Abstract:
Methods of mapping the shading dictionary parameters provided by a postscript language interpreter into device space coordinates and performing shading in the device space is disclosed. For radial shading, the shading transformation matrix representing the coordinate conversion from the shading space to device space is embedded within the shading mechanism parameters to directly compute the shading color for a particular device space pixel, avoiding device coordinate transformations to the shading space. For triangle meshes based shading, the vertexes of the defined triangles are converted to device domain prior to performing interpolation for determining device pixel shading color. For patch based shading, comprising of patches bounded by four Bezier curves, the Bezier curves are converted to the device domain to directly compute the corresponding shading parameter variables for a particular device pixel.

Description:
BACKGROUND OF THE INVENTION 
   This invention relates generally to raster graphics systems and more specifically to shading operations. 
   In a raster graphics system, smooth shading in colour is used for two main operations. The first is to fill directly in a geometrically defined area, and the second is to produce shading patterns and then paint a defined area with the smooth colour transition. There is a plurality of methods to define such shading mechanisms. In a raster image processor related to a printer, the Postscript Language Reference Manual defines seven types of shading and parameters via shading dictionaries. The Postscript Language Reference, Third Edition, Adobe Systems, Inc., published by Addison Wesley Publishing, is hereby incorporated by reference. 
   Shading provides a smooth transition between colours across an area to be painted, independent of the resolution of any particular output device and without specifying the number of steps in the colour transition. If the shading area is a simple geometric shape, the shading operations can be performed directly to paint the corresponding device pixels with the shading colour defined in a shading dictionary. However, in most cases, the required shading area is comprised of complicated geometric shapes, which warrant the definition of a shading pattern. In the Postscript Language reference manual, a Type 2 shading pattern dictionary defines this type of shading patterns. Such a pattern dictionary contains the entry of a shading dictionary. The shading dictionary specifies details of a particular gradient fill, including the type of shading to be used, the geometry of the area to be shaded, and the geometry of the gradient fill itself. In addition, the shading dictionary can contain a function dictionary—which is required for some types of shading and optional for others—that defines how the colour or colours vary across the area to be shaded. The seven types of shading can be grouped in to several categories including function-based shading, axial shading, radial shading, triangle meshes based shading and patch meshes based shading. 
   In a raster graphics system, the output device contains a predefined number of pixels arranged in a predefined layout, with an inherent device resolution. A particular device&#39;s coordinate system is named the device space. A device space origin can be anywhere on the output, depending on the usage of the device. When painting pixels on the device, each device pixel needs to be painted as defined by the painting instruction. Coordinates specified in a Postscript program refer to locations within a coordinate system, named user space, which always bears the same relationship to the current page, regardless of the output device. 
   The coordinate transformation from the user space to the device space is achieved using the current transformation matrix (CTM). This matrix contains information related to translation, rotation and scaling. It is represented as a six-element array given by:
 
└a b c d t x  t y ┘
 
The coordinate transformation from user space coordinate pair (x, y) to device coordinates (x′, y′) is given by the following equations.
 
 x′=ax+cy+t   x 
 
 y′=bx+dy+t   y 
 
   In the case of shading operations, each device pixel coordinate needs to be transferred to the coordinate system, named as the shading space, that is used to define the shading operation, in order to compute the colour values corresponding to that particular device pixel. This coordinate mapping involves computing a shading space coordinate pair (x″, y″) for a given (x′, y′) using the shading transformation matrix [a″ b″ c″ d″ t″ x  t″ y ], where its inverse matrix is given by,
 
[a i ″ b i ″ c i ″ d i ″ t ix ″ t iy ″]=[a″ b″ c″ d″ t″ x  t″ y  ] −1 
 
and the correspondent mapping is defined as below:
 
 x″=a   i   ″x′+c   i   ″y′+t   ix ″
 
 y″=b   i   ″x′+d   i   ″y′+t   iy ″
 
   In the case that a direct shading operation is invoked, such a shading space is the current user space. That is, for a pair (x″, y″), x″=x, y″=y and [a″ b″ c″ d″ t″ x  t″ y ]=CTM; However, in the case that a shading operation is used with a Type 2 pattern dictionary, the shading space is the pattern coordinate system (x p , y p ) established at the time the shading pattern is instantiated with the PostScript operator makepattern. That means x″=x p , y″=y p  and [a″ b″ c″ d″ t″ x  t″ y ]=[a p  b p  c p  d p  t px  t py ] (the compound pattern transformation matrix). Since each shaded device pixel needs to be mapped to the shading space to compute the corresponding pixel colour, shading operations are inherently computation intensive and thus inefficient. 
   Radial (Type 3) shading, as described by the Postscript Language reference manual, defines a colour blend that varies between two circles. They are commonly used to depict 3D spheres and cones. The blend circles are defined in terms of a subsidiary parametric variable, which varies linearly between 0.0 and 1.0. The parametric variable normally needs to be calculated first from a unique circle in shading coordinate space on that a particular point (x″, y″) resides. The resulting parametric variable is then used to compute the corresponding shading colour by use of a defined mathematical function in the Type 3 shading dictionary. When shading a device pixel, it is necessary to transform the device pixel to the shading coordinates, and identify the circle in which the mapped pixel resides. From the identified circle, the corresponding parametric variable can be found to determine the colour that should be painted on the current device pixel. Notice that the function dictionary is used in this type of shading operation. There are three types of function dictionaries supported in PostScript language level  3 : the sampled function (Type 0), the exponential interpolation function (Type 2) and the stitching function (Type 3). These Function dictionaries are tightly associated with smooth shading operation, providing close control over the shape of the colour transitions across the geometry of the shading. The sampled function (Type 0) provides an approximation of any desired mathematical function based on a sequence of sampled values, using either linear or cubic interpolation as defined in the function dictionary. For a given function input, the surrounding nearest neighbor sample values of the input need to be located for performing any specified interpolation scheme. 
   Triangle meshes based (Types 4 and 5) shading, as described by the Postscript Language reference manual, defines the area to be shaded by a path composed entirely of triangles. The colour at each vertex of the triangle is specified, and a technique known as Gouraud interpolation is used to colour the interiors. The function dictionary is the optional entry of these types of shading dictionaries. When shading the device pixels, it is necessary to transform the device pixels to the shading coordinates and identify the location within the defined triangles in the shading space to determine the colours of the pixels to be painted. 
   Patch meshes based (Types 6 and 7) shading as described by the Postscript Language reference manual, defines the area to be shaded by a path composed entirely of patches bounded by four cubic Bezier curves. The colour at four intersection points of the boundary Bezier curves is specified, and two linear parameters (u,v) spanning 0.0 to 1.0 inclusive are used to interpolate the colour of interior of the patch. The interpolation is defined in the shading domain rather than in the device domain. The function dictionary is optional for these types of shading operation. When shading the device pixels, it needs to transform the device pixels to the shading coordinates and identify the location within the defined patches in the shading space to perform the corresponding colour interpolation. 
   In the above shading types, the necessity to transform the device coordinates to shading coordinates, simply to determine the colour at the device pixel is computationally expensive. Coordinate transformations also incur round-off errors when used with fixed-point arithmetic. In order to avoid such drawbacks, the colour computations should be performed in the device space. 
   SUMMARY OF INVENTION 
   The present invention provides a method of computing the parameter variable that is used to calculate the shading colour of a function dictionary that is specified in the shading dictionary, or computing the correspondent shading colour if no function dictionary is defined for a particular device pixel directly, using the device coordinate system. The shading mechanism defined by the shading dictionary is re-mapped from the shading coordinates to the device space using a plurality of parameters. These parameters are computed once per shading dictionary. Interpolation algorithms are performed on the device space to determine the colour to be painted. 
   In accordance with an aspect of the present invention, there is described a method of merging the shading transform matrix [a″ b″ c″ d″ t″ x  t″ y ] with the radial shading equations of parametric circles, to produce a quadratic equation in terms of the parameter variable corresponding to the shading colour. The determinant of the quadratic equation indicates the existence of a defined colour if positive (or zero). The solution of the quadratic equation, within the interval 0.0 to 1.0 inclusive, represents the parameter variable corresponding to the shading colour. 
   A further aspect of this invention related to radial shading includes derivation of a plurality of parameters from the shading dictionary to rapidly compute the aforementioned determinant of the quadratic equation. 
   Another aspect of the present invention is related to radial shading that includes the systematic determination of the surrounding neighborhood for an input of the sampled function (Type 0), facilitating any underlying interpolation scheme (either linear or cubic spline interpolation) defined in the function dictionary for shading operations. Based on the assumption of the forwarding interpolation strategy is used for the implementation of all possible interpolation schemes required for shading operations, the number of the sample points in the surrounding neighborhood is computed. Accordingly, the index in the sample table and the position in the sequential DataSource of the nearest sample point—that is always referred as the left-bottom sample point of the surrounding neighborhood—is determined, based on the candidate index value in the sample table correspondent to the input. Subsequently, the index in the sample table and the position in sequential Datasource of the rest of the sample points in the surrounding neighborhood are calculated relative to the left-bottom point. The correspondent sample values of all surrounding neighbors of the input are retrieved. The information of the surrounding neighborhood is used for any underlying interpolation computation that might be involved in shading operation, such as in the shading Type 3 operation. 
   In accordance with another aspect of the present invention, there is described a method of performing triangle meshes based shading directly in the device domain. The vertices of the triangles defined by the shading dictionary are converted to device space once per triangle. Each mapped triangle is then shaded according to the sequence defined in the shading dictionary by applying Gouraud interpolation scheme in the device space. When a mapped triangle in the device space is in shading process, it is referred as the current triangle. In order to achieve a correct and fast shading colour computation, a pair of edges of the current triangle (in the device space)—that intersects the scanline on which the current device pixel resides—is accurately located in the left-to-right order. Subsequently, only the shading colour difference from previous shading colour (when the function dictionary does not present in the shading dictionary), or the difference from the previous value of the parametric variable (when the function dictionary presents) that is used to compute the correspondent shading colour, is calculated according to the increment in the device coordinate y′ (once per scanline) and x′ (per pixel), and then used to determine the correspondent shading colour of the device pixel. 
   In accordance with another aspect of the present invention, there is described a method of performing patch meshes based shading directly in the device domain. The control points of the patch defined by the shading dictionary and bounded by four cubic Bezier curves are converted to device domain once per patch. Each the mapped patch in the device space is then shaded according to the sequence defined in the shading dictionary, by mapping the four-sided device patch onto the unit square that has two parametric variable (u,v) respectively varied horizontally and vertically from 0.0 to 1.0 inclusive, applying multi-dimensional linear interpolation on the dvice Bezier curves to locate each surface point of the device-domain patch for (u,v) varying over the unit space and employing bi-linear interpolation algorithm in the unit space to compute the shading colour for the resulting device surface pixel at its correspondent parametric value (u, v). When a mapped patch in the device space is being shaded, it is referred as the current patch. The pair of C boundary curves (correspondent to the two horizontal sides of the unit space for v=0.0 and v=1.0) and the pair of the D boundary curves (correspondent to the two vertical sides of the unit space for u=0.0 and u=1.0) are correctly identified among the four Bezier boundary curves of the current patch before the patch is mapped onto its correspondent unit space. Implicit control points residing on the device space patch are computed for the shading Type 6 and are used, together with the twelve mapped control points of the boundary Bezier curves of the patch, to divide the C curves (correspondent to v varying vertically across the parametric space, i.e., the unit square) and the D curves (correspondent to the u varying horizontally across the unit square) into a plurality of subsections. Whereas for shading Type 7, its four mapped internal control points in the device space are used directly, together with the twelve mapped boundary control points of the mapped patch, to derive such a plurality of subsections of the C and D curves within the device-domain patch. As a result, the u and v domain of 0.0 to 1.0 is subdivided into number of steps that equals to the product of a experimental factor (misPT=1.3) and the number of device curve pixels on the longest C device-domain control curve among the device-domain boundary C curves and the device-domain horizontal control curves defined by the device implicit (for shading Type 6) or internal (for shading Type 7) control points in horizontal direction, but in vertical direction is equivalent to the number of device curve pixel of the longest control D device-domain curve among the device-domain D curves and the device domain vertical control curves defined by the device implicit (for shading Type 6) or internal (for shading Type 7) control points in the vertical direction. Accordingly, the number of steps in the u and v domain of 0.0 to 1.0 is equivalent to the number of subsection of the C and D curves of the mapped patch respectively. The length of each subdivided Bezier curve of the mapped patch is approximated in the device space using the linear subsection with antialias consideration. Therefore, any device pixel within the specified shading patch can be directly attributed to the corresponding u and v values, enabling the direct computation of shading colour. 
   A further aspect of this invention related to the patch mesh based shading includes the patch fold-over handling. When a patch appears to fold-over on itself, the cubic Bezier curves (the C curves or D curves) on the patch can be intersecting or self-intersecting. That means the location of the pixels on the device space Bezier curve may change direction so that pixels are mapped onto previously mapping pixels as the value of u or v increases in the parametric space. For the case that more than one parametric space location (u, v) is mapped to the same device pixel, only the shading colour correspondent to the largest value of u will be chosen to paint the device pixel if these parametric coordinates have the same value of v; Otherwise, only the shading colour correspondent to the largest value of v will be used. In the related invention, each device-domain patch is painted with the solid device white colour first, except from the shading operation whose Background colour is not defined and that is under the process of caching the shading bitmap (where the initialization of the bitmap mask is equivalent to painting the patch white). After that, the patch is shaded as the parametric variables (u, v) varying from 1.0 to 0.0, instead of from 0.0. to 1.0, across the unit square. For each device-domain curve subsection, the end point of the curve subsection is traced as the last shading point to prevent from any duplicated mapping onto the previous mapped device pixel, ensuring one-to-one mapping between the parametric space (u, v) to the device space. Also, within each device-domain curve subsection, each device curve pixel is then painted with the shading colour only if the current colour of the pixel is white in the device domain, achieving correct one-to-one mapping between the unit space and the device patch. 
   Still other aspects of the present invention will become readily apparent to those skilled in this art from the following description, wherein there is shown and described a preferred embodiment of this invention, simply by way of illustration of one of the best modes suited for to carry out the invention. As it will be realised, the invention is capable of other different embodiments and its several details are capable of modifications in various obvious aspects all without departing from the invention. Accordingly, the drawing and descriptions will be regarded as illustrative in nature and not as restrictive. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The accompanying drawings incorporated in and forming a part of the specification illustrate several aspects of the present invention, and together with the description serve to explain the principles of the invention. 
       FIG. 1  is a block diagram of shading module. 
       FIG. 2  is an illustration of radial shading in device space. 
       FIG. 3  is an illustration of triangle mesh based shading in device space. 
       FIG. 4  is an illustration of patch mesh based shading in device space. 
       FIG. 5  is an illustration of a computer system for implementing the various aspects of the present invention. 
       FIG. 6  is a block diagram of a methodology for performing image shading accordance with an aspect of the present invention. 
       FIG. 7  is a block diagram of a methodology for performing radial shading in accordance with an aspect of the present invention. 
       FIG. 8  is a block diagram of a methodology for performing triangular mesh shading in accordance with an aspect of the present invention. 
       FIG. 9  is a block diagram of a methodology for performing patch meshes shading in accordance with an aspect of the present invention. 
       FIG. 10  an illustration of the surrounding neighbourhood for an input of Type 0 function dictionary (under the forwarding interpolation strategy) used by four types of underlying interpolation scheme that might be involved in shading operations. 
       FIG. 11  is a block diagram of a systematic mechanism for determining the surrounding neighbourhood of a given input of Type 0 function dictionary for shading operations in accordance with an aspect of the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   Referring to  FIG. 1 , in a raster graphics output device such as a printer or a display, the rasterisation of input visual information is performed via a raster image processor  102  contained within a control unit  100  of the device. Shading operations required for rasterisation of input data is usually performed within a shading module  104 . Such a module contains predominantly two sub-modules, one for performing shading operation on areas bounded by simple geometric shapes and another for accomplishing shading pattern rendering operations on coverage areas with complicated geometrical boundaries. When painting a defined area with a shading pattern, a pattern dictionary is normally used, that describes the pattern type, unique identification number for identifying the particular pattern, Implementation key for specifying any implementation-dependent entries that are associated with the pattern and a shading definition, i.e., a shading dictionary. There are seven types of shading dictionaries specified in PostScript language level 3. Each type of shading dictionary can contain one of three types of function dictionaries for shading Types 1, 2 and 3, and optionally for shading Types 4, 5, 6 and 7 to define how the shading colour or colours vary across the area to be shaded. 
   After the image rendering job is processed by the raster image processor  102 , it is input into shading module  104 . The shading rendering operation instructions together with relevant parameters from raster image processor  102  is input into both the Shading Fill Operation sub-module  110 , which is used for performing shading operations on areas bounded by simple geometric shapes, and the Shading Pattern Operation  120  sub-module which is used for accomplishing shading pattern rendering operations on coverage areas with complicated geometrical boundaries. The Shading Pattern Operation sub-module  120  then sends its parameters, i.e., the Type 2 pattern dictionary, to the Shading Pattern Dictionary Operation routines  122 . The Shading Fill Operation sub-module  110  accepts the shading dictionary as the input argument data of the sub-module. The Seven types of Shading Dictionary Preparation routines  112  then receive input (i.e., the shading dictionary) from both the Shading Fill Operation sub-module  110  and the Shading Pattern Dictionary Operation routines  122 . The Seven types of Shading Dictionary preparation routines  112  then sends data to the Shading Pattern Preparation routines  114 , and the Three types of Function Dictionary Preparation routines  124  if the function dictionary presents in the shading dictionary. The Three types of Function Dictionary Preparation routines  124  subsequently forwards the data to the Shading Pattern Preparation routines  114 . Data from Shading Pattern Preparation routines  114  is then input to the Seven types of Shading Gradient Fill routines  116 . Data from the Seven types of Shading fill routines is sent to Three types of Colour Interpolation Function routines  126 , as well as to Painting Operations  106 , if the function dictionary is defined. In addition, data from Three types of Colour Interpolation Function routines  126  is sent to Painting Operations  106 . Although, there are Seven Types of Shading Preparation routines  112 , Three types of Function Dictionary Preparation routines  122  and Three Types of Colour Interpolation  126  Function routines and Seven types of Shading Gradient fill routines  126  shown in  FIG. 1  corresponding to the number of shading types, function types and colour interpolation routines currently defined by Postscript, the number of routines may be varied to any suitable value to correspond to the implementation for control unit  100 . An aspect of the present invention is directed to techniques for shading gradient fill routines to be employed by the Seven types of Shading Gradient fill routines  116 . 
   Table 1 below illustrates the general contents of a shading dictionary. 
   
     
       
             
           
             
             
             
           
         
             
               TABLE 1 
             
           
           
             
                 
             
             
               General contents of a shading dictionary 
             
           
        
         
             
               KEY 
               TYPE 
               VALUE 
             
             
                 
             
             
               ShadingType 
               Integer 
               1- Function-based shading 
             
             
               (Required) 
                 
               2- Axial shading 
             
             
                 
                 
               3- Radial shading 
             
             
                 
                 
               4- Free-form Gouraud-shaded triangle 
             
             
                 
                 
                   mesh 
             
             
                 
                 
               5- Lattice-form Gouraud-shaded triangle 
             
             
                 
                 
                   mesh 
             
             
                 
                 
               6- Coons patch mesh 
             
             
                 
                 
               7- Tensor-product patch mesh 
             
             
               ColourSpace 
               Name or Array 
               The colour space in which shading 
             
             
               (Required) 
                 
               colour values are expressed. May be 
             
             
                 
                 
               any device, CIE-based, or special colour 
             
             
                 
                 
               space except a Pattern space. 
             
             
               Backgorund 
               Array 
               An array of colour components 
             
             
               (Optional) 
                 
               corresponding to the colour space, 
             
             
                 
                 
               specifying a single background colour 
             
             
                 
                 
               value. 
             
             
               BBox 
               Array 
               An array of four numbers giving the 
             
             
               (Optional) 
                 
               left, bottom, right and top coordinates 
             
             
                 
                 
               respectively, of the shading bounding 
             
             
                 
                 
               box. 
             
             
               Anti Alias 
               Boolean 
               A flag indicating whether to filter the 
             
             
               (Optional) 
                 
               shading function to prevent aliasing 
             
             
                 
                 
               artifacts. 
             
             
                 
             
           
        
       
     
   
   The information contained within a shading dictionary is essential for performing shading operations either as a shading pattern or as a colour filling of a specified geometric boundary. According to the Postscript Language Reference Manual, the contents of a shading dictionary are described as shown in Table 1. Depending on the type of shading, further information specific to each shading mechanism is included in each shading dictionary. 
   Referring to  FIG. 6 , there is shown a method  600  for computing the shading colour (if no function dictionary is specified) or the value of a parameter variable (if the function dictionary is defined in the shading dictionary) corresponding to the shading colour for a particular device pixel directly, using the device coordinate system. At  602 , the shading mechanism defined by the shading dictionary is re-mapped from the shading coordinates to the device space using a plurality of parameters. These parameters are computed once per shading dictionary. For example, the shading mechanism can be a specified closed surface in a shading domain. Additionally, colours for points located along the perimeter of the closed surface in the shading domain can be defined, thus when the closed surface is mapped from the shading domain to a device domain, forming a device domain closed surface, the colours corresponding to the points located along the perimeter of the closed surface are also mapped into the device domain. As shown at  604 , interpolation algorithms for points bounded by the closed surface are performed in the device space to determine the shading colour to be painted. 
   Radial (Type 3) shading defines a colour blend that varies between two circles. They are commonly used to depict 3D spheres and cones. In addition to the dictionary entries in Table 1, Type 3 shading dictionary contains the entries shown in Table 2. 
                                 TABLE 2                   Additional entries in a type 3 shading dictionary            KEY   TYPE   VALUE               Coords   Array   An array of six numbers [x     0   , y     0   , r     0   , x     1   , y     1   , r     1     ]         (Required)       specifying the centres and radii of the starting               and ending circles, expressed in the shading               coordinate space       Domain   Array   Array of two numbers [t     0,     t     1     ]  specifying the       (Optional)       limiting values of a parametric variable t,               which is considered to vary linearly between               these two values, as the colour gradient varies               between the starting and ending circles.       Function   Dictionary or   A 1-in n-out function dictionary or an array of       (Required)   Array   n 1-in, 1-out function dictionaries, where n is               the number of colour components as defined by               the colour space. Entries represent colour               values corresponding to values of parameter t.       Extend   Array   An array of two boolean values specifying       (Optional)       whether to extend the shading beyond the               starting and ending circles.                    
The blend circles for shading are defined in terms of a subsidiary parametric variable, s, given by:
 
           s   =       t   -     t   0           t   1     -     t   0               
Where s varies between 0.0 and 1.0 inclusive, as the variable t varies between t 0  to t 1 , for shading circles between the starting and ending circles, as shown in  FIG. 2 . As illustrated in  FIG. 2 , the starting circle  202  is for the case when s=0.0 (as t=t 0 ) and the ending circle  206  is for the case where s=1.0 (as t=t 1 ). Circle  204  illustrates when s is greater than or equal to 0.0 and less than or equal to 1.0. The device pixel to be pained  208  is located on circle  204 .
 
   The centre [x″ c (s), y″ c (s)] and radius [r″(s)] of each blend circle in the shading space can be represented as follows:
 
 x″   c ( s )= x″   0   +s ×( x″   1   −x″   0 )
 
 y″   c ( s )= y″   0   +s ×( y″   1   −y″   0 )
 
 r ″( s )= r″   0   +s ×( r″   1   −r″   0 )
 
However, when rendering a shaded area in the device space, the rendering operations are performed one raster scanline at a time. Therefore, usually, it is necessary to map a particular pixel within a given raster line into the shading coordinate space using the inverse of the shading transformation matrix [a″ b″ c″ d″ t″ x  t″ y ], to identify the shading circle on that the mapped pixel resides. Hence determining the value of corresponding s, and the parameter variable t, which produces the colour to be painted with the help of the entry of the function dictionary defined in this type of shading dictionary?
 
   However, according to an aspect of the present invention, it is possible to compute the corresponding s value directly for a given device pixel. Referring to  FIG. 7 , with continued reference to  FIG. 2 , the process  700  involves, as shown at  702 , combining the shading transformation matrix [a″ b″ c″ d″ t″ x  t″ y ] with the blend circle equations to derive a quadratic equation in terms of the variable s, as shown below:
 
As 2 +Bs+C0 Where,
 
A=P0
 
 B=dx′×P 1 +dy′×P 2 +P 3
 
 C=dx′×dx′×P 5 +dy′×dy′×P 6 +dx′×dy′×P 7 +P 4
 
In which,
 
dx′=x′−x′ 0  and dy′=y′−y′ 0 , where (x′, y′) is the current device pixel coordinates that needs to be painted and (x′ 0 , y′ 0 ) represents the centre coordinates of the starting circle in the device domain. The constants P0 to P8 are computed as shown below:
 
 P 0=( x″   1   −x″   0 )×( x″   1   −x   0 )+( y″   1   −y″   0 )×( y″   1   −y″   0 )−( r″   1   −r″   0 )×( r″   1   −r″   0 )
 
 P 1=2 ×[d ″×( x″   1   −x″   0 )− b″ ×( y″   1   −y″   0 )]/ P 8
 
 P 2=−2×[ a″× ( y″   1   −y″   0 )− c″× ( x″   1   −x″   0 )]/ P 8
 
 P 3=−2×[ r″   0 ×( r″   1   −r″   0 )]
 
 P 4=−( r″   1   −r″   0 )
 
 P 5=( d″×d″+b″×b ″)/( P 8 ×P 8)
 
 P 6=( c″×c″+a″×a ″)/( P 8 ×P 8)
 
 P 7=−2×( c″×d″+a″×b″ )/( P 8× P 8)
 
 P 8 =a″×d″−b″×c″ 
 
It should be noted that all constants P0 to P8 and (x′ 0 , y′ 0 ) should be computed only once per each shading dictionary, thus as shown at  704  constants P0 to P8 are stored, which reduces the computation that should be performed per each rendering pixel. Therefore, per pixel computations are limited to computing dx′ and dy′ values.
 
   At  706 , the determinant of the above quadratic equation is computed. The determinant (Δ) of the above quadratic equation is given by:
 
Δ= B×B− 4 ×A×C 
 
If Δ is negative, the device pixel is outside the blending circle domain and need not be painted. Otherwise, a solution to the quadratic equation exists, and is given by:
 
 s =(− B ±√{square root over (Δ)})/(2× A ),
 
which is computed at  708 .
 
The solution within 0.0≦s≦1.0 interval is considered as the valid solution for s. The corresponding parameter t is then simply computed at  710  using the equation below,
 
 t=t   0   +s ×( t   1   −t   0 )
 
Therefore, using the resulting t value and the function dictionary contained within the shading dictionary, it is possible to determine the shading colour for the device pixel (x′, y′) directly, without involving coordinate space transformations per each pixel. Hence, this method is faster in rendering plurality of shaded device pixels compared to the methods known by prior art.
 
   The function dictionary is employed by Radial (Type 3) shading, as indicated in Table 2, for performing the shading colour computation respect to the parametric variable t. The function dictionary used here specifies the colour function&#39;s representation, the set of attributes that parameterize that representation and the additional data needed by that representation. Three types of function dictionary are supported in the PostScript language level 3. The function dictionaries are solely used by smooth shading dictionary to provide close control over the shape of the colour transitions across the geometry of the shading. All types of function dictionary share the keys listed in Table 3. 
                                 TABLE 3                   General contents of a function dictionary            KEY   TYPE   VALUE               FunctionType   Integer   0- Sampled function       (Required)       2- Exponential interpolation function               3- Stitching function       Domain   Array   An array of 2 × m numbers for clipping       (Required)       input values, where m is the number of input               values.       Range (required   Array   An array of 2 × n numbers for clipping       for Type 0,       input values, where n is the number of output       Optional for       values.       others)                    
For shading operations, the parameter m=2 for shading Type 1 or 1 for others; n=1 or the number of colour components defined in shading dictionary. Moreover, each type of function dictionary must include attributes appropriate to the particular function type.
 
   Sampled (Type 0) function uses a sequence of sample values to approximate any described mathematic function whose input domain and output range are bounded. This type of function is the most general type of function. In addition to the entries in table 3, the Type 0 function includes the entries listed in Table 4. 
                                 TABLE 4                   Additional entries in a Type 0 function dictionary            KEY   TYPE   VALUE               Order   Integer   The order of interpolation between       (Optional)       samples.               1 - linear interpolation               3 - cubic spline interpolation       DataSource   String or file   A string or positional file providing the       (Required)       sequence of sample values that specifies               the function.       BitPerSample   Interger   The number of bits used to represent each       (Required)       component of each sample value. The               number must be 1, 2, 4, 8, 12, 16, 24               or 32.       Encode   Array   An array of 2 × m numbers specifying the       (Optional)       liner mapping of input values into the               domain of the function&#39;s sample table.       Decode   Array   An array of 2 × n numbers specifying the       (Optional)       liner mapping of sample values into the               range of the values appropriate for the               function&#39;s output values.       Size   Array   An array of m positive integers specifying       (Optional)       the number of samples in each input               dimension of the sample table.                    
The sample values used by the Type 0 function are usually organized as an m-dimensional table in which each entry has n components. The output values of the function is calculated based on the sample table, using the required interpolation scheme (i.e., either the liner interpolation or cubic spline interpolation scheme). As mentioned before, for shading operation, the input dimension of a function dictionary could be 2 for shading Type 1, or 1 for others. Therefore, the 1-D linear interpolation, 2-D linear interpolation, 1-D cubic spline interpolation or 2-D cubic spline interpolation scheme could be involved to compute shading colour(s) for shading operation. In order to perform any these possible interpolation scheme for an input value of the function Type 0 dictionary, the nearest surrounding sample values of the input need to be located in the sample table of the function dictionary. Such determination is usually accomplished using different mechanisms suitable to different underlying interpolation scheme respectively, because the number of surrounding sample values and their position relative to the input value in the sample table are different. However, in the described invention, a systematic mechanism for determining the surrounding neighborhood in the sample table for an input of Type 0 function dictionary is provided under the assumption of the forwarding interpolation strategy is applied, facilitating any above mentioned types of interpolation scheme that might be involved in shading operations.
 
   Under the forwarding interpolation strategy, the surrounding neighborhood in sample value table for an input of Type 0 function dictionary can be depicted as shown in  FIG. 10 , for the above mentioned four types of interpolation schemes that might be defined in the function dictionary for shading operations. Referring to  FIG. 11 , with continued reference to  FIG. 10 , given an input of Type 0 function dictionary, at  1102  the input is clipped within the Domain of the function, and at  1102  encoded using the Encode definition of the function and clipped according to the input dimension specified in the Size entry of the function. The result is used as the candidate input of the sample table of the function. According to the candidate, at  1106  the index (Idxlb) of the nearest sample point in the sample table is determined, that is always treated as the left-bottom point of the surrounding neighborhood of the given input as shown in  FIG. 11 . Subsequently, at  1108  the number of sample value in the surrounding neighborhood in sample table is computed as below: 
   For Order=1 or Size i &lt;4 (i=1, . . . , m), the number of sample values in the surrounding neighborhood=C×P, where the parameter C=2 and P=m; 
   For Order=3 or all Size i ≧4 (i=1, . . . , m), the number of sample values in the surrounding neighborhood=C×p, where C=3 and P=2×m−1; 
   Notice that the forwarding interpolation strategy is applied. At  1110 , the determined index in the sample table of the nearest sample point is then examined in order to clip the surrounding neighborhood around boundary of sample table as shown below: 
   If  Idxlb   i ≧Size i   −C+ 1 then  Idxlb   i   =Idxlb   i −1; where  i= 1 , . . . , m.    
   Subsequently, at  1112  the index of the other sample points of the surrounding neighborhood in the sample table, as well as the position in the sequential DataSource of all surrounding neighborhood points are computed relative to the nearest sample point. At  1114 , the correspondent sample values of the surrounding neighborhood are retrieved. At  1116 , all the information about the surrounding neighborhood of the input is sent to the underlying interpolation scheme specified in the function dictionary, to calculate the correspondent output function value, e.g., the shading colour in shading Type 3 operations. It should be noted that the number of sample points in the surrounding neighborhood is calculated once for the sampled (Type 0) function used by shading operations. 
   Triangle mesh based shading is commonly used to paint complex coloured and shaded 3D shapes. The area to be shaded is defined by a path composed entirely of triangles. The shading dictionary DataSource entry specifies the colour at each vertex of the triangles. 
   According to the present invention, each triangle  300  defined by the given vertex coordinates, C A (x″ a ,y″ a ), C B (x″ b ,y″ b ), C C (x″ c , y″ c ) in the shading space as illustrated in  FIG. 3 , are mapped to the device domain coordinates C A (x′ a ,y′ a ), C B (x′ b ,y′ b ), C C (x′ c , y′ c ) from the shading space. A closed path is constructed in the device domain by connecting the vertices via lines  302 ,  304  and  306 . As shown in  FIG. 3 , the colour of any device pixel  310  within the constructed closed path, that needs to be painted is directly computed in the device domain. The raster scanline  308  in which the device pixel resides is identified by its y-coordinate. The two edges Edge 1 , Edge 2 , where this scanline meets the constructed device-domain triangular path are determined correctly in the left-to-right order. For the instance shown in  FIG. 3 , the colour at device coordinates (x′, y′) is computed using a method  800  as described in  FIG. 8  as follows: 
   Step  1  at  802 : 
   
       
       Initialize the last shading point and last pair of intersected edges for a device triangle, for example: LastP.x′=−1, LastP.y′=−1, LastEdge 1 =−1 and LastEdge 2 =−1. At  804  it is determined if the pixel (x′,y′) is on the previous scanline.
 
If at  804 , if  dy′=y′− Last P.y′= 0 (yes);
       then at  806  C(x′,y′) is computed along the intersected scanline as
 
 dx′=x ′−Last P.x ′, Last C+=dx′×IncrmC   scanline ;
   and at  808  LastP.x′=x′ and return C(x′, y′)=LastC;   
     
       However, if at  804  dy′−y′−LastP.y′ does not equal 0 (no), then at  810  for dy′&gt;0, identify the left Edge 1  and right Edge 2  among three triangle edges; If at  812  the pixel (x′, y′)=(x′ a , y′ a ), (x′ b , y′ b ) or (x′ c , y′ c ), (yes) return the correspondent vertex colour C A , C B , or C C  as shown at  814 ; otherwise (no), at  816  if the Edge 1 =Edge 2 , the pixel line is on one of the edges of the triangle (yes), then at  818 
       C edge1 =the colour of the left vertex of the edge,   x′ edge1 =the x-coordinator of the left vertex of the edge,   C edge2 =the colour of the right vertex of the edge,   x′ edge2 =the x-coordinator of the right vertex of the edge,
 
 ddx′=x′   edge2   −x′   edge1   , IncrmC   scanline =( C   edge2   −C   edge1 )/ ddx ′ and
 
Last C=[C   edge1 ×( x′   edge2   −x′ )+ C   edge2 ×( x′−x′   edge1 )]/ ddx′;  
    return C(x′, y′)=LastC; otherwise (no),   at  820 , If Edge 1 =LastEdge 1  and Edge 2 =LastEdge 2  (yes) then at  822 ,
 
 C   edge1   +=dy′×IncrmC   edge1  and  C   edge2   +=dy′×IncrmC   edge2 ; otherwise (no)
   
     
       at  812  for Edge 1 ≠LastEdge 1  or Edge 2 ≠LastEdge 2 ,
 
 ddy′   left   =y′   a   −y′   b   , IncrmC   edge1 =( C   a   −C   b )/ ddy′   left , and
 
 C   edge1   =[C   a ×( y′−y′   b )+ C   b ×( y′   a   −y ′)/ ddy′   left ,
 
 ddy′   right   =y′   a   −y′   c   , IncrmC   edge2 =( C   a   −C   c )/ ddy′   right , and
 
 C   edge2   =[C   a ×( y′−y′   c )+ C   c ×( y′   a   −y ′)/ ddy′   right .
 
       At  824 :
 
 x′   edge1   =x′   a +( x′   b   −x′   a )×( y′−y′   a )/( y′   b   −y   a ),
 
 x′   edge2   =x′   a +( x′   c   −x′   a )×( y′−y′   a )/( y′   c   −y′   a ),
 
 ddx′=x′   edge2   −x′   edge1   , IncrmC   scaline =( C   edge2   −C   edge1 )/ ddx′. 
 
       At  826 :
 
Last C=[C   edge1 ×( x′   edge2   −x ′)+ C   edge2 ×( x′−x′   edge1 )/ ddx′, 
 
LastP.x′=x′, LastP.y′=y′, LastEdge1=Edge1 and LastEdge2=Edge2;
 
return  C ( x′, y ′)=Last C ; End;
 
Where colour represented at the scanline intersection points with the triangular path edges are given by C edge1  and C edge2 , the x-coordinates of the intersection points are given by x′ edge1  and. x′ edge1 , C(x′,y′) represents the shading colour or the parametric value coorespondent to the shading colour at device coordinates (x′,y′). The varible IncrmC scanline  represents the constant increment in the colour or the parametric value along the scanline, whereas IncrmC edge1  and IncrmC edge2  for the increment in the colour or parametric value along the left Edge 1  and the right Edge 2  respectively. Step  802  is executed only once per device triangle. Also, for a particular device scanline, the y-coordinate remains unchanged. Therefore, only the steps  806  and  808  are repeated for every pixel on the scanline, where dx′=1. The results from steps  812 ,  816 ,  820  and  822  remains unchanged, and do not need re-computing along each scanline. As indicated at  820 , when the intersected scanline moves along the same pair of the intersected edges of the device triangle, only the constant increment in the colour or the parametric value, i.e., IncrmC edge1  and IncrmC edge2 , are used for calculating C edge1  and C edge2 . It should be noted that the value of dy′=y′−LastP.y′ always is 1 for the device-domain triangle and the constant increments IncrmC edge1  and IncrmC edge2  are only calculated once per pair of intersected edges.
 
     
  
   Referring to  FIG. 4 , there is illustrated an exemplary shading area  400  to be shaded using patch mesh based shading bounded by four cubic Bezier curves C 1 , C 2 , D 1 , D 2 . The boundary C curves, C 1 , C 2 , and the boundary D curves, D 1 , D 2 , are correspondent to the two horizontal sides of the unit square for u=0.0 and 1.0 and the two vertical sides of the unit square for v=0.0 and 1.0 respectively. As illustrated by graph  450 , parametric variables u and v vary between 0.0 and 1.0. The device pixel to be painted  402  can be represented by a point at the intersection of curves  404  and  406 . The curve  404  is parallel and between curves C 1 , C 2 , and the curve  406  is parallel and lies between curves, D 1 , D 2 . Such curves  404  and  406  are then referred as the C and D curves of the patch corresponding to the parametric variable u and v varying horizontally and vertically across the unit square respectively. 
   Patch mesh based shading is commonly used for graphical special effects. The shading area is constructed using a plurality of colour patches, each bounded by four cubic Bezier curves, C 1 , C 2 , D 1 , D 2  as shown in  FIG. 4 . The C curves and D curves are defined over parametric variable u and v respectively. Colours are specified for each of the four-corners of the patch. The coordinate mapping for the shading is usually given by the surface S defined as below, 
   For shading Type 6
 
 S =(1− v )× C   1 ( u )+ v×C   2 ( u )+(1 −u )× D   1 ( v )+ u×D   2 ( v )−(1 −v )×[(1− u )× C   1 (0)+ u×C   1 (1)]− v ×[(1 −u )× C   2 (0)+ u×C   2 (1)]
 
For shading Type 7
 
           S   =       ∑     i   =   0     3     ⁢       ∑     j   =   0     3     ⁢       P   ij     ⨯       B   i     ⁡     (   u   )       ⨯       B   j     ⁡     (   v   )                   
where P ij  is the control point in row i and column j of the patch, B i  and B j  are Bernstein polyniminals
   B   0 ( t )=(1− t ) 3     B   1 ( t )=3 ×t ×(1 −t ) 2     B   2 ( t )=3× t   2 ×(1− t )   B   3 ( t )= t   3   
It should be noted that the four control points (P 11 , P 12 , P 21 , P 22 ) are implicitly defined by the boundary curves of the patch in shading Type 6 and given by equations:
   P   11   =S (1/3, 2/3)   P   12   =S (2/3, 2/3)   P   21   =S (1/3, 1/3)   P   22   =S (2/3, 1/3) 
Where S is the surface over the patch of shading Type 6. The four control points is hence called implicit control points in Shading Type 6. Furthermore, the span of the parametric variable u and v is limited to 0.0 to 1.0 inclusive for both shading Type 6 and Type 7.
 
   Using the above equation, finding the device pixel coordinate(x′, y′) with respect to the surface coordinate (S x″ , S y″ ) in the shading space with the shading colour represented by a particular value of the parametric variables (u,v) involves computing the Bezier curve points (in the shading space) correspondent to the parametric variables, computing the correspondent surface coordinates in the shading space, converting the surface coordinates to the device space, and computing the correspondent shading colour (if the function dictionary is not defined in the shading dictionary), or the parametric value correspondent to the shading colour (if the function dictionary is defined), according to the parametric variables (u, v), by using linear interpolation algorithm. In raster image processing, such a requirement, i.e., to find the colour at a particular device pixel coordinate, is often encountered. According to an aspect of the present invention, this can be performed directly on the device space using a methodology  900  as shown in  FIG. 9  as follows: 
   At  902 : Map the control points of a patch to device domain using the shading transformation matrix [a″ b″ c″ d″ t″ x  t″ y ]. There are twelve control points for four boundary curves in a patch of shading Type 6, but sixteen control points, including twelve boundary curve control points and four internal surface control points, in a patch of shading Type 7. The mapped control points constructs a closed path in the device space bounded by the transformed Bezier curves. 
   At  904 : Identify the C boundary curves (correspondent to the two horizontal sides of the unit space for v=0.0 and v=1.0) and the D boundary curves (correspondent to two vertical sides for u=0.0 and u=1.0) among the four boundary curves of the device-domain patch, before the device-domain patch is mapped onto its correspondent unit space. 
   At  906 : Paint the device space patch with solid device white first, expect from the shading operation whose Background colour is not defined and that is under the process of the internal caching scheme of the shading bitmap (where the initialization of the bitmap mask is equivalent to painting the patch white). 
   At  908  it is determined if the shading is type 6, if it is (yes) then at  910 , find the four implicit control points (P 11 , P 12 , P 21 , P 22 ) residing on the device-domain patch for shading Type 6. The two points on each device-domain C Bezier boundary curves at known u=1/3 and 2/3, and the two points on each device-domain D Bezier curves at known v=1/3 and 2/3 are hence calculated. An algorithm similar to De Casteljau algorithm is used to estimate these points. After that, the four surface points P 11 =S( 1 / 3 ,  2 / 3 ), P 12 =S( 2 / 3 ,  2 / 3 ), P 21 =S( 1 / 3 ,  1 / 3 ) and P 22 =S( 2 / 3 ,  1 / 3 ) in the device domain are computed as defined previously (but in the device space). 
   At  912 : Divide all device space C and D curves of the mapped patch, respectively corresponding to the parameter v and u varying vertically and horizontally across the unit square, into a plurality of subsections, by exploiting the twelve mapped boundary control points as well as the four computed implicit control points in the device space for shading Type 6 or the four mapped internal control points for shading Type 7. The length of the C and D boundary Bezier curves in the device space, and the C and D Bezier control curves defined by the implicit control points for shading Type 6or the internal control points are estimated by using linear approximation as the sum of linear distances between start boundary control point, implicit or internal control points and end boundary control point [i.e., (l 1 +l 2 +l 3 +l 4 +l 5 +l 6 ) as shown in  FIG. 4 ]. This provides the approximate span of the patch in device pixels, along u and v directions only. As a result, the unit parametric space with the parametric variable u and v in 0.0 to 1.0 inclusive is subdivided into number of steps along u and v axis respectively as given by one of the following two schemes: 
   Scheme I: 
   
       
       
         
           (1) In the horizontal direction, the u domain from 0.0 to 1.0 is divided into the number (nu) of steps that equals the product of the experimental factor (misPT=1.3) and the number of device curve pixels on the longest C control curves among the device-domain boundary C curves and the device horizontal C control curves defined by the device implicit control points for shading Type 6 or the mapped internal control points for shading Type 7 in the horizontal direction. 
           (2) In the vertical direction, the v domain from 0.0 to 1.0 is divided into the number (nv) of steps that equivalent to the number of device curve pixels on the longest D control curves among the device-domain boundary D curves and the device vertical D control curves defined by the device implicit control points for shading Type 6 or the mapped internal control points for shading Type 7 in the vertical direction.
 
Scheme II:
 
           (1) In the vertical direction, the v domain from 0.0 to 1.0 is divided into the number (nv) of steps that equals the product of the experimental factor (misPT=1.3) and the number of device curve pixels on the longest D control curves among the device-domain boundary D curves and the device-domain vertical D control curves defined by the device implicit control points for shading Type 6 or the mapped internal control points for shading Type 7 in the vertical direction. 
           (2) In the horizontal direction, the u domain from 0.0 to 1.0 is divided into the number (nu) of steps that equivalent to the number of device curve pixels on the longest C control curves among the device-domain boundary C curves and the device horizontal C control curves defined by the device implicit control points for shading Type 6 or the mapped internal control points for shading Type 7 in the horizontal direction. 
         
       
     
  
   At  914 : there are nu steps along u direction and nv steps along v direction, the step sizes in u and v directions are set to 1/nu and 1/nv respectively. The device space surface pixels, for each combination of u and v values for u=1, 1−1/nu, 1−2/nu . . . 1−u − step/nu . . . 0 and v=1, 1−1/nv, 1−2/nv . . . 1−v − step/nv . . . 0, are then computed, using De Casteljau algorithm for finding Bezier curve points corresponding to each u and v, and the previously defined equation for computing  S  (in the device space). 
   At  916 : the coordinates (S x′ , S y′ ) in the device domain of the resulting surface point S of the mapped patch are examined against the traced last shaded surface point in the device domain, preventing from any duplicated mapping onto the previous mapped device pixel and ensuring one-to-one mapping between the parametric space (u, v) to the device space (x′, y′). If the correspondent device pixel of the surface point has not been shaded, the device D (for the above division scheme I of the parametric domain) or C (for the above division in scheme II of the parametric domain) curve subsection from the previous shaded surface point to the current one is approximated using the linear subsection with anti-alias consideration as shown at  918 . Otherwise (yes), go to step  914 . 
   At  920  it is determined whether the current colour of the device curve pixel is white. If the current colour of the device pixel is white (yes), at  922  the shading colour is simply computed using bilinear interpolation of colour values defined for the corners of the patch and the current u and v values. Processing then returns to  914 . 
   The above steps are repeated in the device space for all the patches contained in the shading geometrical shape defined in the shading dictionary. Therefore the shading operations are performed entirely in the device domain for rapid rendering. It should be noted that a patch (defined in shading Type 6 or 7) can appear to fold-over on itself. When a patch folds-over on itself, the cubic C and D Bezier curves on the patch can be intersecting or self-intersecting. That means the location of the pixels on the device space Bezier curve may change direction so that pixels are mapped onto previously mapped pixels as the value of u or v increases in the parametric space. For the case that more than one parametric value (u, v) is mapped to the same device pixel, only the shading colour correspondent to the largest value of u will be chosen to paint the device pixel if these parametric coordinates have the same value of v; Otherwise, only the shading colour correspondent to the largest value of v will be used. In the related invention, in order to deal with such situations, after a patch is mapped into the device domain the device-domain patch is painted with solid device white at  904 . The parametric variable (u, v) are forced to be varied from 1.0 to 0.0, instead of from 0.0 to 1.0 at  914 . Each device curve subsection is examined against the traced last shaded device coordinates at  916 , and each device curve pixel with each device curve subsection is checked against the current device colour of the device pixel at step  920 . Therefore, the correct one-to-one mapping between the unit space and the device patch are achieved. It should also be noted that the division scheme I and II of the above parametric domain are both tested with the same quality for all complicated cases using the patch based shading. The division scheme I of the unit space is recommended for Shading Type 6 and 7 operations due to higher performance. 
     FIG. 5  is a block diagram that illustrates a computer system  500  upon which any embodiment of the present invention may be implemented. A single computer system  500  can be used to implement the various functions of control unit  100 , such as RIP operations  102  ( FIG. 1 ), the Shading Module  104  ( FIG. 1 ), and/or Painting Operations  106  ( FIG. 1 ). Alternatively multiple computer systems  500  can be used to implement the functions of control unit  100 . 
   Computer system  500  includes a bus  502  or other communication mechanism for communicating information and a processor  504  coupled with bus  502  for processing information. Computer system  500  also includes a main memory  506 , such as random access memory (RAM) or other dynamic storage device coupled to bus  502  for storing information and instructions to be executed by processor  504 . Main memory  506  also may be used for storing a temporary variable or other intermediate information during execution of instructions to be executed by processor  504 . Computer system  500  further includes a ready only memory (ROM)  508  or other static storage device coupled to bus  502  for storing static information and instructions for processor  504 . A storage device  550 , such as a magnetic disk or optical disk, is provided and coupled to bus  502  for storing information and instructions. 
   The invention is related to the user of computer system  500  for image shading. According to one embodiment of the invention, image shading is provided by computer system  500  in response to processor  504  executing one or more sequences of one or more instructions contained in main memory  506 . Such instructions may be read into main memory  506  from another computer-readable medium, such as storage device  510 . Execution of the sequence of instructions contained in main memory  506  causes processor  504  to perform the process steps described herein. One or more processors in a multi-processing arrangement may also be employed to execute the sequences of instructions contained in main memory  506 . In alternative embodiments, hard-wired circuitry may be used in place of or in combination with software instructions to implement the invention. Thus, embodiments of the invention are not limited to any specific combination of hardware circuitry and software. 
   The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to processor  504  for execution. Such a medium may take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Non-volatile media include for example optical or magnetic disks, such as storage device  510 . Volatile media include dynamic memory such as main memory  506 . Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise bus  502 . Transmission media can also take the form of acoustic or light waves such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media include for example floppy disk, a flexible disk, hard disk, magnetic cards, paper tape, any other physical medium with patterns of holes, a RAM, a PROM, an EPROM, a FLASHPROM, any other memory chip or cartridge, a carrier wave as described hereinafter, or any other medium from which a computer can read. 
   Various forms of computer-readable media may be involved in carrying one or more sequences of one or more instructions to processor  504  for execution. For example, the instructions may initially be borne on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system  500  can receive the data on the telephone line and use an infrared transmitter to convert the data to an infrared signal. An infrared detector coupled to bus  502  can receive the data carried in the infrared signal and place the data on bus  502 . Bus  502  carries the data to main memory  506  from which processor  504  retrieves and executes the instructions. The instructions received by main memory  506  may optionally be stored on storage device  510  either before or after execution by processor  504 . 
   Computer system  500  also includes a communication interface  508  coupled to bus  502 . Communication interface  518  provides a two-way data communication coupling to a network link  520  that is connected to a local network  522 . For example, communication interface  518  may be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface  518  may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface  518  sends and receives electrical, electromagnetic, or optical signals that carry digital data streams representing various types of information. 
   Network link  520  typically provides data communication through one or more networks to other data devices. For example, network link  520  may provide a connection through local network  522  to a host computer  524  or to data equipment operated by an Internet Service Provider (ISP)  526 . ISP  526  in turn provides data communications through the worldwide packet data communication network, now commonly referred to as the “Internet”  528 . Local networks  522  and Internet  528  both use electrical, electromagnetic, or optical signals that carry the digital data to and from computer system  500 , are exemplary forms of carrier waves transporting the information. 
   Computer system  500  can send messages and receive data, including program codes, through the network(s), network link  520 , and communication interface  518 . In the Internet example, a server  530  might transmit a requested code for an application program through Internet  528 , ISP  526 , local network  522 , and communication interface  518 . In accordance with the invention, one such downloaded application provides for image shading as described herein. 
   The received code may be executed by processor  504  as it is received, and/or stored in storage device  510 , or other non-volatile storage for later execution. In this manner, computer system  500  may obtain application code in the form of a carrier wave. 
   The image data to be processed by computer system  500  can be retrieved from storage device  510 , main memory  506  or any non-volatile memory associated with computer system  500 . In addition, image data can also be obtained from a remote location via communication interface  518 . Remote devices coupled to local networks  522  or Internet  528  send the image data to network link  520  to be received by remote communication interface  518 . 
   What has been described above includes exemplary implementations of the present invention. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing the present invention, but one of ordinary skill in the art will recognize that many further combinations and permutations of the present invention are possible. Accordingly, the present invention is intended to embrace all such alterations, modifications and variations that fall within the spirit and scope of the appended claims interpreted in accordance with the breadth to which they are fairly, legally and equitably entitled.