Abstract:
The present invention discloses a method, apparatus and computer program product for positioning a plurality of sub-images such as spheres or circles within a bounded image (such as a character). The size (eg radius) and/or packing density of the sub-images can be varied in accordance with the distance of a point where the sub-image is to be located from a reference line ( 131, 171 ) which can be internal or external of the bounded image. The use of actual and virtual dimensions (r a , r v ) enables overlapping and spaced apart sub-images to be created. A threshold test is also able to prevent display of sub-images below a certain size.

Description:
FIELD OF THE INVENTION 
     The present invention relates to computer graphics and, in particular, to creating a bounded image including a number of sub-images. Although the present invention finds particular application to the creation of computer fonts in which the boundary is the exterior of a character, the invention is not limited thereto and also finds application in the printing of large scale images on substrates such as textiles paper, sheet polymers and the like. 
     The preferred embodiment of the present invention discloses a method of packing circles in arbitrary regions where packing density and properties of component circles are subject to variation according to their position with respect to reference objects (eg points, lines or curves) or goals. 
     BACKGROUND OF THE INVENTION 
     In the field of computer graphics it is often desirable to decorate or enhance given shapes to make them more visually interesting, eg for the purposes of advertising or web page design. This is done by adding texture to the region defined by a shape, such as filling the region with multiple instances of smaller objects, resulting in a more complex composite image. Circle packing provides such a texture. 
     Traditional circle packing methods obey the following general conditions: 
     (a) the packing is reasonably dense and evenly distributed across the region, 
     (b) circles are positioned so as to give the impression of random placement and avoid noticeable locally repeating patterns, 
     (c) circles do not intersect neighbouring circles, 
     (d) circles are tangential to and touching their nearest neighbours, 
     (e) the packing is obsculatory, ie any available area is always covered by the largest possible circle. 
     The simplest prior art method of circle packing is the random placement of circles within the region. This is inefficient and does not satisfy conditions (a), (c), (d) or (e) above. 
     Apollonian packing is also known. This is achieved by placing at random unfilled points within the region, the largest circle that touches at least three other circles. It satisfies all of the above conditions but does not allow control over properties of the individual circles, whose diameters are defined deterministically. 
     The related 1-tangent method (Pickover C. 1990, “Computers, Pattern, Chaos and Beauty”. St Martin&#39;s Press, New York, p 332-6) is a prior art method which relaxes the condition that the filling circle touch at least three neighbours, and requires only that the filling circle touch at least 1 neighbour. This method is more efficient than Apollonian packing, satisfies all of the above conditions except (e), and does not necessarily result in a deterministic packing. However, again 1-tangent packing does not provide control over properties of individual circles. 
     DISCLOSURE OF THE INVENTION 
     The object of the present invention is to create such composite images from sub-images within a boundary (the sub-images not being restricted to circles) but for aesthetic reasons to relax the above general conditions and/or to provide a spatial relationship followed by a parameter of the sub-images within the image boundary. 
     For example, to further enhance the artistic effect it is often desirable to manipulate the packing density or properties of component circles dependent on their position with respect to reference objects or goals. For instance, it may be desired to vary the size of circles so that they increase in diameter in accordance with their vertical position, or to vary the density of the circles so that they are more densely packed towards tile centre of the boundary. 
     The existing techniques are limited in that although they address the problem of even-distribution packings within a space, they do not provide for further manipulations of density or properties of components such as circles that the user may wish to apply. Given the computational nature of these algorithms, it is desired that such manipulations occur with a minimum of interaction from the user, but that the user also have reasonably precise control over the final result. 
     In accordance with one aspect of the present invention there is disclosed a method of creating a bounded image from a plurality of sub-images, said method comprising the steps of: 
     (1) defining a boundary for said image, 
     (2) specifying parameters of each said sub-image, and 
     (3) creating said sub-images within said boundary whilst maintaining a spatial relationship of at least one parameter within said boundary. 
     In accordance with a second aspect of the present invention there is disclosed an apparatus for creating a bounded image from a plurality of sub-images, said apparatus comprising: 
     first means to define a boundary for said image, 
     second means to specify parameters for each said sub-image, and 
     third means to create sub-image within said boundary whilst maintaining a spatial relationship of at least one parameter within said boundary. 
     In accordance with a third aspect of the present invention there is disclosed a computer program product for creating a bounded image from a plurality of sub-images, said product comprising: 
     first means to define a boundary for said image, 
     second means to specify parameters for each said sub-image, and 
     third means to create sub-images within said boundary whilst maintaining a spatial relationship of at least one parameter within said boundary. 
     In accordance with a fourth aspect of the present invention there is disclosed a bounded image having a plurality of sub-images and being formed by any of the first, second or third aspects. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Some embodiments of the present invention will now be described with reference to the drawings in which: 
     FIG. 1 illustrates a closed region defined by a set of outline curves; 
     FIG. 2 is a perspective view of the Euclidean distance map (EDM) and medial axis transform (MAT) of the closed region; 
     FIG. 3 illustrates a typical prior art circle packing of R; 
     FIG. 4 illustrates how in an initial embodiment the radius of tangential circles is constrained by their proximity to a line; 
     FIG. 5 is a flow diagram describing the preferred overall method of circle packing with position-dependent variation; 
     FIG. 6 illustrates a circle packing of the closed region with fixed radius circles; 
     FIG. 7 illustrates the clamping of tangential radii; 
     FIG. 8 illustrates a uniform decrease in density across the object; 
     FIG. 9 illustrates virtual radii and actual radii resulting in decreased density; 
     FIG. 10 illustrates uniform increase in density across the object and corresponding neighbour overlap; 
     FIG. 11 illustrates virtual radii and actual radii resulting in increased density and overlap; 
     FIG. 12 illustrates size variation dependent on distance from a reference line L; 
     FIG. 13 illustrates the relationship between a point P within the object and an external line; 
     FIG. 14 illustrates density variation dependent on distance from the line; 
     FIG. 15 illustrates the relationship between a point P in the object and an external reference point C; 
     FIG. 16 illustrates a size variation dependent on distance from the MAT; 
     FIG. 17 illustrates the relationship between a point P in the object and the MAT; 
     FIG. 18 illustrates a size variation dependent on distance from the boundary; 
     FIG. 19 illustrates the relationship between a point P in the object and the boundary; 
     FIG. 20 illustrates a size variation dependent on normal direction; 
     FIG. 21 illustrates the normal at a point P within the object, surface normals at intervals around the boundary, and a reference angle theta; 
     FIG. 22 illustrates a density variation by horizontal compression; 
     FIG. 23 illustrates circle packing in a horizontally compressed region; 
     FIG. 24 illustrates a density variation by vertical compression; 
     FIG. 25 illustrates circle packing in a vertically compressed region; 
     FIG. 26 illustrates a size variation dependent on distance from a reference line; 
     FIG. 27 illustrates a size variation dependent on distance from a reference line with deletion of circles whose radii fall below a threshold; 
     FIG. 28 illustrates the application of the invention to three dimensions; and 
     FIG. 29 illustrates a general purpose computer system by means of which an example of the invention can be put into practice. 
    
    
     DETAILED DESCRIPTION 
     In the preferred embodiment, a method is provided for packing circles within bounded regions of arbitrary shape where packing density and properties of component circles are subject to variation according to their position with respect to reference objects or goals. Reproducibility can be guaranteed if required. 
     Although the principles of the preferred embodiment have general applicability to arbitrarily shaped 2-dimensional regions defined by a set of closed curves, and may be extended without major modification to n-dimensional regions, the preferred embodiment in describes the method in terms of 2-dimensional regions that describe character shapes. 
     FIG. 1 illustrates a typical 2-dimensional region  11  formed into the character R defined by a set of closed 2-dimensional outline curves  12 . That the region defies a character shape is only by way of example. The region  11  could simply be a rectangle in the case of a textile pattern. 
     FIG. 2 shows a perspective view of the Euclidean distance map (EDM)  21  of R, with internal points elevated according to their distance from the outline  12 . Contours  22 ,  23  on the EDM are shown to demonstrate the principle of the EDM, and the ridge  24  of the EDM is also shown. The ridge  24  can be converted directly to a set of curves that describe the medial axis transform (MAT)  25  of the region  11 . Techniques of determining the EDM and MAT are standard image processing algorithms and are well documented elsewhere (eg Gonzales &amp; Woods “Digital Image Processing” or any image processing textbook). The Euclidean distance map is used in the preferred embodiment, however, any suitable distance metric can be used in practice. The MAT need not necessarily be derived from the EDM, and can be derived independently, and need only be generated if necessary: if the user selects a packing variation that does not require the MAT it need not be generated. 
     Turning now to FIG. 3 there is shown the result of circle packing the region R  31  without position dependent variation. This result is similar to the result that would be achieved with the standard 1 tangent technique. Notice that each circle  32  is tangential with at least one neighbour  32 , and that packing conditions (a), (b), (c), (d) above are satisfied, but that condition (e) is not satisfied, i.e. the packing is not necessarily obsculatory. 
     FIG. 4 demonstrates the principle of tangential packing in accordance with a first embodiment. A circle  41  with centre C  44  exists within the region R relative to the outline  42 . Contour lines  43  radiating inward from  42  indicate distance from  42  within R. A random direction is chosen indicated by the vector v  48  radiating from C. The vector v intersects the circle  41  at point P  46  and intersects line  42  at the point Q  45 . The point M  47  mid way between P and Q is selected, and is adjusted along the line PQ towards P until M is a suitable centroid for a neighbouring circle. M is a suitable centroid for a new circle if the distance d  49  between M and P is less than or equal to the distance from M to Q (deduced directly from the EDM). Once the distance d has been determined, a new circle of radius d and centred at point M is created, which is tangential to the circle  41 . 
     Turning now to FIG. 5, there is illustrated a flow diagram of the preferred method for packing circles in arbitrary regions where packing density and properties of component circles are subject to variation according to their position with respect to reference objects or goals. The preferred method is described with reference to examples consisting of 2-dimensional regions described by closed outline curves that describe character shapes. However, the principles of this invention have general applicability to arbitrarily shaped n-dimensional regions and the invention is not limited to 2-dimensional cases. 
     In initial step  500  of the proposed method, the process commences, and the following information  501  is input by the user  502 : 
     (a) a set of closed 2-dimensional curves that describe a region. Specifically, FIG. 1 illustrates a character glyph of a font described by its outline, and its filled interior region. 
     (b) parameters that affect the final packing result. These parameters can include (but are not limited to) the size range between which circle radii can vary, a preferred direction, a packing density threshold, a measure of allowed overlap between neighbouring circles, a minimum radii threshold below which circles are deleted, etc, and 
     (c) one or more reference objects that govern the position-dependent variation. These objects can include (but are not limited to) points, line segments, lines, the set of outline curves, the MAT, an arbitrary curve or set of curves, etc. 
     In step  503 , the Euclidean distance map (EDM) of the closed region is determined. If the region&#39;s medial axis transform (MAT) is required for position-dependent variation then it is extracted directly from the EDM in step  504 , otherwise the MAT is not generated. This concludes the pre-packing steps ( 500  to  504 ) that must occur before packing takes place. 
     The next step  505  is the first step of the packing loop ( 505  to  512 ). A random point P within the closed region R is chosen. If P is already covered by an existing circle. or if the EDM distance value for P is below the minimum allowed circle radius then another random point is chosen. This process continues until either a suitable point P is found, or a predefined number of iterations is reached, in which case the packing loop terminates and step  513  is performed. 
     In step  506  a virtual radius r v  is determined. The virtual radius is the radius used for packing but is not necessarily the final radius of the circle. This radius r v  embodies density variation and is determined by: the EDM distance at P, the packing density and overlap parameters, and the position of P relative to the reference object(s). 
     In decision block  507  a check is made as to whether a circle centred at P with radius r v  call be added to the packing without violating the conditions defined by the user&#39;s choice of parameters. If decision block  507  returns false (No), then processing returns to step  505  and a new random point P is chosen. Otherwise processing continues to step  508 . 
     In step  508  the virtual radius r v  is converted to a corresponding actual radius r a  that is then assigned to be the circle&#39;s radius. The actual radius r a  embodies both density and size variation and is determined by: the EDM distance at P, packing density and overlap parameters, and the position of P relative to the reference object(s). The actual radius r a  can be further adjusted in the post-packing steps. 
     In step  509  a circle centred at point P with radius r a  is added to the packing and the relevant data structures are updated. 
     In decision block  510  a check is made as to whether the area covered by the existing, circles as a percentage of the total area enclosed by R is greater than or equal to a density threshold specified by the user. If the decision block  510  returns false (No), then processing continues to decision block  511  and the packing loop continues. Otherwise the packing loop terminates and processing continues to step  513 . 
     In decision block  511  a check is made to determine whether a new neighbouring circle can be placed tangential with the circle most recently created. This process continues until either it is determined that a tangential neighbour can be placed, or a predefined number of iterations is reached, in which case the packing loop terminates and step  513  is performed. If decision block  511  returns false (No) then processing returns to step  505  at the start of the packing loop, and a new random point is chosen. Otherwise processing continues to step  512 . 
     Step  512  determines the point that is the centroid of the neighbouring tangential circle whose existence was proven by decision block  511 . This point is passed on and processing returns to step  506 . 
     Step  513  is the first of the post-packing steps ( 513 - 514 ). In step  513  post-packing manipulations are applied to the packed circles. This can include, but is not limited to, varying the actual radii r a  of circles based on their proximity to the reference object(s). 
     In step  514  the process terminates. The circle packing is now complete, and can be displayed on a monitor, printed, and/or passed on as data for further processing by other programs. 
     FIG. 6 shows a tangential packing of circles  61  with equal radius in accordance with the process of FIG.  5 . This results in a different style of packing that is less dense than 1-tangent packing and is most definitely not obsculatory. 
     This preferred method of performing tangential packing is achieved as explained in FIG. 7. A circle  701  with centre C  702  and radius r  707  exists within the region R relative to the outline or boundary  703 . Contour lines  709  radiating inward from the boundary  703  indicate distance from the boundary within R. A random direction is chosen indicated by the vector v  704  radiating from C. The vector v intersects C at point P  705  and intersects the boundary  703  at the point Q  706 . The point C′  708  lies on the line PQ a distance r from P, and is the candidate for a neighbouring tangential circle with radius equal to the radius of C. If the distance from C′ to the boundary  703  (deduced directly from the EDM) is less than or equal to r then the neighbouring circle can be placed, else it is discarded. 
     Turning now to FIG. 8 there is shown a circle packing with uniformly decreasing packing density  81 . This is achieved by assigning each circle a different virtual and actual radius as explained in FIG.  9 . Given two neighbouring tangential circles C 1   91  with virtual radius r v1    93  and actual radius r a1    94 , and C 2   92  with virtual radius r v2    95  and actual radius r a2    96 , it is seen that the circles are tangential on their virtual radii which touch at the point T  97 . This is because packing is performed on the virtual radii as described in steps  506  and  507  of the circle packing loop of FIG.  5 . In the case of FIG. 9 the actual radii are less than the virtual radii, resulting in a less dense packing. 
     FIG. 10 shows a circle packing with uniformly increased packing density  101  and the resulting intersection between neighbouring circles. This is achieved by assigning each circle a different virtual and actual radius as illustrated in FIG.  11 . Given two neighbouring tangential circles C 1   111  with virtual radius r v1    113  and actual radius r a1    114 , and C 2   112  with virtual radius r v2    115  and actual radius r a2    116 , it is shown that the circles are tangential on their virtual radii which touch at the point T  117 . This is because packing is performed on the virtual radii as described in steps  506  and  507  of the circle packings loop of FIG.  5 . In the case of FIG. 11 the actual radii are greater than the virtual radii, resulting in a denser packing and overlap of circles. This conflicts with packing conditions (c) and (d) but is allowed as it is often desired as an artistic effect. 
     There is shown in FIG. 12 a packing of circles whose size varies with respect to their proximity to a reference object. The circles in the upper right region  121  have greater radii than those circles in the lower left region  122 , with a smooth variation in radii size between these extremes. FIG. 13 illustrates the reference object that results in the size variation shown in FIG. 12, which is the line segment AB  131 . Contour lines  133  indicate distance from this reference object, for instance the point P  132  within region R is approximately 2.5 units from AB. In this case, circle radius is decreased according to distance from AB. This size variation is applied in the post-packing step  513 . The density of virtual radii of circles does not vary in this case. 
     Turning now to FIG. 14 there is shown a packing of circles whose density varies with respect to their proximity to a reference object. The circles in the upper right region  141  are packed more densely than those circles in the lower left region  142 , with a smooth variation in density between these extremes. This density variation is determined during the packing loop steps  506 ,  507  and  508  of FIG.  5 . The reference object used in this case is the same reference object illustrated in FIG.  13 . 
     FIG. 15 illustrates a different type of reference object, the point source reference object C  151  . Curved contour lines  153  radiate outwards from C. For instance the point P  152  within region R is approximately 3.3 units from C. Multiple point sources can also be used to give a more complex pattern of variation if desired. 
     There is shown in FIG. 16 a packing of circles whose size varies with respect to their proximity to a reference object, in this case the MAT of region R. The circles towards the centre of the region R  161  have greater radii than those circles towards the boundary  162 , with a smooth variation in size between these extremes. FIG. 17 illustrates the reference object that results in the size variation shown in FIG. 16, which is the MAT  171  of region R derived from the EDM. For any point P  172  within region R the closest skeleton point S  173  is found, and the amount of variation is determined by the distance between P and S. This size variation is applied in the post-packing step  513 . The density of virtual radii of circles does not vary in this case. Alternatively, if desired, a density variation based on proximity to MAT can be applied during the packing loop steps  506 .  507  and  508  of FIG.  5 . 
     FIG. 18 illustrates a packing of circles, the size of which varies with respect to their proximity to a reference object, in this case the boundary. The circles  181  towards the boundary have greater radii than those circles  182  towards the centre of the region R, with a smooth variation in size between these extremes. FIG. 19 illustrates the reference object that results in the size variation shown in FIG. 18, which is the outline or boundary  191 . For any point  192  within region R the smallest distance to the boundary  192  can be determined directly from the EDM, effectively giving the distance to the closest point S  193  that lies on the boundary  191 , and the amount of variation is determined by this distance. This size variation is applied in the post-packing step  513  of FIG.  5 . The density of virtual radii of circles does not vary in this case. Alternatively, density variation based on proximity to the boundary  191  can be applied during the packing loop steps  506 ,  507  and  508  of FIG.  5 . 
     FIG. 20 illustrates a packing of circles, the size of which varies with respect to their proximity to a reference goal, in this case a preferred direction. The circles  201 ,  202  are positioned such that their direction to the closest outline point most closely agrees with some preferred direction, have greater radii than those circles  203  positioned such that their direction to the closest outline point least closely agrees with some preferred direction. There is a smooth variation in circle size between these extremes. 
     FIG. 21 shows the outline  211  and the outward normals at regular intervals along this curve  212 . For any point P  213  within region R the vector n  214  indicating the direction to the closest outline point is determined. This vector is compared for agreement with a preferred vector v  215  derived from an angle theta  216  specified by the user in the pre-packing user input step  502 . The dot product of vectors v and n can be used to provide a measure of the agreement between the two vectors, and the amount of variation is determined by the measure of agreement. This size variation is applied in the post-packing step  513 . The density of virtual radii of circles does not vary in this case. Alternatively, density variation based on agreement with a preferred direction can be applied during the packing loop steps  506 ,  507  and  508  of FIG.  5 . 
     Turning now to FIG. 22, there is shown a circle packing with variation in packing density  221  caused by directional compression. The packed circles are of uniform size, but the packing density varies differently in the horizontal and vertical directions. FIG. 23 illustrates this process, which can be achieved by compressing the outline or boundary  231  horizontally during the pre-packing step  501 , performing the packing as described above to result in uniformly packed circles  232 , and then decompressing the boundary  231  during the post-packing step  513  of FIG.  5 . The circle&#39;s centres are repositioned during the decompression stage, resulting in a directionally biased density variation. 
     FIG. 24 shows a circle packing with variation in packing density  241  caused by directional expansion rather than compression, that is similar to the example of FIGS. 22 and 23 but in another direction. The packed circles are of uniform size, but the packing density varies differently in the horizontal and vertical directions. FIG. 25 illustrates this process, which can be achieved by compressing the outline  251  vertically during the pre-packing step  501 , performing the packing as described above to create uniformly distributed circles  252 , and then decompressing the outline  251  during the post-packing step  513 . The circle&#39;s centres are repositioned during the decompression stage, resulting in directionally biased density variation. It will be apparent that compression need not be restricted to the horizontal and vertical directions, and can be applied along any direction. 
     FIGS. 26 and 27 demonstrate the process of clipping circles whose radii fall below some threshold specified by the user in the user input step  502 . FIG. 26 shows a circle packing with size variation such that larger circles occur in the lower right region  261  and smaller circles occur in the top left region  262 . FIG. 27 shows the same packing, but with circles below some threshold removed. The threshold in this case is approximately the radius of the smallest circle,  273  in this case. Larger circles are still present in the bottom right region  271 , but the top left region  272  is now devoid of circles. 
     In FIG. 28 a three dimensional boundary is filled with circles in a manner analogous to that of FIG.  12 . 
     The method of FIG. 5 is preferably practiced using a conventional general-purpose computer system  2900 , such as that shown in FIG. 29 wherein the processes of FIG. 5 may be implemented as software, such as an application program executing within the computer system  2900 . In particular, the steps of the method of FIG. 5 are effected by instructions in the software that are carried out by the computer. The software may be divided into two separate parts; one part for carrying out the FIG. 5 methods: and another part to manage the user interface between the latter and the user. The software may be stored in a computer readable medium, including the storage devices described below, for example. The software is loaded into the computer from the computer readable medium, and then executed by the computer. A computer readable medium having such software or computer program recorded on it is a computer program product. The use of the computer program product in the computer preferably effects an advantageous apparatus for providing a user interface in accordance with the embodiments of the invention. 
     The computer system  2900  comprises a computer module  2901 , input devices such as a keyboard  2902  and mouse  2903 , output devices including a printer  2915  and a display device  2914 . A Modulator-Demodulator (Modem) transceiver device  2916  is used by the computer module  2901  for communicating to and from a communications network  2920 , for example connectable via a telephone line  2921  or other functional medium. The modem  116  can be used to obtain access to the Internet, and other network systems, such as a Local Area Network (LAN) or a Wide Area Network (WAN). 
     The computer module  2901  typically includes at least one processor unit  2905 . a memory unit  2906 , for example formed from semiconductor random access memory (RAM) and read only memory (ROM), input/output (I/O) interfaces including a video interface  2907 , and an I/O interface  2913  for the keyboard  2902  and mouse  2903  and optionally a joystick (not illustrated), and an interface  2908  for the modem  2916 . A storage device  2909  is provided and typically includes a hard disk drive  2910  and a floppy disk drive  2911 . A magnetic tape drive (not illustrated) may also be used. A CD-ROM drive  2912  is typically provided as a non-volatile source of data. The components  2905  to  2913  of the computer module  2901 , typically communicate via an interconnected bus  2904  and in a manner which results in a conventional mode of operation of the computer system  2900  known to those in the relevant art. Examples of computers on which the embodiments can be practised include IBM-PC&#39;s and compatibles. Sun Sparcstations or alike computer systems evolved therefrom. 
     Typically, the application program of the preferred embodiment is resident on the hard disk drive  2910  and read and controlled in its execution by the processor  2905 . Intermediate storage of the program and any data fetched from the network  2920  may be accomplished using the semiconductor memory  2906 , possibly in concert with the hard disk drive  2910 . In some instances, the application program may be supplied to the user encoded on a CD-ROM or floppy disk and read via the corresponding drive  2912  or  2911 , or alternatively may be read by the user from the network  2920  via the modem device  2916 . Still further, the software can also be loaded into the computer system  2900  from other computer readable medium including magnetic tape, a ROM or integrated circuit, a magneto-optical disk, a radio or infra-red transmission channel between the computer module  2901  and another device, a computer readable card such as a PCMCIA card, and the Internet and Intranets including email transmissions and information recorded on websites and the like. The foregoing is merely exemplary of relevant computer readable mediums. Other computer readable mediums may be practiced without departing from the scope and spirit of the invention. 
     The method of providing a user interface may alternatively be implemented in dedicated hardware such as one or more integrated circuits performing the functions or sub functions of FIG.  5 . Such dedicated hardware may include graphic processors, digital signal processors, or one or more microprocessors and associated memories. 
     Industrial Applicability 
     It is apparent from the above that the embodiment(s) of the invention are applicable to the computer graphics industry and to the font generation aspects thereof, in particular. 
     The foregoing describes only some embodiments of the present invention, and modifications and/or changes can be made thereto without departing from the scope and spirit of the invention, the embodiment being illustrative and not restrictive. 
     In the context of this specification, the word “comprising” means “including principally but not necessarily solely” or “having” or “including” and not “consisting only of”. Variations of the word comprising, such as “comprise” and “comprises” have corresponding meanings. 
     The forgoing describes only some embodiments of the present invention and modifications, obvious to those skilled in the art, that can be made thereto without departing from the scope of the present invention. For example, although circles have been described in detail as the sub-images, the use of apparent dimensions enables other images to be located at the centre of the “circle” determined by the virtual radius.