Patent Publication Number: US-11049222-B2

Title: Smoothing method, smoothing device, and storage medium storing smoothing program

Description:
The present application is based on, and claims priority from JP Application Serial Number 2018-142614, filed on Jul. 30, 2018, the disclosure of which is hereby incorporated by reference herein in its entirety. 
     BACKGROUND 
     1. Technical Field 
     The present disclosure relates to a smoothing technique for correcting color values associated with multiple grid points arranged in a device-dependent color space such as an RGB color space or a CMYK color space, where R indicates red, G indicates green, B indicates blue, C indicates cyan, M indicates magenta, Y indicates yellow, and K indicates black. 
     2. Related Art 
     A colorimetric value of a patch may adversely affect the accuracy of color prediction and may degrade a gradation due to a variation in a colorimeter, a variation in a colorimetric position, a variation in colors of patches, or the like in profile generation executed in a subsequent imaging process. Thus, a smoothing process is executed to improve the gradation. JP-A-2005-094160 describes a colorimetric data correction method of correcting colorimetric data of patches of colors associated with multiple grid points of a lattice cube set in an RGB color space. The colorimetric data is, for example, represented by color values of a CIE L*a*b color space that is a device-independent color space. CIE is the International Commission on Illumination. Hereinafter, “*” of L*a*b is omitted or L*a*b is referred to as Lab. Colorimetric data of grid points located on a ridgeline of the lattice cube is corrected to an average value of colorimetric data associated with grid points included in a predetermined range on the ridgeline, while a concerned grid point on the ridgeline is treated as the center of the predetermined range. Colorimetric data of grid points located on a surface of the lattice cube is corrected to an average value of colorimetric data associated with grid points included in a rectangular range in which a concerned grid point on the surface is the center of gravity. 
     When the aforementioned colorimetric data correction method is executed, a color reproduction range after the correction may be smaller than a color reproduction range before the correction. A method of correcting an outer surface of the lattice cube is important to suppress the reduction, caused by the correction, in the color reproduction range. Regarding the colorimetric data of the grid points on the surface of the lattice cube, it is important to minimize a change, caused by colorimetric before the correction, in the shape of a surface of gamut or color gamut or leave a trajectory of an outer edge of the gamut. It is important to not only minimize the change in the shape of the surface of the color gamut but also execute correction to enable smooth gradation expression. It is important to enable both the minimization and the correction. 
     SUMMARY 
     According to an aspect of the disclosure, a smoothing method is to execute a smoothing process on color values associated with multiple grid points arranged in a device-dependent color space. 
     The multiple grid points include multiple surface grid points arranged on a surface of a grid point region in which the multiple grid points are arranged in the device-dependent color space. 
     The smoothing method includes calculating polynomial approximation coefficients to be used in a polynomial approximation equation for calculating approximate values of color values corresponding to positions in a first processing direction in the device-dependent color space for multiple first target grid points that are among the multiple surface grid points and arranged in the first processing direction in the device-dependent color space. 
     The smoothing method includes smoothing color values associated with the first target grid points using the polynomial approximation equation when the color values associated with the first target grid points are to be smoothed. 
     In addition, according to another aspect of the disclosure, a smoothing device includes units corresponding to the coefficient calculation and smoothing of the smoothing method. 
     Furthermore, according to another aspect of the disclosure, a computer-readable storage medium stores a smoothing program for causing a computer to enable functions corresponding to the coefficient calculation and smoothing of the smoothing method. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram schematically showing an example of a grid point region in which multiple grid points are arranged in a device-dependent color space. 
         FIG. 2  is a diagram schematically showing an example of a cross-section, taken along a line II-II shown in  FIG. 1 , of the grid point region. 
         FIG. 3  is a block diagram schematically showing an example of a configuration of a smoothing system. 
         FIG. 4  is a flowchart of an example of a smoothing process. 
         FIG. 5  is a diagram schematically showing an example in which lines to be subjected to polynomial approximation are set on surfaces of a three-dimensional grid point region. 
         FIG. 6  is a diagram schematically showing an example in which lines to be subjected to the polynomial approximation are set on surfaces of a four-dimensional grid point region. 
         FIG. 7  is a flowchart of an example of a polynomial approximation correction process. 
         FIG. 8  is a flowchart of an example of an outlier exclusion process. 
         FIG. 9  is a flowchart of an example of a weight determination process. 
         FIG. 10  is a diagram schematically showing an example of a weight of an extreme grid point excluding grid points located at both edges. 
         FIG. 11  is a flowchart of an example of a weighted polynomial approximation process. 
         FIG. 12  is a diagram schematically showing an example in which color values are corrected based on a smoothing intensity. 
         FIG. 13  is a diagram schematically showing an example in which lines to be subjected to smoothing are set at positions extending in the three-dimensional grid point region. 
         FIG. 14  is a diagram schematically showing an example of the smoothing. 
         FIG. 15  is a diagram schematically showing an example of a filter for calculating weighted averages of color values based on the smoothing intensity. 
         FIG. 16  is a diagram showing an example of a color reproduction range after the polynomial approximation executed on a ridgeline indicating colors from white to cyan, compared with weighted averaging executed on the ridgeline. 
     
    
    
     DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Hereinafter, an embodiment of the disclosure is described. The embodiment merely exemplifies the disclosure, and not all characteristics described in the embodiment may be necessary for the disclosure. 
     (1) Overview of Technique Included in Disclosure 
     First, an overview of a technique included in the disclosure is described with reference to examples shown in  FIGS. 1 to 16 . The drawings included in the present application schematically show the examples. Enlargement rates of the drawings in directions may vary and the drawings may not be matched. Elements of the present technique are not limited to specific examples indicated by reference symbols. In “the overview of the technique included in the disclosure”, words in brackets indicate supplemental explanation of previous words. 
     In the present application, a range of values “Min to Max” indicates values equal to or larger than the minimum value Min and equal to or smaller than the maximum value Max. 
     First Aspect 
     A smoothing method according to a first aspect of the present technique is to correct color values (for example, component values L i , a i , and b i  in a Lab color space shown in  FIG. 1 ) associated with multiple grid points P 1  arranged in a device-dependent color space CS 1 . The smoothing method according to the first aspect includes a coefficient calculation process ST 1  and a smoothing process ST 2 , as exemplified in  FIG. 4 . Color values z i  exemplified in  FIG. 1  collectively represent the component values L i , a i , and b i . The multiple grid points P 1  are arranged in a grid point region  500  of a lattice cube in the device-dependent color space CS 1 . The grid points P 1  include multiple surface grid points P 2  arranged on a surface  510  of the grid point region  500 . In the coefficient calculation process ST 1 , polynomial approximation coefficients (a 0 , . . . , and a d ) are calculated for multiple first target grid points P 10  that are among the multiple surface grid points P 2  and arranged in the first processing direction D 1  in the device-dependent color space CS 1 . The polynomial approximation coefficients (a 0 , . . . , and a d ) are used in a polynomial approximation equation for calculating approximate values (for example, approximate values y i  shown in  FIG. 1 ) of color values (for example, the color values z i  shown in  FIG. 1 ) corresponding to positions (for example, positions x i  shown in  FIG. 1 ) in the first processing direction D 1  in the device-dependent color space CS 1 . The polynomial approximation coefficients include a constant number a 0  and can be calculated using a value of a determinant A exemplified in  FIG. 1 . When color values (z i ) associated with the first target grid points P 10  are to be smoothed, the color values (z i ) associated with the first target grid points P 10  are smoothed using the polynomial approximation equation. 
     In the first aspect, the polynomial approximation coefficients (a 0 , . . . , and a d ) are calculated for the multiple first target grid points P 10  arranged on the surface  510  of the grid point region  500  and are used in the polynomial approximation equation for calculating the approximate values (y i ) of the color values (z i ) corresponding to the positions (x i ) in the first processing direction D 1 , the polynomial approximation equation is used upon the smoothing, and the color values (z i ) associated with the first target grid points P 10  are smoothed. Thus, in the smoothing of color values associated with the multiple surface grid points P 2 , a variation in the color values (z i ) is reduced, and changes in the shape of a surface of original gamut at the positions (x i ) on the surface  510  of the grid point region  500  can be reduced. As a result, according to the first aspect, the smoothing method can be provided, which enables smooth gradation expression while suppressing a change in the shape of the gamut surface, compared with a case in which the color values associated with the multiple surface grid points are averaged. 
     The first processing direction may be changed and indicates a direction identifying the arrangement of the multiple first target grid points to be processed. Thus, for example, the coefficient calculation process and the smoothing process may be executed on a certain grid point as one of multiple first target grid points arranged in an R axis direction and may be executed on another grid point as one of multiple first target grid points arranged in a G axis direction. 
     In addition, a single direction may be set as the first processing direction, and the coefficient calculation process and the smoothing process may be executed on a single grid point. Alternatively, two directions may be set as the first processing direction, and the coefficient calculation process and the smoothing process may be executed on a single grid point. Thus, for example, the coefficient calculation process and the smoothing process may be executed on a certain grid point as one of the multiple target grid points arranged in the R axis direction and may be executed on the grid point as one of the multiple target grid points arranged in the G axis direction. 
     An effect of enabling smooth gradation expression while suppressing a change in the shape of the gamut surface as much as possible is obtained by executing the coefficient calculation process and the smoothing process on grid points in the first processing direction once. When two directions are set as the first processing direction, and the coefficient calculation process and the smoothing process are executed, smoother gradation expression is enabled. 
     In the device-dependent color space, a color to be perceived by a person may not be identified even when coordinate values are determined, and a color is defined depending on a color reproduction characteristic of a device. The device-dependent color space may include an RGB color space, a CMY color space, a CMYK color space, and the like. 
     A grid point indicates a virtual point arranged in the device-dependent color space as an input color space, and it is assumed that a color value as an output coordinate value corresponding to the position of the grid point in the input color space is stored in the grid point. The present technique includes a case in which multiple grid points may be arranged at even intervals in the input color space and a case in which multiple grid points may be arranged at uneven intervals in the input color space. 
     The color values include colorimetric values that are results of executing colorimetry on patches indicating colors of the grid points, calculated values indicating colors of the patches, and correction values calculated by an interpolation operation from the colorimetric values and the calculated values. 
     The polynomial approximation equation may be calculated by polynomial approximation executed to give the same weight to the first target grid points or may be calculated by weighted polynomial approximation executed to give different weights to some of the multiple first target grid points. Specifically, the polynomial approximation equation may include a weighted polynomial approximation equation. In addition, the polynomial approximation equation may be calculated by polynomial approximation using all the multiple first target grid points or may be calculated by polynomial approximation using a first target grid point selected from the multiple first target grid points. 
     When correction values associated with the first target grid points are referred to as first correction values, the first correction values may be approximate values or may be values calculated using the approximate values and color values. 
     The surface of the gamut is referred to as outer edge of the gamut in some cases, and the surface of the grid point region is referred to as outer edge of the grid point region in some cases. 
     The above additional remarks are applied to the following aspects. 
     Second Aspect 
     The first processing direction may be different from multiple axes included in the device-dependent color space as far as multiple surface grid points are arranged in the first processing direction. However, the first processing direction may be along any of axes AX 1  included in the device-dependent color space CS 1 , as exemplified in  FIG. 1  and the like.  FIG. 1  shows that the first processing direction D 1  is the R axis direction. The first processing direction D 1 , however, may be set to the G axis direction or a B axis direction. The first processing direction D 1  may be set to a C axis direction, an M axis direction, a Y axis direction, or a K axis direction as far as the first processing direction D 1  is set in the CMYK color space. According to a second aspect, a suitable technique for enabling smooth gradation expression while suppressing a change in the shape of the gamut surface as much as possible can be provided. 
     Third Aspect 
     As exemplified in  FIG. 1  and the like, the device-dependent color space CS 1  may be a D-dimensional color space with a number D of axes AX 1 , where the number D is 3 or more. The grid point region  500  may include a number 2 D  of vertices  520 . The surface  510  of the grid point region  500  may include multiple ridgelines  530  connecting the vertices  520  to each other and may include multiple sectioned surfaces  540  sectioned by the multiple ridgelines  530 . Multiple sectioned surface grid points P 4  that are among the multiple surface grid points P 2  and arranged on the sectioned surfaces  540  may be arranged in a first axis direction and a second axis direction different from the first axis direction. The first axis direction and the second axis direction are determined based on the positions of the sectioned surfaces  540 . For example, when the sectioned surfaces  540  are along an RG plane, the R axis direction may be set to the first axis direction and the G axis direction may be set to the second axis direction. As exemplified in  FIG. 5  and the like, in the smoothing method, the first processing direction D 1  may be set to the first axis direction, and the coefficient calculation process ST 1  and the smoothing process ST 2  may be executed on the multiple sectioned surface grid points P 4 . In the smoothing method, the first processing direction D 1  may be set to the second axis direction, and the coefficient calculation process ST 1  and the smoothing process ST 2  may be executed on the multiple sectioned surface grid points P 4 . 
     According to a third aspect, a suitable technique for enabling smooth gradation expression while suppressing a change in the shape of the gamut surface as much as possible can be provided. 
     Fourth Aspect 
     As exemplified in  FIGS. 1, 9, and 10 , the polynomial approximation coefficients may include weighted polynomial approximation coefficients to be used in the weighted polynomial approximation equation for calculating the approximate values of the color values (z i ) corresponding to the positions (x i ) in the first processing direction D 1  in the device-dependent color space CS 1 . In the coefficient calculation process, weights (for example, weights w i ) of the first target grid points P 10  may be determined based on the color values (z i ) associated with the multiple first target grid points P 10 , and the weighted polynomial approximation coefficients may be calculated based on the determined weights (w i ). According to a fourth aspect, the smoothing method can be provided, which is suitable to enable smooth gradation expression while suppressing a change in the shape of the original gamut surface by the weighted polynomial approximation process using the weighted polynomial approximation coefficients, compared with the case in which the color values associated with the multiple surface grid points are averaged. 
     Fifth Aspect 
     As exemplified in  FIGS. 1, 9, and 10 , in the coefficient calculation process, when a single extreme grid point P 13  having a color value (z i ) that serves as an extreme and is any of the color values (z i ) corresponding to the positions (x i ) and is larger or smaller than color values (z 1  and z n ) associated with edge grid points P 11  that are among the multiple first target grid points P 10  and located at both edges in the first processing direction D 1  exists among the multiple first target grid points P 10 , a weight (w m ) of the extreme grid point P 13  may be the largest among the weights (for example, w i ) of the first target grid points P 10 , and the weighted polynomial approximation coefficients may be calculated. Thus, on the surface  510  of the grid point region  500 , a change in the shape of the original gamut surface can be reduced. As a result, according to a fifth aspect, the smoothing method can be provided, which is suitable to enable smooth gradation expression while suppressing a change in the shape of the original gamut surface, compared with the case in which the color values associated with the multiple surface grid points are averaged. 
     Extremes include a local maximum value and a local minimum value. This additional remark is applied to the following aspects. 
     Sixth Aspect 
     As exemplified in  FIG. 8 , in the coefficient calculation process, whether a grid point P 1  that is included in the multipole first target grid points P 10  is a grid point P 15  that is to be excluded and is not to be used for the calculation of the polynomial approximation coefficients may be determined based on the color values (z i ) associated with the multiple first target grid points P 10 , the grid point P 15  to be excluded may be excluded from the multiple first target grid points P 10 , and the polynomial approximation coefficients may be calculated. In a sixth aspect, an approximate curve that is close to true values can be calculated by excluding a grid point with an inappropriate color value. Thus, according to the sixth aspect, the smoothing method can be provided, which is suitable to enable smooth gradation expression by a smoothing process executed by the polynomial approximation (including the weighted polynomial approximation) in a subsequent process while suppressing a change in the shape of the original gamut surface as much as possible. 
     Seventh Aspect 
     As exemplified in  FIG. 1  and the like, the device-dependent color space CS 1  may be a D-dimensional color space with a number D of axes AX 1 , where the number D is 3 or more. The grid point region  500  may include a number 2 D  of vertices  520 . The surface  510  of the grid point region  500  may include multiple ridgelines  530  connecting the vertices  520  to each other and may include multiple surfaces  540  sectioned by the multiple ridgelines  530 . As exemplified in  FIG. 5 , according to a seventh aspect, in the smoothing method, after the coefficient calculation process and the smoothing process are executed on multiple ridgeline grid points P 3  that are among the multiple surface grid points P 2  and arranged on the ridgelines  530 , the coefficient calculation process and the smoothing process may be executed on the multiple sectioned surface grid points P 4  that are among the multiple surface grid points P 2  and arranged on the sectioned surfaces  540 . Color values associated with the sectioned surface grid points P 4  are smoothed after the smoothing of color values associated with the multiple ridgeline grid points P 3  in the seventh aspect. Thus, according to the seventh aspect, the smoothing method can be provided, which is suitable to enable smooth gradation expression while suppressing a change in the shape of the original gamut surface, compared with the case in which the color values associated with the multiple surface grid points are averaged. 
     Eighth Aspect 
     As exemplified in  FIG. 2  and the like, the multiple grid points P 1  may include multiple internal grid points P 5  arranged in an internal region  550  included in the grid point region  500 . As exemplified in  FIG. 4 , according to an eighth aspect, the smoothing method may further include an internal smoothing process ST 3  of smoothing color values (z i ) associated with the internal grid points P 5  by a smoothing process different from the smoothing process using the polynomial approximation equation. Smoothing that is different from polynomial approximation is executed on the color values associated with the internal grid points P 5  and the smoothing is executed by the polynomial approximation (including the weighted polynomial approximation) on color values associated with the surface grid points in the eighth aspect. Thus, according to the eighth aspect, the smoothing method can be provided, which is suitable to enable smooth gradation expression while suppressing a change in the shape of the original gamut surface as much as possible. 
     Ninth Aspect 
     As exemplified in  FIGS. 2 and 14  and the like, in the internal smoothing process, weighted averages (y i ) of color values associated with grid points including grid points adjacent to each other in a second processing direction D 2  may be associated with multiple second target grid points P 20  that are among the multiple internal grid points P 5  and arranged in the second processing direction D 2  in the device-dependent color space CS 1 . In a ninth aspect, the color values associated with the internal grid points P 5  are corrected to the weighted averages (y i ) including the color values associated with the grid points including the grid points P 1  adjacent to each other in the second processing direction D 2 . 
     The weighted averages of the color values associated with the grid points may be values calculated when a coefficient of 0 is included. The weighted averages of the color values associated with the grid points may be values calculated when all the coefficients are the same. Thus, the ninth aspect includes the case where the original color values associated with the internal grid points to be processed are corrected to the weighted averages since the coefficients for the color values associated with the internal grid points to be processed are 1. The ninth aspect also includes the case where simple averages of the original color values associated with the grid points are the weighted averages since all the coefficients are the same. The color values associated with the internal grid points P 5  are smoothed using the weighted averages and the smoothing process is executed to use the polynomial approximation (including the weighted polynomial approximation) to smooth the color values associated with the surface grid points in the ninth aspect. Thus, according to the ninth aspect, the smoothing method can be provided, which is suitable to enable smooth gradation expression while suppressing a change in the shape of the original gamut surface as much as possible. 
     The second processing direction may be changed and indicates a direction identifying the arrangement of the multiple second target grid points to be processed. Thus, for example, the internal smoothing process may be executed on a certain grid point as one of multiple second target grid points arranged in the R axis direction and may be executed on another grid point as one of multiple second target grid points arranged in the G axis direction. 
     In addition, a single direction may be set as the second processing direction and the internal smoothing process may be executed on a single grid point. Alternatively, two or more directions may be set as the second processing direction and the internal smoothing process may be executed on a single grid point. Thus, for example, the internal smoothing process may be executed on a certain grid point as one of the second target grid points arranged in the R axis direction, and may be executed on the certain grid point as one of the second target grid points arranged in the G axis direction, and may be executed on the certain grid point as one of second target grid points arranged in the B axis direction. 
     The above additional remarks are applied to the following aspects. 
     Tenth Aspect 
     The second processing direction may be different from the multiple axes included in the device-dependent color space as far as multiple internal grid points are arranged in the second processing direction. As exemplified in  FIG. 2  and the like, the second processing direction may be along any of the multiple axes AX 1  included in the device-dependent color space CS 1 .  FIG. 2  shows that the second processing direction D 2  is the R axis direction. The first processing direction D 1 , however, may be set to the G axis direction or the B axis direction. When the first processing direction D 1  is set in the CMYK color space, the first processing direction D 1  may be set to the C axis direction, the M axis direction, the Y axis direction, or the K axis direction. According to a tenth aspect, a suitable technique for enabling smooth gradation expression while suppressing a change in the shape of the original gamut surface as much as possible can be provided. 
     Eleventh Aspect 
     As exemplified in  FIG. 1  and the like, the device-dependent color space CS 1  may be a D-dimensional color space with a number D of axes AX 1 , where the number D is 3 or more. As exemplified in  FIG. 2  and the like, the multiple internal grid points P 5  may be arranged in axis directions that are along the number D of axes AX 1 . According to an eleventh aspect, in the smoothing method, the second processing direction D 2  may be sequentially set to the number D of axes AX 1 , each of the axis directions may be sequentially treated as the second processing direction D 2 , and the internal smoothing process ST 3  may be executed on the multiple internal grid points P 5 . 
     According to the eleventh aspect, a suitable technique for enabling smooth gradation expression while suppressing a change in the shape of the original gamut surface as much as possible can be provided. 
     Twelfth Aspect 
     As exemplified in  FIG. 4 , according to a twelfth aspect, the smoothing method may further include a smoothing intensity reception process ST 4  of receiving a setting of an intensity (for example, a rate c shown in  FIG. 12 ) of the smoothing. In the smoothing process ST 2 , a weight for the approximate values obtained using the polynomial approximation equation may be treated as the rate (c) based on the intensity, weighted averages (for example, first correction values r i  shown in  FIG. 12 ) of approximate values (y i ) of the first target grid points P 10  and the color values (z i ) associated with the first target grid points P 10  may be associated with the first target grid points P 10 . According to the twelfth aspect, a technique for smoothing color values of the internal grid points based on user&#39;s preference can be provided. 
     Thirteenth Aspect 
     A smoothing device (for example, a host device  100  shown in  FIG. 3 ) according to a thirteenth aspect of the present technique includes a coefficient calculating unit U 1  and a smoothing unit U 2 . The coefficient calculating unit U 1  corresponds to the coefficient calculation process ST 1 . The smoothing unit U 2  corresponds to the smoothing process ST 2 . According to the thirteenth aspect, the smoothing device can be provided, which enables smooth gradation expression while suppressing a change in the shape of the original gamut surface as much as possible. The smoothing device may include an internal smoothing unit U 3  and a smoothing intensity receiving unit U 4 . The internal smoothing unit U 3  corresponds to the internal smoothing process ST 3 . The smoothing intensity receiving unit U 4  corresponds to the smoothing intensity reception process ST 4 . 
     Fourteenth Aspect 
     A smoothing program PRO according to a fourteenth aspect of the present technique causes a computer (for example, the host device  100  shown in  FIG. 3 ) to enable a coefficient calculating function FU 1  and a smoothing function FU 2 . The coefficient calculating function FU 1  corresponds to the coefficient calculation process ST 1 . The smoothing function FU 2  corresponds to the smoothing process ST 2 . According to the fourteenth aspect, the smoothing program PRO can be provided, which enables smooth gradation expression while suppressing a change in the shape of the original gamut surface as much as possible. The smoothing program PRO may cause the computer to enable an internal smoothing function FU 3  and a smoothing intensity receiving function FU 4 . The internal smoothing function FU 3  corresponds to the internal smoothing process ST 3 . The smoothing intensity receiving function FU 4  corresponds to the smoothing intensity reception process ST 4 . 
     In addition, the present technique is applicable to a complex device including the smoothing device, a method of controlling the smoothing device, a method of controlling the complex device, a program for controlling the smoothing device, a program for controlling the complex device, a computer-readable storage medium storing the smoothing program and the control programs, and the like. Each of the devices may be composed of multiple units. 
     (2) Overview of Smoothing Method According to Specific Example 
       FIG. 1  schematically exemplifies the grid point region  500  in which the multiple grid points P 1  are arranged in the device-dependent color space CS 1 . A lower portion of  FIG. 1  schematically exemplifies the first target grid points P 10  arranged in the first processing direction D 1  on the surface  510  of the grid point region  500 . The first processing direction D 1  shown in  FIG. 1  is the R axis direction but may be changed to the G axis direction or the B axis direction.  FIG. 2  schematically exemplifies a cross-section, taken along a line II-II shown in  FIG. 1 , of the grid point region  500 . A lower portion of  FIG. 2  schematically exemplifies the multiple second target grid points P 20  arranged in the second processing direction D 2  at positions extending in the internal region  550  included in the grid point region  500 . The second processing direction D 2  shown in  FIG. 2  is the R axis direction but may be changed to the G axis direction or the B axis direction. In  FIGS. 1 and 2 , R indicates the R axis, G indicates the G axis, and B indicates the B axis. In  FIGS. 1 and 2 , K indicates a black point at which the amounts of R, G, and B components are the smallest, and W indicates a white point at which the amounts of R, G, and B components are the largest. 
     The device-dependent color space CS 1  shown in  FIG. 1  is a three-dimensional RGB color space with  3  axes AX 1 . The three axes AX 1  are the R axis, the G axis, and the B axis. A number n of grid points P 1  are arranged at substantially even intervals in each of the axis directions or in each of directions of the axes AX 1 . The number n of grid points P 1  arranged in each of the axis directions is not limited but may be three or more or may be in a range of 9 to 64. 
     When a point of origin is indicated by 0 and the positions of the grid points P 1  arranged in the axis directions are indicated by integers, the maximum value among the integers indicating the positions may not be exactly divisible by (n−1). In this case, an interval between adjacent grid points P 1  may be a quotient of the division of the maximum value among the integers indicating the positions by (n−1) or may be a value obtained by adding 1 to the quotient. In this case, the grid points P 1  are arranged at substantially even intervals in the axis directions. 
     Since the number n of grid points P 1  are arranged in each of the axis directions, a number n D  of grid points P 1  or a number n 3  of grid points P 1  are included in the grid point region  500 . Color values (Lp, ap, and bp) in the Lab color space, which is a device-independent color space, are associated with the grid points P 1 . A component L in the Lab color space indicates lightness. Components a and b in the Lab color space indicate chromaticity. The color values (Lp, ap, and bp) of the grid points P 1  are stored in a color conversion table  400  that is a lookup table. The color conversion table  400  includes data indicating correspondence relationships between coordinate values (Rp, ap, and bp) in the RGB color space and coordinate values or color values (Lp, ap, and bp) in the Lab color space. The RGB color space is an input color space. The Lab color space is an output color space. The color values (Lp, ap, and bp) are, for example, obtained by causing a colorimeter to execute colorimetry on patches of colors corresponding to input coordinate values (Rp, ap, and bp) of the grid points P 1 . The patches are also referred to as color charts. The color values (Lp, ap, and bp) are not limited to colorimetric values that are results of executing the colorimetry on the patches. The color values (Lp, ap, and bp) may be values calculated from the colorimetric values, values calculated from colorimetric values of other patches, simulation values, or the like. When an ICC profile is generated or adjusted, an A 2 B table included in the ICC profile is applicable to the color conversion table  400 . ICC is an abbreviation for International Color Consortium. 
     The colorimetric values may adversely affect the accuracy of color prediction and degrade a gradation due to a variation in the colorimeter, a variation in a colorimetric position, a variation in colors of the patches, or the like in the generation of a profile in a subsequent imaging process. In a specific example, since color values, each of which may vary, are smoothed, polynomial approximation is applied to the color values (L i , a i , and b i ) associated with the multiple first target grid points P 10  arranged in the first processing direction D 1  along an axis direction on the surface  510  of the grid point region  500 . Thus, the smoothing method can be provided, which enables smooth gradation expression while suppressing a change in the shape of the gamut surface as much as possible, compared with the case in which the color values associated with the multiple surface grid points are averaged. 
     The device-dependent color space CS 1  to which the present technique is applicable is not limited to the RGB color space and may be the CMY color space, the four-dimensional CMYK color space with four axes, or the like. 
     An overview of a method of correcting a color value is described below. In the following description, a color value associated with a grid point is referred to as color value of the grid point in some cases. 
     The surface  510  of the grid point region  500  shown in  FIGS. 1 and 2  includes the multiple ridgelines  530  extending in the axis directions and includes the multiple surfaces  540  sectioned by the multiple ridgelines  530 . The vertices  520  of the grid point region  500  are located at positions at which the ridgelines  530  intersect with each other. Specifically, the ridgelines  530  connect the vertices  520  to each other. For the convenience of a process described later, it is assumed that the ridgelines  530  include the vertices  520  and that the sectioned surfaces  540  do not include the ridgelines  530 . In the D-dimensional device-dependent color space CS 1 , a number 2 D  of vertices  520  of the grid point region  500  exist. In the three-dimensional RGB color space, the grid point region  500  is treated as a rectangular parallelepiped and includes 8 (=2 3 ) vertices  520 , 12 ridgelines  530 , and 8 sectioned surfaces  540 . Although not shown, the grid point region  500  includes 16 (=2 4 ) vertices  520 , 32 ridgelines  530 , and 24 sectioned surfaces  540  in the four-dimensional CMYK color space. 
     For the convenience of the description, the multiple grid points P 1  included in the grid point region  500  are classified as follows. 
     The multiple grid points P 1  include the multiple surface grid points P 2  arranged on the surface  510  of the grid point region  500  and the multiple internal grid points P 5  arranged in the internal region  550  included in the grid point region  500 . The multiple surface grid points P 2  include the multiple ridgeline grid points P 3  arranged on the ridgelines  530  and the multiple sectioned surface grid points P 4  arranged on the sectioned surfaces  540 . It is assumed that multiple first target grid points P 10 , which are among the multiple surface grid points P 2  and to be subjected to the polynomial approximation, are arranged in the first processing direction D 1  that is any of the multiple axis directions. The multiple first target grid points P 10  shown in  FIG. 1  include the two edge grid points P 11  located at the edges in the R axis direction and multiple intermediate grid points P 12  located between the edge grid points P 11 . The R axis direction is an example of the first processing direction D 1 . It is assumed that the multiple second target grid points P 20  that are located at the positions extending in the internal region  550  of the grid point region  500  and to be subjected to weighted averaging are arranged in the second processing direction D 2  that is any of the multiple axis directions. The multiple second target grid points P 20  shown in  FIG. 2  include two sectioned surface grid points P 4  located at edges in the R axis direction and multiple internal grid points P 5  located between the sectioned surface grid points P 4 . The R axis direction is an example of the second processing direction D 2 . 
     As shown in  FIG. 1 , it is assumed that the positions of the number n of the first target grid points P 10  arranged in the first processing direction D 1  are x i  in the first processing direction D 1  and that the color values associated with the first target grid points P 10  are (L i , a i , and b i ). In this case, i is a variable identifying a first target grid point P 10  and is in a range of integers of 1 to n. Each of color values z i  indicates any of a lightness component L, a chromaticity component a, and a chromaticity component b. When fourth-order polynomial approximation is executed, an approximation equation for calculating the approximate values y i  of the color values z i  corresponding to the positions x i  is expressed by the following Equation (1).
 
 y   i   =a   4   x   i   4   +a   3   x   i   3   +a   2   x   i   2   +a   1   x   i   +a   0   (1)
 
     In this case, a 0 , a i , a 2 , a 3 , and a 4  indicate polynomial approximation coefficients for x i   0 , x i   1 , x i   2 , x i   3 , and x i   4 .  FIG. 1  shows that the degree d of the polynomial approximation is 4. The degree d may be 3 or may be 5 or more. 
     Generally, weighted polynomial approximation coefficients a 0 , . . . , and a d  can be calculated according to the following Equation (2).
 
 A =( X′WX ) −1   X′WZ   (2)
 
     In Equation (2), X indicates a matrix of a number n of rows and a number (d+1) of columns. The matrix X has a number (d+1) of components in an i-th row. For example, the matrix X has x i   0 , x i   1 , x i   2 , x i   3 , and x i   4  in the i-th row. X′ indicates a transpose of the matrix X. Z indicates a matrix of a number n of rows and one column. The matrix Z has a color value z i  in an i-th row. W indicates a matrix of a number n of rows and a number n of columns. The matrix W has a weight in an i-th row and an i-th column. Other components of the matrix W indicate 0. When all weights are 1, the polynomial approximation is executed without a weight. When a weight that is not 1 exists, the weighted polynomial approximation is executed. 
     The calculated approximate values y i  may be associated with the first target grid points P 10  and may serve as the first correction values. Alternatively, the calculated approximate values y i  may be converted to the first correction values based on a set smoothing intensity and may be associated with the first target grid points P 10 . 
     As shown in  FIG. 2 , it is assumed that the positions of the number n of second target grid points P 20  arranged in the second processing direction D 2  are x i  in the second processing direction D 2  and that color values associated with the second target grid points P 20  are (L i , a i , and b i ). In this case, i is a variable identifying a second target grid point P 20  and is in a range of integers of 1 to n. Each of the color values z i  indicates any of a lightness component L, a chromaticity component a, and a chromaticity component b. When color values z i  of the internal grid points P 5  are to be corrected, the color values z i  of the internal grid points P 5  are corrected to the weighted averages of the color values of the grid points including the grid points adjacent to each other in the second processing direction D 2 . When the weighted averages are y i , the weighted averages y i  are expressed by the following Equation (3). 
     
       
         
           
             
               
                 
                   
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                           w 
                           j 
                         
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                           z 
                           
                             i 
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                           j 
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                             s 
                           
                         
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     In Equation (3), s indicates a positive integer indicating a range of grid points to be subjected to the weighted averaging. For example, when s=1, three grid points including grid points located at both edges in the second processing direction D 2  are used. When s=2, five grid points including the grid points located at the edges in the second processing direction D 2  are used. In Equation (3), j is a variable that identifies a grid point to be used and is in a range of integers of −s to s. In Equation (3), z i+j  indicates a color value of a grid point to be used for the weighted averaging, and w j  indicates a weight for a grid point to be used for the weighted averaging. When all weights w j  are the same or are, for example, 1, obtained weighted averages y i  are simple averages. The simple averages are included in the weighted averages y i . 
     The obtained weighted averages y i  are associated with the second target grid points P 20  and treated as second correction values. 
     (3) Specific Example of Configuration of Smoothing System 
       FIG. 3  schematically exemplifies a configuration of a smoothing system SY 1 . The smoothing system SY 1  includes the host device  100 , a display device  130 , a colorimeter  120 , and an ink jet printer  200 . The host device  100  is an example of the smoothing device. In the host device  100 , a CPU  111 , a ROM  112 , a RAM  113 , a storage device  114 , an input device  115 , a communication I/F  118 , a colorimeter I/F  119 , and the like are connected to each other and receive and output information from and to each other. CPU is an abbreviation for central processing unit. ROM is an abbreviation for read only memory. RAM is an abbreviation for random access memory. I/F is an abbreviation for interface. The ROM  112 , the RAM  113 , and the storage device  114  are memories. At least the ROM  112  and the RAM  113  are semiconductor memories. The display device  130  displays, based on display data from the host device  100 , a screen corresponding to the display data. As the display device  130 , a liquid crystal display panel or the like may be used. 
     The storage device  114  stores an OS (not shown), the smoothing program PRO, the color conversion table  400 , and the like. The OS, the smoothing program PRO, the color conversion table  400 , and the like are read into the RAM  113  and used for a smoothing process. OS is an abbreviation for operating system. As the storage device  114 , a nonvolatile semiconductor memory such as a flash memory, a magnetic storage device such as a hard disk, or the like may be used. 
     As the input device  115 , a pointing device, hardware keys including a keyboard, a touch panel attached to a surface of a display panel, or the like may be used. The communication I/F  118  is connected to a communication I/F  210  of the printer  200  and receives and outputs information such as printing data from and to the printer  200 . The colorimeter I/F  119  is connected to the colorimeter  120  and receives colorimetric data including colorimetric values from the colorimeter  120 . As a standard for the I/Fs  118 ,  119 , and  210 , USE, a near-field communication standard, or the like may be used. USB is an abbreviation for Universal Serial Bus. Communication of the I/Fs  118 ,  119 , and  210  may be wired communication, wireless communication, or network communication via a LAN, the Internet, or the like. LAN is an abbreviation for local area network. 
     The colorimeter  120  executes colorimetry on color patches PA 1  formed on a print substrate ME 1 . The print substrate ME 1  is an example of a medium on which a color chart CH 1  is formed. The colorimeter  120  executes the colorimetry on color patches of a color chart displayed by a display device not shown and outputs colorimeter values. The patches are also referred to as color charts. The colorimetric values indicate lightness L and colorimetric coordinates a and b in the CIE Lab color space. The host device  100  acquires colorimetric data from the colorimeter  120  and executes various processes on the colorimetric data. 
     The smoothing program PRO shown in  FIG. 3  causes the host device  100  to enable the coefficient calculating function FU 1 , the surface smoothing function FU 2 , the internal smoothing function FU 3 , and the smoothing intensity receiving function FU 4 . The coefficient calculating function FU 1 , the surface smoothing function FU 2 , and the internal smoothing function FU 3  are included in a smoothing function of smoothing a color value of a grid point. 
     The CPU  111  of the host device  100  reads information stored in the storage device  114  into the RAM  113  and executes the read program to execute various processes. The CPU  111  executes the smoothing program PRO read and stored in the RAM  113 , thereby executing the processes corresponding to the functions FU 1  to FU 4 . The smoothing program PRO causes the host device  100  to function as the coefficient calculating unit U 1 , the surface smoothing unit U 2 , the internal smoothing unit U 3 , and the smoothing intensity receiving unit U 4 . The host device  100  is a computer. The host device  100  executes the smoothing program PRO and executes the coefficient calculation process ST 1 , the surface smoothing process ST 2 , the internal smoothing process ST 3 , and the smoothing intensity reception process ST 4 . The processes ST 1  to ST 4  are included in the smoothing method of causing the computer to execute a process of correcting the color values associated with the multiple grid points P 1  arranged in the device-dependent color space CS 1 . A computer-readable storage medium storing the smoothing program PRO for causing the computer to enable the functions FU 1  to FU 4  is not limited to the internal storage device of the host device and may be an external storage device of the host device. 
     The host device  100  may be a computer such as a tablet terminal or a personal computer. For example, when a body of a laptop personal computer is applied to the host device  100 , the display device  130 , the colorimeter  120 , and the printer  200  are normally connected to the body. When a computer that is a laptop personal computer or the like and is provided with a display device is applied to the host device  100 , the colorimetric  120  and the printer  200  are normally connected to the computer. When the host device  100  is provided with a display device, display data is output to the display device included in the host device  100 . The host device  100  may include all the constituent elements  111  to  119  in a single casing. Alternatively, the constituent elements  111  to  119  may be configured in multiple devices separated from each other so that the constituent elements  111  to  119  can communicate with each other. In addition, even when at least one of the display device  130 , the colorimeter  120 , and the printer  200  is included in the host device  100 , the present technique is enabled. 
     The printer  200  shown in  FIG. 3  is an ink jet printer that discharges at least C ink, M ink, Y ink, and K ink as color materials from a recording head  220  and forms an output image IM 0  corresponding to printing data. The C, M, Y, and K ink is supplied from ink cartridges Cc, Cm, Cy, and Ck to the recording head  220 . Nozzles Nc, Nm, Ny, and Nk jet C, M, Y, and K ink drops  280 . When the ink drops  280  land on the print substrate ME 1 , ink dots are formed on the print substrate ME 1 . As a result, the print substrate ME 1  has the output image IM 0  thereon. When a profile that indicates a color production characteristic of the printer  200  is to be generated, the color chart CH 1  with the patches PA 1  corresponding to colors of grid points may be formed by the printer  200  on the print substrate ME 1 . The color conversion table including colorimetric values of the patches PA 1  as color values (Lp, ap, and bp) is used as the A 2 B table for the generation of the profile. 
     (4) Specific Example of Smoothing Process 
       FIG. 4  exemplifies the smoothing process to be executed by the host device  100  shown in  FIG. 3 . The host device  100  executes multiple processes in parallel by multiple tasks. Step S 102  corresponds to the smoothing intensity reception process ST 4 , the smoothing intensity receiving function FU 4 , and the smoothing intensity receiving unit U 4 . Steps S 104  to S 106  correspond to the coefficient calculation process ST 1 , the coefficient calculating function FU 1 , and the coefficient calculating unit U 1 . Step  3106  corresponds to the surface smoothing process ST 2 , the surface smoothing function FU 2 , and the surface smoothing unit U 2 . Steps S 110  to S 112  correspond to the internal smoothing process ST 3 , the internal smoothing function FU 3 , and the internal smoothing unit U 3 . Hereinafter, descriptions of “steps” are omitted. 
     In the smoothing process, after the color values of the surface grid points P 2  are smoothed in S 104  to S 108 , the color values of the internal grid points P 5  are smoothed in S 110  to S 114 .  FIG. 4  shows that processes of smoothing the color values of the surface grid points P 2  in S 104  to S 108  are surrounded by a broken line, processes of smoothing the color values of the internal grid points P 5  in S 110  to S 114  are surrounded by a broken line, and the broken lines are surrounded by a solid line. 
     When the smoothing process is started, the host device  100  receives input color values corresponding to colors of the grid points P 1  arranged in the device-dependent color space CS 1  and input positional data of the grid points P 1  in S 100 . When the positional data of the grid points P 1  is represented by gradation values of 256 gradations and the number of grid points arranged in each of the axis directions is 17, the positional data of the grid points P 1  is set to 0, 16, 32, . . . . In S 100 , the host device  100  executes a process of receiving the input gradation values as the positional data of the grid points P 1 . 
     Regarding the input of the color values, when the color conversion table  400  including the color values (Lp, ap, and bp) associated with the grid points P 1  is stored in the storage device  114 , the host device  100  reads the color conversion table  400  into the RAM  113  from the storage device  114 . When a color conversion table  400  is to be newly generated, the host device  100  forms patches corresponding to the colors of the grid points P 1  in the printer or the display device, causes the colorimeter  120  to execute the colorimetry on the patches, acquires color values or colorimetric values from the colorimeter  120 , and associates the grid points P 1  with the color values, thereby newly forming the color conversion table. The host device  100  may acquire the color conversion table  400  from an external device or a recording medium. 
     In subsequent S 102 , a smoothing intensity setting screen  800  shown in  FIG. 4  is displayed by the display device  130 . The smoothing intensity setting screen  800  includes slider control  810  that receives, as a single setting amount, a setting of a smoothing intensity for the color values of the surface grid points P 2  and the color values of the internal grid points P 5 . The slider control  810  is an operational section for sensibly setting a smoothing intensity. An operation of moving a slider  814  along a slider bar  812  is received by the input device  115 . The host device  100  sets the smoothing intensity based on the position of the slider  814  operated by a user, corrects the color values of the surface grid points P 2  based on the set smoothing intensity, and corrects the color values of the internal grid points P 5  to obtain the set smoothing intensity. The smoothing intensity corresponds to a rate c shown in  FIG. 12 . The rate c is changed at six stages from 0 indicating “low” to 1 indicating “high” and can be set to, for example, 0, 0.2, 0.4, 0.6, 0.8, and 1. The host device  100  holds the rate c corresponding to the received setting in at least one of the RAM  113  and the storage device  114 . 
     In S 102 , the smoothing intensity setting corresponding to the correction rate c based on the polynomial approximation coefficients a 0 , . . . , and a d  is received and the smoothing intensity setting for the color values of the internal grid points P 5  is received. 
     In subsequent S 104 , the host device  100  sets lines that are among multiple lines settable on the surface  510  of the grid point region  500  and correspond to the arrangement of the multiple first target grid points P 10  that are among the multiple surface grid points P 2  and to be subjected to the polynomial approximation. The lines are referred to as lines to be processed. 
       FIG. 5  schematically exemplifies a state in which the lines to be processed are set on the surface  510  of the three-dimensional grid point region  500 . The lines to be processed are ridgelines  530  and lines extending on the sectioned surfaces  540 . In the specific example, the host device  100  sequentially sets all the ridgelines  530  in S 122  and sequentially sets the multiple lines extending on the sectioned surfaces  540  in subsequent S 124 . 
     When the device-dependent color space CS 1  is a three-dimensional RGB color space, 12 ridgelines  530  and  6  sectioned surfaces  540  exist. In this case, in S 122 , the host device  100  sequentially sets the lines to be processed from the 12 ridgelines  530 . The ridgelines  530  extend in the first processing direction D 1  in which the multiple first target grid points P 10  are arranged. A number (2×(n−2)) of lines are settable at positions extending on each of the sectioned surfaces  540 . For example, a number (n−2) of lines extending in the R axis direction are settable on a sectioned surface  540  including vertices of K, R, and B, and a number (n−2) of lines extending in the B axis direction are settable on the sectioned surface  540  including the vertices of K, R, and B. Thus, in S 124 , the host device  100  sequentially sets sectioned surfaces  540  to be processed from the  6  sectioned surfaces  540  and sequentially sets lines to be processed from a number (2×(n−2)) of lines on each of the sectioned surfaces  540  to be processed. 
     Although not shown, when the device-dependent color space CS 1  is a four- or higher-dimensional color space, lines to be processed can be set in the same manner as a case in which the device-dependent color space CS 1  is a three-dimensional color space. 
       FIG. 6  schematically exemplifies a state in which lines to be processed are set on a surface  510  of a four-dimensional grid point region  500 . When the device-dependent color space CS 1  is a four-dimensional CMYK color space, 32 ridgelines  530  and 24 sectioned surfaces  540  exist. In this case, in S 122 , the host device  100  sequentially sets lines to be processed from the 32 ridgelines  530 .  FIG. 6  shows coordinates of the 32 ridgelines  530 . It is assumed that components C, M, Y, and K of the CMYK color space are in a range of values of 0 to 100 and change between 0 and 100 in a coordinate system. In subsequent S 124 , the host device  100  sequentially sets sectioned surfaces to be processed from the 24 sectioned surfaces  540  and sequentially sets lines to be processed from a number (2×(n−2)) of lines on each of the sectioned surfaces  540  to be processed.  FIG. 6  shows coordinates of the 24 sectioned surfaces  540 . 
     When the lines to be processed are ridgelines  530 , all the multiple first target grid points P 10  are ridgeline grid points P 3 . When the lines to be processed extend on the sectioned surfaces  540 , the two edge grid points P 11  among the multiple first target grid points P 10  are ridgeline grid points P 3 , and multiple intermediate grid points P 12  are sectioned surface grid points P 4 . 
     In S 106  after S 104  shown in  FIG. 4 , the host device  100  executes polynomial approximation on the multiple surface grid points P 2  located on the lines to be processed and executes a surface smoothing process of smoothing color values. 
       FIG. 7  exemplifies the surface smoothing process executable in S 106 . S 202  to S 206  correspond to the coefficient calculation process ST 1 , the coefficient calculating function FU 1 , and the coefficient calculating unit U 1 . A process of S 202  corresponds to a process ST 1 - 1  included in the coefficient calculation process ST 1 . A process of S 204  corresponds to a process ST 1 - 2  included in the coefficient calculation process ST 1 . A process of S 206  corresponds to a process ST 1 - 3  included in the coefficient calculation process ST 1 . S 210  corresponds to the internal smoothing process ST 2 , the internal smoothing function FU 2 , and the internal smoothing unit U 2 . When a polynomial approximation correction process is started, the host device  100  executes an outlier exclusion process on the lines to be processed in S 202 , executes a weight determination process on the lines to be processed in S 204 , calculates the weighted polynomial approximation coefficients in a weighted polynomial approximation process in S 206 , and calculates approximate values. The outlier exclusion process of S 202  is executed in order to suppress increases in weights of surface grid points having largely varying color values. The weight determination process of S 204  is executed in order to maintain the shape of the gamut surface as much as possible. The weight determination process of S 204  may not be executed. An effect is obtained by the outlier exclusion process not only when the weighted polynomial approximation is used but also when the polynomial approximation is used without a weight. 
       FIG. 8  exemplifies the outlier exclusion process executable in S 202 . The outlier exclusion process corresponds to the process ST 1 - 1  included in the coefficient calculation process ST 1 . The coefficients a 0 , . . . , and a d  of the approximation equation for color values (L i , a i , and b i ) corresponding to the positions x i  in the first processing direction D 1  are calculated for each of the components L, a, and b. In S 302  immediately after the start of the outlier exclusion process, the host device  100  sets components to be processed from the components L, a, and b of the color values. The color values z i  indicate color values of the components to be processed. 
     In subsequent S 304 , the host device  100  calculates polynomial approximation coefficients a 0 , . . . , and a d  of a tentative approximation equation indicating tentative approximate values y i  of the color values z i  corresponding to the positions x i . When four-order polynomial approximation is to be executed, the tentative approximation equation is the aforementioned Equation (1) or y i =a 4 x i   4 +a 3 x i   3 +a 2 x i   2 +a 1 x i +a 0 . The polynomial approximation coefficients a 0 , . . . , and a 4  that are not weighted can be calculated according to the following equation.
 
 A =( X′X ) −1   X′Z   (4)
 
     In subsequent S 306 , the host device  100  uses the tentative approximation equation including the polynomial approximation coefficients a 0 , . . . , and a 4  to calculate tentative approximate values y i  of the first target grid points P 10 . 
     In subsequent S 308 , the host device  100  calculates distances Δz i  from the tentative approximate values y i  to the color values z i  for the first target grid points P 10 .
 
Δ z   i   =|z   i   −y   i |  (5)
 
     In subsequent S 310 , the host device  100  branches the process based on whether all the components L, a, and b of the color values are already set. When a component remains unset, the hose device  100  repeatedly executes the processes of S 302  to S 310 . When all the components L, a, and b are already set, the host device  100  acquires the maximum value among the distances Δz i  based on all the components L, a, and b and determines whether the maximum value is smaller than a threshold T 1  in S 312 . The threshold T 1  is not limited but may be a positive integer in a range of 2 to 5. When the maximum value among the distances Δz i  is equal to or larger than the threshold T 1 , the host device  100  excludes, from the multiple first target grid points P 10 , a grid point P 15  that is to be excluded and has a color value whose distance Δz i  from a tentative approximate value is the maximum value in S 314 . After that, the host device  100  causes the process to return to S 302 . When the maximum value among the distances Δz i  is smaller than the threshold T 1  in S 312 , the host device  100  terminates the outlier exclusion process. Thus, the outlier exclusion process is executed until the maximum value among the distances Δz i  is smaller than the threshold T 1 . 
     As described above, whether the first target grid points P 10  are grid points P 15  to be excluded is determined based on the color values of the multiple first target grid points P 10 . A grid point P 15  to be excluded is excluded from the multiple first target grid points P 10  upon the weighted polynomial approximation described later. 
       FIG. 9  exemplifies the weight determination process executable in S 204  shown in  FIG. 7 . The weight determination process corresponds to the process ST 1 - 2  included in the coefficient calculation process ST 1 . The weight determination process is executed to determine weights w i  of the first target grid points P 10  for the weighted polynomial approximation to be executed after the weight determination process.  FIG. 10  schematically exemplifies a weight w m  of an extreme grid point P 13  appearing at a first target grid point P 10  among the multiple first target grid points P 10  excluding the grid points at both edges. 
     In S 402  immediately after the start of the weight determination process, the host device  100  sets, to 1, weights w i  of first target grid points P 10  excluding a grid point P 15  to be excluded. At this time, the grid point P 15  to be excluded is excluded from the multiple first target grid points P 10 . However, when a weight of the grid point P 15  to be excluded is set to 0, the grid point P 15  may not be excluded and the subsequent weighted polynomial approximation process may be executed. 
     In subsequent S 404 , the host device  100  acquires color values z 0  and z n  of the edge grid points P 11  that are among the multiple first target grid points P 10  and located at the edges in the first processing direction D 1 . S 404  is executed in order to maintain, as much as possible, the shape of a surface that is included in the surface of the gamut and on which a color value changes to a value larger than a range of the color values z 0  and z n .  FIG. 10  shows color values z i  corresponding to the positions x i  in the first processing direction D 1  when z 0 &lt;z n . 
     In subsequent S 406 , the host device  100  searches an extreme grid point P 13  having a color value z i  larger than the range of the color values z 0  and z n .  FIG. 10  shows extreme grid points P 13  having local maximum values z m  larger than the color value z n  in results  1  and  4 . In S 408 , the host device  100  searches an extreme grid point P 13  having a color value z i  lower than the range of the color values z 0  and z n .  FIG. 10  shows extreme grid points P 13  having local minimum values z m  lower than the color value z 0  in results  2 ,  3 , and  5 . 
     In subsequent S 410 , the host device  100  branches the process based on whether the number of extreme grid points P 13  found in S 406  and S 408  is 1. When the number of extreme grid points P 13  found in S 406  and S 408  is 1, the host device  100  increases a weight w m  of the found extreme grid point P 13  in S 412 . After that, the host device  100  terminates the weight determination process. The weight w m  is not limited but may be a value larger than 1 or may be in a range of 2 to 20. On the other hand, when the number of extreme grid points P 13  found in S 406  and S 408  is 0 or 2 or more, the host device  100  terminates the weight determination process without changing the weights of the first target grid points P 10 . 
     For example, in the result  1  shown in  FIG. 10 , the single extreme grid point P 13  that is among the multiple intermediate grid points P 12  exists at a position x m  corresponding to a color value z m  serving as a local maximum value and larger than the color values z 0  and z n  of the edge grid points P 11 . In this case, a weight w m  of the extreme grid point P 13  is larger than 1. 
     In the result  2  shown in  FIG. 10 , the single extreme grid point P 13  that is among the multiple intermediate grid points P 12  exists at a position x m  corresponding to a color value z m  serving as a local minimum value and smaller than the color values z 0  and z n  of the edge grid points P 11 . In this case, a weight w m  of the extreme grid point P 13  is larger than 1. 
     In the result  3  shown in  FIG. 10 , two extreme grid points P 13  that are among the multiple intermediate grid points P 12  exist. A weight w m  of an extreme grid point that is among the extreme grid points P 13  and has a color value z m  smaller than the color value z 0  is larger than 1, while a weight w m  of an extreme grid point that is among the extreme grid points P 13  and has a color value z m  smaller than the color value z n  and larger than z 0  is 1. 
     In the result  4  shown in  FIG. 10 , an extreme grid point P 13  that is among the multiple intermediate grid points P 12  exists at a position corresponding to a color value z m  serving as a local minimum value and smaller than the color value z 0  that is smaller than the color value z n , and an extreme grid point P 13  that is among the multiple intermediate grid points P 12  exists at a position corresponding to a color value z m  serving as a local maximum value and larger than the color value z n  that is larger than the color value z 0 . In this case, weights w m  of the extreme grid points P 13  are 1. 
     In the result  5  shown in  FIG. 10 , two extreme grid points P 13  exist among the multiple intermediate grid points P 12 . Since color values z m  of the two extreme grid points P 13  are smaller than the color value z n  and larger than the color value z 0 , weights w m  of the two extreme grid points P 13  are 1. 
     In the aforementioned manner, the host device  100  determines weights w i  of the first target grid points P 10  based on the color values z i  of the multiple first target grid points P 10 . The weights w i  of the first target grid points P 10  are used to calculate the weighted polynomial approximation coefficients a 0 , . . . , and a 4 . When a single extreme grid point P 13  exists among multiple first target grid points P 10  excluding a grid point P 15  to be excluded and has a color value z i  corresponding to a position x i  and serving as an extreme larger or smaller the range of the color values z 0  and z n  of the edge grid points P 11  located at the edges in the first processing direction D 1 , a weight w m  of the extreme grid point P 13  is the largest among the weights w i  of the first target grid points P 10 . 
       FIG. 11  exemplifies the weighted polynomial approximation process executable in S 206  shown in  FIG. 7  or a process of calculating the weighted polynomial approximation coefficients and the approximate values. The weighted polynomial approximation process corresponds to the process ST 1 - 3  included in the coefficient calculation process ST 1 . The weighted polynomial approximation process is executed to calculate the approximate values y i  as a preprocess of correcting the color values z i  of the first target grid points P 10 . 
     In S 502  immediately after the start of the weighted polynomial approximation process, the host device  100  sets components to be processed from the components L, a, and b of the color values. The color values z i  indicate color values of the components to be processed. 
     In subsequent S 504 , the host device  100  calculates the weighted polynomial approximation coefficients a 0 , . . . , and a d  of the approximation equation indicating the approximate values of the color values z i  corresponding to the positions x i . When a grid point P 15  to be excluded is included in the multiple first target grid points P 10  arranged in the first processing direction D 1 , the process of S 504  is executed on the first target grid points P 10  excluding the grid point P 15  to be excluded. When the four-order polynomial approximation is to be executed, the approximation equation is the aforementioned Equation (1) or y i =a 4 x i   4 +a 3 x i   3 +a 2   x   i   2 +a 1 x i +a 0 . 
     The weighted polynomial approximation coefficients a 0 , . . . , and a 4  are calculated according to the aforementioned Equation (2) or A=(X′WX) −1 X′WZ. 
     In subsequent S 506 , the host device  100  uses the approximation equation (1) including the weighted polynomial approximation coefficients a 0 , . . . , and a 4  to calculate the approximate values y i  of the first target grid points P 10 . Since an approximate value y i  needs to be obtained for the grid point P 15  to be excluded, the process of S 506  is executed on the multiple first target grid points P 10  including the grid point P 15  to be excluded. 
     In subsequent S 508 , the host device  100  branches the process based on whether all the components L, a, and b of the color values are already set. When a component remains unset, the host device  100  repeatedly executes the processes of S 502  to S 508 . When all the components L, a, and b are already set, the host device  100  terminates the weighted polynomial approximation process. 
     In the aforementioned manner, the host device  100  calculates the weighted polynomial approximation coefficients a 0 , . . . , and a d  used in the approximation equation for calculating the approximate values y i  of the color values z i  corresponding to the positions x i  in the first processing direction D 1  for the multiple first target grid points P 10  arranged in the first processing direction D 1 . 
     After the weighted polynomial approximation process, the host device  100  executes a process of maintaining the color values z 0  and z n  of the edge grid points P 11  that are among the multiple first target grid points P 10  and located at the edges in the first processing direction D 1  in S 208  shown in  FIG. 7 . The process of S 208  may be a process of replacing approximate values y 0  and y n  corresponding to the positions x 0  and x n  of the edge grid points P 11  with the original color values z 0  and z n . 
     In subsequent S 210 , the host device  100  associates the first correction values r i  based on the correction rate c corresponding to the smoothing intensity set in S 102  shown in  FIG. 4  with the first target grid points P 10 . The first correction values r i  are based on the polynomial approximation coefficients a 0 , . . . , and a d  in a case in which the color values of the first target grid points P 10  are corrected. After that, the host device  100  terminates the polynomial approximation correction process. 
       FIG. 12  schematically exemplifies a state in which the color values z i  are corrected based on the correction rate c in S 210  shown in  FIG. 7 . A correction process shown in  FIG. 12  is a process of calculating weighted averages of the approximate values y i  based on the polynomial approximation coefficients a 0 , . . . , and a d  and the original color values z i  using the ratio of c: (1−c). An approximate curve C 1  shown in  FIG. 12  indicates the approximate values y i  for the positions x i  of the first target grid points P 10 . The first correction values r i  are calculated according to the following Equation (6).
 
 r   i   =c×y   i +(1− c )× z   i   (6)
 
     When the correction rate c is 1, the first correction values r i  of the multiple first target grid points P 10  excluding the edge grid points P 11  are equal to the approximate values y i . When the correction rate c is 0, the first correction values r i  of the multiple first target grid points P 10  are equal to the original color values z i . 
     As described above, the host device  100  uses, as the correction rate c, a weight for the approximate values y i  obtained by the polynomial approximation to associate the weighted averages, serving as the first correction values r i , of the approximate values y i  and the color values z i  with the first target grid points P 10 . 
     In S 108  after S 106  shown in  FIG. 4 , the host device  100  branches the process based on whether all the lines settable on the surface  510  of the grid point region  500  are already set on the surface  510 . When a line remains unset on the surface  510 , the host device  100  repeatedly executes the processes of S 104  to S 108 . When all the lines are already set on the surface  510 , the host device  100  causes the process to proceed to S 110 . 
     As shown in  FIGS. 5 and 6 , lines to be processed are sequentially set from the multiple ridgelines  530 . In this case, a line to be processed is set for each of the vertices  520  multiple times. Since the color values of the edge grid points P 11  located at vertices  520  are maintained, correction values for the color values of the ridgeline grid points P 3  do not depend on the order in which the lines that are the multiple ridgelines  530  and to be processed are set. 
     After the polynomial approximation coefficients a 0 , . . . , and a d  are calculated for the multiple ridgeline grid points P 13  by setting all the ridgelines  530  as lines, and the color values are corrected by the polynomial approximation equation, the polynomial approximation coefficients a 0 , . . . , and a d  are calculated for the multiple sectioned surface grid points P 4 , and the color values are corrected by the polynomial approximation equation. Since the color values of the edge grid points P 11  located on the ridgelines  530  are maintained, the correction values for the color values of the ridgeline grid points P 3  on the sectioned surfaces  540  and correction values for the color values of the sectioned surface grid points P 4  on the sectioned surfaces  540  do not depend on the order in which the lines are set. A reduction in the color reproduction range is appropriately suppressed by setting all the ridgelines  530  as the lines. 
     In subsequent S 110 , the host device  100  sets a line that is among multiple lines settable at positions extending in the internal region  550  of the grid point region  500  and corresponds to the arrangement of multiple second target grid points P 20  that are among the multiple grid points P 1  and to be subjected to smoothing. 
       FIG. 13  schematically exemplifies a state in which lines that are to be processed and on which the smoothing is executed at positions extending in the internal region  550  of the three-dimensional grid point region  500  are set. When the device-dependent color space CS 1  is a three-dimensional RGB color space, cross-sections that are perpendicular to the R axis and extend through multiple internal grid points P 5  are a number (n−2) of planes, cross-sections that are perpendicular to the G axis and extend through multiple internal grid points P 5  are a number (n−2) of planes, and cross-sections that are perpendicular to the B axis and extend on multiple internal grid points P 5  are a number (n−2) of planes. A number (2×(n−2)) of lines are settable on each of the cross-sections. For example, a number (n−2) of lines extending in the G axis are settable on each of the cross-sections perpendicular to the R axis, and a number (n−2) of lines extending in the B axis are settable on each of the cross-sections perpendicular to the R axis. Thus, the host device  100  sequentially sets cross-sections to be processed from a number (3×(n−2)) of cross-sections and sequentially sets lines to be processed from a number (2×(n−2)) of lines for the cross-sections to be processed. 
     Although not shown, when the device-dependent color space CS 1  is a four- or higher-dimensional color space, lines to be processed can be set in the same manner as a case in which the device-dependent color space CS 1  is a three-dimensional color space. 
     When a line to be processed extends in the internal region  550  of the grid point region  500 , surface grid points P 2  that are among the multiple second target grid points P 20  and located at the edges in the second processing direction D 2  are sectioned surface grid points P 4 , and remaining grid points among the multiple second target grid points P 20  are internal grid points P 5 , as shown in  FIG. 2 . 
     In S 112  after S 110  shown in  FIG. 4 , the host device  100  smoothes the color values z i  of the multiple internal grid points P 5  on lines to be processed. The internal smoothing process executed in S 112  is different from the polynomial approximation and can be treated as a process of executing the weighted averaging described with reference to  FIGS. 2 and 14 . 
       FIG. 14  schematically exemplifies a state in which weighted averages y i  as a result of the smoothing of the color values z i  of the multiple internal grid points P 5  included in the multiple second target grid points P 20  are calculated.  FIG. 14  shows a filter F 1  for calculating the weighted averages y i .  FIG. 15  schematically exemplifies the filter F 1  for calculating the weighted averages y i  of the color values z i  based on weights associated with the set smoothing intensity. The filter F 1  shown in  FIGS. 14 and 15  has weights w j  applied to color values z i+j  of 5 internal grid points including an internal grid point located at a processing position x i  and treated as the center of the 5 internal grid points. In this case, j is a variable identifying an internal grid point P 5  to which a weight is applied in a case in which the processing position x i  is treated as the center. The number of weights w j  of the filter F 1  is not limited to 5 and may be 3 or may be 7 or more. The number of weights w j  of the filter F 1  may be set based on a number n of grid points arranged in each of the axis directions. 
     The weighted averages y i  are calculated according to the aforementioned Equation (3) or the following equation. 
     
       
         
           
             
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     The weights w j  of the filter F 1  change in a stepwise manner based on the smoothing intensity set in S 102  shown in  FIG. 4 , as shown in  FIG. 15 . The smoothing intensity corresponds to the rate c shown in  FIG. 12  as a value. For example, when the smoothing intensity is 1, the smoothing intensity is highest and all the weights w j  is 1. In this case, the weighted averages y i  are simple averages of the color values z i+j . The simple averages are included in the weighted averages y i . When the smoothing intensity is 0.8 that is second highest, weights w −1 , w 0 , and w 1  are 1, and weights w −2  and w 2  are 0.5. When the smoothing intensity is 0 that is lowest, the weight w 0  is 1, and the weights w −2 , w −1 , w 1 , and w 2  are 0. In this case, the weighted averages y i  are equal to the original color values z i . The color values z i  are included in the weighted averages y i . 
     As described above, the host device  100  treats, as the second correction values, weighted averages y i  of color values z i+j  associated with grid points including grid points adjacent to each other in the second processing direction D 2  and associates the weighted averages y i  as the second correction values with the multiple second target grid points P 20  that are among the multiple internal grid points P 5  and arranged in the second processing direction D 2 . Thus, the color values z i  associated with the internal grid points P 5  are smoothed with the set smoothing intensity. 
     In S 114  after S 112  shown in  FIG. 4 , the host device  100  branches the process based on whether all the lines settable at the positions extending in the internal region  550  of the grid point region  500  are already set. When a line extending in the internal region  550  of the grid point region  500  remains unset, the host device  100  repeatedly executes the processes of S 110  to S 114 . When all the lines are already set at the positions extending in the internal regions  550 , the host device  100  terminates the smoothing process. 
       FIG. 16  shows results of comparing a case in which color values of surface grid points are corrected by the weighted polynomial approximation with a case in which the color values of the surface grid points are corrected by the weighted averaging for the color conversion table that is before the correction and includes, as color values (Lp, ap, and bp), colorimetric values of patches corresponding to colors of grid points arranged in the CMYK color space. In  FIG. 16 , circles indicate chromaticity coordinates a and b on a ridgeline indicating colors from white to cyan in the case in which the color values are corrected by the weighted polynomial approximation, crosses indicate the chromaticity coordinates a and b on the ridgeline indicating the colors from white to cyan in the case in which the color values are corrected by the weighted averaging, and a broken line indicates the chromaticity coordinates a and b on the ridgeline indicating the colors from white to cyan before the correction. 
     Ridgelines in the grid point region included in the CMYK color space correspond to the surface of the gamut in the Lab color space. Thus, the chromaticity coordinates a and b shown in  FIG. 16  correspond to gamut surface shapes indicated in the color conversion table. 
     As shown in  FIG. 16 , when the color values are corrected by the weighted averaging, the chromaticity coordinates b on the ridgeline indicating the colors from white to cyan are smaller than those before the correction. This is considered to be due to the fact that a bulge of the surface of the gamut is reduced by the weighted averaging executed on the color values. When the color values are corrected by the weighted polynomial approximation, the chromaticity coordinates b on the ridgeline indicating the colors from white to cyan are the same as or almost the same as those before the correction and larger than those in the case in which the color values are corrected by the weighted averaging. This is considered to be due to the fact that a variation in the color values is reduced and a change in the shape of the gamut surface can be reduced.  FIG. 16  shows that smooth gradation expression can be enabled while a change in the shape of the gamut surface is damaged as little as possible. 
     As described above, this specific example can provide the smoothing method for enabling smooth gradation expression while suppressing a change in the shape of the original gamut surface, compared with the case in which the color values associated with the multiple surface grid points are averaged. 
     Color values in the gamut are smoothed by correcting the color values of the internal grid points P 5  after the correction of the color values of the surface grid points P 2  by the polynomial approximation, while a change in the shape of the gamut surface is damaged as little as possible. Thus, smooth gradation expression is enabled, while a change in the shape of the gamut surface is suppressed as much as possible. In addition, the color values of the internal grid points P 5  are quickly corrected by applying weighted averages on the color values of the internal grid points P 5 . 
     (5) Modified Examples 
     Various modified examples are considered in the disclosure. 
     For example, the output device is not limited to the ink jet printer  200  and may be an electrophotographic printer such as a laser printer using toner as color materials, a three-dimensional printer, a display device, or the like. 
     Color materials that form an image are not limited to the C, M, Y, and K ink and may include not only the C, M, Y, and K ink but also Lc, Lm, DY, Or, Gr, and Lk ink and an uncolored material for image quality improvement. The Lc ink has a lower color density than the C ink, the Lm ink has a lower color density than the M ink, the DY ink has a higher color density than the Y ink, and the Lk ink has a lower color density than the K ink. Lc indicates light cyan, Lm indicates light magenta, DY indicates dark yellow, Or indicates orange, Gr indicates green, and Lk indicates light black. 
     The aforementioned processes may be changed. The order in which the processes are executed may be changed, one or more of the processes may be omitted, and another process may be added to the processes. For example, in the weight determination process shown in  FIG. 9 , the process of S 408  may be executed before the process of S 406 . 
     In the embodiment, the processes may be executed in a state in which the smoothing intensities for the color values of the surface grid points P 2  do not include the intensity corresponding to 0. The process of S 102  shown in  FIG. 4  may be omitted. The approximate values y i  may be treated as the first correction values and associated with the surface grid points P 2 . A weight w i+j  of the filter F 1  may be a fixed value. 
     In the embodiment, the smoothing intensity for the color values of the surface grid points P 2  is associated with the smoothing intensity for the color values of the internal grid points P 5 . However, the smoothing intensity for the surface grid points and the smoothing intensity for the internal grid points may be separately received. 
     In the polynomial approximation correction process shown in  FIG. 7 , the process of maintaining the color values z 0  and z n  of the edge grid points P 11  in S 208  may be omitted, and the first correction values may be associated with the multiple first target grid points P 10  excluding the edge grid points P 11  in S 210 . 
     In addition, the polynomial approximation coefficients a 0 , . . . , and a d  may be determined so that the approximate values y 0  and y n  corresponding to the positions x 0  and x n  of the edge grid points P 11  are equal to the original color values z 0  and z n . 
     When approximate values are calculated by the polynomial approximation process based on the polynomial approximation coefficients for color values associated with at least some of the multiple surface grid points P 2  and are smoothed, an effect of enabling smooth gradation expression while suppressing a change in the shape of the gamut surface as much as possible is obtained by the smoothing of the color values associated with the multiple grid points. 
     Thus, the aforementioned effect is obtained even when approximate values are calculated for the color values of the multiple internal grid points P 5  by the polynomial approximation process based on the polynomial approximation coefficients and are smoothed or even when the color values of the multiple internal grid points P 5  are not corrected. 
     In addition, the aforementioned effect is obtained even when the weighted polynomial approximation is not executed on the color values of the multiple surface grid points P 2  and the polynomial approximation is executed on the color values of the multiple surface grid points P 2  without using a weight. 
     (6) Conclusion 
     As described above, according to the disclosure, each of the aspects can provide a technique for enabling smooth gradation expression while suppressing a change in the shape of the gamut surface as much as possible. Even in a technique enabled by only configuration requirements according to each of independent claims, the aforementioned basic effects are obtained. 
     In addition, a configuration obtained by replacing configurations among the configurations described in the aforementioned examples with each other, a configuration obtained by changing a combination of configurations among the configurations described in the aforementioned examples, a configuration obtained by replacing a configuration among the configurations described in the aforementioned examples with a configuration of a known technique, a configuration obtained by changing a combination of a configuration among the configurations described in the aforementioned examples and a configuration of a known technique, and the like are enabled. The disclosure includes these configurations.