Abstract:
A method and apparatus is disclosed for converting raster images into a vector format by identifying and converting the borders of image features into a mathematical format referred to as string sequences. To enable the string sequencing of color images, the present invention identifies and resolves contrast conflicts in the image features in order to avoid misperceiving the vectorized image when converted back into a raster format. A contrast conflict occurs when there is a “contrast tie” between overlapping features of the image. The feature that would normally be perceived as the dominant feature breaks the contrast tie so that when the vector image is reconstructed, the dominate feature appears in the foreground of the image while the recessive feature appears in the background. A contrast tie detector (CTD) performs a set of comparisons on the raw pixel data to detect the occurence of a contrast tie, and a contrast tie breaker (CTB) performs another set of comparisons on the raw pixel data to break the contrast tie. A contrast tie is broken by modifying a color identifier of a perimeter pixel relative to the color identifier of a target pixel, thereby enabling the correct string sequencing of color borders.

Description:
This application includes a Microfiche Appendix comprised of 3 microfiche including 212 total frames. 
     FIELD OF INVENTION 
     The present invention relates to the conversion of a raster image into a vector format, particularly to an apparatus and method for breaking contrast ties within a localized area of the image which enables the further step of defining color borders. 
     BACKGROUND OF THE INVENTION 
     Image processing generally refers to the manipulation of pictorial data using computers. Computer Aided Design (CAD) is one example of how computers are used to draft complex engineering drawings such as mechanical, architectural or electrical drawings. Other examples of image processing include the manipulation of still photographs or cinema to achieve various effects such as feature enhancement, three-dimensional rendering, or animation. However, the term “image processing” is not limited to pictures—it generally refers to the digitization and computer processing of any analog signal that represents something physical and perceptible in the world. An audio signal can be digitized and processed with computers to perform manipulations such as noise reduction or voice recognition; modulated electrical signals, such as telecommunications or cardiograms, can be digitized and processed by computers to extract pertinent information, and so on. The common factor in image processing is the need to digitize the continuous-time analog signal into a discrete-time digital signal that can be operated on by computers using boolean mathematics implemented in transistor logic circuits. 
     A common format for storing and manipulating digitized signals is simply to represent the digital data as a bit-map of pixels, where each pixel represents a particular characteristic (e.g., magnitude or color) of the analog signal. Typically, a picture is digitized into a bit-map of individual pixels where each pixel represents a color of the image in a very small, localized area. When the pixels are displayed as a congregation, the original picture appears without loss of perception to the human eye as long as there is sufficient resolution (number of pixels) both in spatial and color contexts. A black and white picture can be represented with one-bit pixels, where the state of the bit (0 or 1) represents the two colors, black and white. To digitize a picture comprising multiple colors, each pixel is represented using n-bits such that each pixel can take on one of 2 n  different colors. The process of converting an analog image into a bit-map image is referred to as “rasterizing” the image. 
     There are several well known problems with raster images that limit or inhibit the full potential of computer manipulation. The amount of memory necessary to store a large photograph having many colors (thousands or even millions of colors) can be immense, and the problem is exacerbated when attempting to digitize a series of photographs such as a scene in a movie. Not only do raster images require large amounts of storage memory, but processing such a large amount of data can be slow, particularly when attempting to transfer raster images over a network such as the internet. 
     Another problem inherent with the raster format is an inability to perform many of the desired image processing functions, such as three-dimensional rendering or animation. Even a simple operation such as magnifying the image results in a granular distortion caused by enlarging the pixels of the image as illustrated in FIG.  1 . It is possible to perform digital signal processing when magnifying a raster image by effectively re-sampling the image to attenuate the granular distortion that occurs due to the fixed resolution. However, this process only reduces the granular distortion rather than eliminate it. Furthermore, resampling the image is time consuming and not practical in many applications. Other operations, such as electronic focusing and automatic imaging, are simply not possible when the image is stored in a raster format. 
     The alternative is to convert the raster image into a mathematical format known as vectors. A vector is a mathematical description of the image which is not constrained by a fixed resolution as with raster data. Furthermore, vectors allow for a much more diverse range of image manipulations due to the mathematical representation of the image. A simple example of a vector is a line beginning at a particular X,Y point in a cartesian plane and ending at another X,Y point in that plane. To date, however, converting raster images into a series of linear vectors has been largely unsuccessful. This is due to the immense number of different possible patterns that raster images can form. For example, a 3 by 3 pixel matrix of a nine color picture can take on about 21,000 different possible patterns. Each possibility must be accounted for when converting the image into a set of linear vectors. If just one more pixel is added to the matrix, the complexity increases exponentially. Consequently, techniques for converting raster images into linear vectors normally operate on only a small section of the image at a time. The problem with this approach is that the resulting vectorized image appears fragmented due to discontinuities when the individual sections are combined. 
     There have been other attempts to convert a raster image using vectors that are more abstract mathematical representations. For example, a technique referred to as wavelets attempts to represent the raster image as a series of interconnected mathematical equations. With wavelets, a particular feature of the image might be represented using a number of polynomials which approximate the feature contours. The problem with this technique, however, is that the equations become extremely complex unless the operation is constrained to only a small group of pixels. But again, distortion due to fragmentation occurs when attempting to combine the individual equations into a complete image. A similar distortion occurs with using fractals, a method where the image is represented using a library of different shapes which approximate the various features of an image. Similar to wavelets, however, discontinuities occur with fractals when attempting to combine the individual shapes which causes distortion due to fragmentation. 
     Another known method for raster to vector conversion, referred to as the Automatic Bezier Curve technique, draws Bezier curves through tags that are placed somewhat arbitrarily on the image. Unlike fractals or wavelets, Bezier techniques cannot convert complicated pictures without losing information. On the other hand, the Bezier technique can sequence equations together over a large number of pixels which reduces the amount of fragmentation typically associated with other techniques. 
     The Bezier method is illustrated in FIG.  1 B. The first step is to identify the color borders for each feature in the image, and then to place somewhat arbitrarily tags on the borders. These tags then become the new representation of the image. When the image is to be displayed, these tags are converted into a rasterized format by drawing Bezier curves through the tags as shown in FIG.  1 B. The problem with the Bezier conversion technique is that it has a hard time dealing with complicated images. When tags are placed some definition is lost. In a complex image, many tags are placed resulting in a great deal of loss in picture definition. Even more severe is that the Bezier technique only looks at simple borders. In complex images, there is usually an interaction of shades that catch the eye. The Bezier curve technique treats all of these different shades as either borders or points. This often results in a vector representation that is more complicated than a raster representation; in effect, the Bezier solution can actually be worse than the problem. 
     There have been other attempts to represent the feature contours of an image using mathematics capable of spanning more than just a small area on the image. For example, U.S. Pat. No. 4,777,651 discloses a method of pixel-to-vector conversion where the line and edge features of the image are converted into a series of continous line segments. The line segments are generated by connecting a series of slope elements which follow the contour of a line or edge in the image. The slope elements are generated by scanning a 3×3 array of pixels through the raster image data and generating a slope vector for each pixel. The 3×3 array of pixels are evaluated to determine the slope of the image contour at the current pixel. The slope elements are then connected to form continous line segments that represent the contours of the image. The line segments are further processed to generate the vectors that can be understood and manipulated using computers. 
     The process of sequencing a series of slope elements into line segments which represent the contours of an image has been known in the art as early as 1961. See H. Freeman, “On the Encoding of Arbitrary Geometric Configurations,”  IRE Transactions,  EC-10(2), June 1961, 260-268. Freeman describes the technique of scanning a square matrix through raster data to generate a slope element for each pixel, and then connecting the slope elements into line segments which represent the image contours. 
     There are drawbacks with the aforementioned prior art methods of sequencing slope elements into line segments to represent the image contours. Namely, U.S. Pat. No. 4,777,651 is capable of sequencing line segments for contours of only black and white images. Thus, this technique cannot be applied to color images without first converting the image into black and white which means the color information is lost. Furthermore, the list of square matrices for generating the slope elements shown in FIG. 9 is not exhaustive and, consequently, fragmentation can occur when attempting to generate and connect the line segments. Still further, sequencing the line segments using only the slope values does not provide optimal information for generating the final vectors which ultimately represent the image. In other words, the vectors generated using this technique are not optimal because there is not enough information provided in line segments comprised of only slope values. Also, these techniques do not address the optical illusion which can manifest in images where the contrast in certain features determines or enables perception. 
     There is, therefore, a need for an improved method and apparatus for converting the contours or borders of a raster image into a more abstract mathematical representation which can be converted into a vector format without fragmenting the image. More specifically, it is an object of the present invention to convert the color borders of a raster image into a mathematical representation, wherein the raster image is comprised of pixels representing more colors than just black and white. Another object of the present invention is to provide enough information in the border representation of the image so as to optimize the vectors ultimately generated from this information. Still another object of the present invention is to identify and resolve contrast conflicts in the various features of an image in order to avoid misperceiving the vectorized image when converted back into a raster format. 
     SUMMARY OF THE INVENTION 
     The present invention discloses a method and apparatus for converting the color borders of a raster image into a more abstract mathematical representation that can be converted into a vector format without significantly fragmenting the image. The invention involves scanning a neighborhood of cells in the form of a square matrix through the raster image in order to generate a surface string (or an indicia thereof) representing the color border at each pixel, and then to sequence the surface strings into string sequences which represent the color borders. 
     The neighborhood of cells comprise a target cell (e.g., a center cell) and a number of surrounding perimeter cells. Each cell corresponds to a pixel in the raster image. A color value is assigned to the perimeter cells indicating whether the pixel represented by the perimeter cell is the same or different in color from the pixel represented by the target cell. Thus, there is a limited number of different color combinations that are possible in a neighborhood (e.g., 256 in a 3×3 neighborhood). 
     As the neighborhood is scanned through the raster image, the color values of the cells are changed according to the color of the perimeter pixels relative to the target pixel. The resulting neighborhood is then used to index a neighborhood table comprising all the possible different color combinations, that is, all the different possible neighborhoods. The neighborhood table outputs up to six different surface strings which represent the color border or borders which pass through the current neighborhood of pixels comprising the current target pixel. There is a limited number of possible surface strings that are necessary to account for all the different possible color combinations in a neighborhood. However, a significant aspect of the present invention is to provide an exhaustive set of surface strings that can account for all the possible arrangements of color borders in a raster image. In the preferred embodiment, a neighborhood comprises a 3×3 square matrix of cells which results in an exhaustive set of 58 surface strings. 
     Each surface string has associated with it a receive slope and a send slope which represent the slope of a corresponding color border passing through the neighborhood. The receive and send slopes are used by a string sequencer to connect the surface strings into a contiguous string sequence representing a color border in the image. Although the receive and send slopes are used to connect the surface strings, the string sequencer actually sequences numbers which identify the surface strings rather than merely sequencing slope values as is done in the aforementioned prior art techniques. Sequencing surface string IDs, as opposed to merely sequencing slopes, provides additional information about each surface string which can be used to generate vectors that better represent the raster image. 
     Another enabling aspect of the present invention is to sequence the surface strings according to a predetermined criteria. In the preferred embodiment, a priority is assigned to surface strings based on their location in the image and their type. Northern surface strings have the highest priority meaning that a string sequence will begin with the most northern surface string in the image, and the sequencing of strings will always progresses from north to south—never from south to north. The next highest priority is assigned to a special type of surface string referred to as a “protruding corner” surface string. That is, a “protruding corner” surface string has priority over a “square corner” surface string which has priority over a “step” surface string. So if two surface strings are at the same northern position in an image, the surface string type determines which surface string to begin a string sequence and in which direction to sequence. If two surface strings at the same northern location are of the same type, then the western surface string has priority over the eastern surface string. 
     To enable the string sequencing of color images, the present invention identifies and resolves contrast conflicts in the image features in order to avoid misperceiving the vectorized image when converted back into a raster format. A contrast conflict occurs when there is a “contrast tie” between overlapping features of the image. The feature that would normally be perceived as the dominant feature breaks the contrast tie so that when the vector image is reconstructed, the dominate feature appears in the foreground of the image while the recessive feature appears in the background. A contrast tie detector (CTD) performs a set of comparisons on the raw pixel data to detect the occurrence of a contrast tie, and a contrast tie breaker (CTB) performs another set of comparisons on the raw pixel data to break the contrast tie. A contrast tie is broken by modifying a color identifier of a perimeter pixel relative to the color value of a target pixel, thereby enabling the correct string sequencing of color borders. These and other aspects and advantages of the present invention will be better understood by reading the following detailed description of the invention in conjunction with the following drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1A illustrates the granular distortion that occurs when magnifying a raster image. 
     FIG. 1B illustrates a prior art Bezier curve technique for converting raster images into vectors. 
     FIG. 2 is a block diagram of the elements for defining the color borders of a raster image according to the aspects of the present invention. 
     FIG. 3 shows an exhaustive set of surface strings corresponding to a neighborhood of cells representing pixels of the raster image, wherein the surface strings are connected into a string sequence to define a color border. 
     FIG. 4A shows a neighborhood of cells comprising a target cell surrounded by perimeter cells with color values of X, Y, or Z indicating that the color of the representative pixel in the image is either not considered (X), the same color (Y) as the pixel represented by the target cell, or a different color (Z) than the pixel representing the target cell. 
     FIG. 4B shows the possible slope values for the send and receive slopes of a surface string. 
     FIG. 5A shows a color raster image comprising a simple feature and the corresponding string sequences generated by the present invention. 
     FIG. 5B shows a color raster image with more complex features and the corresponding string sequences generated by the present invention. 
     FIGS. 6A-6C illustrate a contrast conflict (contrast tie) in a raster image and how the present invention resolves the conflict. 
     FIG. 7A shows the four quadrants in a 3×3 array of pixels that are processed to detect a contrast tie. 
     FIG. 7B shows the four quadrants in a 3×3 array of pixels and surrounding pixels that are processed to break a contrast tie. 
     FIGS. 8A-8D illustrate the pixel bit maps evaluated to break a contrast tie in an example raster image. 
     FIG. 9 shows a perceived embodiment of the present invention comprising a system memory for storing a computer program which implements the aspects of the present, and a data processor for executing the steps of the computer program. 
     FIG. 10 illustrates an alternative embodiment of the present invention: an image processing integrated circuit comprising logic circuits for implementing the aspects of the present invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     FIG. 2 shows a block diagram of the elements used to define the color borders of a raster image according to the aspects of the present invention. The raster image data  2  that is to be converted into a vector format  4  can come from many different alternative sources. For example, it may be generated by illuminating a photograph with a light source and detecting the reflected light with a charge-coupled-device (CCD). A CCD is typically formed of an array of individual photosensitive cells which detect an intensity or brightness of the light. Thus, the output of the CCD is an array of pixels which represent a portion of the raster image bit map. As the light source is scanned over the photograph, the pixels output by the CCD are concatenated to form the entire raster image. Color in the photograph can be detected by employing red-green-blue (RGB) filters in front of three respective CCDs, and then combining the outputs of the CCDs into an n-bit pixel that can represent one of 2 n  colors. 
     The aspects of the present invention are not limited to raster images that represent photographs. The present invention can also convert other digitized signals, such as audio or communication signals, into a vector format for more efficient and effective processing by computers. Such signals are typically generated by a transducer (e.g., a microphone, antenna, recording head, etc.) which senses and transduces into an electrical signal some physical manifestation, such as an acoustic or electromagnetic wave. Typically, the electrical signal is then digitized and converted into a raster format using an analog-to-digital (A/D) converter. 
     The raster image data  2  processed by the present invention can be generated in real time meaning that the raster data can be converted directly into a vector format as the analog signal is being digitized. Alternatively, the raster image data  2  may be stored on a recording medium, such as a magnetic or optical disc, and processed at a later time by the present invention to define the color borders and ultimately convert the raster data into a vector format. 
     The raster image data  2 , whether processed in real time or read from a storage medium, is processed by a contrast breaker  3  which resolves “contrast ties” in the image features. This aspect of the present invention is described in greater detail below with reference to FIGS. 6-8. After breaking contrast ties, the raster data is input into a neighborhood scanner  8  which scans through the raster data and process the image pixels in a square array referred to as a neighborhood. In the preferred embodiment, the neighborhood represents a 3×3 array of image pixels. 
     The scanning processes is understood with reference to FIG. 5A which shows the pixels of a raster image and a neighborhood  10   a  overlaying the image such that the first pixel  12  of the image is in the center of the neighborhood  10   a  (the cells of the neighborhood  10   a  laying outside the image are assigned the color Y). After processing this array of pixels, the neighborhood  10   a  is shifted to the right by one pixel so that the next pixel  14  of the image is in the center of the neighborhood  10   a.  This process continues until the neighborhood has passed over every pixel in the image ending at neighborhood  10   d.    
     In the present invention, the neighborhood of pixels are not processed directly as in the prior art. Instead, the pixels are converted into corresponding cells which comprise a color value as shown in FIG.  4 A. The neighborhood of cells comprises a target cell (in this case the center cell) denoted T and a number of perimeter cells denoted P 0-7 . The color values that can be assigned to each cell are Y and Z. The color Y indicates that the color of the pixel represented by the cell is the same color as the pixel represented by the target cell (the color of the target cell T is always Y), and the color Z indicates that the color of the pixel represented by the cell has a different color than the pixel represented by the target cell. Thus, in a 3×3 array of cells with the target cell always having a color value of Y, there are 2 8  or 256 different possible color combinations for the perimeter cells P 0-7.    
     Referring again to FIG. 2, the neighborhood of cells  16  generated for each pixel in the image is used as an address into a neighborhood lookup table  18 . The content of the neighborhood lookup table is shown below: 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
               
                 Neighborhood Lookup Table 
               
             
          
           
               
                 SURFACE 
                   
                   
                   
                   
               
               
                 STRINGS 
                 ADR 
                 P 0 P 1 P 2   
                 P 3  T P 4   
                 P 5 P 6 P 7   
               
               
                   
               
             
          
           
               
                 00, 00, 00, 00, 00, 00; 
                 0 
                 Z Z Z 
                 Z Y Z 
                 Z Z Z 
               
               
                 01, 02, 00, 00, 00, 00; 
                 1 
                 Z Z Z 
                 Z Y Z 
                 Z Z Y 
               
               
                 03, 04, 00, 00, 00, 00; 
                 2 
                 Z Z Z 
                 Z Y Z 
                 Z Y Z 
               
               
                 03, 02, 00, 00, 00, 00; 
                 3 
                 Z Z Z 
                 Z Y Z 
                 Z Y Y 
               
               
                 05, 06, 00, 00, 00, 00; 
                 4 
                 Z Z Z 
                 Z Y Z 
                 Y Z Z 
               
               
                 05, 07, 46, 02, 00, 00; 
                 5 
                 Z Z Z 
                 Z Y Z 
                 Y Z Y 
               
               
                 05, 04, 00, 00, 00, 00; 
                 6 
                 Z Z Z 
                 Z Y Z 
                 Y Y Z 
               
               
                 05, 02, 00, 00, 00, 00; 
                 7 
                 Z Z Z 
                 Z Y Z 
                 Y Y Y 
               
               
                 10, 00, 00, 00, 00, 00; 
                 8 
                 Z Z Z 
                 Z Y Y 
                 Z Z Z 
               
               
                 09, 00, 00, 00, 00, 00; 
                 9 
                 Z Z Z 
                 Z Y Y 
                 Z Z Y 
               
               
                 15, 00, 00, 00, 00, 00; 
                 10 
                 Z Z Z 
                 Z Y Y 
                 Z Y Z 
               
               
                 15, 00, 00, 00, 00, 00; 
                 11 
                 Z Z Z 
                 Z Y Y 
                 Z Y Y 
               
               
                 12, 13, 00, 00, 00, 00; 
                 12 
                 Z Z Z 
                 Z Y Y 
                 Y Z Z 
               
               
                 12, 07, 46, 00, 00, 00; 
                 13 
                 Z Z Z 
                 Z Y Y 
                 Y Z Y 
               
               
                 12, 00, 00, 00, 00, 00; 
                 14 
                 Z Z Z 
                 Z Y Y 
                 Y Y Z 
               
               
                 12, 00, 00, 00, 00, 00; 
                 15 
                 Z Z Z 
                 Z Y Y 
                 Y Y Y 
               
               
                 14, 00, 00, 00, 00, 00; 
                 16 
                 Z Z Z 
                 Y Y Z 
                 Z Z Z 
               
               
                 17, 16, 00, 00, 00, 00; 
                 17 
                 Z Z Z 
                 Y Y Z 
                 Z Z Y 
               
               
                 18, 00, 00, 00, 00, 00; 
                 18 
                 Z Z Z 
                 Y Y Z 
                 Z Y Z 
               
               
                 16, 00, 00, 00, 00, 00; 
                 19 
                 Z Z Z 
                 Y Y Z 
                 Z Y Y 
               
               
                 41, 00, 00, 00, 00, 00; 
                 20 
                 Z Z Z 
                 Y Y Z 
                 Y Z Z 
               
               
                 07, 46, 16, 00, 00, 00; 
                 21 
                 Z Z Z 
                 Y Y Z 
                 Y Z Y 
               
               
                 18, 00, 00, 00, 00, 00; 
                 22 
                 Z Z Z 
                 Y Y Z 
                 Y Y Z 
               
               
                 16, 00, 00, 00, 00, 00; 
                 23 
                 Z Z Z 
                 Y Y Z 
                 Y Y Y 
               
               
                 56, 54, 00, 00, 00, 00; 
                 24 
                 Z Z Z 
                 Y Y Y 
                 Z Z Z 
               
               
                 17, 54, 00, 00, 00, 00; 
                 25 
                 Z Z Z 
                 Y Y Y 
                 Z Z Y 
               
               
                 54, 00, 00, 00, 00, 00; 
                 26 
                 Z Z Z 
                 Y Y Y 
                 Z Y Z 
               
               
                 54, 00, 00, 00, 00, 00; 
                 27 
                 Z Z Z 
                 Y Y Y 
                 Z Y Y 
               
               
                 13, 54, 00, 00, 00, 00; 
                 28 
                 Z Z Z 
                 Y Y Y 
                 Y Z Z 
               
               
                 07, 46, 54, 00, 00, 00; 
                 29 
                 Z Z Z 
                 Y Y Y 
                 Y Z Y 
               
               
                 54, 00, 00, 00, 00, 00; 
                 30 
                 Z Z Z 
                 Y Y Y 
                 Y Y Z 
               
               
                 54, 00, 00, 00, 00, 00; 
                 31 
                 Z Z Z 
                 Y Y Y 
                 Y Y Y 
               
               
                 20, 21, 00, 00, 00, 00; 
                 32 
                 Z Z Y 
                 Z Y Z 
                 Z Z Z 
               
               
                 22, 23, 00, 00, 00, 00; 
                 33 
                 Z Z Y 
                 Z Y Z 
                 Z Z Y 
               
               
                 24, 25, 00, 00, 00, 00; 
                 34 
                 Z Z Y 
                 Z Y Z 
                 Z Y Z 
               
               
                 24, 23, 00, 00, 00, 00; 
                 35 
                 Z Z Y 
                 Z Y Z 
                 Z Y Y 
               
               
                 53, 48, 00, 00, 00, 00; 
                 36 
                 Z Z Y 
                 Z Y Z 
                 Y Z Z 
               
               
                 53, 07, 46, 23, 00, 00; 
                 37 
                 Z Z Y 
                 Z Y Z 
                 Y Z Y 
               
               
                 53, 25, 00, 00, 00, 00; 
                 38 
                 Z Z Y 
                 Z Y Z 
                 Y Y Z 
               
               
                 53, 23, 00, 00, 00, 00; 
                 39 
                 Z Z Y 
                 Z Y Z 
                 Y Y Y 
               
               
                 28, 00, 00, 00, 00, 00; 
                 40 
                 Z Z Y 
                 Z Y Y 
                 Z Z Z 
               
               
                 22, 00, 00, 00, 00, 00; 
                 41 
                 Z Z Y 
                 Z Y Y 
                 Z Z Y 
               
               
                 24, 00, 00, 00, 00, 00; 
                 42 
                 Z Z Y 
                 Z Y Y 
                 Z Y Z 
               
               
                 24, 00, 00, 00, 00, 00; 
                 43 
                 Z Z Y 
                 Z Y Y 
                 Z Y Y 
               
               
                 53, 13, 00, 00, 00, 00; 
                 44 
                 Z Z Y 
                 Z Y Y 
                 Y Z Z 
               
               
                 53, 07, 46, 00, 00, 00; 
                 45 
                 Z Z Y 
                 Z Y Y 
                 Y Z Y 
               
               
                 53, 00, 00, 00, 00, 00; 
                 46 
                 Z Z Y 
                 Z Y Y 
                 Y Y Z 
               
               
                 53, 00, 00, 00, 00, 00; 
                 47 
                 Z Z Y 
                 Z Y Y 
                 Y Y Y 
               
               
                 30, 27, 00, 00, 00, 00; 
                 48 
                 Z Z Y 
                 Y Y Z 
                 Z Z Z 
               
               
                 30, 23, 17, 00, 00, 00; 
                 49 
                 Z Z Y 
                 Y Y Z 
                 Z Z Y 
               
               
                 30, 25, 00, 00, 00, 00; 
                 50 
                 Z Z Y 
                 Y Y Z 
                 Z Y Z 
               
               
                 30, 23, 00, 00, 00, 00; 
                 51 
                 Z Z Y 
                 Y Y Z 
                 Z Y Y 
               
               
                 30, 48, 00, 00, 00, 00; 
                 52 
                 Z Z Y 
                 Y Y Z 
                 Y Z Z 
               
               
                 30, 07, 46, 23, 00, 00; 
                 53 
                 Z Z Y 
                 Y Y Z 
                 Y Z Y 
               
               
                 30, 25, 00, 00, 00, 00; 
                 54 
                 Z Z Y 
                 Y Y Z 
                 Y Y Z 
               
               
                 30, 23, 00, 00, 00, 00; 
                 55 
                 Z Z Y 
                 Y Y Z 
                 Y Y Y 
               
               
                 30, 56, 00, 00, 00, 00; 
                 56 
                 Z Z Y 
                 Y Y Y 
                 Z Z Z 
               
               
                 30, 17, 00, 00, 00, 00; 
                 57 
                 Z Z Y 
                 Y Y Y 
                 Z Z Y 
               
               
                 30, 00, 00, 00, 00, 00; 
                 58 
                 Z Z Y 
                 Y Y Y 
                 Z Y Z 
               
               
                 30, 00, 00, 00, 00, 00; 
                 59 
                 Z Z Y 
                 Y Y Y 
                 Z Y Y 
               
               
                 30, 13, 00, 00, 00, 00; 
                 60 
                 Z Z Y 
                 Y Y Y 
                 Y Z Z 
               
               
                 30, 07, 46, 00, 00, 00; 
                 61 
                 Z Z Y 
                 Y Y Y 
                 Y Z Y 
               
               
                 30, 00, 00, 00, 00, 00; 
                 62 
                 Z Z Y 
                 Y Y Y 
                 Y Y Z 
               
               
                 30, 00, 00, 00, 00, 00; 
                 63 
                 Z Z Y 
                 Y Y Y 
                 Y Y Y 
               
               
                 32, 52, 00, 00, 00, 00; 
                 64 
                 Z Y Z 
                 Z Y Z 
                 Z Z Z 
               
               
                 34, 35, 00, 00, 00, 00; 
                 65 
                 Z Y Z 
                 Z Y Z 
                 Z Z Y 
               
               
                 58, 57, 00, 00, 00, 00; 
                 66 
                 Z Y Z 
                 Z Y Z 
                 Z Y Z 
               
               
                 58, 35, 00, 00, 00, 00; 
                 67 
                 Z Y Z 
                 Z Y Z 
                 Z Y Y 
               
               
                 36, 37, 00, 00, 00, 00; 
                 68 
                 Z Y Z 
                 Z Y Z 
                 Y Z Z 
               
               
                 36, 07, 46, 35, 00, 00; 
                 69 
                 Z Y Z 
                 Z Y Z 
                 Y Z Y 
               
               
                 36, 57, 00, 00, 00, 00; 
                 70 
                 Z Y Z 
                 Z Y Z 
                 Y Y Z 
               
               
                 36, 35, 00, 00, 00, 00; 
                 71 
                 Z Y Z 
                 Z Y Z 
                 Y Y Y 
               
               
                 38, 00, 00, 00, 00, 00; 
                 72 
                 Z Y Z 
                 Z Y Y 
                 Z Z Z 
               
               
                 34, 00, 00, 00, 00, 00; 
                 73 
                 Z Y Z 
                 Z Y Y 
                 Z Z Y 
               
               
                 58, 00, 00, 00, 00, 00; 
                 74 
                 Z Y Z 
                 Z Y Y 
                 Z Y Z 
               
               
                 58, 00, 00, 00, 00, 00; 
                 75 
                 Z Y Z 
                 Z Y Y 
                 Z Y Y 
               
               
                 36, 13, 00, 00, 00, 00; 
                 76 
                 Z Y Z 
                 Z Y Y 
                 Y Z Z 
               
               
                 36, 07, 46, 00, 00, 00; 
                 77 
                 Z Y Z 
                 Z Y Y 
                 Y Z Y 
               
               
                 36, 00, 00, 00, 00, 00; 
                 78 
                 Z Y Z 
                 Z Y Y 
                 Y Y Z 
               
               
                 36, 00, 00, 00, 00, 00; 
                 79 
                 Z Y Z 
                 Z Y Y 
                 Y Y Y 
               
               
                 33, 00, 00, 00, 00, 00; 
                 80 
                 Z Y Z 
                 Y Y Z 
                 Z Z Z 
               
               
                 17, 35, 00, 00, 00, 00; 
                 81 
                 Z Y Z 
                 Y Y Z 
                 Z Z Y 
               
               
                 57, 00, 00, 00, 00, 00; 
                 82 
                 Z Y Z 
                 Y Y Z 
                 Z Y Z 
               
               
                 35, 00, 00, 00, 00, 00; 
                 83 
                 Z Y Z 
                 Y Y Z 
                 Z Y Y 
               
               
                 37, 00, 00, 00, 00, 00; 
                 84 
                 Z Y Z 
                 Y Y Z 
                 Y Z Z 
               
               
                 07, 46, 35, 00, 00, 00; 
                 85 
                 Z Y Z 
                 Y Y Z 
                 Y Z Y 
               
               
                 57, 00, 00, 00, 00, 00; 
                 86 
                 Z Y Z 
                 Y Y Z 
                 Y Y Z 
               
               
                 35, 00, 00, 00, 00, 00; 
                 87 
                 Z Y Z 
                 Y Y Z 
                 Y Y Y 
               
               
                 56, 00, 00, 00, 00, 00; 
                 88 
                 Z Y Z 
                 Y Y Y 
                 Z Z Z 
               
               
                 17, 00, 00, 00, 00, 00; 
                 89 
                 Z Y Z 
                 Y Y Y 
                 Z Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 90 
                 Z Y Z 
                 Y Y Y 
                 Z Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 91 
                 Z Y Z 
                 Y Y Y 
                 Z Y Y 
               
               
                 13, 00, 00, 00, 00, 00; 
                 92 
                 Z Y Z 
                 Y Y Y 
                 Y Z Z 
               
               
                 07, 46, 00, 00, 00, 00; 
                 93 
                 Z Y Z 
                 Y Y Y 
                 Y Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 94 
                 Z Y Z 
                 Y Y Y 
                 Y Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 95 
                 Z Y Z 
                 Y Y Y 
                 Y Y Y 
               
               
                 32, 21, 00, 00, 00, 00; 
                 96 
                 Z Y Y 
                 Z Y Z 
                 Z Z Z 
               
               
                 34, 23, 00, 00, 00, 00; 
                 97 
                 Z Y Y 
                 Z Y Z 
                 Z Z Y 
               
               
                 58, 25, 00, 00, 00, 00; 
                 98 
                 Z Y Y 
                 Z Y Z 
                 Z Y Z 
               
               
                 58, 23, 00, 00, 00, 00; 
                 99 
                 Z Y Y 
                 Z Y Z 
                 Z Y Y 
               
               
                 36, 48, 00, 00, 00, 00; 
                 100 
                 Z Y Y 
                 Z Y Z 
                 Y Z Z 
               
               
                 36, 07, 46, 23, 00, 00; 
                 101 
                 Z Y Y 
                 Z Y Z 
                 Y Z Y 
               
               
                 36, 25, 00, 00, 00, 00; 
                 102 
                 Z Y Y 
                 Z Y Z 
                 Y Y Z 
               
               
                 36, 23, 00, 00, 00, 00; 
                 103 
                 Z Y Y 
                 Z Y Z 
                 Y Y Y 
               
               
                 38, 00, 00, 00, 00, 00; 
                 104 
                 Z Y Y 
                 Z Y Y 
                 Z Z Z 
               
               
                 34, 00, 00, 00, 00, 00; 
                 105 
                 Z Y Y 
                 Z Y Y 
                 Z Z Y 
               
               
                 58, 00, 00, 00, 00, 00; 
                 106 
                 Z Y Y 
                 Z Y Y 
                 Z Y Z 
               
               
                 58, 00, 00, 00, 00, 00; 
                 107 
                 Z Y Y 
                 Z Y Y 
                 Z Y Y 
               
               
                 36, 13, 00, 00, 00, 00; 
                 108 
                 Z Y Y 
                 Z Y Y 
                 Y Z Z 
               
               
                 36, 07, 46, 00, 00, 00; 
                 109 
                 Z Y Y 
                 Z Y Y 
                 Y Z Y 
               
               
                 36, 00, 00, 00, 00, 00; 
                 110 
                 Z Y Y 
                 Z Y Y 
                 Y Y Z 
               
               
                 36, 00, 00, 00, 00, 00; 
                 111 
                 Z Y Y 
                 Z Y Y 
                 Y Y Y 
               
               
                 27, 00, 00, 00, 00, 00; 
                 112 
                 Z Y Y 
                 Y Y Z 
                 Z Z Z 
               
               
                 17, 23, 00, 00, 00, 00; 
                 113 
                 Z Y Y 
                 Y Y Z 
                 Z Z Y 
               
               
                 25, 00, 00, 00, 00, 00; 
                 114 
                 Z Y Y 
                 Y Y Z 
                 Z Y Z 
               
               
                 23, 00, 00, 00, 00, 00; 
                 115 
                 Z Y Y 
                 Y Y Z 
                 Z Y Y 
               
               
                 48, 00, 00, 00, 00, 00; 
                 116 
                 Z Y Y 
                 Y Y Z 
                 Y Z Z 
               
               
                 07, 46, 23, 00, 00, 00; 
                 117 
                 Z Y Y 
                 Y Y Z 
                 Y Z Y 
               
               
                 25, 00, 00, 00, 00, 00; 
                 118 
                 Z Y Y 
                 Y Y Z 
                 Y Y Z 
               
               
                 23, 00, 00, 00, 00, 00; 
                 119 
                 Z Y Y 
                 Y Y Z 
                 Y Y Y 
               
               
                 56, 00, 00, 00, 00, 00; 
                 120 
                 Z Y Y 
                 Y Y Y 
                 Z Z Z 
               
               
                 17, 00, 00, 00, 00, 00; 
                 121 
                 Z Y Y 
                 Y Y Y 
                 Z Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 122 
                 Z Y Y 
                 Y Y Y 
                 Z Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 123 
                 Z Y Y 
                 Y Y Y 
                 Z Y Y 
               
               
                 13, 00, 00, 00, 00, 00; 
                 124 
                 Z Y Y 
                 Y Y Y 
                 Y Z Z 
               
               
                 07, 46, 00, 00, 00, 00; 
                 125 
                 Z Y Y 
                 Y Y Y 
                 Y Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 126 
                 Z Y Y 
                 Y Y Y 
                 Y Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 127 
                 Z Y Y 
                 Y Y Y 
                 Y Y Y 
               
               
                 42, 43, 00, 00, 00, 00; 
                 128 
                 Y Z Z 
                 Z Y Z 
                 Z Z Z 
               
               
                 47, 45, 00, 00, 00, 00; 
                 129 
                 Y Z Z 
                 Z Y Z 
                 Z Z Y 
               
               
                 44, 40, 00, 00, 00, 00; 
                 130 
                 Y Z Z 
                 Z Y Z 
                 Z Y Z 
               
               
                 44, 45, 00, 00, 00, 00; 
                 131 
                 Y Z Z 
                 Z Y Z 
                 Z Y Y 
               
               
                 39, 31, 00, 00, 00, 00; 
                 132 
                 Y Z Z 
                 Z Y Z 
                 Y Z Z 
               
               
                 39, 07, 46, 45, 00, 00; 
                 133 
                 Y Z Z 
                 Z Y Z 
                 Y Z Y 
               
               
                 39, 40, 00, 00, 00, 00; 
                 134 
                 Y Z Z 
                 Z Y Z 
                 Y Y Z 
               
               
                 39, 45, 00, 00, 00, 00; 
                 135 
                 Y Z Z 
                 Z Y Z 
                 Y Y Y 
               
               
                 29, 11, 00, 00, 00, 00; 
                 136 
                 Y Z Z 
                 Z Y Y 
                 Z Z Z 
               
               
                 47, 11, 00, 00, 00, 00; 
                 137 
                 Y Z Z 
                 Z Y Y 
                 Z Z Y 
               
               
                 44, 11, 00, 00, 00, 00; 
                 138 
                 Y Z Z 
                 Z Y Y 
                 Z Y Z 
               
               
                 44, 11, 00, 00, 00, 00; 
                 139 
                 Y Z Z 
                 Z Y Y 
                 Z Y Y 
               
               
                 39, 13, 1l, 00, 00, 00; 
                 140 
                 Y Z Z 
                 Z Y Y 
                 Y Z Z 
               
               
                 39, 07, 46, 11, 00, 00; 
                 141 
                 Y Z Z 
                 Z Y Y 
                 Y Z Y 
               
               
                 39, 11, 00, 00, 00, 00; 
                 142 
                 Y Z Z 
                 Z Y Y 
                 Y Y Z 
               
               
                 39, 11, 00, 00, 00, 00; 
                 143 
                 Y Z Z 
                 Z Y Y 
                 Y Y Y 
               
               
                 19, 00, 00, 00, 00, 00; 
                 144 
                 Y Z Z 
                 Y Y Z 
                 Z Z Z 
               
               
                 17, 45, 00, 00, 00, 00; 
                 145 
                 Y Z Z 
                 Y Y Z 
                 Z Z Y 
               
               
                 40, 00, 00, 00, 00, 00; 
                 146 
                 Y Z Z 
                 Y Y Z 
                 Z Y Z 
               
               
                 45, 00, 00, 00, 00, 00; 
                 147 
                 Y Z Z 
                 Y Y Z 
                 Z Y Y 
               
               
                 31, 00, 00, 00, 00, 00; 
                 148 
                 Y Z Z 
                 Y Y Z 
                 Y Z Z 
               
               
                 07, 46, 45, 00, 00, 00; 
                 149 
                 Y Z Z 
                 Y Y Z 
                 Y Z Y 
               
               
                 40, 00, 00, 00, 00, 00; 
                 150 
                 Y Z Z 
                 Y Y Z 
                 Y Y Z 
               
               
                 45, 00, 00, 00, 00, 00; 
                 151 
                 Y Z Z 
                 Y Y Z 
                 Y Y Y 
               
               
                 11, 56, 00, 00, 00, 00; 
                 152 
                 Y Z Z 
                 Y Y Y 
                 Z Z Z 
               
               
                 17, 11, 00, 00, 00, 00; 
                 153 
                 Y Z Z 
                 Y Y Y 
                 Z Z Y 
               
               
                 11, 00, 00, 00, 00, 00; 
                 154 
                 Y Z Z 
                 Y Y Y 
                 Z Y Z 
               
               
                 11, 00, 00, 00, 00, 00; 
                 155 
                 Y Z Z 
                 Y Y Y 
                 Z Y Y 
               
               
                 13, 11, 00, 00, 00, 00; 
                 156 
                 Y Z Z 
                 Y Y Y 
                 Y Z Z 
               
               
                 07, 46, l1, 00, 00, 00; 
                 157 
                 Y Z Z 
                 Y Y Y 
                 Y Z Y 
               
               
                 11, 00, 00, 00, 00, 00; 
                 158 
                 Y Z Z 
                 Y Y Y 
                 Y Y Z 
               
               
                 11, 00, 00, 00, 00, 00; 
                 159 
                 Y Z Z 
                 Y Y Y 
                 Y Y Y 
               
               
                 42, 08, 26, 21, 00, 00; 
                 160 
                 Y Z Y 
                 Z Y Z 
                 Z Z Z 
               
               
                 47, 08, 26, 23, 00, 00; 
                 161 
                 Y Z Y 
                 Z Y Z 
                 Z Z Y 
               
               
                 44, 08, 26, 25, 00, 00; 
                 162 
                 Y Z Y 
                 Z Y Z 
                 Z Y Z 
               
               
                 44, 08, 26, 23, 00, 00; 
                 163 
                 Y Z Y 
                 Z Y Z 
                 Z Y Y 
               
               
                 39, 08, 26, 48, 00, 00; 
                 164 
                 Y Z Y 
                 Z Y Z 
                 Y Z Z 
               
               
                 39, 08, 26, 07, 46, 23; 
                 165 
                 Y Z Y 
                 Z Y Z 
                 Y Z Y 
               
               
                 39, 08, 26, 25, 00, 00; 
                 166 
                 Y Z Y 
                 Z Y Z 
                 Y Y Z 
               
               
                 39, 08, 26, 23, 00, 00; 
                 167 
                 Y Z Y 
                 Z Y Z 
                 Y Y Y 
               
               
                 29, 08, 26, 00, 00, 00; 
                 168 
                 Y Z Y 
                 Z Y Y 
                 Z Z Z 
               
               
                 47, 08, 26, 00, 00, 00; 
                 169 
                 Y Z Y 
                 Z Y Y 
                 Z Z Y 
               
               
                 44, 08, 26, 00, 00, 00; 
                 170 
                 Y Z Y 
                 Z Y Y 
                 Z Y Z 
               
               
                 44, 08, 26, 00, 00, 00; 
                 171 
                 Y Z Y 
                 Z Y Y 
                 Z Y Y 
               
               
                 39, 08, 26, 13, 00, 00; 
                 172 
                 Y Z Y 
                 Z Y Y 
                 Y Z Z 
               
               
                 39, 08, 26, 07, 46, 00; 
                 173 
                 Y Z Y 
                 Z Y Y 
                 Y Z Y 
               
               
                 39, 08, 26, 00, 00, 00; 
                 174 
                 Y Z Y 
                 Z Y Y 
                 Y Y Z 
               
               
                 39, 08, 26, 00, 00, 00; 
                 175 
                 Y Z Y 
                 Z Y Y 
                 Y Y Y 
               
               
                 08, 26, 27, 00, 00, 00; 
                 176 
                 Y Z Y 
                 Y Y Z 
                 Z Z Z 
               
               
                 08, 26, 17, 23, 00, 00; 
                 177 
                 Y Z Y 
                 Y Y Z 
                 Z Z Y 
               
               
                 08, 26, 25, 00, 00, 00; 
                 178 
                 Y Z Y 
                 Y Y Z 
                 Z Y Z 
               
               
                 08, 26, 23, 00, 00, 00; 
                 179 
                 Y Z Y 
                 Y Y Z 
                 Z Y Y 
               
               
                 08, 26, 48, 00, 00, 00; 
                 180 
                 Y Z Y 
                 Y Y Z 
                 Y Z Z 
               
               
                 08, 26, 07, 46, 23, 00; 
                 181 
                 Y Z Y 
                 Y Y Z 
                 Y Z Y 
               
               
                 08, 26, 25, 00, 00, 00; 
                 182 
                 Y Z Y 
                 Y Y Z 
                 Y Y Z 
               
               
                 08, 26, 23, 00, 00, 00; 
                 183 
                 Y Z Y 
                 Y Y Z 
                 Y Y Y 
               
               
                 08, 26, 56, 00, 00, 00; 
                 184 
                 Y Z Y 
                 Y Y Y 
                 Z Z Z 
               
               
                 08, 26, 17, 00, 00, 00; 
                 185 
                 Y Z Y 
                 Y Y Y 
                 Z Z Y 
               
               
                 08, 26, 00, 00, 00, 00; 
                 186 
                 Y Z Y 
                 Y Y Y 
                 Z Y Z 
               
               
                 08, 26, 00, 00, 00, 00; 
                 187 
                 Y Z Y 
                 Y Y Y 
                 Z Y Y 
               
               
                 08, 26, 13, 00, 00, 00; 
                 188 
                 Y Z Y 
                 Y Y Y 
                 Y Z Z 
               
               
                 08, 26, 07, 46, 00, 00; 
                 189 
                 Y Z Y 
                 Y Y Y 
                 Y Z Y 
               
               
                 08, 26, 00, 00, 00, 00; 
                 190 
                 Y Z Y 
                 Y Y Y 
                 Y Y Z 
               
               
                 08, 26, 00, 00, 00, 00; 
                 191 
                 Y Z Y 
                 Y Y Y 
                 Y Y Y 
               
               
                 42, 52, 00, 00, 00, 00; 
                 192 
                 Y Y Z 
                 Z Y Z 
                 Z Z Z 
               
               
                 47, 35, 00, 00, 00, 00; 
                 193 
                 Y Y Z 
                 Z Y Z 
                 Z Z Y 
               
               
                 44, 57, 00, 00, 00, 00; 
                 194 
                 Y Y Z 
                 Z Y Z 
                 Z Y Z 
               
               
                 44, 35, 00, 00, 00, 00; 
                 195 
                 Y Y Z 
                 Z Y Z 
                 Z Y Y 
               
               
                 39, 37, 00, 00, 00, 00; 
                 196 
                 Y Y Z 
                 Z Y Z 
                 Y Z Z 
               
               
                 39, 07, 46, 35, 00, 00; 
                 197 
                 Y Y Z 
                 Z Y Z 
                 Y Z Y 
               
               
                 39, 57, 00, 00, 00, 00; 
                 198 
                 Y Y Z 
                 Z Y Z 
                 Y Y Z 
               
               
                 39, 35, 00, 00, 00, 00; 
                 199 
                 Y Y Z 
                 Z Y Z 
                 Y Y Y 
               
               
                 29, 00, 00, 00, 00, 00; 
                 200 
                 Y Y Z 
                 Z Y Y 
                 Z Z Z 
               
               
                 47, 00, 00, 00, 00, 00; 
                 201 
                 Y Y Z 
                 Z Y Y 
                 Z Z Y 
               
               
                 44, 00, 00, 00, 00, 00; 
                 202 
                 Y Y Z 
                 Z Y Y 
                 Z Y Z 
               
               
                 44, 00, 00, 00, 00, 00; 
                 203 
                 Y Y Z 
                 Z Y Y 
                 Z Y Y 
               
               
                 39, 13, 00, 00, 00, 00; 
                 204 
                 Y Y Z 
                 Z Y Y 
                 Y Z Z 
               
               
                 39, 07, 46, 00, 00, 00; 
                 205 
                 Y Y Z 
                 Z Y Y 
                 Y Z Y 
               
               
                 39, 00, 00, 00, 00, 00; 
                 206 
                 Y Y Z 
                 Z Y Y 
                 Y Y Z 
               
               
                 39, 00, 00, 00, 00, 00; 
                 207 
                 Y Y Z 
                 Z Y Y 
                 Y Y Y 
               
               
                 33, 00, 00, 00, 00, 00; 
                 208 
                 Y Y Z 
                 Y Y Z 
                 Z Z Z 
               
               
                 17, 35, 00, 00, 00, 00; 
                 209 
                 Y Y Z 
                 Y Y Z 
                 Z Z Y 
               
               
                 57, 00, 00, 00, 00, 00; 
                 210 
                 Y Y Z 
                 Y Y Z 
                 Z Y Z 
               
               
                 35, 00, 00, 00, 00, 00; 
                 211 
                 Y Y Z 
                 Y Y Z 
                 Z Y Y 
               
               
                 37, 00, 00, 00, 00, 00; 
                 212 
                 Y Y Z 
                 Y Y Z 
                 Y Z Z 
               
               
                 07, 46, 35, 00, 00, 00; 
                 213 
                 Y Y Z 
                 Y Y Z 
                 Y Z Y 
               
               
                 57, 00, 00, 00, 00, 00; 
                 214 
                 Y Y Z 
                 Y Y Z 
                 Y Y Z 
               
               
                 35, 00, 00, 00, 00, 00; 
                 215 
                 Y Y Z 
                 Y Y Z 
                 Y Y Y 
               
               
                 56, 00, 00, 00, 00, 00; 
                 216 
                 Y Y Z 
                 Y Y Y 
                 Z Z Z 
               
               
                 17, 00, 00, 00, 00, 00; 
                 217 
                 Y Y Z 
                 Y Y Y 
                 Z Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 218 
                 Y Y Z 
                 Y Y Y 
                 Z Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 219 
                 Y Y Z 
                 Y Y Y 
                 Z Y Y 
               
               
                 13, 00, 00, 00, 00, 00; 
                 220 
                 Y Y Z 
                 Y Y Y 
                 Y Z Z 
               
               
                 07, 46, 00, 00, 00, 00; 
                 221 
                 Y Y Z 
                 Y Y Y 
                 Y Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 222 
                 Y Y Z 
                 Y Y Y 
                 Y Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 223 
                 Y Y Z 
                 Y Y Y 
                 Y Y Y 
               
               
                 42, 21, 00, 00, 00, 00; 
                 224 
                 Y Y Y 
                 Z Y Z 
                 Z Z Z 
               
               
                 47, 23, 00, 00, 00, 00; 
                 225 
                 Y Y Y 
                 Z Y Z 
                 Z Z Y 
               
               
                 44, 25, 00, 00, 00, 00; 
                 226 
                 Y Y Y 
                 Z Y Z 
                 Z Y Z 
               
               
                 44, 23, 00, 00, 00, 00; 
                 227 
                 Y Y Y 
                 Z Y Z 
                 Z Y Y 
               
               
                 39, 48, 00, 00, 00, 00; 
                 228 
                 Y Y Y 
                 Z Y Z 
                 Y Z Z 
               
               
                 39, 07, 46, 23, 00, 00; 
                 229 
                 Y Y Y 
                 Z Y Z 
                 Y Z Y 
               
               
                 39, 25, 00, 00, 00, 00; 
                 230 
                 Y Y Y 
                 Z Y Z 
                 Y Y Z 
               
               
                 39, 23, 00, 00, 00, 00; 
                 231 
                 Y Y Y 
                 Z Y Z 
                 Y Y Y 
               
               
                 29, 00, 00, 00, 00, 00; 
                 232 
                 Y Y Y 
                 Z Y Y 
                 Z Z Z 
               
               
                 47, 00, 00, 00, 00, 00; 
                 233 
                 Y Y Y 
                 Z Y Y 
                 Z Z Y 
               
               
                 44, 00, 00, 00, 00, 00; 
                 234 
                 Y Y Y 
                 Z Y Y 
                 Z Y Z 
               
               
                 44, 00, 00, 00, 00, 00; 
                 235 
                 Y Y Y 
                 Z Y Y 
                 Z Y Y 
               
               
                 39, 13, 00, 00, 00, 00; 
                 236 
                 Y Y Y 
                 Z Y Y 
                 Y Z Z 
               
               
                 39, 07, 46, 00, 00, 00; 
                 237 
                 Y Y Y 
                 Z Y Y 
                 Y Z Y 
               
               
                 39, 00, 00, 00, 00, 00; 
                 238 
                 Y Y Y 
                 Z Y Y 
                 Y Y Z 
               
               
                 39, 00, 00, 00, 00, 00; 
                 239 
                 Y Y Y 
                 Z Y Y 
                 Y Y Y 
               
               
                 27, 00, 00, 00, 00, 00; 
                 240 
                 Y Y Y 
                 Y Y Z 
                 Z Z Z 
               
               
                 17, 23, 00, 00, 00, 00; 
                 241 
                 Y Y Y 
                 Y Y Z 
                 Z Z Y 
               
               
                 25, 00, 00, 00, 00, 00; 
                 242 
                 Y Y Y 
                 Y Y Z 
                 Z Y Z 
               
               
                 23, 00, 00, 00, 00, 00; 
                 243 
                 Y Y Y 
                 Y Y Z 
                 Z Y Y 
               
               
                 48, 00, 00, 00, 00, 00; 
                 244 
                 Y Y Y 
                 Y Y Z 
                 Y Z Z 
               
               
                 07, 46, 23, 00, 00, 00; 
                 245 
                 Y Y Y 
                 Y Y Z 
                 Y Z Y 
               
               
                 25, 00, 00, 00, 00, 00; 
                 246 
                 Y Y Y 
                 Y YZ 
                 Y Y Z 
               
               
                 23, 00, 00, 00, 00, 00; 
                 247 
                 Y Y Y 
                 Y Y Z 
                 Y Y Y 
               
               
                 56, 00, 00, 00, 00, 00; 
                 248 
                 Y Y Y 
                 Y Y Y 
                 Z Z Z 
               
               
                 17, 00, 00, 00, 00, 00; 
                 249 
                 Y Y Y 
                 Y Y Y 
                 Z Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 250 
                 Y Y Y 
                 Y Y Y 
                 Z Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 251 
                 Y Y Y 
                 Y Y Y 
                 Z Y Y 
               
               
                 13, 00, 00, 00, 00, 00; 
                 252 
                 Y Y Y 
                 Y Y Y 
                 Y Z Z 
               
               
                 07, 46, 00, 00, 00, 00; 
                 253 
                 Y Y Y 
                 Y Y Y 
                 Y Z Y 
               
               
                 00, 00, 00, 00, 00, 00; 
                 254 
                 Y Y Y 
                 Y Y Y 
                 Y Y Z 
               
               
                 00, 00, 00, 00, 00, 00; 
                 255 
                 Y Y Y 
                 Y Y Y 
                 Y Y Y 
               
               
                   
               
             
          
         
       
     
     In the above table, the Y and Z color values for the neighborhood cells are shown in the right hand columns. The corresponding binary address represented by a particular color combination of the permitter cells P 0-7  is shown in the middle column denoted ADR. In other words, each of the 256 possible color combinations for the perimeter cells P 0-7  generates a binary addresses into the neighborhood lookup table  18 . For each color combination or address, the neighborhood table outputs a string ID for up to six different surface strings as shown in the left column (as describe below, there are 58 different surface strings). The ordering of the surface strings in each entry of the above table is significant. As explained below, the surface strings are linked together into a string sequence; by ordering the surface strings as shown in each entry of the above table, the linking process is simplified. 
     A surface string defines the general slope of a color border passing through a given neighborhood of cells. It is possible that six different color borders will pass through a single neighborhood of cells which is why the neighborhood lookup table  18  outputs up to six different surface strings for any given neighborhood  16 . FIG. 3 shows the 58 different surface strings provided in the present invention which, as far as is known to the applicant, define every possible permutation of a color border in a raster image. The 58 surface strings shown in FIG. 3 were generated through an exhaustive, heuristic search that required many man hours to complete; the set of surface strings in FIG. 3 is a significant aspect of the present invention. 
     The cell coloring in each surface string shown in FIG. 3 denotes the above-described color value (Y or Z) for each cell. The black cells represent the color Y, the white cells represent the color Z, and the shaded cells represent the color X which means the color of the cell is ignored (i.e., the cell color could be either Y or Z). Referring again to the above neighborhood lookup table, the entry corresponding to neighborhood address  69  is: 
     
       
         
               
               
               
               
               
             
           
               
                   
               
               
                 SURFACE 
                   
                   
                   
                   
               
               
                 STRINGS 
                 ADR 
                 P 0 P 1 P 2   
                 P 3  T P 4   
                 P 5 P 6 P 7   
               
               
                   
               
             
             
               
                 36, 07, 46, 35, 00, 00; 
                 69 
                 Z Y Z 
                 Z Y Z 
                 Y Z Y 
               
               
                   
               
             
          
         
       
     
     The color code for the first surface string of the above entry, surface string  36 , is shown in FIG. 3 to be: 
     
       
         Z Y X |Z Y X |Y X X. 
       
     
     Note that the color code for surface string 36 matches the color code of the neighborhood address 69 except for the don&#39;t care color X. A similar color code which matches the color code of neighborhood  69  can be verified for each of the other surface strings (7, 46 and 35) in the above entry. 
     Each surface string shown in FIG. 3 has associated with it a receive slope and a send slope which define the general slope of a color border as it enters the neighborhood and the general slope as it exits the neighborhood. In the present invention, there are eight possible slope values that can be assigned to the send and receive slope of a surface string. These slope values ( 1 - 8 ) are illustrated in FIG.  4 B. Notice that there are two slope values assigned to each slope direction: diagonal, vertical, and horizontal, e.g., the slope values 1 and 2 are assigned to the diagonal direction downward from left to right. The reason two slope values are assigned to each direction is because each color border is either an east or west border. 
     The concept of east and west color borders is understood from the raster image comprising a simple feature as shown in FIG.  5 A. This raster image comprises only two colors, black and white. The feature depicted in the image has associated with it a west  20 A and an east  20 B white color border, and a west  22 A and an east  22 B black color border. The west white border corresponding to the east white border  20 A is the left edge of the image in FIG. 5A, and the east white border corresponding to the west white border  20 B is the right edge of the image in FIG.  5 A. Notice that in FIG. 5A the west and east borders begin in pairs, which is always the case in the present invention. West and east borders also always end in pairs, although the borders do not always end with the respective border that it began with. As described in greater detail below, the present invention generates surface strings for both the west and east borders, but the tags for generating vectors are placed only on either of the west or east borders. Both west and east strings sequences are used to place the tags, but the tags are placed only on either the west or east borders. 
     Referring again to the surface strings shown in FIG. 3, the receive and send slope of each surface string is assigned one of the values shown in FIG.  4 B. Consider, for example, the eleventh surface string [ 11 ]. This surface string is assigned a receive slope of 2 and a send slope of 7. Similarly, the surface string [ 17 ] is assigned a receive slope of 8 and a send slope of 1. As explained in greater detail below, the receive and send slopes are used to connect the surface strings into a string sequence which represents a color border in the image. The first surface string [ 1 ] is assigned a receive slope of 0 because string sequences start with this surface string. Similarly, the surface string [ 8 ] is assigned a send slope of 0 because string sequences end with this surface string. A lookup table showing each of the 58 surface strings and their corresponding receive and send slopes is shown below: 
     
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Surface Strings Table 
               
             
          
           
               
                   
                   
                   
                   
                 S 
                 R 
                 S 
               
               
                   
                   
                   
                   
                 t 
                 c 
                 n 
               
               
                   
                   
                   
                   
                 r 
                 v 
                 d 
               
               
                   
                   
                   
                   
                 g 
                 — 
                 — 
               
               
                   
                   
                   
                   
                 — 
                 S 
                 S 
               
               
                   
                   
                   
                   
                 I 
                 1 
                 1 
               
               
                   
                 P 0 P 1 P 2   
                 P 3  T P 4   
                 P 5 P 6 P 7   
                 D 
                 p 
                 p 
               
               
                   
                   
               
             
          
           
               
                   
                 Z Z Z 
                 Z Y Z 
                 Z Z Y 
                 01 
                 0 
                 1 
               
               
                   
                 Z Z Z 
                 Z Y Z 
                 X X Y 
                 02 
                 0 
                 2 
               
               
                   
                 Z Z Z 
                 Z Y Z 
                 Z Y X 
                 03 
                 0 
                 3 
               
               
                   
                 Z Z Z 
                 Z Y Z 
                 X Y Z 
                 04 
                 0 
                 4 
               
               
                   
                 Z Z Z 
                 Z Y Z 
                 Y X X 
                 05 
                 0 
                 5 
               
               
                   
                 Z Z Z 
                 Z Y Z 
                 Y Z Z 
                 06 
                 0 
                 6 
               
               
                   
                 X X X 
                 X Y X 
                 Y Z Y 
                 07 
                 0 
                 6 
               
               
                   
                 Y Z Y 
                 X Y X 
                 X X X 
                 08 
                 2 
                 0 
               
               
                   
                 Z Z Z 
                 Z Y Y 
                 Z Z Y 
                 09 
                 9 
                 1 
               
               
                   
                 Z Z Z 
                 Z Y Y 
                 Z Z Z 
                 10 
                 9 
                 8 
               
               
                   
                 Y Z Z 
                 X Y Y 
                 X X X 
                 11 
                 2 
                 7 
               
               
                   
                 Z Z Z 
                 Z Y Y 
                 Y X X 
                 12 
                 9 
                 5 
               
               
                   
                 X X X 
                 X Y Y 
                 Y Z Z 
                 13 
                 10 
                 6 
               
               
                   
                 Z Z Z 
                 Y Y Z 
                 Z Z Z 
                 14 
                 7 
                 10 
               
               
                   
                 Z Z Z 
                 Z Y Y 
                 Z Y X 
                 15 
                 9 
                 3 
               
               
                   
                 Z Z Z 
                 Y Y Z 
                 X X Y 
                 16 
                 7 
                 2 
               
               
                   
                 X X X 
                 Y Y X 
                 Z Z Y 
                 17 
                 8 
                 1 
               
               
                   
                 Z Z Z 
                 Y Y Z 
                 X Y Z 
                 18 
                 7 
                 4 
               
               
                   
                 Y Z Z 
                 Y Y Z 
                 Z Z Z 
                 19 
                 2 
                 10 
               
               
                   
                 Z Z Y 
                 Z Y Z 
                 Z Z Z 
                 20 
                 5 
                 0 
               
               
                   
                 X X Y 
                 Z Y Z 
                 Z Z Z 
                 21 
                 6 
                 0 
               
               
                   
                 Z Z Y 
                 Z Y X 
                 Z Z Y 
                 22 
                 5 
                 1 
               
               
                   
                 X X Y 
                 X Y Z 
                 X X Y 
                 23 
                 6 
                 2 
               
               
                   
                 Z Z Y 
                 Z Y X 
                 Z Y X 
                 24 
                 5 
                 3 
               
               
                   
                 X X Y 
                 X Y Z 
                 X Y Z 
                 25 
                 6 
                 4 
               
               
                   
                 Y Z Y 
                 X Y X 
                 X X X 
                 26 
                 5 
                 0 
               
               
                   
                 X X Y 
                 Y Y Z 
                 Z Z Z 
                 27 
                 6 
                 10 
               
               
                   
                 Z Z Y 
                 Z Y Y 
                 Z Z Z 
                 28 
                 5 
                 8 
               
               
                   
                 Y X X 
                 Z Y Y 
                 Z Z Z 
                 29 
                 1 
                 8 
               
               
                   
                 Z Z Y 
                 Y Y X 
                 X X X 
                 30 
                 5 
                 9 
               
               
                   
                 Y Z Z 
                 X Y Z 
                 Y Z Z 
                 31 
                 2 
                 6 
               
               
                   
                 Z Y X 
                 Z Y Z 
                 Z Z Z 
                 32 
                 3 
                 0 
               
               
                   
                 X Y Z 
                 Y Y Z 
                 Z Z Z 
                 33 
                 4 
                 10 
               
               
                   
                 Z Y X 
                 Z Y X 
                 Z Z Y 
                 34 
                 3 
                 1 
               
               
                   
                 X Y Z 
                 X Y Z 
                 X X Y 
                 35 
                 4 
                 2 
               
               
                   
                 Z Y X 
                 Z Y X 
                 Y X X 
                 36 
                 3 
                 5 
               
               
                   
                 X Y Z 
                 X Y Z 
                 Y Z Z 
                 37 
                 4 
                 6 
               
               
                   
                 Z Y X 
                 Z Y Y 
                 Z Z Z 
                 38 
                 3 
                 8 
               
               
                   
                 Y X X 
                 Z Y X 
                 Y X X 
                 39 
                 1 
                 5 
               
               
                   
                 Y Z Z 
                 X Y Z 
                 X Y Z 
                 40 
                 2 
                 4 
               
               
                   
                 Z Z Z 
                 Y Y Z 
                 Y Z Z 
                 41 
                 7 
                 6 
               
               
                   
                 Y X X 
                 Z Y Z 
                 Z Z Z 
                 42 
                 1 
                 0 
               
               
                   
                 Y Z Z 
                 Z Y Z 
                 Z Z Z 
                 43 
                 2 
                 0 
               
               
                   
                 Y X X 
                 Z Y X 
                 Z Y X 
                 44 
                 1 
                 3 
               
               
                   
                 Y Z Z 
                 X Y Z 
                 X X Y 
                 45 
                 2 
                 2 
               
               
                   
                 X X X 
                 X Y X 
                 Y Z Y 
                 46 
                 0 
                 1 
               
               
                   
                 Y X X 
                 Z Y X 
                 Z Z Y 
                 47 
                 1 
                 1 
               
               
                   
                 X X Y 
                 X Y Z 
                 Y Z Z 
                 48 
                 6 
                 6 
               
               
                   
                 Y Z X 
                 Y Z Z 
                 Y Y Y 
                 49 
                 4 
                 7 
               
               
                   
                 Y Y Y 
                 Z Z Y 
                 X Z Y 
                 50 
                 8 
                 3 
               
               
                   
                 X Z Y 
                 Z Z Y 
                 Y Y Y 
                 51 
                 3 
                 9 
               
               
                   
                 X Y Z 
                 Z Y Z 
                 Z Z Z 
                 52 
                 4 
                 0 
               
               
                   
                 Z Z Y 
                 Z Y X 
                 Y X X 
                 53 
                 5 
                 5 
               
               
                   
                 Z Z Z 
                 Y Y Y 
                 X X X 
                 54 
                 7 
                 7 
               
               
                   
                 Y Y Y 
                 Y Z Z 
                 Y Z X 
                 55 
                 10 
                 4 
               
               
                   
                 X X X 
                 Y Y Y 
                 Z Z Z 
                 56 
                 8 
                 8 
               
               
                   
                 X Y Z 
                 X Y Z 
                 X Y Z 
                 57 
                 4 
                 4 
               
               
                   
                 Z Y X 
                 Z Y X 
                 Z Y X 
                 58 
                 3 
                 3 
               
               
                   
                   
               
             
          
         
       
     
     In the above table, the left columns represent the color code for each surface string shown in FIG. 3, the middle column is a string ID, and the right columns are the respective receive and send slopes assigned to each surface string. Note that in the above surface strings table there are entries which comprise slopes of 9 and 10. These slopes correspond to the horizontal slopes of 7 and 8 shown in FIG. 4B, only the direction of the slopes are reversed for sequencing surface strings from east to west as described below. 
     Referring again to FIG. 2, each of the surface string IDs 24 output by the neighborhood lookup table  18  are used as an address into the surface strings table 26 shown above. Notice that surface strings  49 ,  50 ,  51  and  55  in the above table are special cases; they are not output by the neighborhood lookup table  18  but instead are generated by combining adjacent surface strings according to the following table: 
     
       
         
               
             
               
               
               
             
           
               
                   
               
               
                 Special Case Surface Strings Table 
               
             
          
           
               
                   
                 Special Case 
                 Surface Strings 
               
               
                   
                 Strings 
                 Combination 
               
               
                   
                   
               
               
                   
                 49 
                 35, 11 
               
               
                   
                 50 
                 17, 44 
               
               
                   
                 51 
                 36, 30 
               
               
                   
                 55 
                 13, 25 
               
               
                   
                   
               
             
          
         
       
     
     In the above table, the special case surface strings are generated when the string sequencer  30  of FIG. 2 encounters two adjacent surface strings which make up a special case surface string. An example of when a special case surface string is generated is described below with reference to FIG.  5 A. 
     The surface strings table  26  outputs the receive and send slope 28 for each surface string ID  24  output by the neighborhood lookup table  18 . The string sequencer  30  then uses the pixel location  32  of the current target pixel, the surface string IDs  24 , and the surface string slopes  28  to generate string sequences  34  which define the color borders of the raster image. A string sequence  34  comprises a sequence of connected surface string IDs  24  in an order that follows the contour of the color border. An example of a string sequence  34  is shown in FIG. 2 as comprising the following sequence of surface string IDs:  44 → 34 → 44 → 39 . Note that from the above surface strings table and the slopes shown in FIG. 4B, the receive and send slopes of the surface strings in this sequence match. For example, the send slope of string [ 44 ] is  3 , and the receive slope of string [ 34 ] is  3 . Referring again to the above neighborhood lookup table, the surface strings in each entry are listed in a particular order such that when the string sequencer  30  processes the surface strings at a given pixel, the surface strings are automatically linked into to the appropriate string sequences without having to check that the send and receive slopes match. Ultimately, the string sequences  34  are processed by a vector generator  36  which places tags on one of the west or east borders which are then used to generate the vectors that make up the vector image  4 . 
     The operation of the string sequencer  30  is understood with reference to FIG. 5A which shows a black and white raster image comprising a simple black feature and the corresponding string sequences that are generated by the present invention. As described above, the neighborhood scanner  8  of FIG. 2 scans a 3×3 neighborhood of cells  10   a  through the image starting with the first pixel  12 . Thus, for the first pixel  12  no surface strings are generated by the neighborhood lookup table  18  because the color code for the neighborhood of cells  10   a  (YYY YYY YYY) corresponds to address  255  in the above neighborhood lookup table, which has no surface strings for that entry. In fact, a surface string is not generated until the neighborhood of cells reaches pixel  38  where the color code for the neighborhood of cells  10   b  is (YYY YYY YZZ) corresponding to address  252  in the above neighborhood lookup table which comprises only surface string [ 13 ] in that entry. That surface string [ 13 ] is the correct and only surface string for this neighborhood  10   b  can be verified by examining the colors of the neighborhood  10   b  in FIG. 5A with the colors of surface string [ 13 ] shown in FIG.  3 . 
     As described above, the surface string [ 13 ] indexes the surface strings table  26  of FIG. 2 which outputs the corresponding receive and send slopes of  10  and  6 . The string sequencer  30  processes the location of the pixel  32 , the surface string ID[ 13 ]  24 , and the receive and send slopes 28 in order to sequence the east white border  20 A in FIG.  5 A. As explained in greater detail below, there are specific priorities that establish where to begin and end a string sequence. In FIG. 5A, the east white border  20 A starts with pixel  40  based on these priorities, and sequences around the black feature resulting in a string sequence of: 
     
       
         17→56→56→56→13→ . . . →51. 
       
     
     The last surface string in the above sequence (surface string [51]) is a “corner” surface string which ends the current string sequence. A string sequence is generated for the black west border  22 A starting at pixel  42  and sequencing around the black feature resulting in a string sequence of: 
     
       
         18→54→54→54→12→ . . . →10. 
       
     
     The above string sequence  22 A ends at pixel  41  because surface string [10] is a “protruding corner” surface string which has priority over the “corner” surface string at pixel  43 . The surface string priorities are described in greater detail below. When the neighborhood  10   c  reaches target pixel  42 , it begins a string sequence for both the west black border  22 A and the east black border  22 B. Thus, the resulting string sequence for the east black border  22 B is: 
     
       
         18→33→56→56→ . . . →10. 
       
     
     Similarly, when the neighborhood of cells reaches target pixel  44 , the string sequencer  30  begins the string sequence for the west white border  20 B and generates the simple string sequence of 44→36. 
     How the string sequencer  30  of FIG. 2 determines which pixel to begin and end a string sequence is also a significant aspect of the present invention. Basically, the string sequencer  30  operates according to a set of priority rules which are shown in the following table: 
     
       
         
               
             
               
               
               
             
           
               
                   
               
               
                 Priority Rules 
               
             
          
           
               
                   
                 Surface String 
                 Priority 
               
               
                   
                   
               
               
                   
                 North over South 
                 Highest 
               
               
                   
                 Protruding Corner 
                 Second Highest 
               
               
                   
                 Corner 
                 Third Highest 
               
               
                   
                 Step 
                 Fourth Highest 
               
               
                   
                 West over East 
                 Lowest 
               
               
                   
                   
               
             
          
         
       
     
     What the above priority rules mean is that a string sequence will always begin with the surface string representing the most northern pixel and sequence from north to south. If two surface strings represent pixels at the same latitude, then the surface string&#39;s type determines the priority. In the above table, “protruding corner” surface strings, which in FIG. 3 are surface strings [ 9 ], [ 10 ], [ 14 ], [ 19 ], [ 28 ], and [ 41 ], have the next highest priority. Next in priority are “corner” surface strings which in FIG. 3 are surface strings [ 15 ], [ 18 ], [ 33 ], and [ 38 ], followed by “step” corner strings which in FIG. 3 are surface strings [ 13 ], [ 17 ], [ 27 ], and [ 29 ]. The special case surface strings [ 49 ], [ 50 ], [ 51 ] and [ 55 ] discussed above are also all corner surface strings. Finally, the lowest priority is to start string sequences with surface strings representing western pixels over eastern pixels. In other words, if two surface strings of the same type represent pixels at the same latitude, then the western surface string has priority over the eastern surface string and the sequencing will progress from west to east. 
     These priority rules can be understood with reference to FIG.  5 A. In evaluating the surface strings of the east white border  20 A the string sequencer  30  of FIG. 2 will begin the string sequence with surface string [ 17 ] representing pixel  40  because it is the most northern surface string, and because it is a corner surface string which has priority over the step surface string [ 13 ] representing pixel  38 . Surface string [ 17 ] at pixel  40  is considered a corner surface string because it is adjacent to surface string [ 44 ] and the combination of [ 17 ] and [ 44 ] results in the special case surface string [ 50 ] according to the above table (again, the special case surface strings are all corner surface strings). 
     Thus, when the string sequencer  30  encounters pixel  38 , it begins to build a string sequence starting with the step surface string [ 13 ] and sequencing from west to east. However, when the string sequencer  30  encounters pixel  40 , the string sequence is reversed so that it begins with surface string [ 17 ] and sequences from east to west. The string sequence for the east white border  20 A continues sequencing around the black feature as shown in FIG. 5A until it reaches surface string [ 30 ]. Surface string [ 30 ] ends the string sequence because when combined with surface string [ 36 ] of the west white border  20 B it forms the special case surface string [ 51 ] which is a corner surface string. Similarly, the string sequence for the west black border  22 A begins with surface string [ 18 ] at pixel  42  because it is a corner surface string which has priority over step surface string [ 12 ]. The string sequence  22 A ends at pixel  41  because surface string [ 10 ] is a protruding corner surface string which has priority over the corner surface string [ 33 ] at pixel  43  (string sequences always end with the highest priority surface string, where east ending surface strings have priority over west ending surface strings). 
     In the preferred embodiment of the present invention as implemented in the computer source code attached hereto as Appendix A, the string sequencer  30  generates string sequences for both the east and west borders for each feature in the image. However, it has been determined that the vectors can be generated by placing tags only on one of the east or west borders. Both the west and east string sequences are used to place the tags, but the tags are placed only on one of the west or east borders. The 32 surface strings identified by an * in FIG. 3 are surface strings that can be part of a string sequence representing a west border. Notice that some surface strings, such as the corner surface strings [ 15 ] and [18], can be part of a string sequence representing either a west or an east border. 
     A better understanding of how the present invention uses the above priority rules to sequence strings is understood with reference to FIG. 5B which shows a raster image having more complex features and more colors than that of FIG.  5 A. In FIG. 5B, only the string sequences representing the west borders are shown; the string sequences for the east borders in FIG. 5B are generated, but for clarity they are not shown. FIG. 5B also illustrates the situation where the neighborhood lookup table  18  of FIG. 2 will output multiple surface strings for a single neighborhood. 
     Consider, for example, the neighborhood  46  near the center of the image which has the following color pattern:                           
     Referring to the above neighborhood lookup table, this color pattern corresponds to entry: 
     
       
         
               
               
               
               
               
             
           
               
                   
               
               
                 SURFACE 
                   
                   
                   
                   
               
               
                 STRINGS 
                 ADR 
                 P 0 P 1 P 2   
                 P 3  T P 4   
                 P 5 P 6 P 7   
               
               
                   
               
             
             
               
                 05, 02, 00, 00, 00, 00; 
                 7 
                 Z Z Z 
                 Z Y Z 
                 Y Y Y 
               
               
                   
               
             
          
         
       
     
     Thus, this entry outputs two surface string IDs [ 05 ] and [ 02 ]. The surface string [ 05 ] begins the string sequence 48 for the west border of the pyramid, and the surface string [ 02 ] begins the string sequence (not shown) for the east border of the pyramid. 
     The string sequence  50  begins with surface string [ 46 ] at pixel  52  because it is the most northern pixel, and sequences to surface string [ 38 ] which is a corner surface string and thus ends the sequence. Similarly, string sequence  54  begins with surface string [ 3 ] at pixel  56  because it is the most northern pixel, and sequences to surface string [ 36 ] which, when combined with surface string [ 30 ], forms special surface string [ 51 ] which is a corner surface string and thus ends the sequence. 
     String sequence 58 of FIG. 5B begins with surface string [ 13 ] at pixel  63  and sequences west to east as shown. The reason string sequence  58  begins with step surface string [ 13 ] at pixel  63  is because it has priority over the step surface string [ 17 ] at pixel  60  (pixel  63  is more western). The string sequence  64  starting at pixel  66  begins with surface string [ 41 ] because it is a protruding corner surface string which has priority over the corner surface string [ 15 ] at pixel  68 . Thus, string sequence  64  sequences from east to west as shown and ends at protruding corner surface string [ 10 ] at pixel  70 . 
     String sequence  72  begins with corner surface string [ 15 ] at pixel  76  because it has priority over corner surface string [ 18 ] at pixel  74  (i.e., it is the more western corner surface string). Similarly, string sequence  78  starts with surface string [ 13 ] at pixel  65  because together with surface string [ 25 ] at pixel  69  it forms a special case corner surface string [ 55 ] which has priority over step surface string [ 17 ] at pixel  67 . It should be apparent how the remainder of the string sequences in FIG. 5B are generated using the priorities rules listed in the table above. 
     Contrast Tie Breaker 
     A necessary step in vectorizing a color image is to identify and resolve contrast ties in the image features. A contrast tie occurs when two features overlap such that one feature should appear in the background and the other in the foreground. This phenomenon is illustrated in FIG. 6A which shows a bit map of a raster image with two overlapping diagonal lines. The two possible representations of the diagonal lines in vector format are shown in FIGS. 6B and 6C; however, a human&#39;s perception of FIG. 6A will favor the representation shown in FIG. 6C over FIG.  6 B. If the contrast ties are not resolved “correctly”, the image will appear distorted when reconstructed from the vector data. 
     The contrast breaker  3  of FIG. 2 comprises two components: a contrast tie detector (CTD) for detecting the occurrence of a contrast tie in the raster data, and a contrast tie breaker (CTB) for breaking the contrast ties in a manner that parallels human perception. The general idea is to identify the presence of a contrast tie, and then to resolve or break the tie by changing the color value of a perimeter pixel relative to the target pixel in order to interrupt the string sequence of a recessive image feature, thereby allowing a dominant image feature to appear in the foreground. 
     In order to reduce the number of comparison permutations, both the CTD and the CTB operate on quadrant subsections of a 3×3 array of pixels as shown in FIG.  7 A. In the preferred embodiment, the contrast tie breaker  3  operates concurrently with the neighborhood scanner  8  described above. For each 3×3 array of pixels processed, the CTD performs a set of comparisons on the pixel data in the four quadrant subsections shown in FIG. 7A. A contrast tie is detected when the color of the A pixel equals the color of the D pixel (A=D), the color of the B pixel equals the color of the C pixel (B=C), and the color of the A pixel does not equal the color of the B pixel (A!=B). The raster bit map shown in FIG. 6A demonstrates when a contrast tie is detected in each of the four quadrants of the 3×3 array of pixels, as is further illustrated in FIGS. 8A-8D. 
     In order to break a contrast tie, the CTB performs several progressive tests over the pixel data. These tests are performed over additional pixels surrounding the 3×3 array of pixels as illustrated in FIG.  7 B. The first test, referred to as the contrast test, performs the following comparisons on the colors of the pixels 
     contrastl=(A!=E)+(A!=F)+(D!=K)+(D!=L), 
     contrast 2 =(B!=G)+(B!=H)+(C!=I)+(C!=J), 
     A_loses=contrast 1 &gt;contrast 2 . 
     If A_loses is true, then the color identifier for the D pixel is set to Z. That is, the color identifier of the D pixel is set different from the target pixel (the A pixel) so that the string sequence will terminate rather than sequence through the A and D pixels. This provides the desired effect of placing a recessive image feature in the background by breaking the feature into two string sequences as illustrated by the diagonal line sequencing from left to right in FIG.  6 C. If A_loses is false, then the color identifier of the D pixel is left unchanged. This provides the desired effect of placing the dominate image feature in the foreground by allowing the string sequence to sequence through the A and D pixels as illustrated by the diagonal line sequencing from right to left in FIG.  6 C. If A_loses is neither true nor false (i.e., if contrast 1 =contrast 2 ), then the contrast test is indeterminate and the CTB performs a further test, referred to as the “brightness” test, to break the contrast tie. 
     The brightness test compares the brightness of pixel A to that of pixel B. In the preferred embodiment, each pixel comprises a red value, a green value and a blue value (RGB), and a pixel&#39;s brightness is generated by summing the RGB values. With the function get_whiteness (x) returning the sum of RGB for pixel x, the brightness test performs the following comparison 
     contrast 1 =get_whiteness(A), 
     contrast 2 =get_whiteness(B), 
     A_loses=contrast 1 &lt;contrast 2 . If A_loses is true, then the color identifier for the D pixel is set to Z, and if A_loses is false, then the color identifier for the D pixel is not modified. If the brightness of the A and D pixels are equal (contrast 1 =contrast 2 ), then the CTB performs yet another test, referred to as the “boldness” test, to break the contrast tie. 
     The boldness test compares the boldness of pixel A to that of pixel B, where a pixel&#39;s boldness is defined as its largest RGB value. With the function get_boldness(x) returning the largest RGB value for pixel x, the boldness test performs the following comparison 
     contrast 1 =get_boldness(A), 
     contrast 2 =get_boldness(B), 
     A_loses=contrast 1 &lt;contrast 2 . 
     If A_loses is true, then the color identifier for the D pixel is set to Z, and if A_loses is false, then the color identifier for the D pixel is not modified. If the boldness of the A and D pixels are equal (contrast 1 ==contrast 2 ), then the CTB performs a second boldness test by comparing the second largest RGB values of the A and B pixels. 
     With the function get — 2nd_boldness(x) returning the second largest RGB value for pixel x, the second boldness test performs the following comparisons 
     contrast 1 =get — 2nd_boldness (A), 
     contrast 2 =get — 2nd_boldness (B), 
     A_loses=contrast 1 &lt;contrast 2 . 
     If A_loses is true, then the color identifier for the D pixel is set to Z, and if A_loses is false, then the color identifier for the D pixel is not modified. If the second boldness of the A and D pixels is equal (contrast 1 ==contrast 2 ), then the contrast tie is not broken and the color identifier for the D pixel is left unmodified. 
     The above CTB tests are further understood with reference to FIGS. 8A-8D which correspond to the raster bit map of FIG.  6 A. Referring to FIG. 8A, the CTD detects a contrast tie in quadrant one because the color of pixels A and D are equal, the color of pixels B and C are equal, and the color of pixel A is different from the color of pixel B. Turning to the tests performed by the CTB, the contrast test is indeterminate because contrast 1 ==contrast 2 . The brightness test will indicate that pixel A is brighter than pixel B, therefore the color identifier for pixel D is not modified (it remains Y) so that the string sequence will sequence through the A and D pixels. Referring to FIG. 8B, the CTD will again detect a contrast tie in quadrant two and again the first test performed by the CTB, the contrast test, will be indeterminate because contrast 1 ==contrast 2 . However, in this case the brightness test will indicate that pixel A is not brighter than pixel B so the color identifier for pixel D will be set to Z (i.e., different from pixel A). Consequently, the string sequence will terminate at pixel D and begin again at pixel A so that the diagonal line sequencing from left to right in FIG. 6C will appear in the background. 
     Preferred Embodiments 
     Referring now to FIG. 9, shown is an embodiment of the present invention wherein the raster-to-vector (R/V) converter is implemented as a computer program  82  executed by a data processor  80  running under a particular operating system (O/S)  84 . A non-volatile memory, such as a disc drive  86 , stores the operating system  84  and the R/V converter  82  when the computer system is powered down. When the computer system is powered up, the O/S  84  and R/V converter  82  are loaded into a system memory (RAM)  88 , and the data processor  80  reads and executes the O/S  84  and R/V converter  82  from the system memory  88 . The raster image to be converted into a vector image may be stored on the disc drive  86 , or it may alternatively be loaded in real time from some other external device such as a scanner. An end user may interface with the computer system through a keyboard  90  and mouse  92  in order to direct which raster images are to be converted, and to display the vector images on a video display  94 . The source code attached hereto as Appendix A illustrates the composition of the computer program embodiment of the present invention. The source code in Appendix A is written in a programming language referred to as C, but other programming languages, such as Assembly, Pascal or Basic, could easily be employed to implement the various aspects of the present invention. 
     An alternative embodiment of the present invention is shown in FIG.  10 . In this embodiment, the raster-to-vector converter is implemented as part of an image processing integrated circuit. Some of the components shown in FIG. 2 are shown in FIG. 10 as combinatorial logic circuits and lookup tables implemented in transistor circuitry. It is within the ability of those skilled in the art to convert the source code shown in Appendix A into equivalent logic circuits for incorporation into an image processing integrated circuit. 
     The objects of the invention have been fully realized through the embodiments disclosed herein. Those skilled in the art will appreciate that the aspects of the invention can be achieved through various other embodiments without departing from the essential function. The particular embodiments disclosed are illustrative and not meant to limit the scope of the invention as appropriately construed by the following claims.