Abstract:
A technique developed for simulation of constrained particle systems is applied to the visualization and manipulation of colors, both in isolation and relation to each other, in one or more color spaces. Utilizing the technique, a first color can be adjusted to a second color in a first color space subject to one or more objectives between the first and second colors in the first color space; one or more constraints for adjustment of first and second colors in the first color space; and/or one or more constraints and/or objectives between the adjustment of the first and second colors in the first color space and an adjustment of a corresponding first color in a second color space to a second color in the second color space.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority from U.S. Provisional Patent Application Serial No. 60/111,875, filed Dec. 11, 1998, entitled “Method and Apparatus for Working with Constrained Color on a Computer Terminal Display”. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a color picker utilized in computer graphic art design. 
     2. Description of the Prior Art 
     Graphic art design software, such as desktop publishing software, uses advanced techniques for accurately reproducing colors. However, tools utilized by graphic art design software for editing colors, both single colors and sets of related colors, are still fairly primitive. This is true despite the existence of sophisticated color design principles in the graphic arts. 
     Graphic art design software for working with colors on a computer system is configured to maintain color fidelity between an input device and an output device. A color in such graphic art design software is specified as a set of coordinates in a color space. A color space is typically defined by three attributes, namely, source transform, destination transform and boundary. The source transform is an algorithm that converts a color from a first color space to a second device-neutral color space. The destination transform is an algorithm that converts a color in the second device-neutral color space into a third color space. The source transform and the destination transform coact to convert a color from the first color space to the third color space, and vice versa, via the second device-neutral color space. The most commonly used second device-neutral color space is CIE XYZ, a device-independent color space configured to represent every color perceivable by the human visual system. The boundary is an imaginary surface defining the limit of legal colors in the color space. The boundary of each color space is related to the range of colors producible by a device, such as a color monitor, color printer or color scanner, represented by the color space of the device or the limitation of human vision. 
     Graphic art design software includes a color picker which enables each color in an image to be adjusted. Preferably, the color picker includes an interface that displays a range of producible colors in terms of a perceptual color space. The perceptual color space can be depicted as a three-dimensional, double or dual-cone having the color white represented by a point of one cone and having the color black represented by a point of the other cone. The cone having the point representing the color white diverges conically therefrom toward the color black and the cone having the point representing the color black diverges conically therefrom toward the color white. The conical divergence of the two cones meet intermediate the points representing the colors white and black. 
     A color within the perceptual color space can be characterized in polar coordinates. Specifically, the perceptual color space has a central axis representing lightness which extends between the points at opposite ends of the dual-cone. The radial distance from the lightness axis represents saturation and the angle around the lightness axis represents hue. 
     A typical color picker enables an artist to work within the perceptual color space by providing a graphical interface that enables selection of a plane in the perceptual color space. For example, the graphical interface of one color picker enables the artist to select a desired hue by moving a slider on a hue color bar representing hue. Adjacent the hue color bar, the color picker displays the range of colors of the perceptual color space in a saturation-lightness plane for the selected hue. The artist then moves a computer icon in the saturation-lightness plane and selects a point therein corresponding to a color to be displayed in a select part of an image. 
     The graphical interface of another color picker includes a hue circle surrounding a triangle representing a saturation-lightness plane. In this color picker, the artist selects a desired hue by moving a computer icon to a desired position in the hue circle. Thereafter, the artist moves another computer icon in the saturation-lightness triangle and selects a point therein corresponding to a color to be displayed in a select part of the image. 
     The graphical interface of yet another color picker includes a slider on a lightness bar and an adjacent hue-saturation circle. In this color picker, the artist moves the slider to a desired position on the lightness bar. Then the artist moves a computer icon in the saturation-hue circle and selects a point therein corresponding to a color to be displayed in a select part of the image. 
     As discussed above, a second device-independent color space is utilized to transform colors from a first color space to a third color space. To change a color in the third color space utilizing a graphical interface in the first color space, two color transforms are required. Namely, a first color transform between the first color space and the second device-independent color space, and a second color transform between the second device-independent color space and the third color space. Moreover, in order to depict colors represented in the third color space in the user interface of the first color space, two additional color transforms are necessary between the third color space and the first color space. Hence, when editing colors in a third color space utilizing a graphic interface in a first color space, four color transforms are necessary between the various color spaces in order to affect a change of a color in a third color space based upon a graphical interface representation of this color in the first color space. 
     A problem with using prior art color pickers is that each color transform introduces numerical error into the color transformations. This error affects the fidelity of the colors displayed in the first color space or produced in the third color space. Another problem with using prior art color pickers is that they do not enable one color in a color space to be changed as a function of another color in the color space, thereby maintaining a desired relationship, i.e., hue, saturation and/or lightness, therebetween. Still another problem with prior art color pickers is that their graphical user interfaces do not permit visualization of relationships between colors in an image. 
     It is, therefore, an object of the present invention to overcome the above problems and others by providing a method for adjusting one or more colors in one or more color spaces utilizing techniques for modeling physical systems, e.g., a particle simulator. It is an object of the present invention to provide a method for reducing the number of color transforms over prior art color pickers. It is an object of the present invention to provide a method for relating two or more colors in one or more color spaces so that a change to one color produces a desired change to the other related colors. It is an object of the present invention to provide an apparatus for performing the foregoing methods. Still other objects of the present invention will become apparent to those of ordinary skill in the art upon reading and understanding the following detailed description. 
     SUMMARY OF THE INVENTION 
     Accordingly, I have invented a method of adjusting color in a color space of a device. The method includes adjusting a first color to a second color in a first color space and subjecting the adjustment from the first color to the second color in the first color space to (i) one or more objectives for the adjustment of the first color in the first color space to the second color in the first color space; (ii) one or more constraints for the adjustment of the first color in the first color space to the second color in the first color space; and/or (iii) one or more constraints between the adjustment from the first color to the second color in the first color space and an adjustment of a corresponding first color in a second color space to a second color in the second color space. 
     The step of subjecting the adjustment can include the steps of converting the adjustment from the first color to the second color in the first color space to an adjustment of the first color to the second color in the second color space and determining for the adjustments in the first color space and/or the second color space a change of the first and second colors thereof that does not violate the one or more constraints. 
     I have also invented a method of adjusting a color in a color space. The method includes determining a state vector including a first plurality of scalar values which define a first color in a first color space. An objective function is determined for desired change of the first plurality of scalar values. A partial derivative of the objective function with respect to a derivative of the state vector is determined to obtain an objective vector. The objective vector defines a change to the first plurality of scalar values which minimizes the objective function. The objective vector and the state vector are combined to obtain a change of the first color related to the desired change. 
     The method can also include mapping the first plurality of scalar values in the first color space to a first plurality of scalar values in a second color space to define a second color therein. The second color corresponds to the first color. The objective function is determined for the desired change in the first plurality of scalar values in the second color space. 
     The method can also include imposing a constraint on changes to the first plurality of scalar values and determining a constraint function and a constraint vector corresponding to the imposed constraint. A partial derivative of the constraint vector with respect to the state vector can be determined. The partial derivative defines a change to the state vector which does not violate the constraint function. The first plurality of scalar values can be changed as a function of the desired change to the first plurality of scalar values and as a function of the constraint vector. 
     The method can also include the steps of including in the state vector a second plurality of scalar values which define a second color in a second color space. A constraint function is determined between the first plurality of scalar values and the second plurality of scalar values. A partial derivative of the constraint function with respect to the state vector is determined to obtain a constraint vector which defines a change to the state vector which does not violate the constraint function. The second plurality of scalar values is changed as a function of the desired change to the first plurality of scalar values and as a function of the constraint vector whereby the second color changes as a function of the change to the first color. The first plurality of scalar values and the second plurality of scalar values can be mapped into a third color space. The constraint function can be determined between the first plurality of scalar values and the second plurality of scalar values in the third color space. The partial derivative of the constraint function is determined with respect to the first and second pluralities of scalar values in the respective first and second color spaces to obtain therefor the constraint vector which defines the change to the state vector which does not violate the constraint function between the first and second pluralities of scalar values in the third color space. 
     I have also invented a method of adjusting color in a color space that includes defining a first color in a first color space and defining a second color in a second color space. The first color in the first color space and the second color in the second color space are transformed into a corresponding first and second colors in the third color space. A constraint is defined between the first color and the second color in the third color space. Changing the first color in the first color space causes the second color in the second color space to change as a function of the change to the first color in the first color space and as a function of the constraint. 
     The method can also include determining a partial derivative of the constraint with respect to the first and second colors in the respective first and second color spaces and determining from the partial derivative of the constraint at least one change in the first and second colors in the respective first and second color spaces which does not violate the constraint. The second color in the second color space is changed as a function of the change of the first color in the first color space and as a function of the at least one change. 
     I have also invented a method of adjusting a color in a color space that includes determining a state vector including a first plurality of scalar values corresponding to a first color in a first color space. An objective function is determined for a desired change of the first plurality of scalar values corresponding to a desired change of the first color. An objective vector is determined from the objective function. The objective vector defines a change to the state vector which minimizes the objective function. The state vector and the objective vector are combined to obtain the desired change to the first color. 
     The method can also include imposing a constraint on changes to the first plurality of scalar values and determining a constraint function and a constraint vector corresponding to the imposed constraint. A partial derivative of the constraint vector with respect to the state vector can be determined. The partial derivative defines a change to the state vector which does not violate the constraint function. The first plurality of scalar values can be changed as a function of the desired change to the first plurality of scalar values and as a function of the constraint vector. 
     A color space constraint function can be determined between the first plurality of scalar values and a second plurality of scalar values of the state vector. The second plurality of scalar values define a second color in the first color space. A color space constraint vector is determined from the color space constraint function. The color space constraint vector defines a change in the state vector which does not violate the constraint function. The second plurality of scalar values is changed as a function of the change to the first plurality of scalar values and as a function of the color space constraint vector whereby the second color changes as a function of the desired change to the first color. 
     The state vector can include a third plurality of scalar values which define a third color in a second color space. A movement constraint function can be determined between the first plurality of scalar values and the third plurality of scalar values. A movement constraint vector is determined from the movement constraint function. The movement constraint vector defines at least one change in at least one of the first and third pluralities of scalar values which does not violate the movement constraint function. The third plurality of scalar values is changed as a function of the change to the first plurality of scalar values and as a function of the movement constraint vector whereby the third color changes as a function of the change to the first color. 
     I have also invented an apparatus for adjusting a color in a color space that includes a state vector determining means for determining a state vector including a first plurality of scalar values which define a first color in a first color space. An objective function determining means determines an objective function for a desired change of the first plurality of scalar values. An objective vector determining means determines a partial derivative of the objective function with respect to a derivative of the state vector to obtain an objective vector which defines a change to the first plurality of scalar values which minimizes the objective function. A combining means combines the objective vector and the state vector to obtain the desired change of the first color. 
     The apparatus can include a mapping means for mapping the first plurality of scalar values in the first color space to a first plurality of scalar values in a second color space to define a second color therein. The second color corresponds to the first color. The objective function determining means determines the objective function for the desired change in the first plurality of scalar values in the second color space. 
     The apparatus can include an imposing means for imposing a constraint on changes to the first plurality of scalar values. A constraint determining means determines a constraint function and a constraint vector corresponding to the imposed constraint. A partial derivative determining means determines a partial derivative of the constraint vector with respect to the state vector. The partial derivative defines a change to the state vector which does not violate the constraint function. A changing means changes the first plurality of scalar values as a function of the desired change to the first plurality of scalar values and as a function of the constraint vector. 
     The apparatus can include a state vector including means for including in the state vector a second plurality of scalar values which define a second color in a second color space. A constraint function determining means can determine a constraint function between the first plurality of scalar values and the second plurality of scalar values. A constraint vector determining means can determine a partial derivative of the constraint function with respect to the state vector to obtain therefor a constraint vector which defines a change in the state vector which does not violate the constraint function. A changing means can change the second plurality of scalar values as a function of the desired change to the first plurality of scalar values and as a function of the constraint vector whereby the second color changes as a function of the change of the first color. 
     The constraint function determining means can map the first and second pluralities of scalar values into a third color space. The constraint function can be determined between the first plurality of scalar values and the second plurality of scalar values in the third color space. The constraint vector determining means can determine the partial derivative of the mapping of the first and second pluralities of scalar values in the third color space. 
     I have also invented an apparatus for adjusting a color in a color space that includes a state vector determining means for determining a state vector including a first plurality of scalar values corresponding to a first color in a first color space. An objective function determining means determines an objective function for a desired change of the first plurality of scalar values corresponding to a desired change of the first color. An objective vector determining means determines from the objective function an objective vector which defines a change to the state vector which minimizes the objective function. A combining means combines the state vector and the objective vector to obtain the desired change of the first color. 
     The apparatus can include an imposing means for imposing a constraint on changes to the first plurality of scalar values. A constraint determining means determines a constraint function and a constraint vector corresponding to the imposed constraint. A partial derivative determining means determines a partial derivative of the constraint vector with respect to the state vector. The partial derivative defines a change to the state vector which does not violate the constraint function. A changing means changes the first plurality of scalar values as a function of the desired change to the first plurality of scalar values and as a function of the constraint vector. 
     A color space constraint function determining means can determine a color space constraint function between the first plurality of scalar values and a second plurality of scalar values of the state vector. The second plurality of scalar values define a second color in the first color space. A color space constraint vector determining means can determine for the color space constraint function a color space constraint vector which defines a change in the state vector which does not violate the constraint function. A changing means can change the second plurality of scalar values as a function of the change to the first plurality of scalar values and as a function of the color space constraint vector whereby the second color changes as a function of the change to the first color. 
     The apparatus can have an including means which includes in the state vector a third plurality of scalar values which define a third color in a second color space. A movement constraint determining means can determine at least one movement constraint function between the first plurality of scalar values and the third plurality of scalar values. A movement constraint vector determining means can determine for the movement constraint function a movement constraint vector which defines at least one change in at least one of the first and third pluralities of scalar values which does not violate the movement constraint function. The changing means can change the third plurality of scalar values as a function of the change to the first plurality of scalar values and as a function of the movement constraint vector whereby the third color changes as a function of the change to the first color. 
     The movement constraint determining means can map the first and second pluralities of scalar values into a third color space and can determine the movement constraint function between the first plurality of scalar values and the second plurality of scalar values in the third color space. The movement constraint vector determining means can determine the movement constraint function with respect to the mappings of the first and second pluralities of scalar values in the third color space. 
     I have also invented a method of adjusting a color in a color space that includes determining a state vector including a first plurality of scalar values corresponding to a first color in a first color space. The first plurality of scalar values are mapped to a second plurality of scalar values in a second color space to define a second color therein. An objective function is determined for a desired change of the second plurality of scalar values in the second color space corresponding to a desired change of the second color. An objective vector is determined from the objective function. The objective vector defines a change to the state vector which minimizes the objective function. The state vector and the objective vector are combined to obtain a change to the first color corresponding to the desired change of the second color. 
     Lastly, I have invented an apparatus for adjusting a color in a color space that includes a state vector determining means for determining a state vector which includes a first plurality of scalar values corresponding to a first color in a first color space. A mapping means maps the first plurality of scalar values to a second plurality of scalar values in a second color space to define a second color therein. An objective function determining means determines an objective function for a desired change of the second plurality of scalar values in the second color space corresponding to a desired change of the second color. An objective vector determining means determines from the objective function an objective vector. The objective vector defines a change to the state vector which minimizes the objective function. A combining means combines the state vector and the objective vector to obtain a change to the first color corresponding to the desired change of the second color. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic drawing of a computer system on which a software program operating in accordance with the present invention is installed; 
     FIG. 2 is a schematic drawing of a three-dimensional, dual-cone perceptual color space; 
     FIGS. 3 a  and  3   b  through  13   a  and  13   b  are top and side views, respectively, of the perceptual color space shown in FIG. 2 including various examples of imposed objectives and/or constraints on particles C 1 -C 4  therein which correspond to colors of a device; 
     FIG. 14 is a diagrammatic representation of movement of colors in accordance with a constraint function imposed across two color spaces; and 
     FIG. 15 is a diagrammatic representation of movements of colors in accordance with an objective function imposed across two color spaces. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     With reference to FIG. 1, the present invention is a method for visualizing and manipulating one or more colors in one or more color spaces of devices, such as a CRT, a printer and/or a scanner. The method is embodied in a software program which is installed on a computer  2  which includes the necessary hardware and operating system software to implement the present invention. 
     The computer  2  includes a memory unit  6 , a color CRT  8 , a keyboard  10  and a mouse  12  connected to a central processing unit (CPU)  4  in a manner known in the art. The memory unit  6  stores the operating system software and the software program embodying the present invention. An output device, such as a color printer  14 , and an input device, such as a color scanner  16 , can also be connected to the CPU  4  in a manner known in the art. 
     The color CRT  8 , color printer  14  and color scanner  16  can each produce colors in a color space defined by their respective capabilities. For example, color CRT  8  has a CRT color space  18  which is defined by the red, green and blue (RGB) color-producing characteristics of color CRT  8 . Preferably, CRT color space  18  is an RGB color space, however, CRT color space  18  can be another color space. Similarly, color printer  14  and color scanner  16  define printer color space  20  and scanner color space  22  which preferably are a CMYK color space and an RGB color space, respectively. While the CRT color space  18  and the scanner color space  22  are both RGB color spaces, the RGB color space corresponding to CRT color space  18  can have different boundaries than the RGB color space corresponding to scanner color space  22 . 
     The present invention utilizes well-known techniques for modeling particle systems to visualize and manipulate a color by itself or in relation to one or more other colors in the same color space or different color spaces. To enable such visualization and manipulation, the software program embodying the present invention converts CRT color space  18  from an RGB color space to a perceptual color space  28 . A convenient model of perceptual color space  28  is shown in FIG. 2, however, perceptual color space  28  is not to be construed as limited by the model shown in FIG. 2. A perceptual color space  28  relates the physiological and neurological ways in which the eyes and brain receive and process color to a three-dimensional, dual-cone color space  30  having a point  32  representing the color white at one end thereof and a point  34  representing the color black at the other end thereof. A boundary  36  of the dual-cone color space  30  diverges conically from point  32  toward point  34  and diverges conically from point  34  to point  32 . The diverging cones meet intermediate points  32  and  34 . 
     FIGS. 3 a  and  3   b  through  13   a  and  13   b  show respective top and side views of perceptual color space  28  including particles C 1 -C 4  therein which represent colors of an image (not shown), such as an image received on color scanner  16 , which is preferably displayed, in whole or in part, on color CRT  8 . 
     Referring to FIGS. 3 a  and  3   b , in accordance with the present invention, if it is desired to change the color associated with particle C 1  of the image without changing the colors associated with particles C 2 -C 4 , mouse  12  is manipulated to move a cursor  40  on color CRT  8  to point to the particle C 1  in FIG. 3 a . Particle C 1  is then selected by activating (clicking) one of the buttons of mouse  12 . Thereafter, dragging cursor  40  causes corresponding movement of particle C 1 . 
     FIGS. 4 a  and  4   b  show the result of moving particle C 1  to a position opposite the starting position of particle C 1  in the saturation direction of perceptual color space  28 . As a result of this change, the color associated with particle C 1  changes from, for example, the color yellow in FIGS. 3 a  and  3   b  to the color light blue in FIGS. 4 a  and  4   b.    
     In order to move particles in perceptual color space  28 , particles C 1 -C 4  are represented in the CPU  4  and memory unit  6  of computer  2  as a state vector which includes a plurality of scalar values which define the position particles C 1 -C 4  in perceptual color space  28  and the colors associated with particles C 1 -C 4 . The plurality of scalar values includes groups of scalar values, with each group of scalar values representing the position of one of the particles C 1 -C 4  in perceptual color space  28  and the color associated therewith. 
     After particle C 1  is selected, moving cursor  40  via mouse  12  produces differences between the position of cursor  40  and particle C 1 . In response to any difference between the position of cursor  40  and particle C 1 , the software of the present invention determines an objective function which mathematically describes the difference therebetween. Next, the present invention determines a partial derivative of the objective function with respect to the state vector to obtain an objective vector  86 . The objective vector  86  defines a change in the position of particle C 1  which minimizes the difference between the position of cursor  40  and the position of particle C 1 . For movement of particle C 1  absent a constraint (discussed hereinafter) imposed thereon, the objective vector  86  can be combined directly with the state vector to change the position of particle C 1  to follow the path of cursor  40  in perceptual color space  28 . More specifically, combining the state vector and the objective vector  86  changes a group of scalar values associated with particle C 1 . This change produces a change in the position of particle C 1  and a change of the color represented by this group of scalar values. Hence, by simply pointing, clicking and dragging particle C 1 , the color associated therewith in the original image can be adjusted. 
     In the foregoing example, particle C 1  was moved in the saturation-hue plane shown in FIGS. 3 a  and  4   a . This movement also produced a change in the position of particle C 1  in the saturation-lightness plane shown in FIGS. 3 b  and  4   b . Alternatively, particle C 1  can be moved from the position shown in the saturation-lightness plane of FIG. 3 b  to the position shown in the saturation-lightness plane of FIG. 4 b  by utilizing the point, click and drag method described above. The choice of moving particle C 1  in the hue-saturation plane shown in FIGS. 3 a  and  4   a  or the saturation-lightness plane shown in FIGS. 3 b  and  4   b , or both, will depend on the desired change of the color associated with particle C 1 . 
     With reference to FIGS. 5 a - 6   b , using the point, click and drag method discussed above, particle C 1  is selected in FIG. 5 a . Thereafter, utilizing keyboard  10  or mouse  12 , a constraint is imposed on particle C 1  that maintains the saturation constant in the hue-saturation plane. This constraint is shown in FIGS. 5 a  and  6   a  as a track  42  which is perpendicular to the disallowed direction of motion of particle C 1  in the hue-saturation plane. This constraint is also shown in the saturation-lightness plane of FIGS. 5 b  and  6   b  as a line segment  44  which extends parallel to the lightness axis in perceptual color space  28 . Line segment  44  is likewise perpendicular to the disallowed direction of motion of particle C 1 . In the views shown in FIGS. 5 a - 6   b , track  42  and line segment  44  suggest directions of motion possible for particle C 1  with the active saturation constraint imposed thereon. 
     To change the position of particle C 1 , and hence the color associated therewith, the position of particle C 1  in FIG. 5 a  is changed utilizing the point, click and drag method described above whereby particle C 1  is moved to the position shown in FIG. 6 a . As discussed above, the present invention determines an objective function for the desired change of the group of scalar values corresponding to the change of particle C 1  from the position shown in FIG. 5 a  to the position shown in FIG. 6 a . The present invention also determines an objective vector  86  which defines a change to the group of scalar values for particle C 1  which minimizes the objective function. 
     In addition, the present invention defines, for the constraint shown in FIGS. 5 a - 6   b , a constraint function for the change in the position of particle C 1  in FIGS. 5 a  and  6   a . The constraint function mathematically defines one or more directions of disallowed motion of particle C 1 . The present invention then determines a partial derivative of the constraint function with respect to the first state vector to obtain therefor a constraint vector which defines at least one change in the first state vector which does not violate the saturation constraint. More specifically, the constraint vector is determined from a Jacobian matrix which is the partial derivative of the constraint function with respect to the first state vector and a LaGrange multiplier, in a manner well-known in the art of particle system modeling. Thereafter, in response to changing the position of cursor  40 , the present invention solves the objective vector  86  and the constraint vector for changes in the position of cursor  40 . The solutions of the objective vector  86  and the constraint vector are supplied to a non-linear solver software routine  72 , shown schematically in FIG. 14, which determines a change to the group of scalar values for particle C 1  which minimizes the objective function  84  without violating the saturation constraint. Preferably, the non-linear routine solver  72  is a fourth order Runge-Kutta algorithm which is well-known in the art of particle system modeling. 
     Hence, in response to movement of cursor  40  in the hue-saturation plane, objective vector  86  attempts to have particle C 1  follow the movement of cursor  40  and the constraint vector prevents changes in the saturation direction of the hue-saturation plane shown in FIGS. 5 a  and  6   a . Since particle C 1  moves in the hue-saturation plane shown in FIGS. 5 a  and  6   a , there are no changes to the lightness of the color associated with particle C 1  in the saturation-lightness plane shown in FIGS. 5 b  and  6   b . If it is desired to change the lightness of the color associated with particle C 1 , particle C 1  can be selected in FIG. 5 b  and moved in the lightness direction along line segment  44 . 
     With reference to FIGS. 7 a - 8   b , a hue constraint is imposed on particle C 4  and a saturation constraint is imposed on particle C 1 . In response to imposing these constraints, the present invention determines for particle C 4  a hue constraint function and a hue constraint vector, and determines for particle C 1  a saturation constraint function and a saturation constraint vector. In FIGS. 7 a - 8   b , line segments  44  and  46  show directions of allowed motion of particle C 1  with the saturation constraint imposed thereon and line segment  46  shows the direction of allowed motion of particle C 4  with the hue constraint imposed thereon. Since the saturation-lightness planes shown in FIGS. 7 b  and  8   b  do not show hue, line segment  46  is only shown in the hue-saturation planes shown in FIGS. 7 a  and  7   b.    
     Using the point, click and drag method, particle C 4  is selected. Thereafter, in response to moving cursor  40  from the position shown in FIG. 7 a  to the position shown in FIG. 8 a , the present invention determines an objective function  84  and objective vector  86  corresponding to changes in the position of cursor  40 . Thereafter, the present invention determines for the change in position of cursor  40  solutions for the objective vector  86 , the hue constraint vector and the saturation constraint vector and provides these solutions to the non-linear solver routine  72  which determines therefrom a change to the group of scalar values for particle C 4 . Since the saturation constraint vector is related to changes in the position of particle C 1 , the solution of the saturation constraint vector provided to the non-linear solver routine  72  for changes in the position of particle C 4  does not produce a change to the group of scalar values for particle C 1 . 
     The present invention can also impose a boundary constraint when an attempt is made to move a particle, e.g., C 4 , outside of the perceptual color space  28 . For example, as shown in FIGS. 8 a  and  8   b , when particle C 4  crosses boundary  36  of perceptual color space  28 , the present invention imposes a boundary constraint on further changes to the saturation of the color associated with movement of particle C 4  in a manner that violates the boundary  36  of perceptual color space  28 . This boundary constraint is shown graphically in FIGS. 8 a  and  8   b  by cursor  40  being outside of the hue-saturation plane of FIG. 8 a , and particle C 4  is constrained by the intersection of particle C 4  and boundary  36  in FIG. 8 b  and, thus remains inside the hue-saturation plane of FIG. 8 a . In operation, once the non-linear solver routine  72  determines a change in the position of particle C 4 , the present invention determines if the new position of particle C 4  is outside boundary  36 . If not, the present invention permits the change in the position of particle C 4 . However, if the change in the position of particle C 4  is outside boundary  36 , the present invention determines when cursor  40  crossed boundary  36  and adjusts the group of scalar values for particle C 4  to correspond to the position of cursor  40  when it crossed boundary  36 . The boundary constraint restricts any subsequent attempted changes in the position of particle C 4  outside boundary  36  while permitting changes to the position of particle C 4  on or inside boundary  36 . 
     In FIGS. 9 a - 10   b , a lightness constraint is imposed on particle C 2 , a saturation constraint is imposed on particle C 1  and a hue constraint is imposed on particle C 4 . In response to imposing these constraints, the present invention determines a lightness constraint function and a lightness constraint vector for particle C 2 , determines a saturation constraint function and a saturation constraint vector for particle C 1 , and determines a hue constraint function and a hue constraint vector for particle C 4 . Line segments  44  and  46  show directions of allowed motion of particle C 1  with the saturation constraint imposed thereon, line segment  46  shows the direction of allowed motion of particle C 4  with the hue constraint imposed thereon and line segment  48  shows the allowed direction of motion of particle C 2  with the lightness constraint imposed thereon. Since the hue-saturation plane shown in FIGS. 9 a  and  10   a  do not include lightness, no line segment or track appears in connection with particle C 2  therein. 
     Using the point, click and drag method, particle C 2  can be moved from the position shown in FIG. 9 a  to the position shown in FIG. 10 a . More specifically, in response to changing the position of cursor  40 , the present invention determines therefor an objective function  84  and an objective vector  86 . The present invention determines solutions for the objective vector  86  and the lightness constraint vector for changes in the position of cursor  40  and provides the solutions to the non-linear solver routine  72  which determines therefrom a change of the group of scalar values for particle C 2  which minimizes the objective function  84  without violating the lightness constraint imposed on particle C 2 . 
     Referring to FIGS. 11 a - 13   b , particles C 1 -C 4  are positioned in perceptual color space  28  at positions corresponding to the colors they represent in an image. The present invention determines a state vector which includes a plurality of scalar values which is defined by one or more groups of scalar values, with each group of scalar values associated with the position of one of the particles C 1 -C 4 . More specifically, the state vector includes a first group of scalar values which define the position of particle C 1  in perceptual color space  28  and a second group of scalar values which define the position of particle C 2  in perceptual color space  28 . The positions of particles C 3  and C 4  are likewise defined by third and fourth groups of scalar values of the state vector. 
     In the example shown in FIGS. 12 a - 13   b , a hue constraint is imposed between particles C 1  and C 2 . In response to imposing this constraint, the present invention determines a hue constraint function, shown by angle  50  in FIGS. 12 a  and  13   a , which constrains particles C 1  and C 2  to move simultaneously in the hue direction. Using the point, click and drag method, particle C 2  is moved from the position shown in FIG. 12 a  to the position shown in FIG. 13 a . In response to changing the position of cursor  40 , the present invention determines an objective function  84 , an objective vector  86  and a hue constraint vector. Next, the present invention determines solutions for the objective vector  86  and the hue constraint vector and provides the solutions to the non-linear solver routine  72  which determines changes for the first group of scalar values and the second group of scalar values which minimize the objective function  84  without violating the hue constraint imposed between particles C 1  and C 2 . 
     With reference to FIG. 14, in addition to being applied to a particle in a color space, constraint functions can also be applied between two or more color spaces. As shown in FIG. 14, a first hue-saturation color space  60  includes a particle C 1  corresponding to a color of a first device, such as color printer  14 , and a second hue-saturation color space  62  includes a particle C 2  which corresponds to a color of a second device, such as color scanner  16 . 
     In order to impose a constraint between particle C 1  in first color space  60  and particle C 2  in second color space  62 , the present invention defines a state vector which includes a first plurality of scalar values corresponding to the position of particle C 1  in first color space  60  and a second plurality of scalar values corresponding to the position of particle C 2  in second color space  62 . Next, at least one movement constraint is imposed between particle C 1  in first color space  60  and particle C 2  in second color space  62 . In the example shown in FIG. 14, the movement constraint causes the position of particle C 2  to change as a function of changes to the position of particle C 1 . In response to imposing this movement constraint, the present invention determines a movement constraint function  66  and a movement constraint vector  68  between particles C 1  and C 2 . 
     Next, using the point, click and drag method, particle C 1  is selected. Thereafter, in response to changing the position of cursor  40 , the present invention determines an objective function  84  and an objective vector  86  for the movement of cursor  40 . Next, the present invention solves the objective vector  86  and the constraint vector for changes in position of cursor  40  and provides the solutions to the non-linear solver routine  72 . The non-linear solver routine  72  determines from these solutions changes to the first plurality of scalar values and the second plurality of scalar values which minimize the objective function  84  without violating the movement constraint imposed between particles C 1  and C 2 . Hence, changes to the position of cursor  40  produces changes in the position of particle C 1  and produces corresponding changes to the position of particle C 2  with resulting changes to the colors associated with particles C 1  and C 2  in their respective color spaces. 
     A plurality of movement constraints can be imposed between particles C 1  and C 2  in FIG.  14 . In addition, a hue, saturation and/or lightness constraint can be imposed on particle C 1  or particle C 2 , or both. More specifically, any number of constraints can be imposed on particle C 1  or C 2 , or between particles C 1  and C 2 , or any other particles in the first and second color spaces  60  and  62 , as required by the desired constraints or changes to corresponding colors in an image. 
     Preferably, the movement constraint function  66  is determined in a third color space  64 . Specifically, the present invention includes a mapping function  63  which maps the first plurality of scalar values and the second plurality of scalar values corresponding to the position of particles C 1  and C 2  in the first and second color spaces  60  and  62 , into a corresponding first plurality of scalar values and second plurality of scalar values, respectively, in a third device-neutral color space  64 . For purpose of illustration, the mapping of the first and second pluralities of scalar values into the third color space  64  is shown in FIG. 14 by particles C 1 ′ and C 2 ′ in third color space  64 . In response to imposing the movement constraint between particles C 1  and C 2  in the first and second color spaces  60  and  62 , the present invention determines from the first and second pluralities of scalar values in the third color space  64 , the movement constraint function and the movement constraint vector for particles C 1  and C 2 . 
     Thereafter, in response to changing the position of cursor  40  in first color space  60 , the present invention determines an objective function  84  and an objective vector for the change in position of cursor  40 . Next, the present invention solves the objective vector and the movement constraint vector for the change in position of cursor  40  and provides these solutions to the non-linear solver routine  72  which determines therefrom the change in position of particle C 1  in the first color space  60  and the corresponding change in position of particle C 2  in second color space  62 . 
     Once determined for particles C 1 ′ and C 2 ′, the movement constraint vector  68  can be utilized for subsequent changes in the position of particle C 1 . Specifically, the solutions of the objective vector and the constraint vector  68  are determined for each change in position of cursor  40 . These solutions are then supplied to the non-linear solver routine  72  which determines therefrom the change in position of particle C 1  in the first color space  60  and the corresponding change in position of particle C 2  in second color space  62 . Hence, the multiple color transforms between the first and second color spaces are avoided. 
     Supplying the change in position of cursor  40  to the movement constraint vector  68  is shown in FIG. 14 by line  69  extending from first color space  60  to the movement constraint vector  68 . The present invention solves the movement constraint vector  68  for each change in the position of cursor  40 . Supplying the solutions of the objective vector and the constraint vector to the non-linear solver routine  72  is shown in FIG. 14 by lines  70  and  71  from first color space  60  and constraint vector  68 , respectively, to non-linear solver routine  72 . Lastly, application of the solution of the linear solver routine  72  to change the positions of particles C 1  and C 2  is shown in FIG. 14 by lines  74  and  76  from non-linear solver routine  72  to first color space  60  and second color space  62 , respectively. 
     If changes to the position of cursor  40  would result in particle C 1  or C 2  crossing a boundary  78  of first color space  60  or a boundary  80  of second color space  62 , the present invention would impose a boundary constraint on the movement thereof in order to avoid changes in the position of particle C 1  or C 2  outside color space  60  or  62 . A boundary constraint imposed on one of particles C 1  and C 2  would be likewise imposed on the other of particles C 1  and C 2  regardless of which particle C 1  or C 2  cursor  40  was attempting to move. 
     With reference to FIG. 15, one or more objective functions  84  can also be imposed between two or more color spaces. To impose an objective between particle C 1  in first color space  60  and particle C 2  in second color space  62 , the present invention defines a state vector which includes a first plurality of scalar values corresponding to the position of particle C 1  in first color space  60 . Thereafter, the present invention utilizes mapping function  63  to map the first plurality of scalar values corresponding to the position of particle C 1  in first color space  60  into a second plurality of scalar values in second color space  62  corresponding to the position of a particle C 2  therein. Particle C 2  in second color space  62  defines a second color therein corresponding to the first color in the first color space  60 . 
     Next, using the point, click and drag method, particle C 2  is selected. Thereafter, in response to changing the position of cursor  40  from the position shown in solid line to the position shown in dashed line, the present invention determines an objective function  84  for a change of the second plurality of scalar values corresponding to the change in position of cursor  40 . Next, the present invention determines from the objective function  84  an objective vector  86  which defines a change to the state vector, i.e., the first plurality of scalar values, which minimizes the objective function  84 . Specifically, the present invention determines a partial derivative of the objective function  84  with respect to a derivative of the state vector to obtain the objective vector  86 . Then, the state vector and the objective vector  86  are combined to obtain a change in the position of particle C 1  in first color space  60  corresponding to the change in position of particle C 2  in second color space  62 . More specifically, the present invention solves the objective vector  86  for the change in position of cursor  40  and provides this solution to non-linear solver routine  72  which determines therefrom the change to be applied to the first scalar values to change the position of particle C 1  in first color space  60  corresponding to the change in position of particle C 2  in second color space  62 . 
     Determining the objective function  84  for each change in the position of cursor  40  is shown in FIG. 15 by a line  88  extending between second color space  62  and objective function  84 . Supplying the solution of the objective vector  86  to the non-linear solver routine  72  is shown in FIG. 15 by a line  90  from objective vector  86  to non-linear solver routine  72 . Lastly, application of the solution of the non-linear solver routine  72  to change the position of particle C 1  is shown in FIG. 15 by a line  92  from non-linear solver routine  72  to first color space  60 . 
     Constraints and objectives can be applied in one color space or between two or more color spaces utilizing one or more of the methods discussed above in connection with FIGS. 3 a - 15 . The objectives and/or constraints imposed on one or more particles are chosen based on desired movement of the one or more particles and any constraints to such movement in one or more color spaces. 
     As can be seen, the present invention utilizes techniques applicable to particle system modeling and, more specifically, to the constrained motion of particles in a particle system to produce changes in one or more colors in one or more color spaces represented by a computer. The present invention reduces the number of color transforms needed to change a color in two or more color spaces and provides a method for relating two or more colors in a color space so that a change to one color produces a corresponding change to the other related colors. While the present invention has been described as a method, the software program implementing the present invention is installed and operates on hardware of computer  2 . Hence, computer  2  includes appropriate means for implementing the software program of the present invention. 
     The invention has been described with reference to the preferred embodiments. Obvious modifications and alterations will occur to others upon reading and understanding the preceding detailed description. It is intended that the invention be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.