BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to a profile creation method, a profile creation apparatus, and a medium recording a profile creation program for creating a profile which defines correspondence between a first color space range represented by a device-dependent component coordinate and a second color space range represented by a device-independent component coordinate corresponding to the first color space range. 
   2. Description of Related Art 
   There are known such type of apparatuses that determine polynomial coefficients by the multiple linear regression analysis for extrapolation, optimize 3*3 matrices for extrapolation, and create an input chart for the ease of extrapolation. A similar apparatus is known as disclosed in JP-A No. 27272/2002, for example. 
   The above-mentioned conventional apparatuses cause too large a failure in extrapolated portions due to the multiple linear regression analysis or the 3*3 matrix optimization. Though input charts for easy extrapolation are available, the extrapolation may be forced to use a patch chart that becomes cylindrical in the saturation value space, for example. Charts are limited to such format. In addition, it is very difficult to create an accurate input chart that forms a cylindrical saturation value space. 
   SUMMARY OF THE INVENTION 
   The present invention has been made in consideration of the foregoing. It is therefore an object of the present invention to provide a profile creation method, a profile creation apparatus, and a profile creation program capable of creating a profile with a few failures in an extrapolation region. 
   In order to achieve the above-mentioned object, the present invention provides a profile creation apparatus for creating a profile to define correspondence between a first color space range expressed by a device-dependent component coordinate and a second color space range expressed by a device-independent component coordinate corresponding to the first color space range. For this purpose, a first gamut acquisition step is first used to acquire a first gamut for the first color space range based on a chart supplied from a specified input device. A second gamut acquisition step is then used to measure a color in the chart and acquiring a second gamut for the second color space range. The present invention uses a first reduced region acquisition step to acquire a reduced first color space range by reducing the first color space range into the first gamut and acquire a reduced first gamut by reducing the first gamut based on a corresponding reduction ratio. 
   A second reduced region acquisition step is used to acquire a reduced second color space range and a reduced second gamut corresponding to the reduced first color space range and the reduced first gamut based on a specified interpolation algorithm. A magnification ratio calculation step is used to calculate a first magnification ratio corresponding to the second gamut for each grid in the acquired reduced second gamut and calculate a second magnification ratio from a reduced second color space range to a second color space range based on the calculated first magnification ratio. A profile creation step is used to calculate a second color space range based on the calculated second magnification ratio and create a profile defining correspondence between the first color space range and the second color space range based on a calculation result. Such technique enables creation of the profile hardly subject to a failure due to extrapolation. 
   There is provided an example of the technique to reduce the first color space range into the first gamut. The first reduced region acquisition step reduces the first color space range into a first gamut by allowing at least one point on an outline of the reduced first color space range to correspond to an outline of the first gamut. 
   There is provided an example of the technique to calculate each magnification ratio using the magnification ratio calculation step. The magnification ratio calculation step calculates the first magnification ratio and the second magnification ratio based on a polar coordinate system in a color space for the grid. 
   There is provided a specific technique to calculate the second magnification ratio using the magnification ratio calculation step. The magnification ratio calculation step calculates the second magnification ratio using an interpolation operation corresponding to the first magnification ratio. 
   There is provided an example of the technique to calculate the magnification ratio for a grid available in the reduced second color space range outside the reduced second gamut. The magnification ratio calculation step forms a line by connecting a reduction center to a coordinate of a grid in the reduced second color space range outside the reduced second gamut, allows the line to intersect with a surface of the reduced second gamut at an intersecting point, and uses a magnification ratio obtained from this intersecting point as a second magnification ratio. 
   There is provided an example of solving a case where the calculated second color space range exceeds an allowable range. When the calculated second color space range exceeds an allowable range, the profile creation step creates the profile by compressing the calculated second color space range to the allowable range. 
   It is to be distinctly understood that the above-mentioned profile creation method can be implemented as a profile creation apparatus to embody the method. Further, it is to be distinctly understood that the present invention is embodied as a program allowing a computer to be able to implement a function equivalent to the profile creation apparatus. Media to record the program include various computer-readable media such as a flexible disk, CD-ROM, magnetic optical disk, IC card, ROM cartridge, punched card, printed material with a printed code such as a bar code, computer&#39;s internal storage (memory such as RAM and ROM), and external storage. 


   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a configuration diagram showing the external configuration of a profile creation apparatus; 
       FIG. 2  is a configuration diagram showing the internal configuration of the profile creation apparatus; 
       FIG. 3  is a configuration diagram showing the software configuration of the profile creation apparatus; 
       FIG. 4  is a configuration diagram showing the software configuration of the profile creation apparatus; 
       FIG. 5  is a configuration diagram showing the configuration of RGB gamut and range; 
       FIG. 6  is a configuration diagram showing the configuration of Lab gamut and range; 
       FIG. 7  is a configuration diagram showing the configuration of reduced RGB gamut and range; 
       FIG. 8  is a configuration diagram showing the configuration of reduced Lab gamut and range; 
       FIG. 9  is an explanatory diagram showing polar coordinates; and 
       FIG. 10  is a flowchart showing the contents of a profile creation process. 
   


   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The embodiment of the present invention will be described according to the following topics. 
   (1) Profile creation apparatus configuration 
   (2) Profile creation algorithm 
   (3) Conclusion 
   (1) Profile Creation Apparatus Configuration 
     FIG. 1  is a schematic perspective view showing the hardware configuration of the profile creation apparatus according to an embodiment of the present invention.  FIG. 2  is a block diagram showing major parts. In  FIGS. 1 and 2 , a scanner  40  is connected to a computer  10  via a USB cable. A calorimeter  41  is connected to the computer via a parallel cable. The scanner  40  has a light emitting diode and an image sensor though not shown in  FIGS. 1 and 2 . The scanner  40  performs a scan based on instructions supplied from the computer  10  via the USB cable, generates and outputs image data representing each pixel in three primary colors RGB. During scanning, the scanner  40  activates the light emitting diode by moving it to radiate the light to scan objects such as a color chart mounted on a scanning plane. The image sensor moves to detect reflected light from the scan object. The scanner  40  generates pixel data represented in 256 levels for each of three primary colors RGB based on an output signal from the image sensor. 
   The embodiment uses a color chart  42  compliant with the ANSI standard to create an ICC profile. ANSI-standard color charts include reflective and transparent documents. If the scanner  40  operates in a mode of scanning transparent documents, a similar ICC profile can be created. Of course, the other color charts can be used alike. Nevertheless, ANSI compliant charts are preferable because they provide the proper number of colors, appropriately scatter colors within a color region reproducible by the scanner  40 , and are easily available. The scanner  40  is a preferred example of input apparatuses. However, the present invention is not limited thereto and can be applied to the ICC profile creation for various image data input apparatuses such as digital cameras. 
   The calorimeter  41  has a spectrophotometer given freedom to two-dimensionally move on a mounting plane of the color chart  42 . The calorimeter  41  outputs coordinate values (Lab values) in the CIELab color system for each color patch based on an output signal from the spectrophotometer. The spectrophotometer moves on the mounting plane of the color chart  42  and scans each color patch to acquire coordinate values in the device-independent color space for each color patch in the color chart  42 . Of course, transparent color charts can be also used. For the purpose of creating a profile, it is possible to use color charts not compliant with the ANSI standard. 
   The calorimeter  41  is used to acquire coordinate values in the device-independent color space for color patches of the color chart  42 . In consideration of color changes in the color chart  42 , it is preferable to use the calorimeter  41  each time the ICC profile is created. Considering that the color chart  42  is compliant with a specified standard, it may be possible to allow the ICC profile to be created by a user who does not have the colorimeter  41 . In such case, Lab values corresponding to color patches are supplied by a flexible disk and the like. The colorimeter  41  may measure colors not only in the Lab space, but also in the XYZ space and later perform conversion between the color spaces. When an ICC profile to be created is not compliant with the ICC standard, it is also possible to use a space other than the Lab space as the reference space. 
   The computer  10  has a CPU  11  as a core of operation processes. The CPU  11  can access ROM  13  containing BIOS and RAM  14  via a system bus  12 . The system bus  12  connects with external storage apparatuses such as a hard disk drive  15 , a flexible disk drive  16 , and a CD-ROM drive  17 . The hard disk drive  15  stores an operating system (OS)  20  and an application (APL) that are transferred to the RAM  14 . The CPU  11  accesses the ROM  13  and the RAM  14  as needed to execute the software. According to the embodiment, the profile creation program can be transferred from the hard disk drive  15  to the RAM  14  and executed likewise. 
   A serial communication I/O  19   a  connects with input operation devices such as a keyboard  31  and a mouse  32 . The serial communication I/O  19   a  also connects with a display  18  for display purpose via a video board (not shown). The serial communication I/O  19   a  is capable of connection with the scanner  40  via a USB I/O  19   b  and is capable of parallel connection with the colorimeter  41  via a parallel communication I/O  19   c . The configuration of the computer  10  is simplified but may comply with the general configuration for personal computers and workstations. 
   Of course, the present invention is not limited to personal computers. While the embodiment uses a so-called desktop computer, notebook or mobile computers may be used. The computer  10  can be connected to the scanner  40  and the calorimeter  41  via not only communication interfaces such as the USB I/O 19   b  and the parallel communication I/O 19   c , but also various types of connection modes such as ordinary serial interfaces, SCSI and IEEE1394 connections. The same applies to any connection modes that may be developed in the future. 
   In this example, the hard disk drive  15  stores individual programs constituting the profile creation program. Recording media are not limited thereto. For example, a flexible disk  16   a  or a CD-ROM  17   a  may be used for this purpose. The computer  10  reads programs recorded on these recording media via a flexible disk drive  16  and a CD-ROM drive  17 . The programs are installed on the hard disk drive  15 . The programs are read from the hard disk drive  15  to the RAM  14  to control a host computer. Recording media are not limited thereto and may include magnetic optical disks. Further, it is possible to use nonvolatile memory such as flash cards as a semiconductor device. Even when an access is made to an external file server for downloading via a modem and a communication line, it is obvious that the computer&#39;s storage section can function as a recording medium. 
     FIG. 3  is a block diagram showing the configuration of major parts of the computer  10  for creating the ICC profile. In  FIG. 3 , a scanner driver  21  is installed in an OS  20  of the computer  10 . The scanner driver  21  controls the scanner  40  via the USB I/O  19   b . When the scanner  40  is driven under control of the scanner driver  21 , the scanner  40  outputs RGB pixel data that is then input to the computer  10  via the USB I/O  19   b . The scanner driver  21  stores the data as an RGB file  23  in the hard disk drive  15 . Each color patch of the color chart  42  contains an alphabetic letter and a numeric corresponding to each other. The RGB data file  23  stores data by maintaining the correspondence between the data and the associated color patch. 
   A colorimeter driver  22  is installed in the OS  20  and controls the calorimeter  41  via the parallel communication I/O  19   c . Under control of the colorimeter driver  22 , the colorimeter  41  is driven and outputs coordinate values in the device-independent color space. The values are input via the parallel communication I/O  19   c . The colorimeter driver  22  stores data as a Lab data file  24  in the hard disk drive  15 . The Lab data file  24  also stores data by maintaining the correspondence between the data and the associated color patch in the color chart  42 . 
   According to the embodiment, a profile creation module  25  is executed with the RGB data file  23  and the Lab data file  24  stored in the hard disk drive  15  to create an ICC profile compliant with the ICC standard. The created ICC profile is stored as an ICC profile  26  in the hard disk drive  15 . Operations of the profile creation module  25  will be described later. 
   When capturing image data, the scanner  40  references the ICC profile  26  to perform an interpolation. In this manner, RGB values output from the scanner  40  correspond to coordinate values in the Lab space as a device-independent color space. Accordingly, almost the same color is used for the scan image, display, and printout as long as the printer and the display use ICC standard complaint profiles and the scanner  40  use the ICC profile  26 . At this time, the calorimeter  41  is not needed any more.  FIG. 4  is a block diagram showing an example of the configuration for scan using the ICC profile. In  FIG. 4 , the computer  10  has the same configuration as an ordinary personal computer used for the above-mentioned profile creation. The computer  10  references the ICC profile when using the scanner  40 . 
   An APL  27  is installed in the computer  10 . Using the APL  27 , a user issues instructions to capture images and drives the scanner  40  to acquire intended image data. At this time, the OS  20  uses an ICM (Image Color Management)  28  to process colors of the image data handled by the APL  27 . The ICM  28  is an API (Application Program Interface) provided with a CMM (Color Management Module)  28   a  that converts input and output data from each device into Lab values by referencing the ICC profile. 
   When the APL  27  generates an instruction to capture an image, the scanner is driven under control of the scanner driver  21  to input given RGB data via the USB I/O  19   b . The CMM  28   a  of the ICM  28  references the ICC profile to find Lab values corresponding to the RGB data. Colors are output to the printer and the display based on the Lab values. Therefore, color data for the devices are always handled through the intermediation of the Lab values, i.e., values for the device-independent color space. The color data are independent of devices and operating systems. Some colors are treated as approximately the same. 
   The embodiment discloses a technique to predict colors without failure in the ICC profile creation, especially, in an extrapolation algorithm. Generally, it is almost impossible to cover all device RGB values representable on a device using the saturation value space i.e., the color chart  42  that can be input to the input device such as the scanner  40 . The color chart  42  input to devices is generally available as the IT 8  chart for the scanner  40  or the Macbeth color checker for DSC. Accordingly, it is necessary to predict saturation values corresponding to the entire device RGB by extrapolating outside the input saturation value space. Generally, however, the extrapolation algorithm provides no available data near an expected extrapolation value and causes an extrapolation result to fail in most cases. This tendency becomes more obvious at a point more distant from the input saturation value space. To solve this problem, an algorithm is executed in the profile creation module  25  and aims at minimizing a failure in the extrapolation region. 
   (2) Profile Creation Algorithm 
   The following describes the basic concept of the algorithm executed in the profile creation module  25 . The algorithm implements the extrapolation by once reducing an RGB range and an RGB gamut. The embodiment uses a term “gamut” to represent a region of saturation values acquired by input to an input device such as the scanner  40 . A gamut of saturation values acquired from the input color chart  42  is referred to as a Lab gamut (second gamut according to the present invention). A gamut acquired from the device is referred to as an RGB gamut (first gamut according to the present invention). The range of RGB values representable on input devices is assumed to be 0 to 255. This range is referred to as an RGB range (first color space range according to the present invention). The entire Lab region is generated from color prediction in the RGB range and is referred to as a Lab range (second color space range according to the present invention). 
     FIGS. 5 and 6  exemplify shapes of the RGB range, the RGB gamut, the Lab range, and the Lab gamut. In  FIG. 6 , any points in a Lab gamut G 2  can be found by interpolation using a predetermined RGB gamut G 1  and the Lab gamut G 2 . However, an extrapolation needs to be performed to calculate any Lab values in the Lab range R 2  and outside the Lab gamut G 2 . The RGB values corresponding to an outline of the Lab range R 2  constitute an outline of the RGB range R 1 . As shown in  FIGS. 7 and 8 , an RGB reduction center C 1  is assumed to be a point within the RGB range R 1  and the RGB gamut G 1 . The RGB range R 1  and the RGB gamut G 1  are reduced at a constant reduction ratio against the RGB reduction center C 1  so that the entire RGB range R 1  fits into the RGB gamut G 1 . The reduced regions are referred to as a reduced RGB range R 11  and a reduced RGB gamut G 11 , respectively. Since a minimum reduction ratio is preferable, the outline of the reduced RGB range R 11  contains at least one point corresponding to the outline of the RGB gamut G 1 . 
   As mentioned above, the region fit into the RGB gamut G 1  is capable of relatively accurate color prediction by interpolation. A given interpolation algorithm can be used to find a reduced Lab gamut G 21  and a reduced Lab range R 21  corresponding to the reduced RGB gamut G 11  and the reduced RGB range R 11 . Since the RGB reduction center C 1  also exists in the RGB gamut G 1 , an interpolation can be used to find a Lab reduction center C 2  corresponding to the RGB reduction center C 1 . At this point, the reduced Lab gamut G 21  and the Lab gamut G 2  are known. It is understood that the Lab gamut G 2  results from multiplying the reduced Lab gamut G 21  by a against the Lab reduction center C 2 . 
   It is obvious that the magnification ratio α is not a constant and varies with the corresponding grid. The magnification ratio α cannot be a simple scaling factor, either. The detail will be described later. It should be noted that the Lab range R 2  can be found by determining a magnification ratio α conforming to the magnification ratio α for the reduced Lab range R 21 . Therefore, the magnification ratio α′ just needs to be found to find the Lab range R 2  for an entire object of color prediction. The above-mentioned is the basic concept of the extrapolation algorithm based on the gamut reduction performed by the profile creation module  25  according to the embodiment. Since the embodiment uses the technique based on the reduced gamut extrapolation as mentioned above, a reduced gamut, if used, can be restored by using the magnification ratio α within the Lab gamut G 2 . It is possible to expect almost the same accuracy as for the ordinary interpolation. Failure-free extrapolation can be expected even outside the Lab gamut G 2 . 
   The following describes how to determine a reduction ratio for reducing the RGB range R 1  to the reduced RGB range R 1 , reduce the RGB range R 1  and the RGB gamut G 1  using the scaling factor, and calculate the reduced Lab range R 21  and the reduced Lab gamut G 21 . In this case, the RGB range R 1  is first sampled at a proper interval to determine grids of the RGB range R 1 . For example, each of RGB is configured at the same interval of 17 grids. When the grids are reduced at a specified reduction ratio toward the reduction center (e.g., (R, G, B)=(125.5, 127.5, 127.5)), the range is assumed to reach the surface of the RGB gamut G 1 . All grids of the RGB range R 1  are inspected to use the minimum value of reduction ratios for all the grids. In this manner, at least one point in the reduced RGB range R 11  is found on the surface of the RGB gamut G 1 . The entire reduced RGB range R 11  can be included in the RGB gamut G 1 . Therefore, the minimum reduction ratio is found from the grids&#39; reduction ratios. Then, the found minimum reduction ratio is used to reduce the RGB range R 1  and the RGB gamut G 1  and find the reduced RGB range R 1  and the reduced RGB gamut G 11 . 
   As mentioned above, both the reduced RGB range R 11  and the reduced RGB gamut G 11  exist in the RGB gamut G 1 . Therefore, an interpolation operation can be used to find the reduced Lab range R 21  and the reduced Lab gamut G 21 . Accordingly, both are found by using grid data for the RGB gamut G 1  and the Lab gamut G 2 . The Lab reduction center C 2  corresponding to the RGB reduction center C 1  is found by interpolating the RGB gamut G 1  and the Lab gamut G 2 . 
   The next purpose is to determine a magnification ratio from the reduced Lab gamut G 21  to the Lab gamut G 2 . In this case, as mentioned above, it is inconvenient to allocate a simple constant to the scaling factor for conversion from the reduced Lab gamut G 21  to the Lab gamut G 2 . To solve this problem, the embodiment finds the scaling factor as two rotation angles and one scaling factor in a polar coordinate system. When a position vector P in the Lab space is represented as (Lp, ap, bp) according to the rectangular coordinate system, the position vector P is converted into the polar coordinate system like (Lp, ap, bp)=(rp, θp, φp).  FIG. 9  shows this polar coordinate system. In  FIG. 9 , changes in the coordinate system can be expressed by the following equations. 
   
     
       
         
           
             
               
                 
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   In these equations, Lc, ac, and bc are coordinates of the Lab reduction center C 2  represented in the rectangular coordinate system. A given grid P in the Lab gamut G 2  and a grid P′ in the reduced Lab gamut G 21  are converted from the rectangular coordinate system to the polar coordinate system to find rp, θp, φp, rp′, θp′, and φp′, respectively. These elements are used to find a magnification ratio for conversion from the reduced Lab gamut G 21  to the Lab gamut G 2  indicated by the above-mentioned magnification ratio α. The following equations assume magnification ratios (rates of change) for the components r, θ, and φ to be αrp, αθp, and αφp. The values αrp, αθp, and αφp are magnification ratios from a given P′ in the reduced Lab gamut G 21  to the corresponding P in the Lab gamut G 2  and vary with grids. 
   
     
       
         
           
             
               
                 
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                 ) 
               
             
           
         
       
     
   
   The above-mentioned process is used to find magnification ratios for all the grids in the reduced Lab gamut G 21 . 
   The next purpose is to determine a magnification ratio from the reduced Lab range R 21  to the Lab range R 2  and to calculate the Lab range R 2 . The above-mentioned technique is used to find the magnification ratio α from the reduced Lab gamut G 21  to the Lab gamut G 2 . The magnification ratio α is used to find the magnification ratio α′ from the reduced Lab range R 21  to the Lab range R 2 . Further, the magnification ratio α′ is used to find the Lab range R 2 . With respect to grids inside the reduced Lab gamut G 21  in the reduced Lab range R 21 , the magnification ratio α for each grid of the reduced Lab gamut G 21  is interpolated to find the magnification ratio α′ for the reduced Lab range R 21 . By contrast, with respect to grids outside the reduced Lab gamut G 21  in the reduced Lab range R 21 , a line is formed by connecting each grid coordinate in the reduced Lab range R 21  with the Lab reduction center C 2 . That line crosses the surface of the reduced Lab gamut G 21  at an intersecting point that provides a magnification ratio as the magnification ratio α′. 
   In this manner, the magnification ratio α′Q is found with respect to a given grid Q′ in the reduced Lab range R 21  and can be used to find the corresponding grid Q in the Lab range R 2 . The grid Q in the Lab space is represented as (LQ, aQ, bQ) according to the rectangular coordinate system and is represented as follows according to the polar coordinate system (rQ, θQ, φQ). It is assumed that rQ, θQ, and φQ are found from equations (1) through (3) and are offset against the Lab reduction center C 2 =(Lc, ac, bc). 
   [Equation 7]
 
 L   Q   =r   Q  cos θ Q   +L   c   (7)
 
   [Equation 8]
 
 a   Q   =r   Q  sin θ Q  cos φ Q   +a   c   (8)
 
   [Equation 9]
 
 b   Q   =r   Q  sin θ Q  sin φ Q   +b   c   (9)
 
   The same equations can be used to find LQ′, aQ′, and bQ′ with respect to a given coordinate Q′ in the reduced Lab range R 21 . Accordingly, Q′ is converted into Q by α′Q and can be represented as follows. 
   [Equation 10]
 
 L   Q   =αr′   Q   r   Q′  cos(θ Q′ +αθ′ Q )+ L   c   (10)
 
   [Equation 11]
 
 a   Q   =αr′   Q   r   Q′  sin(θ Q′ +αθ′ Q )cos(φ Q′ +αφ′ Q )+ a   c   (11)
 
   [Equation 12]
 
 b   Q   =αr′   Q   r   Q′  sin(θ Q′ +αθ′ Q )sin(φ Q′ +αφ′ Q )+ b   c   (12)
 
   where αr′Q, αθ′Q, and αφ′Q denote components of the magnification ratio α′ for the grid Q′ in the reduced Lab range R 21 . 
   The addition theorem is used to modify equations (10) through (12) as follows. 
   [Equation 13]
 
 L   Q   =αr′   Q   r   Q′ (cos θ Q′  cos αθ′ Q −sin θ Q′  sin αθ′ Q )+ L   c   (13)
 
   [Equation 14]
 
 a   Q   =αr′   Q   r   Q′ (sin θ Q′  cos αθ′ Q +cos θ Q′  sin αθ′ Q )(cos φ Q′  cos αφ′ Q −sin φ Q′  sin αφ′ Q )+ a   c   (14)
 
   [Equation 15]
 
 b   Q   =αr′   Q   r   Q′ (sin θ Q′  cos αθ′ Q +cos θ Q′  sin αθ′ Q )(sin φ Q′  cos αφ′ Q +cos φ Q′  sin αφ′ Q )+ b   c   (15)
 
   These equations are unified into a vector equation as follows. 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ] 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 
                                   L 
                                   Q 
                                 
                               
                             
                             
                               
                                 
                                   a 
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                         = 
                           
                         ⁢ 
                         
                           
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                 ( 
                 16 
                 ) 
               
             
           
         
       
     
   
   The first equation of equation (16) indicates that the grid Q of the Lab range R 2  is found from the grid Q′ of the reduced Lab range R 21  and the corresponding magnification ratio α′Q. A is a matrix that can be expressed by the second equation of equation (16). The above-mentioned process can be used to find Lab values of the grids in the Lab range R 2 . 
   When the ICC profile  26  is created based on the Lab values obtained from the above-mentioned technique, the Lab space is generally conditioned to be 0≦L*≦100, −128≦a*, and b*≦127. If the found Lab range R 2  exceeds this range, the Lab range R 2  needs to be compressed into the specified range. The method is described below. The first step is to find the magnification ratio α′ and the Lab range R 2  only for grids equivalent to the outline of the reduced Lab range R 21 . These grids naturally contain grids corresponding to pure white and pure black in the RGB range R 1 . The grids are then standardized using coordinate values in the Lab range R 2  corresponding to the grids for pure white and pure black so that the pure white becomes (100, 0, 0) and the pure black becomes (0, 0, 0). The embodiment provides the standardization by converting the Lab space into the XYZ space, standardizing the grids in the XYZ space, and then converting the XYZ space back into the Lab space. 
   Then, the mapping is applied to the obtained grids that belong to the outline of the Lab range R 2  outside the range. This mapping is not a complicated process. According to the embodiment, the mapping aims at preserving the complete brightness. The mapping target is a rectangular solid represented by the range. Accordingly, the embodiment uses the method that can be simply implemented. In this case, the mapped grids for the outline of the Lab range R 2  indicate that the first obtained Lab coordinate values naturally differ from those obtained after the mapping. That is to say, a different scaling factor must be used for the magnification ratio α′ found for the corresponding reduced Lab range R 21 . Otherwise, grids in the reduced Lab range R 21  cannot yield coordinate values for the mapped grids in the Lab range R 2 . To solve this problem, equations (4) through (6) are used to again find the magnification ratio α′ using coordinate values for the Lab range R 2  after mapping and the corresponding coordinate values for the reduced Lab range R 21 . Of course, P and P′ in equations (4) through (6) correspond to the coordinate Q for the Lab range R 2  and the coordinate Q′ for the reduced Lab range R 21 , respectively. 
   Then, a coordinate value is calculated for the grid Q within the Lab range R 2  between the outline of the Lab gamut G 2  and the outline of the Lab range R 2 . The coordinate Q′ corresponds to the grid Q and exists between the outline of the reduced Lab gamut G 21  and the outline of the reduced Lab range R 21 . A line is formed by connecting the coordinate Q′ with the Lab reduction center C 2  and crosses the outline of the reduced Lab gamut G 21  at an intersecting point S 1  to be assumed. The same line crosses the outline of the reduced Lab range R 21  at an intersecting point S 2  to be assumed. The points S 1  and S 2  are assumed to correspond to magnification ratios αS 1  and α′S 2 . Both can be found by interpolating the reduced Lab gamut G 21  and the reduced Lab range R 21 . The magnification ratios αS 1  and α′S 2  are mixed with each other based on a ratio of the distance between the coordinate Q′ and S 1  to the distance between the coordinate Q′ and S 2  to find the magnification ratio α′Q corresponding to the coordinate Q′. Equation (16) is used to find the coordinate value of the grid Q for the Lab range R 2  located between the outline of the Lab gamut G 2  and the outline of the Lab range R 2  using the magnification ratio α′Q and the coordinate Q′ found in this manner. The use of such technique enables compression to an intended region without sacrificing gradation properties as resulting from simple clipping. Since the compressed region is located outside the Lab gamut G 2  where the true value is obtained, the compression is available without degrading the accuracy in the Lab gamut G 2 . 
   The above-mentioned technique can be used to find grids in the Lab range R 2  corresponding to grids in the RGB range R 1 . The following describes the method of acquiring Lab coordinate values corresponding to given RGB in the RGB range R 1 . In this case, there is predetermined correspondence between a grid in the RGB range R 1  and a grid in the Lab range R 2 . An interpolation operation just needs to be used to find a Lab coordinate value corresponding to a given RGB coordinate value. When an input RGB value exists in the RGB gamut G 1 , however, improved accuracy is expected by using the Lab gamut G 2  to find Lab values. Only in such region, an interpolation operation is performed to find Lab values using the Lab gamut G 2  corresponding to the RGB gamut G 1 . 
   (3) Conclusion 
   The above-mentioned process is described with reference to a flowchart in  FIG. 10 . In  FIG. 10 , the scanner  40  acquires the RGB gamut G 1  (step S 105 ). The calorimeter  41  acquires the Lab gamut G 2  (step S 110 ). An interpolation is performed to find the reduced Lab range R 21  and the reduced Lab gamut G 21  corresponding to the reduced RGB range R 11  and the reduced RGB gamut G 11  (steps S 115  and S 120 ). The reduced RGB range R 11  and the reduced RGB gamut G 11  are compressed against the RGB compression center C 1  so as to fit the RGB range R 1  and the RGB gamut G 1  into the RGB gamut G 1 . The magnification ratio α is found for each grid based on the relationship between the reduced Lab gamut G 21  and the Lab gamut G 2  (step S 125 ). Then, the magnification ratio for the surface of the reduced Lab range R 21  is assumed to be that for the surface of the reduced Lab gamut G 21 . The reduced Lab range R 21  is enlarged to provide the Lab range R 2  (step S 130 ). When the value for the surface of the Lab range R 2  exceeds the Lab range, the mapping is performed to fit the value into the Lab range R 2  (step S 135 ). The magnification ratio α′ is recalculated according to the relationship between the mapped surface of the Lab range R 2  and the reduced Lab range R 21  (step S 140 ). When the grid exists in the reduced Lab gamut G 21  (steps S 145  and S 150 ), the magnification ratio α is used to enlarge the Lab value of the grid and provide the Lab range R 2  (step S 155 ). When the grid does not exist in the reduced Lab gamut G 21 , the grid is enlarged by using a magnification ratio found by linear interpolating the magnification ratio α′ (step S 160 ). This process is performed for all grids (steps S 165  and S 170 ) to create the ICC profile  26  based on an enlarged result. In this manner, the embodiment once reduces the RGB range R 1  and the RGB gamut G 1  and then implements the extrapolation. This makes it possible to minimize a failure in the extrapolation region for the ICC profile  26 .
There has been a problem of causing too large a failure in extrapolated portions due to the multiple linear regression analysis or the 3*3 matrix optimization. An interpolation is performed to find a reduced Lab range and a reduced Lab gamut corresponding to a reduced RGB range and a reduced RGB gamut. The reduced RGB range and the reduced RGB gamut are compressed so that an RGB range and an RGB gamut are fit into an RGB gamut. A magnification ratio a is found for each grid according to relationship between the reduced Lab gamut and a Lab gamut. Then, a magnification ratio for the surface of the reduced Lab range is assumed to be that for the surface of the reduced Lab gamut. The reduced Lab range is enlarged to provide a Lab range. A magnification ratio a' is found according to the relationship between the surface of the Lab range and the reduced Lab range. When a grid exists in the reduced Lab gamut, the magnification ratio a is used to enlarge the Lab value of the grid and provide the Lab range. When the grid does not exist in the reduced Lab gamut, the grid is enlarged by using the magnification ratio a' to create an ICC profile.