object can be specified, it is possible to reproduce and display the object color. At the same time, the reflectance of the object corresponding to the displayed color can be specified by processing in the reverse order, and virtual color components for obtaining the displayed color can be determined. It is possible to obtain a desired coating color, if the object is coated with a paint or the like obtained by mixing a plurality of pigments and the like in quantities corresponding to the quantities of these color components.

However, the setting of a ratio of mixing or compounding pigments for obtaining the desired coating color requires the trained skill of a technician, and is very low in productivity. In addition, it does not necessarily follow that the coating color on the finished coated surface can always be reproduced to the coating color intended by the user, the designer, and the like owing to differences and variations in the type of component materials such as pigments.

To overcome this problem, computer color matching (hereafter referred to as CCM) has been widely used in which compounding involving the setting of a mixing ratio of pigments, which requires trained skill, is determined by computation by a computer in compounding basic color materials (coloring agents such as pigments) in accordance with the Kubelka-Munk's theory. In this CCM, the mixing ratio and the like of a plurality of pigments whose reflectances are known are determined by computation by a computer, such that the reflectance will be equal to the reflectance of a color sample measured by a spectrophotometer or the like. In another case, the mixing ratio and the like of a plurality of pigments whose tristimulus values are known are determined by computation by a computer, such that the tristimulus values will be equal to the tristimulus values of the color sample. Thus, a method is known for determining the mixing ratio and the like of coloring agents by using CCM so as to reproduce an intended coating color (Japanese Patent Application Laid-Open No. 149760/1987).

With the conventional methods of reproducing a coating color using CCM, however, since compounding is determined in accordance with the Kubelka-Munk's theory, it is impossible to effect compounding by mixing substances whose surface reflectances do not conform to the Kubelka-Munk's theory. In addition, it is impossible to specify a coating color which includes bright materials such a metallic paint and mica as its component materials.

In addition, although the above-described CCM is effective in obtaining a coating color which coincides with a color sample or the like, reflectance values and tristimulus values for specifying the coating color are not subjective. Therefore, it is difficult for the above-described CCM to reflect trends of sensuous coating colors, such as reddish and glossy colors, which are used by designers and the like as specification for obtaining desired coating colors from already existing coating colors.

SUMMARY OF THE INVENTION

In view of the above-described circumstances, it is a primary object of the present invention to provide a method of color reproduction which makes it possible to reproduce a necessary color only by simple processing by using a small amount of known data without using a huge amount of data or effecting massive data processing.

A second object of the present invention is to provide a method of reproducing a coating color which makes it possible to reproduce a coating color intended by a user or a designer who does not have expert knowledge on such as color science and reflection properties of objects, irrespective of the composition and types of paints, as well as a method of selecting a coating color which makes it possible to select an optimum coating color intended by such as the user or designer.

To attain the primary object, in accordance with a first aspect of the present invention, there is provided a method of color reproduction comprising the steps of: on the basis of a plurality of predetermined relationships of correspondence between first values expressed in a predetermined colorimetric system for outputting a predetermined number of colors and second values expressed in a colorimetric system different from the predetermined colorimetric system for specifying a color to be reproduced, estimating a plurality of the second values other than predetermined ones of the second values, and estimating the first values corresponding to the estimated second values; selecting the second values which are identical to or closest to the second values of an arbitrary color from the estimated second values as well as the first values corresponding to the identical or closest second values when the arbitrary color is to be reproduced; and reproducing the color on the basis of the selected first values.

In this method of color reproduction, the aforementioned plurality of relationships of correspondence can be made nonlinear relationships.

In addition, in accordance with a second aspect of the present invention, there is provided a method of reproducing a coating color, comprising the steps of: with respect to a predetermined coating color on a coated surface which is formed with one or a plurality of layers on an object to be coated and in which each of the layers is formed of at least one component material, determining in advance a plurality of relationships of corespondence between characteristic values constituted by amounts of respective ones of all the component materials constituting the coated surface and a spectral reflectance distribution of the coated surface based on the characteristic values; estimating on the basis of the plurality of relationships of correspondence a plurality of relationships of interpolated correspondence expressing correspondence between characteristic values and spectral reflectance distributions of coating colors in which a quantity of at least one component material of all the component materials that are determined on the basis of the relationships of correspondence is varied; selecting a spectral reflectance distribution which is in the relationships of interpolated correspondence corresponding to a coating color to be reproduced when a coating color other than the predetermined coating color is reproduced; and determining quantities of all the component materials by characteristic values that are determined on the basis of the relationships of interpolated correspondence with respect to the selected spectral reflectance distribution, and reproducing the coating color.

In accordance with a third aspect of the present invention, there is provided a method of selecting a coating color, comprising the steps of: with respect to a predetermined coating color on a coated surface which is formed with one or a plurality of layers on an object to be coated and in which each of the layers is formed of at least one component material, determining in advance a plurality of relationships of corespondence between characteristic values constituted by amounts of respective ones of all the component materials constituting the coated surface and a spectral reflectance distribution of the coated surface based on the characteristic values, and determining in advance tristimulus values based on a spectral reflectance distribution of the coated surface based on the characteristic values; estimating on the basis of the plurality of relationships of correspondence a plurality of relationships of interpolated correspondence expressing correspondence between characteristic values and spectral reflectance distributions of coating colors in which a quantity of at least one component material of all the component materials that are determined on the basis of the relationships of correspondence is varied, and determining the tristimulus values based on the spectral reflectance distribution of the coated surface based on characteristic values of the estimated relationships of interpolated correspondence; determining coordinate values on coordinates of a predetermined colorimetric system with respect to each of the tristimulus values and interpolated tristimulus values, and setting a plurality of coordinate values among the determined coordinate values as reference coordinate values for expressing reference colors; and when a tendency of one of the reference colors is to be reflected on an instructed color instructed for reproducing the coating color, selecting the coating color by consecutively selecting coordinate values in a direction from coordinate values specifying the instructed color to the reference coordinate values, starting with proximate coordinate values.

In accordance with a fourth aspect of the present invention, there is provided a method of selecting a coating color, comprising the steps of: with respect to a predetermined coating color on a coated surface which is formed with one or a plurality of layers on an object to be coated and in which each of the layers is formed of at least one component material, determining in advance a plurality of relationships of corespondence between characteristic values constituted by amounts of respective ones of all the component materials constituting the coated surface and a spectral reflectance distribution of the coated surface based on the characteristic values; estimating on the basis of the plurality of relationships of correspondence a plurality of relationships of interpolated correspondence expressing correspondence between characteristic values and spectral reflectance distributions of coating colors in which a quantity of at least one component material of all the component materials that are determined on the basis of the relationships of correspondence is varied; determining varied-angle characteristics of the coated surface expressing flip-flop relationships between a varied angle when a light-receiving angle is varied during reception of light reflected from the coated surface and brightness at the varied angle, on the basis of the spectral reflectance distributions in the relationships of interpolated correspondence or the spectral reflectance distributions in the relationships of correspondence; and selecting the coated color by selecting the varied-angle characteristic of the coating color to be reproduced from the determined varied-angle characteristics.

In accordance with a fifth aspect of the present invention, there is provided a method of selecting a coating color, comprising the steps of: with respect to a predetermined coating color on a coated surface which is formed with one or a plurality of layers on an object to be coated and in which each of the layers is formed of at least one component material, determining in advance a plurality of relationships of corespondence between characteristic values constituted by amounts of respective ones of all the component materials constituting the coated surface and a spectral reflectance distribution of the coated surface based on the characteristic values; estimating on the basis of the plurality of relationships of correspondence a plurality of relationships of interpolated correspondence expressing correspondence between characteristic values and spectral reflectance distributions of coating colors in which a quantity of at least one component material of all the component materials that are determined on the basis of the relationships of correspondence is varied; determining a particle-size distribution of each of the component materials for each of the characteristic values in the relationships of correspondence and the characteristic values in the relationships of interpolated correspondence, and determining depth indexes specifying the depth of coating colors on the basis of the spectral reflectance distributions in the relationships of interpolated correspondence or the spectral reflectance distributions in the relationships of interpolated correspondence and the determined particle-size distribution; and selecting the coating color by selecting from the selected depth indexes.

In accordance with a sixth aspect of the present invention, there is provided a method of reproducing a coating color, comprising the steps of: estimating on the basis of a plurality of relationships of correspondence determined in advance a relationship of correspondence between a spectral reflectance distribution and a characteristic value of a coating color selected by the method of selecting a coating color in accordance with at least one of the third, fourth, and fifth aspects of the invention; and reproducing the coating color by determining a quantity of each of all the component materials by characteristic values which are determined from the estimated relationship of correspondence.

In accordance with the first aspect of the invention, a plurality of predetermined relationships of correspondence are determined in advance between first values expressed in a predetermined colorimetric system for outputting a predetermined number of colors and second values expressed in a colorimetric system different from the predetermined colorimetric system for specifying a color to be reproduced. These relationships of correspondence may be nonlinear relationships. For example, the predetermined colorimetric system includes an RGB colorimetric system, and the difference colorimetric system includes an XYZ colorimetric system. The first values include data on color materials, and the second values include tristimulus values. A plurality of the second values other than predetermined ones of the second values are estimated by interpolation or the like on the basis of the plurality of relationships of correspondence, and the first values corresponding to the estimated second values are estimated. Accordingly, it is possible to determine a plurality of relationships of correspondence including a desired relationship of correspondence from the predetermined plurality of relationships of correspondence. When an arbitrary color is to be reproduced, the second values which are identical to or closest to the second values of the arbitrary color are estimated from the estimated second values, and the first values corresponding to the identical or closest second values are selected. If the color is reproduced on the basis of the selected first values, it is possible to reproduce and output a color close to the desired color or that color itself.

In accordance with the second aspect of the invention, a plurality of relationships of correspondence are determined in advance with respect to a predetermined coating color on a coated surface. This coated surface is formed with one or a plurality of layers on an object to be coated, and each of its layers is formed of at least one component material. A plurality of relationships of correspondence between characteristic values constituted by amounts of respective ones of all the component materials constituting the coated surface and a spectral reflectance distribution of the coated surface based on the characteristic values are determined in advance. These relationships of correspondence can be determined by, for instance, making use of sample coated plates whose spectral reflectances, pigments and the like are already known. A plurality of relationships of interpolated correspondence, which express correspondence between characteristic values and spectral reflectance distributions of coating colors in which a quantity of at least one component material of all the component materials that are determined on the basis of the relationships of correspondence is varied, are estimated by interpolation or the like on the basis of the plurality of relationships of correspondence. Accordingly, the relationships of interpolated correspondence between characteristic values and spectral reflectance distributions can be determined with respect to a desired coating color on the basis of a plurality of predetermined relationships of correspondence. Here, when a coating color other than the predetermined coating color is to be reproduced, a spectral reflectance distribution which is in the relationships of interpolated correspondence corresponding to the coating color to be reproduced is selected. If quantities of all the component materials, including such as color materials and bright materials, are determined by characteristic values that are determined on the basis of the relationships of interpolated correspondence with respect to this selected spectral reflectance distribution, it is possible to reproduce the composition of the coated surface and a desired coating color on a CRT or by means of a color-material mixing apparatus or the like.

In the method of selecting a coating color in accordance with the third aspect of the invention, a plurality of relationships of corespondence between characteristic values constituted by amounts of respective ones of all the component materials constituting the coated surface in the second aspect of the invention and a spectral reflectance distribution of the coated surface based on the characteristic values are determined in advance. At the same time, tristimulus values based on a spectral reflectance distribution of the coated surface based on the characteristic values are determined in advance. These tristimulus values include values represented by a colorimetric system such as the XYZ colorimetric system, and can be represented by coordinate values on chromaticity coordinates. In addition, the Munsell color system can be also used. on the basis of the plurality of relationships of correspondence, a plurality of relationships of interpolated correspondence, described in relation to the second aspect of the invention, are estimated. At the same time, the tristimulus values based on the spectral reflectance distribution of the coated surface based on characteristic values of the estimated relationships of interpolated correspondence are determined. Coordinate values with respect to these tristimulus values and interpolated tristimulus values are determined on coordinates of a predetermined colorimetric system, such as the XYZ colorimetric system, and a plurality of coordinate values among the determined coordinate values are set as reference coordinate values for expressing reference colors. As these reference colors, it is preferable to set basic colors that are used in coating or printing, such as red, blue, yellow, green, magenta, cyan, white, and black. When a tendency of one of the reference colors is to be reflected on an instructed color instructed for reproducing the coating color, if the coating color is selected by consecutively selecting coordinate values in a direction from coordinate values specifying the instructed color to the reference coordinate values, starting with proximate coordinate values, then the coating colors corresponding to the selected coordinate values gradually come to reflect the tendency of the reference color. Accordingly, if the quantities of all the component materials including such as color materials and bright materials are determined on the basis of the characteristic values of the coating color corresponding to the selected coordinate values, it is possible to reproduce a desired coating color on which the tendency of the reference color is reflected.

Here, there are cases where a sensuous flip-flop texture, such as a modulated texture of light and darkness, is included among the coating colors desired by the designer or the like. Accordingly, in the method of selecting a coating color in accordance with the fourth aspect of the invention, varied-angle characteristics of the coated surface expressing flip-flop relationships between a varied angle when a light-receiving angle is varied during reception of light reflected from the coated surface and brightness at the varied angle, are determined on the basis of the spectral reflectance distributions in the relationships of interpolated correspondence or the spectral reflectance distributions in the relationships of correspondence. Since the sensuous flip-flop texture can be expressed by this varied-angle characteristic, if the varied-angle characteristic of the coating color to be reproduced is selected from the determined varied-angle characteristics, it is possible to select a coating color on which the flip-flop texture is reflected. Accordingly, if the quantities of all the component materials, including such as color materials and bright materials, are determined on the basis of the characteristic values of the coating color corresponding to the selected varied-angle characteristic, it is possible to reproduce a coating color on which the sensuous flip-flop texture desired by the designer or the like is reflected.

In addition, sensuous instructions such as "a color having a texture of depth" are also included among the coating colors desired by the designer or the like. Accordingly, in the method of selecting a coating color in accordance with the fifth aspect of the invention, a particle-size distribution of each of the component materials for each of characteristic values in the relationships of correspondence determined in advance and characteristic values in the relationships of interpolated correspondence. Then, depth indexes specifying the depth of coating colors are determined on the basis of the spectral reflectance distributions in the relationships of interpolated correspondence or the spectral reflectance distributions in the relationships of interpolated correspondence and the determined particle-size distribution. Accordingly, sensuous depths corresponding to the selected depth indexes can be expressed as amounts, and if the plurality of depth indexes thus determined are selected, it is possible to select a coating color exhibiting a desired depth. Hence, if the quantities of all the component materials, including such as color materials and bright materials, are determined on the basis of the characteristic values of the coating color corresponding to the selected depth index, it is possible to reproduce a coating color having a desired depth desired by the designer or the like.

In addition, in the sixth aspect of the invention, at least one of a coating color on which the tendency of a reference color is reflected, a coating color having a varied-angle characteristic expressing a flip-flop relation, and a coating color having a desired texture of depth is selected. Then, a relationship of correspondence between a spectral reflectance distribution and a characteristic value of this selected coating color selected is estimated on the basis of a plurality of relationships of correspondence determined in advance. Accordingly, even in a case where coating colors which are desired by the designer or the like and are expressed sensuously are combined, if the quantity of each of all the component materials, such as color materials and bright materials, are determined by characteristic values which are determined from the estimated relationship of correspondence, it is possible to faithfully reproduce the desired, sensuously expressed coating color.

As described above, in accordance with the first aspect of the invention, it is possible to determine a multiplicity of relationships from a small number of relationships determined in advance by a simple algorithm. As a result, there is an advantage in that input values for obtaining required values can be obtained easily.

In accordance with the second aspect of the invention, it is possible to determine the characteristic values of a surface coated with a coating color constituted by a plurality of component materials including color materials and bright materials. Therefore, there is an advantage in that even in the case of a coated surface containing bright materials, such as metal pearl mica, which do not conform to the Kubelka-Munk's theory, it is possible to accurately reproduce a desired color as a coating color.

In accordance with the third aspect of the invention, colors midway in a direction from an instructed color to a reference color can be selected consecutively. Hence, there is an advantage in that, even if a coating color tinged with a tone is instructed by the designer or the like, such as a more reddish color, it is readily possible to select a coating color matching the sense of the designer or the like. In addition, since the characteristic values of a coated surface of a coating color selected as a coating color matching the sense of the designer or the like can be determined, there is an advantage in that a desired coating color tinged with a tone can be reproduced accurately.

In accordance with the fourth aspect of the invention, since it is possible to determine and select varied-angle characteristics of a coated surface expressing the relationship between the varied angle, which allows the sensuous flip-flop texture to be expressed, and the brightness at the varied angle, there is an advantage in that it is possible to select a coating color on which a modulated texture of light and shade desired by the designer or the like is reflected. In addition, since it is possible to determine the characteristic values of a coated surface of the coated color selected as a coating color matching the sense of flip-flop texture of the designer or the like, there is an additional advantage in that it is possible to accurately reproduce the coating color incorporating the desired flip-flop texture.

In accordance with the fifth aspect of the invention, since the depth indexes expressing the depth of coating colors are determined on the basis of the particle-size distributions of the component materials for each characteristic value, there is an advantage in that it is possible to select a coating color having a desired texture of depth, which is sensuously expressed as depth, as the coating color desired by the designer or the like. Moreover, since a coated surface can be formed on the basis of the characteristic values of the coating color with respect to the depth index, there is an advantage in that a coating color presenting a sensuous texture of depth desired by the designer or the like can be reproduced.

In accordance with the sixth aspect of the invention, since relationships of correspondence with respect to a coating color with the tendency of a reference color reflected thereon, a coating color having a flip-flop texture, or a coating color having a desired texture of depth can be selectively estimated, there is an advantage in that, even if sensuous coating colors desired by the designer or the like are combined, it is possible to faithfully reproduce the desired sensuous color as the coating color.

The above and other objects, features and advantages of the present invention will become more apparent from the following detailed description of the invention when read in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram including a personal computer in accordance with a first embodiment of the present invention;

FIG. 2 is a flowchart illustrating the flow of color reproduction processing in accordance with the first embodiment;

FIGS. 3A to 3C are image diagrams illustrating the flow of the color reproduction processing shown in FIG. 2;

FIG. 4 is a is a flowchart illustrating the details of an interpolation processing routine (Step 200 in FIG. 2) in accordance with the first embodiment;

FIGS. 5A to 5C are characteristic diagrams illustrating relationships between color data and tristimulus values, in which FIG. 5A shows the relationship between a value a.sub.1 (Ye) and a value x.sub.1 (X), FIG. 5B shows the relationship between the value a.sub.1 and a value x.sub.2, and FIG. 5C shows the relationship between the value a.sub.1 and a value x.sub.3 ;

FIGS. 6A to 6E are characteristic diagrams illustrating relationships between the color data and the tristimulus values when the value a.sub.3 (Cy) is varied in the characteristic diagram shown in FIG. 5A, in which FIG. 6A shows the case of a.sub.3 =0; FIG. 6B, the case of a.sub.3 =63; FIG. 6C, the case of a.sub.3 =127; FIG. 6D, the case of a.sub.3 =191; and FIG. 6E, the case of a.sub.3 =255;

FIG. 7 is an image diagram illustrating a scanning range required by the values a.sub.2, a.sub.3 ;

FIG. 8 is a flowchart illustrating the details of a curve-deriving processing routine (Step 204 in FIG. 4) in accordance with the first embodiment;

FIG. 9 is an image diagram illustrating a process for determining x.sub.1 coordinates on the basis of the value a.sub.3 ;

FIG. 10 is an image diagram illustrating a process for determining x.sub.1 coordinates on the basis of the value a.sub.2 ;

FIG. 11 is a flowchart illustrating the details of a computing routine (Step 206 in FIG. 4) for determining correspondence between the color data and the tristimulus values in the accordance with the first embodiment;

FIG. 12 is an image diagram illustrating a process for determining intersection coordinates between each curve and a straight line;

FIG. 13 is an image diagram illustrating a process for determining a plurality of intersection coordinates between a curve and a straight line;

FIGS. 14A to 14C are image diagrams illustrating a process for determining the values x.sub.2, x.sub.3 corresponding to points of intersection, in which FIG. 14A shows a process for determining the value al; FIG. 14B, a process for determining the value x.sub.2 ; and FIG. 14c, a process for determining the value x.sub.3 ;

FIGS. 15A to 15C are image diagrams in which FIG. 15 shows a curve having a plurality of points of intersection with a straight line; FIG. 15B shows a process for determining the value x.sub.2 from a plurality of points of intersection; and FIG. 15C shows a process for determining the value x.sub.3 ;

FIG. 16 is a flowchart illustrating the flow of a computing routine (Step 208 in FIG. 4) for determining a boundary in accordance with the first embodiment;

FIG. 17 is an image diagram illustrating a process for plotting points corresponding to points of intersection on an x.sub.2 -x.sub.3 coordinate plane in accordance with the first embodiment;

FIG. 18 is a diagram illustrating a boundary formed in accordance with the first embodiment;

FIG. 19 is a flowchart illustrating the flow (Step 300 in FIG. 2) of color data corresponding to arbitrary tristimulus values in accordance with the first embodiment;

FIG. 20 is an image diagram for explaining a determination as to whether or not the point is within a boundary of arbitrary tristimulus values in accordance with the first embodiment;

FIG. 21 is a flowchart illustrating the details of an interpolation processing routine (Step 200 in FIG. 2) in accordance with a second embodiment;

FIG. 22 is a flowchart illustrating the details of an curve-deriving processing routine (Step 402 in FIG. 21) in accordance with a second embodiment;

FIG. 23 is a flowchart illustrating the details of a computing routine (Step 404 in FIG. 21) in which correspondence between a first set and a second set is determined in accordance with the second embodiment;

FIGS. 24A to 24C are image diagrams illustrating a process for determining the values x.sub.2 to X.sub.N corresponding to the points of intersection in accordance with the second embodiment, in which FIG. 24A shows a process for determining the value al; FIG. 24B, a process for determining the value x.sub.2 ; and FIG. 24c, a process for determining the value X.sub.N ;

FIG. 25 is a diagram illustrating a boundary in another polygonal region in accordance with the first embodiment;

FIG. 26 a schematic diagram of a color reproducing apparatus for reproducing a coating color, including a personal computer, in accordance with a third embodiment of the present invention;

FIG. 27 is a conceptual diagram explaining a configuration of a gonio-spectrophotometer;

FIG. 28 is a diagram illustrating an orthogonal coordinate system for explaining a varied angle .alpha. used in the third embodiment;

FIG. 29 is a characteristic diagram illustrating a varied-angle characteristic of a spectral reflectance factor of a coated surface;

FIGS. 30A to 30C are image diagrams illustrating configurations of coated surfaces, in which FIG. 30A shows a metallic coated surface, FIG. 30B shows a pearl-mica coated surface, and FIG. 30C shows a solid-coated surface;

FIG. 31 is a characteristic diagram illustrating reflectance characteristics of a plurality of coating colors when the varied angle .alpha. is 45.degree. ;

FIG. 32 is a characteristic diagram illustrating the relationship between the varied angle .alpha. and brightness Y with respect to a plurality of coating colors;

FIG. 33 is an image diagram illustrating reflectance characteristics with respect to coating colors in a three-dimensional coordinate system having reflectance, varied angle, and wavelength as axes;

FIG. 34 is a flowchart illustrating the flow of a control main routine for reproducing a coating color in accordance with the third embodiment;

FIG. 35 is a flowchart illustrating the details of Step 500 in FIG. 34 in accordance with the third embodiment;

FIG. 36 is a flowchart illustrating the flow of color reproduction processing (Step 700) in accordance with the third embodiment;

FIGS. 37A to 37C are image diagrams illustrating the flow of coating-color reproduction processing shown in FIG. 36;

FIG. 38 is a diagram illustrating correspondence between a characteristic value vector VX and a reflectance vector VR;

FIG. 39 is a flowchart illustrating the details of Step in FIG. 34 in accordance with a fourth embodiment;

FIG. 40 is a CIE x-y chromaticity diagram including primary colors determined in accordance with the fourth embodiment;

FIG. 41 is a schematic diagram of a neural network apparatus in accordance with the fourth embodiment;

FIG. 42 is an image diagram illustrating a configuration of the network of the network apparatus;

FIG. 43 is an image diagram illustrating adjacent layers in the network;

FIG. 44 is a flowchart illustrating the flow of a control main routine for reproducing a coating color in accordance with a fifth embodiment;

FIG. 45 is a flowchart illustrating the flow of a coating-color selection routine in accordance with a sixth embodiment;

FIG. 46 is a diagram illustrating the Munsell color system;

FIG. 47 is a diagram illustrating the CIE chromaticity coordinates;

FIG. 48 is a diagram illustrating correspondence between the Munsell color system and the CIE chromaticity coordinates;

FIG. 49 is a diagram for explaining that points other than plotted points are obtained by interpolation;

FIG. 50 is a flowchart illustrating the flow of a process in which a tone is imparted to a coating color as instructed, in accordance with a seventh embodiment;

FIG. 51 is a distribution diagram of coating colors in which a plurality of actual coating colors are plotted on the chromaticity coordinate plane;

FIG. 52 is a diagram illustrating areas of color which can be formed on the chromaticity coordinate plane on a monitor and in a paint;

FIG. 53 is a diagram (chromaticity coordinate diagram) illustrating a process for imparting a tone to an instructed coating color;

FIG. 54 is a flowchart illustrating processing for obtaining a coating color on which metallic material and mica material are reflected in accordance with an eighth embodiment;

FIG. 55 is an image diagram illustrating a three-dimensional space of a coordinate system having as axes quantities of component materials governing a coating color, a quantity of metallic material, and a quantity of mica material;

FIG. 56 is a diagram illustrating a region where each quantity is variable;

FIG. 57 is a flowchart illustrating a main routine for obtaining a coating color having a flip-flop texture in accordance with a ninth embodiment;

FIG. 58 is a diagram concerning a reflectance and illustrates a process for obtaining the reflectance;

FIG. 59 is a diagram illustrating varied-angle characteristics;

FIG. 60 is a flowchart illustrating the flow of processing for obtaining a coating color having a flip-flop texture in accordance with the ninth embodiment;

FIG. 61 is a diagram illustrating spectral reflectance characteristics for explaining mirror reflectance;

FIG. 62A is a diagram illustrating the varied-angle characteristic;

FIG. 62B is an image diagram illustrating a configuration of a coated surface to be measured;

FIG. 63 is a diagram illustrating varied-angle characteristics;

FIGS. 64A and 64B are image diagrams with and without a perspective of an image, respectively;

FIGS. 65A and 65B are diagrams illustrating characteristic curves of particle-size distributions of two bright materials, respectively;

FIG. 66 is a flowchart illustrating the flow of processing for obtaining a coating color having a texture of depth in accordance an 11th embodiment;

FIG. 67 is a flowchart illustrating the flow (Step 502 in FIG. 44) of a second NNW method in accordance with the fifth embodiment; and

FIG. 68 is a flowchart illustrating the flow (Step 502 in FIG. 44) of a third NNW method in accordance with the fifth embodiment.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring now to the accompanying drawings, a description will be given of the preferred embodiments of the present invention. As shown in FIG. 1, a color reproducing apparatus includes a personal computer 16 and a color copying apparatus 18. This personal computer 16 is comprised of a keyboard 10 for entering data and the like, a main unit 12 of the computer for computing and outputting data for outputting a desired color from the color copying apparatus 18 in accordance with a program stored in advance, and a CRT 14 for displaying such as the results of computation by the main unit 12 of the computer. As this color copying apparatus 18, color hard-copying apparatus are known which are based on a thermal transfer process, an ink-jet process, an electrophotographic process, and a silver-halide photographic process for outputting color copy images using color data in the RGB colorimetric system as input values.

In a first embodiment, when a predetermined color is reproduced by the color copying apparatus 18, color data in the RGB colorimetric system to be converted to arbitrary color data in the XYZ colorimetric system required for color reproduction are determined on the basis of combinations of a specific number (5.sup.3 sets) of color data in the XYZ colorimetric system and color data in the RGB colorimetric system (trichromatic system), which are set in advance as physical amounts, so as to effect color reproduction.

It should be noted that, as for the color data that are inputted to the color copying apparatus 18, the color filter densities (yellow, magenta, and cyan) based on subtractive mixture of inks or the like are used as color data (Ye, Ma, and Cy). In addition, it is assumed that the inks or the like have ideal color filter density (absorption) characteristics. When the invention is applied to general inks or the like which undergo secondary absorption, it suffices to reduce in advance unnecessary spectral absorption due to the sum of components of the inks or the like when the inks or the like are mixed (so-called color-correction masking).

FIG. 2 shows a routing for processing a program for reproducing a desired color in accordance with this embodiment. In addition, the combination of the group of color data in the XYZ colorimetric system and the group of color data in the RGB colorimetric system for expressing a predetermined color by the color copying apparatus 18 is considered to be in a mapping relation between a set A representing the group of color data in the RGB colorimetric system and a set XX representing the group of color data in the XYZ colorimetric system. Accordingly, respective relations in this processing routine are shown as images in FIGS. 3A to 3C.

In Step 100 in FIG. 2, an output value Oi (i: 1 to 125) is determined with respect to each of a predetermined number of (in this embodiment, it is assumed that each of Ye, Ma, and Cy is specified by one of five values; hence, all the combinations at that time, i.e., 5.sup.3 =125) samples Si (i: 1 to 125) (FIG. 3A). Namely, relations are determined between 125 kinds of color data (Ye, Ma, and Cy) which are input values Si for the color copying apparatus 18 and data on the tristimulus values X, Y, and Z of the color which are output results (output values Oi) with respect to the color data. In Step 200, correspondence between an interpolating point SIi (i: 1, 2, . . . ) and an estimated output value OIi is calculated by performing interpolation on the basis of the relationship of correspondence between the sample point Si and the output value Oi (FIG. 3B). In Step 300, an output value Oo or an estimated output value OIo, which is identical or closest to an output value to be obtained and corresponding to a desired color (i.e., color data on the color to be reproduced, and indicated by a mark * in FIG. 3C), is selected, and an input value (So or SIo) corresponding to the selected value (Oo or OIo) is determined (FIG. 3C).

More particularly, in Step 100, 125 kinds of color data (Ye, Ma, and Cy) are inputted to the color copying apparatus 18, and the color copying apparatus 18 outputs a color medium which has formed a color corresponding to the inputted color data. This outputted color medium is photometrically measured with a spectrometer such as a spectrophotometer, thereby determining tristimulus values (X, Y, and Z). Consequently, the tristimulus values (X, Y, and Z) corresponding to the 125 kinds of color data (Ye, Ma, and Cy) are determined.

It should be noted that, to simplify the description which follows, as for the five kinds of color data (Ye, Ma, and Cy) that are inputted to the color copying apparatus 18, each of the values of Ye, Ma, and Cy is any one of five equispaced values, 0, 63, 127, 191, and 255.

Referring now to FIG. 4, a description will be given of the interpolation processing routine in Step 200. It should be noted that, in the description that follows, values of the color data (Ye, Ma, and Cy) and the tristimulus values (X, Y, and Z) serving as input values are set as follows:

Color data (Ye, Ma, and Cy): values (al, a.sub.2, a.sub.3)

Tristimulus values (X, Y, and Z) : values (x.sub.1, x.sub.2, x.sub.3)

and a description will be given by using these values ai and xi (i=1, 2, 3).

It should be noted that, in the color copying apparatus 18, an image of 256 gradations is outputted, and a desired color is designated by using tristimulus values in the XYZ colorimetric system, so that the value ai and the value xi can be expressed as follows:

In this embodiment, since each of various values ai (i: 1, 2, 3) at the aforementioned sample points to be determined in advance is any one of the five values as described above, this value will be denoted by j and written as a value aij (i: 1, 2, 3, j: 1, 2, 3, 4, 5). It should be noted that these values at the sample points need not be equispaced, and should preferably be used at dense and sparse intervals so as to cover a range which involves a sudden change.

In Step 202, the range of values which are actually used as the values (a.sub.1, a.sub.2, a.sub.3) is determined. First, a change of the value a.sub.i with respect to the value xi with the values a.sub.2 and a.sub.3 set as parameters is examined. In FIGS. 5A to 5C, relationships between the value a.sub.1 and the value xi when the value a.sub.3 is 0 and the values a.sub.2 are 0, 63, 127, 191, and 255 (a.sub.2 j) are shown on an a.sub.1 -xi coordinate plane in which the value a.sub.1 is taken as the ordinate and the value xi as the abscissa. FIG. 5A shows relationships between the value a.sub.1 and the value x.sub.1 on an a.sub.1 -x.sub.1 coordinate plane. FIG. 5B shows relationships between the value a.sub.1 and the value x.sub.2 on an a.sub.1 -x.sub.2 coordinate plane. FIG. 5C shows relationships between the value a.sub.1 and the value x.sub.3 on an a.sub.1 -x.sub.3 coordinate plane.

Curves in the drawings are determined as follows: The five values (x.sub.1, x.sub.2, x.sub.3) when the values a.sub.1 are 0 (a.sub.11), 63(a.sub.12), 127 (a.sub.13), 191(a14), and 255(a.sub.15) when a.sub.3 =0 and a.sub.2 =0 are determined in Step 100. Accordingly, it is possible to plot five points corresponding to the sample points on the respective a.sub.1 -xi coordinate planes in FIGS. 5A to 5C through correspondence between the values of color data on the sample points and the values (x.sub.1, x.sub.2, x.sub.3). It should be noted that points plotted on the a.sub.1 -x.sub.1 coordinate plane are represented as points Pa.sub.1, a.sub.2, a.sub.3 ; points plotted on the a.sub.1 -x.sub.2 coordinate plane are represented as points Qa.sub.1, a.sub.2, a.sub.3 ; and points plotted on the a.sub.1 -x.sub.3 coordinate plane are represented as points Ra.sub.1, a.sub.2, a.sub.3.

Curves as obtained by performing spline interpolation with respect to the plurality of points plotted on each of these a.sub.1 -xi coordinate planes. Hereafter, these curves will be written as curves [a.sub.2, a.sub.3 ]i (i: 1, 2, 3). For example, a curve on the a.sub.1 -x.sub.1 coordinate plane when the value a.sub.3 is 0 and the value a.sub.2 is 0 becomes a curve [0, 0].sub.1, as shown in FIG. 5A. In addition, the examination of the change of the curve [a.sub.2, a.sub.3 ]i by changing the value ai (i=2, 3) will be referred to as scanning by the value ai.

FIGS. 6A to 6E show relationships between the value a.sub.1 and the value x.sub.1 when the value a.sub.3 is one of 0, 63, 127, 191, and 255, respectively. FIG. 6A shows relationships between the value a.sub.1 and the value x.sub.1 when the value a.sub.3 is 0 in the same was as in FIG. 5A. FIG. 6B shows relationships when the value a.sub.3 is 63. FIG. 6C shows relationships when the value a.sub.3 is 127. FIG. 6D shows relationships when the value a.sub.3 is 191. FIG. 6E shows relationships when the value a.sub.3 is 255.

Here, in a case where each of the values a.sub.2 and a.sub.3 is changed in stages in the order of 0, 1, 2, . . . , 255 with respect to the curve [a.sub.2, a.sub.3 ]i to calculate output values for interpolating points by the interpolation from sample points, as for the curve [a.sub.2, a.sub.3 ]i to be determined, it suffices to determine only the value ai corresponding to a value xi* which falls within the range of the values xi which are predetermined values of the desired color. Namely, only a curve [a.sub.2, a.sub.3 ].sub.1 having a point of intersection with a straight line x.sub.1 =x.sub.1 * on the a.sub.1 -x.sub.1 coordinate plane is required. For instance, if the range 10.ltoreq.x.sub.1 * .ltoreq.100 is considered, as can be appreciated from FIGS. 6A to 6E, it is estimated that scanning is not required for the range 191.ltoreq.a.sub.3 .ltoreq.255.

Accordingly, the range of scanning is determined as described below by using the curves obtained by the above-described spline interpolation. It should be noted that since the curve [a.sub.2, a.sub.3 ]i is required to have at least the aforementioned point of intersection with respect to the value x.sub.1, only the curve [a.sub.2, a.sub.3 ].sub.1 is processed in this processing.

If the curve [a.sub.2, a.sub.3 ].sub.1 and the curve [a.sub.2 -64, a.sub.3 ]1 with the value a.sub.3 varied consecutively in the order of 0, 63, 127, 191, and 255 do not have a point of intersection with the straight line x.sub.1 =x.sub.1 * on the coordinate plane, the scanning by the value a.sub.2 which assumes a value from (a.sub.2 -64) to a.sub.2 is not necessary. Incidentally, if the value a.sub.2 is 63, processing is performed with respect to the curve [63, a.sub.3 ]1 and the curve [0, a.sub.3 ].sub.1.

Similarly, if the curve [a.sub.2, a.sub.3 ].sub.1 and the curve [a.sub.2, a.sub.3 -64].sub.1 with the value a.sub.2 varied consecutively in the order of 0, 63, 127, 191, and 255 do not have a point of intersection with the straight line x.sub.1 =x.sub.1 * on the coordinate plane, the scanning by the value a.sub.3 which assumes a value from (a.sub.3 -64) to a.sub.3 is not necessary. Incidentally, if the value a.sub.3 is 63, processing is performed with respect to the curve [a.sub.2, 63].sub.1 and the curve [a.sub.2, 0].sub.1.

As a result of this processing, if the value a.sub.3 exceeds 191, as shown by the hatched portion in FIG. 7, it can be appreciated that scanning is not required irrespective of the value a.sub.2.

In an ensuing Step 204, a curve [a.sub.2 ', a.sub.3 ']i including arbitrary values (a.sub.1 ', a.sub.2 ', a.sub.3 ') is determined by the computing routine shown in FIG. 8. This curve [a.sub.2 ', a.sub.3 ']i is determined by performing spline interpolation from five points plotted on each a.sub.1 -xi coordinate plane when the values a.sub.1 are 0, 63, 127, 191, and 255. In other words, the curve [a.sub.2 ', a.sub.3 ']i is determined from the respective five points from points Pa.sub.1, a.sub.2 ', a.sub.3 ' on the a.sub.1 -x.sub.1 coordinate plane, points Qa.sub.1,a.sub.2 ',a.sub.3 ' on the a.sub.1 -x.sub.2 coordinate plane, and points Ra.sub.1, a.sub.2 ', a.sub.3 ' on the a.sub.1 -x.sub.3 coordinate plane, respectively. Hereafter, a description will be given by using the points Pa.sub.1, a.sub.2 ', a.sub.3 ' plotted on the a.sub.1 -x.sub.1 coordinate plane as an example.

In Step 210 in FIG. 8, x.sub.1 -coordinates of the point Pa.sub.1,a.sub.2 ',a.sub.3 ' (a.sub.1, a.sub.2 : 0, 63, 127, 191, 255) are determined by scanning by the value a.sub.3 as shown below. First, a description will be given of an example in which the value a.sub.1 is 0, and the value a.sub.2 is 255.

The x.sub.1 -coordinates of the point P.sub.0,255,a.sub.3 ' are found by performing spline interpolation from the x.sub.1 -coordinates of five points including a point P.sub.0,255,0, a point P.sub.0,255,63, a point P.sub.0,255,127, a point P.sub.0,255,191, and a point P.sub.0,255,25. That is, since the x.sub.1 -coordinates of these five points are already known as described above, by using these coordinates, points are plotted on the a.sub.3 -x.sub.1 coordinate plane where the value a.sub.3 and the value x.sub.1 perpendicularly intersect each other, and spline interpolation is performed with respect to these plotted points, thereby obtaining a continuous line 50, as shown in FIG. 9. Then, the coordinate of intersection between the straight line a.sub.3 32 a.sub.3 ' and this continuous line 50 is determined, and this intersection coordinate is set as a solution (a value x.sub.1 ' of the x.sub.1 -coordinate of the point P.sub.0,255,a.sub.3 ').

By setting this value a.sub.2 consecutively to 0, 63, 127, 191, and 255 and performing processing similar to the one described above, it is possible to determine a coordinate value of the x.sub.1 -coordinate of each point Pa.sub.1, a.sub.2 ', a.sub.3 ' (a.sub.1 : 0, 63, 127, 191, 255).

In an ensuing Step 212, scanning is effected by the value a.sub.2 by using the coordinate values of the x.sub.1 -coordinates determined in Step 210, so as to determine the x.sub.1 -coordinates of the points Pa.sub.1, a.sub.2 ', a.sub.3 ' (a.sub.1 : 0, 63, 127, 191, 255). First, a description will be given by citing an example in which the value a.sub.1 is 0.

The x.sub.1 -coordinates of the point P.sub.0, a.sub.2 ', a.sub.3 ' are found by performing spline interpolation from the x.sub.1 -coordinates of five points including a point P.sub.0,0,a.sub.3 ', a point P.sub.0,63,a.sub.3 ', point P.sub.0,127,a.sub.3 ', a point P.sub.0,191, a.sub.3 ', and a point P.sub.0,255, a.sub.3 '. That is, since the x.sub.1 -coordinates of these five points have already been found in Step 204, by using these coordinates, points are plotted on the a.sub.2 -x.sub.1 coordinate plane where the value a.sub.2 and the value x.sub.1 perpendicularly intersect each other, and spline interpolation is performed with respect to these plotted points, thereby obtaining a continuous line 52, as shown in FIG. 10. Then, the coordinate of intersection between the straight line a.sub.2 =a.sub.2 ' and this continuous line 52 is determined, and this intersection coordinate is set as a solution (a value x.sub.1 " of the x.sub.1 -coordinate of the point P.sub.0, a.sub.2 ', a.sub.3 ').

By setting this value a.sub.1 consecutively to 0, 63, 127, 191, and 255 and performing processing similar to the one described above, it is possible to determine a coordinate value of each x.sub.1 -coordinate of the point Pa.sub.1, a.sub.2 ', a.sub.3 '.

In an ensuing Step 214, the curve [a.sub.2 ', a.sub.3 ].sub.1 is determined by using the coordinate values of the points determined above. Namely, since the x.sub.1 -coordinates of the points Pa.sub.1, a.sub.2 ', a.sub.3 ' are determined in Steps 210 and 212, points can be plotted on the respective a.sub.1 -x.sub.1 coordinate plane; hence, by performing spline interpolation with respect to these plotted points, it is possible to obtain an arbitrary curve [a.sub.2 ', a.sub.3 ' ].sub.1.

It should be noted that, with respect to the curve [a.sub.2 ', a.sub.3 '].sub.2 on the a.sub.1 -x.sub.2 coordinate plane and the curve [a.sub.2 ', a.sub.3 '].sub.3 on the a.sub.1 -x.sub.3 coordinate plane as well, it is possible to obtain an arbitrary curve by changing the coordinate plane used above.

Thus, it is possible to determine curves [a.sub.2 ', a.sub.3 ']i (i=1, 2, 3) with respect to arbitrary values (a.sub.2 ', a.sub.3 '). Consequently, it is possible to determine values (x.sub.1, x.sub.2, x.sub.3) corresponding to arbitrary values (a.sub.1 ', a.sub.2 ', a.sub.3 ').

In an ensuing Step 206, correspondence is determined between the values (a.sub.1, a.sub.2, a.sub.3) of the color data (Ye, Ma, Cy) and the values (x.sub.1, x.sub.2, x.sub.3) of the tristimulus values (X, Y, Z) by the computing routine shown in FIG. 11. It should be noted that, to simplify the description, a description will be given hereafter by citing an example in which the value x.sub.1 is a desired predetermined value x.sub.1 *. In Step 220 of FIG. 11, by using the curves [a.sub.2 ', a.sub.3 '].sub.1, scanning is effected by the values a.sub.2, a.sub.3 (by setting the values of a.sub.2, a.sub.3 to 0, 1, 2, . . . , 255) so as to determine a point of intersection between each curve [a.sub.2, a.sub.3 ].sub.1 and the straight line x.sub.1 =x.sub.1 *.

First, the value a.sub.3 is set to a predetermined value a.sub.S3, and the value a.sub.2 is varied consecutively in the order of 0, 1, . . . , 255, so as to determine intersection coordinates (FIG. 12). Namely, a point of intersection between the curve [a.sub.2, a.sub.s3 ].sub.1 with the value a.sub.2 varied and the straight line x.sub.1 =x.sub.1 * is computed, a coordinate value Am (m: 1, 2, . . . . , M; M is a maximum number of intersection), i.e., an a.sub.1 -coordinate value, is determined in the order of these intersection coordinates. In addition, the value a.sub.2 of the curve [a.sub.2, a.sub.s3 ].sub.1 corresponding to this coordinate value Am is set as Bm (m: 1, 2, . . . , M; Bm: 1, 2, . . . , 255).

Here, there are cases where a plurality of points of intersection are present in one curve, as shown in FIG. 13. In this case, it is assumed that one curve has a plurality of points of intersection, and coordinate values Amn (n: 1, 2, . . . , K; K is a maximum number of intersection points, 3 in FIG. 13) of the points of intersection are stored in sequence.

In an ensuing Step 222, values x.sub.2, x.sub.3 corresponding to the points of intersection found in Step 220 are determined.

First, in order to determine the value x.sub.2, the value x.sub.2 is determined by using the curve [a.sub.2, a.sub.s3 ].sub.2 corresponding to the curve [a.sub.2, a.sub.s3 ].sub.1 having a point of intersection. Namely, as shown in FIG. 14A, in a case where the point of intersection on the curve [Bm, a.sub.s3 ].sub.1 in which the value a.sub.2 is Bm is the coordinate value Am, a value x.sub.2 at which the value a.sub.1 is the coordinate value Am is determined in the curve [Bm, a.sub.s3 ].sub.2 (corresponding to the aforementioned curve) on the a.sub.1 -x.sub.2 coordinate plane.

Similarly, the value x.sub.3 can be determined from the curve [a.sub.2, a.sub.s3 ].sub.3 on the a.sub.1 -x.sub.3 coordinate plane, as shown in FIG. 14C.

Here, there are cases where a plurality of points of intersection are present in one curve, as described above (FIG. 15A). In this case as well, in order to determine the value x.sub.2, a value x.sub.2k (k: 1, 2, . . . . , K; K is a maximum number of intersection points) at which the value a.sub.1 is the coordinate value Amn is determined in the curve [Bm, a.sub.s3 ].sub.2 on the a.sub.1 -x.sub.2 coordinate plane, as shown in FIG. 15B. Additionally, a value x.sub.3k is determined from the curve [a.sub.2, a.sub.s3 ].sub.3 on the a.sub.1 -x.sub.3 coordinate plane.

In the above-described manner, correspondence can be determined between values (x.sub.1 *, x.sub.2, x.sub.3) and values (a.sub.1, a.sub.2, a.sub.s3) in a case where the value a.sub.3 is set to a predetermined value a.sub.S3. Accordingly, by varying the value a.sub.3 set in Step 220 consecutively to 0, 1, 2, . . . . , 255 and by executing the processing, it is possible to obtain all correspondences between the values (a.sub.1, a.sub.2, a.sub.3) of the color data (Ye, Ma, Cy) and the values (x.sub.1 *, x.sub.2, x.sub.3) of the tristimulus values (X, Y, Z).

Although, in the above, a description has been given by citing an example in which the value x.sub.1 is a predetermined value x.sub.1 *, if the above-described processing is executed by varying this value x.sub.1 in the range (0.ltoreq.x.sub.1 .ltoreq.100) that the value x.sub.1 can assume, it is possible to obtain all correspondences between the values (a.sub.1, a.sub.2, a.sub.3) of the color data (Ye, Ma, Cy) and the values (x.sub.1, x.sub.2, x.sub.3) of the tristimulus values (X, Y, Z).

In an ensuing Step 208, a boundary serving as a determination region including arbitrary values is determined from the respective values x.sub.2, x.sub.3 of the points of intersection determined by the computing routine of FIG. 16. In this case, an x.sub.1 =x.sub.1 * plane in an x.sub.1 -x.sub.2 -x.sub.3 space, which is a coordinate axis where the respective values (x.sub.1, x.sub.2, x.sub.3) perpendicularly intersect each other, is assumed.

In Step 230 in FIG. 16, the aforementioned values x.sub.2, x.sub.3 of the points of intersection, which are determined in the case where the value a.sub.3 is a predetermined value when x.sub.1 =x.sub.1 *, are plotted on the x.sub.2 -x.sub.3 coordinate plane. In an ensuing Step 232, the respective values x.sub.2, x.sub.3 of the points of intersection, which are determined in the case where the value a.sub.3 is varied consecutively in the order of 0, 1, 2, . . . 255, are plotted on the x.sub.2 -x.sub.3 coordinate plane (see FIG. 17). The points in the drawing are represented by S.sub.Am, a.sub.3 (m: 1, 2, . . . , M).

In an ensuing Step 234, a boundary 70 of a convex polygonal region including a plurality of points, which serves as a contour of the group of points plotted on the x.sub.2 -x.sub.3 coordinate plane when x.sub.1 =x.sub.1 *, is determined. In a simple case where curves which correspond to a continuation of points when the value a.sub.3 (=1, 2, . . . ) is a predetermined value do not intersect each other on the x.sub.2 -x.sub.3 coordinate plane, it suffices to determine the boundary 70 of the convex polygonal region by simply connecting together endpoints which constitute the contour of the group of points plotted on the x.sub.2 -x.sub.3 coordinate plane. In other words, as shown in FIG. 18, all the points when the value a.sub.3 is a maximum value and a minimum value are included in the internal region of the boundary 70. Thus, the boundary 70 constitutes contour lines of the region which include all the points with respect to each value a.sub.3 when the value a.sub.3 is a maximum value and a minimum value.

Combinations of the values (a.sub.1, a.sub.2, a.sub.3) corresponding to all the points included in a region IN within this boundary 70 include all sets concerning the values a.sub.2, a.sub.3, and the value a1 becomes a real number which is determined univalently by the values x.sub.1 and the values a.sub.2, a.sub.3. For this reason, the region IN of this boundary 70 covers all the combinations that can be assumed by the values (a.sub.1, a.sub.2, a.sub.3).

Therefore, if the value x.sub.1 is varied in the range (0.ltoreq.x.sub.1 .ltoreq.100) in which the value x.sub.1 is capable of assuming the value x.sub.1 *, and the aforementioned processing is executed, it is possible to determine a boundary of the region which covers all the combinations that can be assumed by the values (a.sub.1, a.sub.2, a.sub.3). For this region, the boundary forms a closed region in the x.sub.1 -x.sub.2 -x.sub.3 space.

In addition, in a case where the convex polygonal region in Step 234 above is complicated and the curves with respect to the value a.sub.3 (1, 2, . . . ) intersect each other, it is estimated that the region including points S.sub.Am, a.sub.3 (m=1, 2, . . . . , M; and a.sub.3 =1, 2, . . . ) becomes a complicated convex polygon, as shown in FIG. 25. In this case, it is possible to determine the boundary 70 by determining an internal region E of the convex polygon as shown below.

First, an angle of .pi./2 to -.pi./2 is divided equally into N parts by N (a natural number) which is sufficiently large, and an angle .theta.i (i: 1, 2, . . . , N) which is incremented by each predetermined angle at the time of this equal division can be expressed by the following Formula (4):

A straight line having a coordinate value .sigma. and an inclination .theta. when x.sub.2 =0 on the x.sub.2 -x.sub.3 coordinate plane can be expressed by Formula (5) below. The x.sub.3 coordinate .sigma. of a point of intersection between this straight line and the coordinate axis (x.sub.2 =0) on the x.sub.2 -x.sub.3 coordinate plane is defined by the following Formula (6) as a function with respect to each of the aforementioned angles .theta.i by using x.sub.2, x.sub.3 as parameters.

where

i=1, 2, . . . , N-1

When i=N,

With respect to each of the angles .theta.i, a maximum value .sigma..sub.M (.theta.i) and a minimum value .sigma..sub.m (.theta.i) based on the coordinate values of the point S.sub.Am, a.sub.3 concerning the angle .theta.i are determined by comparing the results of computation when the coordinate values (x.sub.2, x.sub.3) of the aforementioned point S.sub.Am, a.sub.3 are substituted by using Formula (6) above (refer to Formulae (7)) .

Accordingly, all the aforementioned points S.sub.Am, a.sub.3 are included in the region sandwiched by straight lines defined by each angle .theta.i as well as the maximum value .sigma..sub.M (.theta.i) and the minimum value .sigma..sub.m (.theta.i) corresponding to each angle .theta.i.

Points E.sub.i (x.sub.2, x.sub.3) which are included in the region sandwiched by the straight lines defined by each angle .theta.i as well as the maximum value .sigma..sub.M and the minimum value .sigma..sub.m corresponding to each angle .theta.i can be expressed by the following Formula (8):

Where,

i=1,2, . . . N-1

.alpha.=x.sub.2 tan.theta.i

.sigma..sub.m=.sigma..sub.m (.theta.i)

.sigma..sub.M=.sigma..sub.M (.theta.i)

E.sub.N (x.sub.2, x.sub.3)={(x.sub.2, x.sub.3).vertline..sigma..sub.m (.theta..sub.N).ltoreq.x.sub.3 .ltoreq..sigma..sub.M (.theta..sub.N)}

Of the points E.sub.i (x.sub.2, x.sub.3) which are included in the region defined by each of these angles .theta.i, points E(x.sub.2, x.sub.3) which are included for all the angles .theta.1 are all the points that are included in the convex polygonal region, so that the convex polygonal region can be defined by the following Formula (9):

where E.sub.i (i=1, 2, . . . , N) is E.sub.i (x.sub.2, x.sub.3) of Formula (8) above.

Therefore, it is possible to determine the boundary 70 in the convex polygonal region by the straight lines that are defined by the aforementioned maximum value .sigma..sub.M and the minimum value .sigma..sub.m at each angle .theta.i when the convex polygonal region is formed in such a manner as to include these points E(x.sub.2, x.sub.3).

Step 300 is a processing step for determining values (a.sub.1 *, a.sub.2 *, a.sub.3 *) of color data corresponding to arbitrary tristimulus values (x.sub.1 *, x.sub.2 *, x.sub.3 *) by setting the boundary found in the above as a determination region. It should be noted that, in this step, the x.sub.1 =x.sub.1 * plane in the x.sub.1 -x.sub.2 -x.sub.3 space is considered.

First, the operation proceeds to Step 302 in FIG. 19 to determine whether or not a point T plotted on the x.sub.2 -x.sub.3 coordinate plane by the values (x.sub.2 *, x.sub.3 *) is included in the aforementioned boundary 70. This determination is made as follows: A point is plotted at the position of the values (x.sub.2 *, x.sub.3 *) on the x.sub.2 -x.sub.3 coordinate plane, a semi-infinite straight line is formed in a predetermined direction by setting the point T at the position of (x.sub.2 *, x.sub.3 *) as a starting point, and a point of intersection between this semi-infinite straight line and the boundary is determined. If there are the number of these points of intersection is an odd number, the points are present within the boundary 70, whereas if it is an even number, the points are present outside the boundary 70. For instance, as shown in FIG. 20, the number of points of intersection between a point 80 at the position of (x.sub.2 *, x.sub.3 *) on the one hand, and a semi-infinite straight line 84 on the other, is an odd number, so that a determination is made that the point 80 is present within the boundary 70. Meanwhile, the number of points of intersection between a point 82 and a semi-infinite straight line 86 is an even number, so that a determination is made that the point 82 is present outside the boundary 70.

In an ensuing Step 304, a determination is made as to whether or not the point T is included in the boundary 70. If it is determined that the point is present within the boundary 70, in Step 306, a Mahalanobis distance Di from the point T is determined with respect to all the points included in the boundary 70 by using the following Formula (10):

where i=1, 2, . . .

In an ensuing Step 308, a minimum value of the Mahalanobis distance Di thus determined is selected, and the value (a.sub.1, a.sub.2, a.sub.3) of the point which is this minimum value is selected as an approximate solution of the value (a.sub.1 *, a.sub.2 *, a.sub.3 *) of the desired color data.

Meanwhile, if NO is the answer in the determination in Step 304, an approximate solution corresponding to the desired color data cannot be selected. Hence, in Step 310, processing is carried out to the effect that there is no solution, and this routine ends.

If minimum values of all the Mahalanobis distances Di are selected by consecutively varying the value x.sub.1 in this processing, it is possible to obtain a closest approximate solution.

It should be noted that a determination may be made as to whether or not the aforementioned semi-infinite straight line passes an odd number of times the boundary serving as the closed region in the x.sub.1 -x.sub.2 -x.sub.3 space, although the computation will be complicated.

In addition, although, in the above-described embodiment, a description has been given of the case where the present invention is applied to color reproduction in the color copying apparatus 18 by the subtractive mixture of color stimuli, the present invention is readily applicable to cases of color reproduction in which color data in the RGB colorimetric system on the basis of the additive mixture of color stimuli are determined so as to display a desired color on the CRT 14.

Next, a description will be given of a second embodiment. In the first embodiment, an arbitrary and desired color is specified on the basis of combinations of color data in the XYZ colorimetric system and color data in the RGB colorimetric system, which are in predetermined small numbers, to effect color reproduction. In the second embodiment, on the other hand, an arbitrary relationship is estimated from a small number of relationships with respect to a first set and a second set respectively having a large number of elements and related to each other as physical amounts, and a relevant relationship is selected from a physical amount of a desired second set.

First, preconditions in this embodiment will be described.

Values of the first set are set as values ai (i=1, 2, . . . , K; natural numbers), values of the second set are set as values xj (j=1, 2, . . . , L; natural numbers), and it is assumed that values xi based on combinations of K values ai are determined by a function f, as shown in the following Formula (11):

where p.ltoreq.ai.ltoreq.q (p, q: real numbers; p<q)

The function f.sub.j is unknown.

Here, e discrete values W obtained by equally dividing the interval between the real number p and the real number q by (e-1) can be expressed by the following Formula (12):

where e: natural number

At this time, if it is assumed that the value ai is one of the discrete values W (ai.epsilon.W), and that combinations of the value xj with respect to all permutations and combinations of the value ai are known, then it follows that e.sup.K sets of correspondence are known between (a.sub.1, a.sub.2, . . . , a.sub.K) and (x.sub.1, x.sub.2, . . . , x.sub.L).

In this embodiment, combinations of K values ai* corresponding to combinations of L arbitrary values xj* under these preconditions are assumed.

Hereafter, the operation of this embodiment will be described. Since the main routine is similar to that of FIG. 2, a description thereof will be omitted.

In Step 200 in FIG. 2 in this embodiment, an interpolation processing routine shown in FIG. 21 is executed.

In Step 402 in FIG. 21, curves [a.sub.2 ', . . . . a.sub.K '].sub.L each including arbitrary values (a.sub.1 ', a.sub.2 ', . . . , a.sub.K ') are determined by the computing routine shown in FIG. 22. Each of these curves [a.sub.1 ', a.sub.2 ', . . . , a.sub.K '].sub.L is determined by carrying out spline interpolation with respect to a specific number of points plotted on each a.sub.1 -xi coordinate plane when the value a.sub.1 is a known value. In other words, the curves are determined on the basis of points Pa.sub.1, a.sub.2 ', a.sub.3 ', . . . , a.sub.K ' on the a.sub.1 -x.sub.1 coordinate plane, points Qa.sub.1, a.sub.2 ', a.sub.3 ', . . . , a.sub.K ' on the a.sub.1 -x.sub.2 coordinate plane, and points Ra.sub.1, a.sub.2 ', a.sub.3 ', . . . , a.sub.K ' on the a.sub.1 -x.sub.3 coordinate plane in the same way as in the foregoing embodiment. Coordinate values of respective x.sub.1 coordinates are determined by effecting scanning consecutively by the known value ai, and interpolation is carried out with respect to the points plotted on the a.sub.1 -x.sub.1 coordinate plane, thereby determining arbitrary curves [a.sub.2 ', . . . , a.sub.K '].sub.L.

More particularly, in Step 410 of FIG. 22, each x.sub.1 coordinate of the point P on the a.sub.1 -x.sub.1 coordinate plane is determined by effecting scanning by a predetermined value a.sub.H ' (any one of natural numbers satisfying 2.ltoreq.H.ltoreq.K). First, values other than the predetermined value a.sub.H ' are set to known values, and e curves are determined in which the predetermined value a.sub.H ' is consecutively increased by an increment A each up to (e-1 times)q, starting with a curve in which the predetermined value a.sub.H ' is set to a predetermined value p. Each x.sub.1 coordinate for the predetermined value a.sub.H ' is determined from these e curves. Namely, since the x.sub.1 coordinates are known as described above, by using these coordinates, coordinates of points of intersection between straight lines a.sub.H =a.sub.H ' and the e curves on the a.sub.H -x.sub.1 coordinate plane where the value a.sub.H ' and the value x.sub.1 perpendicularly intersect each other are set as solutions (values x.sub.1 ' of the x.sub.1 coordinates). Then, the values other than the value a.sub.H ' which have been set to known values are respectively set from the predetermined value p by the increment A each up to (e-1 times)q, and processing similar to the one described above is performed to obtain coordinate values of the x.sub.1 coordinates of the respective points Pa.sub.1, a.sub.2, a.sub.3, . . . , a.sub.H ', . . . , a.sub.K. In an ensuing Step 412, by using the coordinate values of the x.sub.1 coordinates found in Step 410, scanning by a value a.sub.G is carried out in the same way as described above to obtain x.sub.1 coordinates of points Pa.sub.1, a.sub.2, a.sub.3, . . . , a.sub.H ', . . . , a.sub.G ', . . . , a.sub.K. In an ensuing Step 414, the processing in Step 412 is processed by mathematical induction to obtain x.sub.1 coordinates of points Pa.sub.1, a.sub.2 ', a.sub.3 ', . . . , a.sub.K '.

In an ensuing Step 416, curves [a.sub.2 ', . . . , a.sub.K '].sub.1 are determined by using the coordinate values of the points thus obtained. That is, since the respective x.sub.1 coordinates of the points Pa.sub.1, a.sub.2 ', a.sub.3 ', . . . , a.sub.K ' are determined in Steps 410 to 414 above, points can be plotted on the a.sub.1 -x.sub.1 coordinate planes. By interpolating with respect to these plotted points, arbitrary curves [a.sub.2 ', . . . , a.sub.K '].sub.1 are determined.

It should be noted that curves [a.sub.2 ', a.sub.3 '].sub.2 on the a.sub.1 -x.sub.2 coordinate plane and curves [a.sub.2 ', a.sub.3 '].sub.3 on the a.sub.1 -x.sub.3 coordinate plane can also be determined in a similar manner by changing the coordinate planes used above.

Next, in Step 404, correspondence between the values (a.sub.1, a.sub.2, . . . , a.sub.K) of the first set and the values (x.sub.1, x.sub.2, . . . , x.sub.L) of the second set is determined by the computing routine shown in FIG. 23. Incidentally, to simplify the description, a description will be given by citing an example in which the value x.sub.1 is a predetermined value x.sub.1 *.

In Step 420 of FIG. 23, scanning is effected by a predetermined value a.sub.H (any one of natural numbers satisfying 2.ltoreq.H.ltoreq.K) by using the curves [a.sub.2 ', . . . , a.sub.K '].sub.1 obtained above (i.e., by varying the value a.sub.H at arbitrary intervals in the range p.ltoreq.a.sub.H .ltoreq.r) to determine points of intersection between a curve [a.sub.2, . . . , a.sub.H, . . . , a.sub.K ].sub.1 and the straight line x.sub.1 =x.sub.1 *.

At this time, the coordinate value of the value a.sub.1 is set as a coordinate value U.sub.Hi (i: 1, 2, . . . , max; max is a maximum number of points of intersection), and the point of intersection is set as a point of intersection (x.sub.1 *, U.sub.Hi). In addition, the value a.sub.H which gives this point of intersection (x.sub.1 *, U.sub.Hi) is set as a value V.sub.H.

In an ensuing Step 422, values x.sub.2 to x.sub.L corresponding to the points of intersection found in Step 420 are determined.

First, to determine the value x.sub.2, the value x.sub.2 is determined by using a curve [a.sub.2, . . . , a.sub.H (V.sub.H), . . . , a.sub.K ].sub.2 corresponding to a curve [a.sub.2, . . . , a.sub.H (V.sub.H), . . . , a.sub.K ].sub.1 having a point of intersection. Namely, when the point of intersection on the curve [a.sub.2, . . . , a.sub.H (V.sub.H), . . . , a.sub.K ]1 in which the value a.sub.H is V.sub.H is a coordinate value U.sub.Hi as shown in FIG. 24A, a value x.sub.2 is determined at which the value a.sub.1 is a coordinate value U.sub.Hi on the curve [a.sub.2, . . . , a.sub.H (V.sub.H), . . . , a.sub.K ].sub.2 (corresponding to the above curve) on the a.sub.1 -x.sub.2 coordinate plane, as shown in FIG. 15(B).

Similarly, as shown in FIG. 15C, the value x.sub.L is finally determined from a curve [a.sub.2, . . . , a.sub.H (V.sub.H), . . . , a.sub.K ].sub.L on the a.sub.1 -x.sub.L coordinate plane using the curve [a.sub.2, . . . , a.sub.H (V.sub.H), . . . , a.sub.K ].sub.L.

In this manner, the values x.sub.2 to x.sub.L which correspond to the points of intersection ((x.sub.1 *, U.sub.Hi) at which the coordinate value a.sub.1 is the coordinate value U.sub.Hi are determined.

Accordingly, if the above processing is executed by consecutively varying the predetermined value of the value a.sub.H which is set, it is possible to determine all the correspondences between the values (a.sub.1, a.sub.2, a.sub.3, . . . ) of the first set and the values (x.sub.1 *, x.sub.2, x.sub.3, . . . ) of the second set. It should be noted that if the above processing is executed by varying the value x.sub.1 in the range that the value x1 can assume, it is possible to determine all the correspondences between the values (a.sub.1, a.sub.2, a.sub.3, . . . ) and the values (x.sub.1, x.sub.2, x.sub.3, . . . ).

In an ensuing Step 406, a space serving as a determination region including arbitrary values is determined from the respective values x.sub.2 to x.sub.L of the points of intersection determined in the above. Namely, the values x.sub.2 to x.sub.L of the points of intersection defined by x.sub.1 =x.sub.1 * are plotted in the x.sub.2 -x.sub.i -x.sub.L coordinate space (an L-.sub.1 dimensional space) as characteristic points SDi (i: .sub.1 .ltoreq.i.ltoreq.MAX; MAX is the total number of characteristic points). This coordinate space is set as a determination region used below. Incidentally, a closed region including all the points plotted in this coordinate space may be determined, and may be set as a boundary CC.

Next, Step 300 in this embodiment is a processing step for determining values (a.sub.1 *, a.sub.2 *, . . . , a.sub.K *) corresponding to arbitrary values (x.sub.1 *, x.sub.2 *, . . . , x.sub.L *) from the determination region determined in the above. First, values (x.sub.2 *, . . . , x.sub.L *) are extracted from desired arbitrary values (x.sub.1 *, x.sub.2 *, . . . , x.sub.L *), and are set as desired points SSD. Then, the distance Di of each point is determined in the space serving as the aforementioned determination region by using the following Formula (13):

Next, smallest values of the distances thus determined are selected, and respective values x.sub.2 to x.sub.L of the characteristic points SDi corresponding to the selected smallest values are set as solutions. Values (a.sub.1, a.sub.2, . . . ) corresponding to these solutions are selected as approximate solutions.

It should be noted that a determination may be made as to whether or not the characteristic points plotted in the coordinate space in the same way as in the above-described embodiment are included in the aforementioned boundary CC, and that the above processing may be effected if it is determined that the points are present within the boundary CC.

This second embodiment is applicable to character recognition. In this case, a set of characteristic amounts of characters is used as the first set, while a set of characters is used as the second set. Accordingly, if characters xj (j=1, 2, . . . , L; natural numbers), i.e., the second set, with respect to characteristic amounts ai (i=1, 2, . . . , K; natural numbers) of characters in the first set are determined with respect to a predetermined number of characters, and their relation is determined from the function f of Formula (11) above, it is possible to specify corresponding (approximate) characters from combinations of the characteristic amounts ai of characters in an arbitrary first set.

Thus, in the above-described embodiment, since desired data can be converted on the basis of a small number of data, and multi-intersection processing, which has conventionally been difficult to perform, is discriminated by the intersection number, a determination as to whether or not desired data is included can be easily made by determining whether the data is located inside or outside a region.

Although, in the above-described embodiment, a description has been given of an example in which curves are determined by spline interpolation, it is possible to use other interpolation methods.

Next, a description will be given of a third embodiment. In this embodiment, the present invention is applied to a color reproducing apparatus for reproducing a coating color. In this embodiment, as shown in FIG. 26, a color-material mixing apparatus 20 having an automatic measuring device is used instead of the color copying apparatus 18. The personal computer 16 is comprised of the keyboard 10 for entering data and the like, the main unit 12 of the computer for computing and outputting relevant data for generating a desired coating color in accordance with a program stored in advance, and the CRT 14 for displaying a coating color or the like which is the result of computation by the main unit 12 of the computer. The color-material mixing apparatus 20 generates a paint by mixing a plurality of color materials after measuring the color materials such as pigments by a measuring instrument, e.g., an electronic force balance, in response to signals outputted from the personal computer 16.

Here, in this embodiment, to reproduce a coating color, physical amounts for specifying the coating color are set as follows.

As already mentioned in the description of the related art, if the spectral reflectance of the coated surface can be specified, the tristimulus values and the like of the color can be determined, with the result that its surface color can be specified. Therefore, in this embodiment, the spectral reflectance of an original or object surface is used to realize color reproduction for displaying a color image or the like and for specifying a faithful coating color on the object surface. It should be noted that values of this spectral reflectance, when measured with respect to surfaces of samples having complicated configurations, such as fibers and metallic coatings, can vary depending on the direction of light reception of the measuring instrument. In this embodiment, therefore, a spectral reflectance factor is used which is a three-dimensional reflectance obtained by varying the angle of incidence upon a sample and the light-receiving angle of a light-receiving element for receiving the light reflected by the sample.

The reflectance of a sample having a flat surface can be usually measured (photometrically measured) by a gonio-spectrophotometer 24. This measured reflectance is referred to as the spectral reflectance factor, which will be simply referred to hereafter as the reflectance R.

As shown in FIG. 27, the gonio-spectrophotometer 24 has a light source 28 and a light-receiving unit 26. In the case of the gonio-spectrophotometer 24, a plane which includes an incident optical axis 32 of the light directed from the light source 28 toward a measuring point Ob of a sample 30 and a optical axis 34 of reflection in the direction of regular reflection when the light of the incident optical axis 32 is regularly reflected at the measuring point Ob, is defined as an incident plane D.sub.1. In this gonio-spectrophotometer 24, the axis connecting the light-receiving unit 26 and the measuring point Ob is set as a measuring optical axis 36. This gonio-spectrophotometer 24 has a mechanism (not shown) in which the light-receiving unit 26 is moved three-dimensionally such that the measuring optical axis 36 is included within the incident plane D.sub.1.

The reflectance R is a function of an angle .alpha. (unit: degree; hereafter referred to as a varied angle .alpha.) formed by the reflection optical axis 34 and the measuring optical axis, i.e., an angle .alpha. of the direction of regular reflection with respect to the light-receiving unit, and a wavelength .lambda. (unit: nm) of light, and can be expressed by the following Formula (14):

where the varied angle .alpha. is 0.degree. when the reflection optical axis 34 and the measuring optical axis 36 coincide with each other. In addition, the sign of the varied angle .alpha. which is obtained from the position of the light-receiving unit 26 rotated clockwise from the direction of regular reflection toward the light source (in the direction of arrow indicating the varied angle .alpha. in FIG. 27) will be set as a positive sign.

As shown in FIG. 28, the varied angle .alpha. can be determine in a rectangular coordinate system using the incident plane and the like. In other words, a normal direction $N of the sample 30, an incident direction $L which is an azimuth between the sample 30 and the light source 28, a light-receiving direction $R in which the light is directed from the sample 30 toward the light-receiving unit 26, and a regularly reflecting direction $P in which the light is regularly reflected from the sample 30 are set. Then, a plane which includes the normal direction $N and the regularly reflecting direction $P is set as the incident plane D.sub.1, while a plane which includes the normal direction $N and the light-receiving direction $R is set as a light-receiving plane D.sub.2. As a result, an angle .theta..sub.1 formed by the normal direction $N and the incident direction $L, an angle .theta..sub.2 formed by the normal direction $N and the light-receiving direction $R, and an angle .theta..sub.3 formed by the incident plane D.sub.1 and the light-receiving plane D.sub.2 are set. In addition, in a case where the surface of the sample 30 is directional (e.g., in the case of fabrics and brushing-finished surfaces), an angle at which the reference direction of the sample surface (coated surface) (a direction $A in FIG. 28) moves away from the incident plane D.sub.1 with the measuring point Ob set as a center is set as an angle .theta..sub.4. Accordingly, the reflectance R in Formula (14) above can be expressed as a general formula by the following Formula (15):

where

.theta..sub.1 : incident angle of the light source (deg)

.theta..sub.2 : light-receiving angle (deg)

.theta..sub.3 : azimuth angle (deg)

.theta..sub.4 : rotational angle (deg)

Formula (15) above has four angular parameters denoted respectively by .theta..sub.1, .theta..sub.2, .theta..sub.3, and .theta..sub.4. It is known that the distribution of intensity of reflected light (a distribution in which the intensity of reflected light is expressed by the distance with an irradiating point set as a center) from a surface coated with a general paint always shows spherical symmetry having similar figures with the regularly reflecting direction $P as an axis, irrespective of the incident angle .theta..sub.1 of the incident light.

FIG. 29 shows a varied-angle characteristic diagram of the spherical reflectance factor and illustrates the spherical symmetry of the light reflected from a surface coated with a general paint (a metallic coated surface). In the drawing, as shown in Table 1 below, the varied-angle characteristic when the varied angle .alpha. is varied in the positive direction when the incident angle .theta..sub.1 is 0.degree. is set as a characteristic AP, while the varied-angle characteristic when the varied angle .alpha. is varied in the negative direction is set as a characteristic AN. Similarly, when the incident angle .theta..sub.1 is 15.degree., 30.degree., 45.degree., and 60.degree., varied-angle characteristics when the varied angle .alpha. is varied in the positive direction are set as characteristics BP, CP, DP, EP, and FP, while varied-angle characteristics when the varied angle .alpha. is varied in the negative direction are set as characteristics BN, CN, DN, and EN.

TABLE 1 ______________________________________ Incident angle .theta..sub.1 0.degree. 15.degree. 30.degree. 45.degree. 60.degree. 75.degree. ______________________________________ Varied angle .alpha. + direction AP BP CP DP EP FP - direction AN BN CN DN EN -- ______________________________________

As can be appreciated from FIG. 29, the varied-angle characteristics are substantially symmetrical irrespective of the incident angle. It should be noted that, when the incident angle was 75.degree., a measurement error occurred due to a sheen phenomenon caused by a reference white plate, so that the listing was omitted here.

Accordingly, the reflectance of the surface coated with the paint can be expressed by the reflectance R(.alpha., .lambda.) as a function of the varied angle .alpha. between the regularly reflecting direction $P and the light-receiving direction $R, as shown in Formula (14) above. For instance, if the angular conditions other than the light-receiving angle .theta..sub.2 are fixed to predetermined values (.theta..sub.1 =60.degree., .theta..sub.3 =0.degree.m, and .theta..sub.4 =0.degree.), and the varied angle .alpha. is varied in the range 0.degree. to 90.degree. (in this case, .alpha.=.theta..sub.1 -.theta..sub.2), and if the reflectance R(.alpha., .lambda.) is measured by the gonio-spectrophotometer, the reflectance R(.alpha., .lambda.) can be determined in the angular range 0.degree.<.alpha.<90.degree..

Also, if the reflectance R(.alpha., .lambda.) is set under the angular conditions listed below, the reflectance R(.alpha., .pi.) can be determined in the angular range -30.degree.<.alpha.<150.degree..

Angular Conditions: ##EQU2##

It should be noted, in the description that follows, the reflectance R(.alpha., .lambda.) in which the varied angle .alpha. is computed from the relation between the regularly reflecting direction $P and the light-receiving direction $R is used even in cases other than the aforementioned angular conditions (.theta..sub.1 =60.degree., .theta..sub.3 =0.degree., and .theta..sub.4 =0.degree.).

As shown in FIGS. 30A to 30C, the coated surface of a sample whose surface is coated is comprised of various substances including color pigments governing the color, bright materials such as metal pearl mica, and clear coat materials on the surfaces.

As shown in FIG. 30A, a coated surface formed by a metallic coating is comprised of a clear coat layer 40, a metallic base layer 42, an intermediate coat layer 44, and an electrodeposited layer 46. The metallic base layer 42 includes a pigment 54 and aluminum 56. As shown in FIG. 30B, a coated surface formed by a pearl mica coating is comprised of a clear coat layer 40, a mica base layer 48, a color base layer 51, an intermediate coat layer 44, and an electrodeposited layer 46. The mica base layer 48 includes a titanized mica pigment 58. As shown in FIG. 30C, a coated surface formed by a solid coating is comprised of a top coat layer 53, an intermediate coat layer 44, and an electrodeposited layer 46. The top coat layer 53 includes a coloring pigment 60.

FIG. 31 shows the relationship between the varied angle .alpha. and brightness Y (Y is determined by Formula (38) which will be described later). As can be appreciated from the drawing, the rate of change of the reflectance R becomes slower in the order of maroon, crystal coral, camel beige, pale green, and grape blue which are used as coating colors. In addition, FIG. 32 shows the relationship of the reflectance R(45.degree., .lambda.) when the varied angle .alpha. is 45.degree.. It can be seen that brightness at a predetermined wavelength varies depending on the kind of the coated surface.

Thus, the characteristic of the reflectance R(.alpha., .lambda.) varies due to the difference in the arrangement of the coated surface, and is, at the same time, affected by the kind and quantity of pigment and bright material. Accordingly, to specify these coated surfaces, in this embodiment, by assuming that component materials which make up the coated surface are x.sub.1, x.sub.2, . . . and that the size of each component material x.sub.i (i=1, 2, . . . ) is a quantity q.sub.1 (kg), a characteristic value vector VX representing the coated surface is defined as shown in the following Formula (16):

Since the reflectance R(.alpha., .lambda.) of the coated surface formed by this characteristic value vector VX is related to the characteristic value vector VX, the reflectance R(.alpha., .lambda.) can be expressed by the following Formula (17):

In this embodiment, since the case in question is the reproduction of a coating color, the characteristic value vector VX shown in the following Formula (18) is considered by taking into consideration only the component materials (pigment and the like) Governing the color among the elements of the characteristic value vector VX and by assuming only e component materials related to the color:

In this embodiment, it is basically assumed that a bright material which, although essentially achromatic, is imparted a color to the extent of substantially changing the color of the pigment, such as some special colored mica, is not used as a component material.

Although, in the above, a description has been given of the reflectance R(.alpha., .lambda., VX), which is based on the continuous characteristics of the varied angle .alpha. and the wavelength .lambda. as elements related to the characteristic value vector VX, the reflectance R(.alpha., .lambda., VX) can be handled approximately, as will be described below.

First, the varied angle .alpha. (0.degree. to 90.degree.) is appropriately divided such as by dividing it into [n-1] parts at equal intervals by a boundary value .alpha..sub.j (j=1, 2, . . . , n, 0.degree.=.alpha..sub.1 <.alpha..sub.2 < . . . <.alpha..sub.n =90.degree.) or by dividing into small parts the range thereof where the change of reflectance is abrupt. It should be noted that it is preferable to provide this appropriate division at intervals of 1.degree. to 5.degree. in such a manner as to obtain 19 to 91 pieces of data.

Similarly, with respect to the wavelength .lambda. as well, the visible w