Abstract:
A fast decoding technique for decoding a position of a bit in a pattern provided on a media surface that can generate large amounts of solution candidates quickly by switching or flipping bits and utilizing a recursion scheme. The fast decoding technique may be employed to simultaneously decode multiple dimensions of a pattern on the media surface.

Description:
FIELD OF THE INVENTION 
   The present invention relates to identifying sections of a linear code. Various aspects of the present invention are particularly applicable to identifying the location of marks on a document that make up sections of a linear code printed on the document. 
   BACKGROUND OF THE INVENTION 
   While electronic documents stored on computers provide a number of advantages over written documents, many users continue to perform some tasks with printed versions of electronic documents. These tasks include, for example, reading and annotating the documents. With annotations, the paper version of the document assumes particular significance, because the annotations typically are written directly onto the printed document. One of the problems, however, with directly annotating a printed version of a document is the difficulty in later converting the annotations into electronic form. Ideally, electronically stored annotations should correspond with the electronic version of the document in the same way that the handwritten annotations correspond with the printed version of the document. 
   Storing handwritten annotations in electronic form typically requires a user to review each handwritten annotation and personally enter it into a computer. In some cases, a user may scan the annotations written on a printed document, but this technique creates a new electronic document. The user must then reconcile the original version of the electronic document with the version having the scanned annotations. Further, scanned images frequently cannot be edited. Thus, there may be no way to separate the annotations from the underlying text of the original document. This makes using the annotations difficult. 
   To address this problem, pens have been developed to capture annotations written onto printed documents. In addition to a marking instrument, this type of pen includes a camera. The camera captures images of the printed document as a user writes annotations with the marking instrument. In order to associate the images with the original electronic document, however, the position of the images relative to the document must be determined. Accordingly, this type of pen often is employed with specialized media having a pattern printed on the writing surface. The pattern represents a code that is generated such that, the different section of the pattern that occur around a location on the media will uniquely identify that location. By analyzing or “decoding” this pattern, a computer receiving an image from the camera can thus determine what portions of the code (and thus what portion of a document printed on the paper) were captured in the image. 
   While the use of such patterned paper or other media allows written annotations on a paper document to be converted into electronic form and properly associated with the electronic version of the document, this technique presents its own difficulties. For example, the printed document itself may obscure areas of the pattern printed on the writing surface of the media. If the pen captures an image of one of these areas, then the computer may not be able to use the pattern to accurately determine the location of the document portion captured by the image. Also, the computer may not accurately recognize the code from the image. For example, if the code is binary, then the computer may erroneously recognize a portion of the pattern representing a “0” value as a “1” value, or vice versa. 
   Further, in some situations the unique positioning properties of the code cannot be utilized because the code values detected from the pattern are not consecutive, or the values do not have sufficient bits to uniquely identify a section of the code. Moreover, in order to stay in synchronism with the movement of the pen, the pattern captured in an image must be decoded within fixed time period. For example, if the pen captures about 100 images per second, the decoding time for each frame cannot exceed 10 ms. The possible solution candidates must therefore be generated and analyzed at a fast rate. 
   BRIEF SUMMARY OF THE INVENTION 
   Advantageously, various implementations of the invention provide a fast decoding technique that can generate large amount of solution candidates quickly by switching bits and utilizing a recursion scheme. Some implementations may further simplify the decoding technique so that the solution candidates are generated by bit reversal or “flipping.” Still further, various implementations may be employed to simultaneously decode several sets or “dimensions” of patterns printed on the same media surface. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  illustrates an example of a programmable computing device that may be employed to implement various examples of the invention. 
       FIGS. 2A and 2B  illustrate the configuration of a camera device that may be employed to capture images according to various examples of the invention. 
       FIG. 3  illustrates the arrangement of bits in an m-array that may be employed according to various examples of the invention. 
       FIGS. 4A-4E  and  6  illustrate various code symbols for forming that may be used to form m-array patterns on a media surface according to various examples of the invention. 
       FIGS. 4F-4I  illustrate various examples of “corner” combinations of pixels. 
       FIG. 5  illustrates one example of a multi-dimensional m-array. 
       FIG. 7  illustrates an example of a decoding tool that may be implemented according to various examples of the invention. 
       FIGS. 8-14  illustrate the creation of various arrays that may be employed during pattern decoding according to various examples of the invention. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Operating Environment 
   While some examples of the invention may be implemented using analog circuits, many examples of the invention may conveniently be implemented using a programmable computing device executing software instructions to perform various functions.  FIG. 1  shows a functional block diagram of an example of a conventional general-purpose digital computing environment that may therefore be used to implement various aspects of the present invention. In  FIG. 1 , a computer  100  includes a processing unit  110 , a system memory  120 , and a system bus  130  that couples various system components including the system memory to the processing unit  110 . The system bus  130  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The system memory  120  includes read only memory (ROM)  140  and random access memory (RAM)  150 . 
   A basic input/output system  160  (BIOS), containing the basic routines that help to transfer information between elements within the computer  100 , such as during start-up, is stored in the ROM  140 . The computer  100  also includes a hard disk drive  170  for reading from and writing to a hard disk (not shown), a magnetic disk drive  180  for reading from or writing to a removable magnetic disk  190 , and an optical disk drive  191  for reading from or writing to a removable optical disk  192  such as a CD ROM or other optical media. The hard disk drive  170 , magnetic disk drive  180 , and optical disk drive  191  are connected to the system bus  130  by a hard disk drive interface  192 , a magnetic disk drive interface  193 , and an optical disk drive interface  194 , respectively. The drives and their associated computer-readable media provide nonvolatile storage of computer readable instructions, data structures, program modules and other data for the personal computer  100 . It will be appreciated by those skilled in the art that other types of computer readable media that can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, random access memories (RAMs), read only memories (ROMs), and the like, may also be used in the example operating environment. 
   A number of program modules can be stored on the hard disk drive  170 , magnetic disk  190 , optical disk  192 , ROM  140  or RAM  150 , including an operating system  195 , one or more application programs  196 , other program modules  197 , and program data  198 . A user can enter commands and information into the computer  100  through input devices such as a keyboard  101  and pointing device  102 . Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner or the like. These and other input devices are often connected to the processing unit  110  through a serial port interface  106  that is coupled to the system bus, but may be connected by other interfaces, such as a parallel port, game port or a universal serial bus (USB). Further still, these devices may be coupled directly to the system bus  130  via an appropriate interface (not shown). A monitor  107  or other type of display device is also connected to the system bus  130  via an interface, such as a video adapter  108 . In addition to the monitor, personal computers typically include other peripheral output devices (not shown), such as speakers and printers. In a preferred embodiment, a pen digitizer  165  and accompanying pen or stylus  166  are provided in order to digitally capture freehand input. Although a direct connection between the pen digitizer  165  and the serial port is shown, in practice, the pen digitizer  165  may be coupled to the processing unit  110  directly, via a parallel port or other interface and the system bus  130  as known in the art. Furthermore, although the digitizer  165  is shown apart from the monitor  107 , it is preferred that the usable input area of the digitizer  165  be co-extensive with the display area of the monitor  107 . Further still, the digitizer  165  may be integrated in the monitor  107 , or may exist as a separate device overlaying or otherwise appended to the monitor  107 . 
   The computer  100  can operate in a networked environment using logical connections to one or more remote computers, such as a remote computer  109 . The remote computer  109  can be a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer  100 , although only a memory storage device  111  has been illustrated in  FIG. 1 . The logical connections depicted in  FIG. 1  include a local area network (LAN)  112  and a wide area network (WAN)  113 . Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
   When used in a LAN networking environment, the computer  100  is connected to the local network  112  through a network interface or adapter  114 . When used in a WAN networking environment, the personal computer  100  typically includes a modem  115  or other means for establishing a communications over the wide area network  113 , such as the Internet. The modem  115 , which may be internal or external, is connected to the system bus  130  via the serial port interface  106 . In a networked environment, program modules depicted relative to the personal computer  100 , or portions thereof, may be stored in the remote memory storage device. 
   It will be appreciated that the network connections shown are illustrative and other techniques for establishing a communications link between the computers can be used. The existence of any of various well-known protocols such as TCP/IP, Ethernet, FTP, HTTP, Bluetooth, IEEE 802.11x and the like is presumed, and the system can be operated in a client-server configuration to permit a user to retrieve web pages from a web-based server. Any of various conventional web browsers can be used to display and manipulate data on web pages. 
   Image Capturing Device 
   Various implementations of the invention may be employed to determine the location of an image taken of a portion of a surface area displaying a non-repeating pattern. As noted above, the surface may be the writing surface of a document printed on paper. It should be noted, however, that surface may also be the surface of a document rendered on a display screen or other medium displaying a document. Thus, according to some examples of the invention, the images may be obtained by an ink pen used to write ink on paper. With other examples of the invention, the pen may be a stylus used to “write” electronic ink on the surface of a digitizer displaying the document. Still further, the surface may be the surface of any media, regardless of whether a document is displayed thereon. 
     FIGS. 2A and 2B  show an illustrative example of a pen  201  that may be employed according to various examples of the invention used with paper media or the like. The pen  201  includes a tip  202  and a camera  203 . The tip  202  that may or may not include an ink reservoir. The camera  203  captures an image  204  from surface  207 . The pen  201  may further include additional sensors and/or processors as represented in broken box  206 . These sensors and/or processors  206  may also include the ability to transmit information to another pen  201  and/or to a personal computer (for example, via a wired connection or via Bluetooth transmissions, infrared transmission, Wi-Fi transmission or other wireless protocol transmissions). 
     FIG. 2B  represents an image as viewed by the camera  203 . In one illustrative example, the resolution of an image captured by the camera  203  is N×N pixels (where, for example, N=32). Accordingly,  FIG. 2B  shows an example image 32 pixels long by 32 pixels wide. It should be appreciated that the size of N may vary with different implementations of the invention. A higher value of N will provide a higher image resolution. Also, while the image captured by the camera  203  is shown as a square for illustrative purposes, the field of view of the camera may be of any other desired shape as is known in the art. 
   The images captured by camera  203  may be defined as a sequence of image frames {I i }, where I i  is captured by the pen  201  at sampling time t i . The sampling rate may be large or small, depending on the system configuration and performance requirement. The size of the captured image frame also may be large or small, depending on the system configuration and performance requirement. Further, it should be appreciated that an image captured by camera  203  may be used directly by a processing system, discussed in more detail below, which decodes a portion of a pattern captured in the image. Alternately, an image captured by the camera  203  may undergo pre-filtering before it is analyzed by the processing system. This pre-filtering may occur in the pen  201  or it may occur outside of the pen  201  (for example, in a personal computer). 
     FIG. 2A  also shows the image plane  209  on which an image  210  of the pattern from location  204  is formed. Light received from the pattern on the object plane  207  is focused by lens  208 . According to various embodiments of the invention, the lens  208  may be a single lens or a multi-part lens system, but is represented in  FIG. 2A  as a single lens for simplicity. Image capturing sensor  211  captures the image  210 . The image sensor  211  may be large enough to capture the image  210 . Alternatively, the image sensor  211  may be large enough to capture an image of the pen tip  202  at location  212 . For reference, the image at location  212  will be referred to as the virtual pen tip. It should be noted that the virtual pen tip location is fixed with respect to image sensor  211 , because of the constant relationship between the pen tip, the lens  208 , and the image sensor  211 . 
   Generating and Displaying a Pattern for Identifying a Position on a Medium 
   As previously noted, various examples of the invention are employed to determine the portion of a document captured in a pen image. With these examples, the medium displaying the document also displays a location pattern for identifying different positions on the medium. Thus, the pattern may be considered to be an encoded data stream in a displayed form. The medium displaying the pattern may be printed paper (or other type of permanent or semi-permanent medium). Alternately, the medium may be a display rendering the encoded data stream together with the image or set of images making up the document. With some examples of the invention, the encoded data stream may even be represented as a permanent or semi-permanent pattern overlaying a display screen (so that the position of any image captured by a pen is locatable with respect to the display screen). 
   In order to be useful for identifying a location in a document, the pattern should be sufficiently non-repetitive so that each portion of the document will have a unique portion of the pattern. One technique for providing such as pattern is to create a binary sequence, referred to herein as an “m-sequence,” that can be arrayed over the area of the document without repeating. 
   An m-sequence may be generated by division of polynomials. More particularly, for every two polynomials Q(x) and P n (x) over the finite field F 2 , where P n (x) is a primitive polynomial of order n, and the order of Q(x) is less than n, the division Q(x)/P n (x) generates an m-sequence m of the order n. For example, supposing that P n (X)=1+x+x 4 , Q 1 (x)=1+x+x 2 , the division Q 1 (x)/P n (x) is shown below. For simplicity, only coefficients of the polynomials are shown. Here, P n (x) and Q 1 (x) are represented as (11001) and (11100) respectively, which are the coefficients of x 0 , x 1 , x 2 , x 3  and x 4  in the two polynomials. 
   
     
               
       
           
           
       
    
   
   The result is an m-sequence m 1 =101100100011110 . . . , with an order of 4 and a period of 15. It should be noted that the polynomials are over the finite field F 2 . This means that the addition and multiplication of the polynomial coefficients follow the addition and multiplication of the finite field F 2 , i.e. addition is XOR and multiplication is AND. 
   Next, the bits in an m-sequence can be regularly arranged over the writing surface of the document such that each bit in the m-sequence corresponds to a specific position in the document. One of the approaches for bit arrangement folds the m-sequence in the following manner, i.e., such that the bits of the m-sequence are arranged diagonally and continue from the opposite side whenever a boundary of the page area is met, so that the whole page is covered, as illustrated in  FIG. 3 . 
     FIG. 4A  shows one example of encoding techniques for encoding a bit with a value of “1” and a bit with a value of “0” into a pattern for identifying positions on a medium. A code symbol for a first bit  401  (for example, with a value of “1”) is represented by vertical column of dark ink or pixels. A code symbol for a second bit  402  (with, for example, a value of “0”) is represented by a horizontal row of dark ink or pixels. It should be appreciated, however, that any color ink or pixels may be used to represent various pattern values. It should be appreciated, however, that the color of the chosen ink should provide sufficient contrast with the background of the medium to be differentiable by an image capturing system. In this example, each of the bit values illustrated in  FIG. 4A  is represented by a 3×3 matrix of dots. The size of the matrix may be modified to be any desired size, however, based upon the size and resolution of the image capture system being used to capture images of the medium.  FIG. 4B  illustrates how a pattern  403  can be formed that represents the various bit values  404 - 411  making up a data stream. 
   Alternative representations of bits with 0 and 1 values are shown in  FIGS. 4C-4E . It should be appreciated that the representation of a one or a zero for the sample encodings of  FIGS. 4A-3E  may be switched without effect.  FIG. 4C  shows bit representations occupying two rows or columns in an interleaved arrangement.  FIG. 4D  shows an alternative arrangement of the pixels in rows and columns in a dashed form. Finally  FIG. 4E  show pixel representations in columns and rows in an irregular spacing format (e.g., two dark dots followed by a blank dot). 
   It should be noted that alternative grid alignments are also possible, including a rotation of the underlying grid to a non-horizontal and non-vertical arrangement (for example, where the correct orientation of the pattern is 45 degrees). Using a non-horizontal and vertical arrangement may, with some examples of the invention, help eliminate visual distractions for the user, as users may tend to notice horizontal and vertical patterns before other pattern orientations. For purposes of simplicity, however, the orientation of the grid (horizontal, vertical and any other desired rotation of the underlying grid) is referred to collectively as the predefined grid orientation. 
   Referring back to  FIG. 4A , if a bit is represented by a 4 by 4 matrix of elements and an imaging system detects a dark row and two white rows in a 4×3 region, then that region is determined to have a value of zero (or, with a reverse arrangement, a value of one). If a 4×3 region is detected with dark column and two white columns, then that region is determined to have a value of one (or, with a reverse arrangement, a value of zero). Accordingly, if the size of the image  210  in  FIG. 2B  is 32×32 pixels and each bit encoding unit size is 4×3 pixels, then the number of captured encoded units should be approximately 100 units. If the bit encoding unit size is 5×5, then the number of captured encoded units should be approximately 46. 
   As previously noted, the graphical pattern  403  of  FIG. 4B  represents a specific bit stream. Graphical pattern  403  includes 12 rows and 18 columns. More particularly, the rows and columns are formed by a bit stream being converted into the graphical pattern  403  using bit representations  401  and  402 . Thus, the pattern  403  of  FIG. 4B  may be viewed as having the following bit representation: 
   
     
       
         
             
           
             [ 
             
               
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
               
               
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
               
               
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
               
             
             ] 
           
         
       
     
   
   Various bit streams may be used to create a pattern like the pattern  403  shown in  FIG. 4B . As previously noted, a random or pseudo-random sequence of ones and zeros, such as an m-sequence, may be used. The bit sequence may be arranged in rows, in columns, diagonally, or following any other formulaic ordering. For example, the above matrix may be formed by the following bit stream if run left to right then down:
         0100 0111 0110 0100 1000 1110 0111 0100 1100.       

   Alternately, the above matrix may be formed by the following bit stream if run top to bottom then right:
         0101 1100 0011 0010 0110 1001 1001 1110 0010.       

   Still further, the above matrix may represent the following bit stream if run diagonally, and then wrapped:
         0110 0000 0101 0101 1000 0011 1111 1010 1010.       

     FIG. 4B  also includes enlargements of pixel blocks from image  403 . The enlargements  404 - 411  show 5×5 pixel blocks. Pixel block  404  shows a dark row between white rows. Pixel block  405  shows a dark column between white columns. Pixel block  406  shows a bottom left corner. Pixel block  407  shows a top right corner. The pixel block  408  shows a dark column with half a dark row on the left. Pixel block  409  shows a dark row with half a dark column above the row. The pixel block  410  shows half a dark row. Pixel block  411  shows half a dark column. Analyzing the combination of pixel blocks, it should be appreciated that all combinations of pixels may be formed by the image segments found in pixel blocks  404 - 411 . The type of pattern shown in  FIG. 4B  may be referred to as a “maze” pattern, as the line segments appear to form a maze with no area being completely enclosed on all four sides by the maze. 
   Upon initial consideration, it would be expected that each of the four “corner” combinations of pixels shown in  FIGS. 4F-4I  would be found in the maze pattern shown in the image  403 . However, as seen in  FIG. 4B , only three types of corners actually exist in the eight pixel blocks  404 - 411 . In this example, there is no corner combination of pixels as shown in  FIG. 4F . By choosing the image segments  401  and  402  to eliminate a type of corner in this manner, the orientation of a captured image based on the missing type of corner can be determined. 
     FIG. 4B  also includes enlargements of pixel blocks from image  403 . The enlargements  404 - 411  show 5×5 pixel blocks. Pixel block  404  shows a dark row between white rows. Pixel block  405  shows a dark column between white columns. Pixel block  406  shows a bottom left corner. Pixel block  407  shows a top right corner. The pixel block  408  shows a dark column with half a dark row on the left. Pixel block  409  shows a dark row with half a dark column above the row. The pixel block  410  shows half a dark row. Pixel block  411  shows half a dark column. Analyzing the combination of pixel blocks, it should be appreciated that all combinations of pixels may be formed by the image segments found in pixel blocks  404 - 411 . The type of pattern shown in  FIG. 4B  may be referred to as a “maze” pattern, as the line segments appear to form a maze with no area being completely enclosed on all four sides by the maze. 
   Upon initial consideration, it would be expected that each of the four “corner” combinations of pixels shown in  FIGS. 4F-41  would be found in the maze pattern shown in the image  403 . However, as seen in  FIG. 4B , only three types of corners actually exist in the eight pixel blocks  404 - 411 . In this example, there is no corner combination of pixels as shown in  FIG. 4F . By choosing the image segments  401  and  402  to eliminate a type of corner in this manner, the orientation of a captured image based on the missing type of corner can be determined. 
   Multidimensional Arrays 
     FIGS. 3-41  relate to one-dimensional arrays, where each bit corresponds to a single position in the array. Various examples of the invention, however, may employ multi-dimensional arrays. With multi-dimensional arrays, each position in the array includes a group of bits. For example, in the multi-dimensional array  501  shown in  FIG. 5 , each of the bit elements in the bit group  501  will have a different array value. More particularly, the first bit in the group, with the value “0,” will have the array value (7, 4, 1) (represented E 7,4,1 ). The second bit in the group, also with the value “0,” will have the array value (6, 4, 1) (represented E 6,4,1 ). The last bit in the group, with the value “1,” will have the array value (0,4,1) (represented E 0,4,1 ). 
     FIG. 6  then illustrates one example of a code symbol  601  that can be used to represent a multidimensional value in an array forming a location pattern. As seen in this figure, the code symbol has four black dark dots  603  that represent the border of the symbol  605 . It also includes data dots  607  that can be either marked black or left white (or blank) to represent data bits. Still further, the illustrated code symbol includes orientation dots  607  that are always left white (or blank) to allow the decoding process to determine an orientation of the symbol. 
   As discussed herein, a code symbol is the smallest unit of visual representation of a location pattern. Generally, a code symbol will include the pattern data represented by the symbol. As shown in the illustrated example, one or more bits may be encoded in one code symbol. Thus, for a code symbol with 1 bit represented, the represented data may be “0” or “1”, for a code symbol representing 2 bits, the represented data may be “00”, “01”, “10” or “11.” Thus, a code symbol can represent any desired amount of data for the location pattern. The code symbol also will have a physical size. When the location pattern is, for example, printed on paper, the size of a code symbol can be measured by printed dots. For example, the illustrated code symbol is 16×16 printed dots. With a 600 dpi printer, the diameter of a printed dot will be about 0.04233 mm. 
   Still further, a code symbol will have a visual representation. For example, if a code symbol represents 2 bits, the visual representation refers to the number and position distribution of the black dots used to represent the data values “00”, “01”, “10” or “11”. Thus, the code symbol illustrated in  FIG. 3C  may be referred to as a “8-a-16” symbol, since it represents 8 data bits using a 16×16 array of discrete areas. Of course, symbols having a variety of different represented pattern data values, sizes, and visual representation configurations will be apparent to those of ordinary skill in the art upon consideration of this description. 
   The bit values for the additional dimensions in a multidimensional array may conveniently be generated by cyclically shifting an original m-sequence to create a multidimensional m-array. More particularly, multiplying Q(x)/P n (x) by x k  will result in an m-sequence that is the k-th cyclical shift of m. That is, letting Q′(x)=x k Q(x), if the order of Q′(x) is still less than n, then the m-sequence m′ generated by Q′(x)/P n (x) is the k-th cyclic shift of m, i.e. m 2 =σ k (m). Here σ k (m) means cyclically-shifting m to the right by k times. For example, referring to the generation of the m-sequence described in detail above, if Q 2 (x)=x+x 2 +x 3 =xQ 1 (x), the division Q 2 (x)/P n (x) will generate an m-sequence m 2 =010110010001111, which is the first cyclical shift of m, i.e. m 2 =σ 1 (m 1 ). 
   Accordingly, cyclically shifted m-sequences may be formed into a multidimensional m-array. That is, the first bit in each group of bits may belong to a first m-sequence. The second bit in each group may then belong to a second m-sequence that is cyclically shifted by a value k 1  from the first m-sequence. The third bit in each group may then belong to a third m-sequence that is cyclically shifted by a value k 2  from the first m-sequence, and so on to form a multidimensional m-array. 
   As shown in  FIG. 6 , the physical positions of the bits of different m-sequences of a multidimensional m-array on the page are slightly different. Among the m-arrays, one dimension of the m-array (i.e., one cyclic shift of an m-sequence) is used for determining the pen position. The remaining m-arrays can then advantageously be used to embed some information, called metadata. For example, a document may have an identification number d. The second m-sequence used in a multidimensional m-array may then be a cyclic shift from the first m-array used for position determination, with the number of shifts being exactly d. Thus, when the values of the first and second m-sequences in the multidimensional m-array are decoded, the shift difference between can be determined to obtain the identification number d of the document. Of course, as will be appreciated by those of ordinary skill in the art, any desired information can be embedded as metadata in a multidimensional m-array as described above. 
   Decoding an M-Array 
   In order to determine the position of an image relative to a document using an m-array, it is necessary to determine the position of a bit captured in the bit relative to the m-array. That is, it is necessary to determine if the bit is the first bit, second bit, etc. in the m-sequence to determine the position of the bit in the m-array. 
   For any number s, where 0≦s&lt;2 n −1, there exists a unique polynomial r(x), where 
             r   ⁡     (   x   )       =       ∑     i   =   0       n   -   1       ⁢       r   i     ⁢     x   i               
whose order is less than n, such that x s ≡r(x)(mod P n (x)), and vice versa. In other words, there is a one-to-one relationship between s and r(x). Thus, x s /P n (x) and r(x)/P n (x) will generate the same m-sequence. For convenience, setting Q(x)=1, m can be assumed to be the m-sequence generated by 1/P n (x). If a bit is the s′-th bit of m, where 0≦s′&lt;2 n −1, the m-sequence that starts from that bit is R=σ −s′ (m)=σ 2     n     −1−s′ (m)=σ s (m), where s=2 n −1−s′. R corresponds to division x s /P n (x).
 
   As previously noted, there exists 
               r   ⁡     (   x   )       =       ∑     i   =   0       n   -   1       ⁢       r   i     ⁢     x   i           ,         
that satisfies r(x)=x s (mod P n (x)). R also corresponds to division r(x)/P n (x). Letting m=(m 0  m 1  . . . m i  . . . m 2     n     −3  m 2     n     −2 ) t  (where the superscript t stands for vector or matrix transpose), and σ i (m t )=(m 2     n     −1−i  m 2     n     −3  . . . m 0  . . . m 2     n     −3−i  m 2     n     −2−i ), r(x)/P n (x) and 1/P n (x) will have the following relationship:
 
   
     
       
         
           
             
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   With R corresponding to the division r(x)/P n (x), and σ i (m) corresponding to x i ·1/P n (x), then,
 
R t =r t {circumflex over (M)}
 
   where R is the m-sequence that starts from the s′-th bit of m, r=(r 0  r 1  r 2  . . . r n−1 ) t  are the coefficients of r(x), and 
   
     
       
         
           
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   Again, the addition and multiplication operations are binary operations, i.e. addition is XOR and multiplication is AND. 
   If an image captures K bits b=(b 0  b 1  b 2  . . . b K−1 ) t  of m (K≧n), and the relative distances between the positions of the bits in the m-sequence are: s i =d(b i ,b 0 ), where i=0, 1, . . . , K−1 and s 0 =0, selecting the s i +1-th bits of R and the s i +1-th columns of {circumflex over (M)} will result in:
 
b t =r t M
 
   where b t  is the transpose of b, M is a sub-matrix of {circumflex over (M)} and consists of the s i +1-th columns of {circumflex over (M)}, where i=0, 1, 2, . . . , K−1. 
   If M is a non-degenerate matrix and b does not contain error bits, then r can be solved by selecting n bits from b by solving for:
 
r t ={tilde over (b)} t {tilde over (M)} −1  
 
   where {tilde over (M)} is any non-degenerate n×n sub-matrix of M, and {tilde over (b)} is the corresponding sub-vector of b consisting of the selected n bits. 
   Stochastic Decoding of an M-Array 
   In most cases, however, an image cannot capture a set of bits b that do not contain error bits. For example, improper illumination, document content, dust and creases can all obscure the visual representation of bits in an image, preventing these bits from being recognized or causing the value of these bits to be improperly recognized. The solution of r becomes difficult when there are error bits in b. Further, decoding becomes even more difficult because the coefficient matrix M is not fixed when the pen moves, changing the image from frame to frame. Moreover, the structure of M is irregular. Therefore, traditional decoding algorithms cannot effectively be applied to solve r under practical circumstances. 
   To address these difficulties, various embodiments of invention provide stochastic solution techniques that provide a high decoding accuracy under practical conditions. As will be described in more detail, these techniques solve the equation b t =r t M incrementally so that many solution candidates are readily available without having to solve this equation exactly. 
   According to various examples of the invention, independent n bits (i.e., the sub-matrix consisting of the corresponding columns of M is non-degenerate) are randomly selected from the group of b that are captured in an image of a document. Supposing that b (0)  are the n bits chosen, a solution for r can then be obtained as:
 
[r (0) ] t =[b (0) ] t [M (0) ] −1  
         where M (0)  contains the corresponding columns of the array M for the chosen bits.       

   For simplicity, the n bits chosen from b to make up b (0)  can be moved to the beginning of b, with the remaining bits making up b moved to the end of b. This leads to the relationship
 
([ b   (0) ] t   ,[  b     (0) ] t )=[ r   (0) ] t ( M   (0)   ,  M     (0) )+(0 n   t   ,[e   (0) ] t )
         where b (0)  are the chosen n bits,  b   (0)  are the remaining bits from the set b, M (0)  is the corresponding columns of M for the chosen bits,  M   (0)  is the corresponding columns of M for the remaining bits, 0 n   t =(0 0 . . . 0) 1×n ,[r (0) ] t =[b (0) ] t [M (0) ] −1 , and [e (0) ] t =[  b   (0) ] t +[r (0) ] t   M   (0) .       

   The value (0 n   t ,[e (0) ] t ) refers to the “difference vector” between ([b (0) ] t ,[  b   (0) ] t ) and [r (0) ] t (M (0) ,  M   (0) ), or simply the different vector of r (0) , and the number of 1&#39;s in (0 n   t ,[e (0) ] t ) is called the number of different bits. The vector containing different bits between ([b (0) ] t ,[  b   (0) ] t ) and [r (0) ] t (M (0) ,  M   (0) ) alternately can be identified as D (0) . If D (0) =(0 n   t ,[e (0) ] t ), then the number d (0)  of 1&#39;s in D (0)  is d (0) =HammingWeight(D (0) )=HammingWeight(e (0) ). That is, d (0)  is the number of different bits between ([b (0) ] t ,[  b   (0) ] t ) and [r (0) ] t (M (0) ,  M   (0) ). 
   Next, some of the chosen bits n from the set b are switched with some of the remaining bits from the set b. In particular, J bit pairs (k j ,l j ) are switched between the original chosen bits n and the remaining bits from the set of bits b, where k 1 ≠k 2 ≠ . . . ≠k J ≦n, n&lt;l 1 ≠l 2 ≠ . . . ≠l J ≦K. It should be noted that the bit order is redefined in ([b (0) ] t ,[  b   (0) ] t ), and these bits are not maintained in their original order. The relationship between the bits before and after switching is: 
   
     
       
         
           
             
               
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   If the choice of (k j , l j ) is to make:
 
[ e   (0) ] t   E   l−n   [P   R     J     (0) ] −1 =1 J   t ,
 
   where 1 J   t =(1 1 . . . 1) 1×J , then
 
[ e   (1) ] t   =[e   (0) ] t +1 J   t ( E   k   t   P   (0)   +E   l−n   t )
 
[ r   (1) ] t   =[r   (0) ] t +1 J   t   E   k   t   [M   (0) ] −1 .
 
   In view of [e (0) ] t E l−n [P R     J     (0) ] −1 =1 J   t  given k 1 ≠k 2 ≠ . . . ≠k J ≦n, the choice of n&lt;l 1 ≠l 2 ≠ . . . ≠l J ≦K is as follows: {l 1 , l 2 , . . . l J }⊂{p 1 , . . . , p m }, where {p 1 , . . . , p m } are the indices of the 0-bits of [e (0) ] t +1 J   t E k   t P (0) , and P R     J     (0)  is invertible. Therefore, if the rank of E k   t P (0) E p−n  is less than J, then such l 1 , l 2 , . . . , l J  cannot be chosen, where E p−n =(e p     1     −n  e p     2     −n  . . . e p     m     −n ) (K−n)×m . Choosing other l 1 , l 2 , . . . , l J  is equivalent to switching a smaller number of bit pairs, and therefore does not conform to the goal of switching J bits. It should be noted that, as long as the rank of E k   t P (0) E p−n  is J, the choice of l 1 , l 2 , . . . , l J  will result in the identical location vector. Therefore, choosing one combination is sufficient. Moreover, as long as P R     J     (0)  is invertible, the newly selected n bits are also independent. 
   With the above choice of l 1 , l 2 , . . . , l J , the number of different bits in e (i+1)  is:
 
The number of 1&#39;s in ([e (0) ] t +1 J   t E k   t P (0) )+J
 
   It should be noted that E k   t P (0) E l−n  actually means choosing the k 1 , . . . , k j -th rows and l 1 −n, l J −n-th columns of P (0) , while 1 J   t E k   t P (0)  actually means summing the k 1 , . . . k j -th rows of P (0) . No matrix computation is needed. 
   Thus, the decoding steps can be summarized as follows. First, an independent n-bit combination is generated from the group of bits b captured in an image. It should be noted that, with various embodiments of the invention, the selection of the n-bits can be combined with bit recognition confidence techniques, to help ensure that the most accurately recognized bits are selected for the n-bit combination. 
   Next, the relationship ([b (0) ] t ,[  b   (0) ] t )=[r (0) ] t (M (0) ,  M   (0) )+(0 n   t ,[e (0) ] t ) is solved to determine d (0) =HammingWeight(D (0) )=HammingWeight(e (0) ). If the number of different bits d (0)  is 0, then the process is stopped and the solution r (0)  is output. Otherwise, all J (=1 and 2) bit pairs are switched, and the number of different bits d is again determined using the relationship ([e (0) ] t +1 J   t E k   t P (0) )+J. It should be noted, however, that this relationship can only be evaluated when the rank of E k   t P (0) E p−n  is J. In this case there is no need to specify l 1 , l 2 , . . . , l J . Next, the minimal number d of different bits is determined. 
   The above process has to be repeated for several times in order to ensure a high enough probability of successful decoding. To estimate the times of selecting the n-bit b (0)  from b, the number r of the error bits in b is first predicted to be d. If r is changed, then 
               p   s     =         C   r   s     ⁢     C     K   -   r       n   -   s           C   K   n         ,         
is computed, which is the probability of the chosen n bits contain s error bits, where
 
             C   a   b     =       a   !         b   !     ⁢       (     a   -   b     )     !               
is the combinatory number, and
 
               P   s     =       ∑     i   =   0     s     ⁢     p   i         ,         
is the probability if the chosen n bits contain less than s+1 error bits. In practice, s=2 in order to minimize the computation load. Next, s 2  is computed, such that 1−(1−P 2 ) s     2   ≧P e , where P e  is the expected probability of successful decoding. If the times S of chosen b (0)  is equal to or larger than s 2 , then the process is stopped and the results are output. Otherwise, the process is repeated with a new independent n-bit combination b (0)  generated from the group of bits b captured in an image. Using this process, as long as the chosen n bits contain less than J+1 error bits, the correct solution is found.
 
   Decoding Using “Bit-Flipping” 
   While the above-described technique can be used to determine the number of a bit in an m-sequence, this technique can be further simplified using “bit-flipping.” As used herein, the term “bit flipping” refers to changing a bit with a value of “1” to a new value of “0,” changing a bit with a value of “0” to a new value of “1.” 
   Supposing [b (1) ] t  is [b (0) ] t  with J bits flipped, and the k i -bits are the k i -th bits of [b (0) ] t , where i=1, 2, . . . , J, 1≦k 1 ≦k 2 &lt; . . . &lt;k J ≦n, then the relationship.
 
[r (1) ] t =[b (1) ] t [M (0) ] −1  
 
   can be used to solve for a new r. It can be proven that:
 
([ b   (1) ] t   ,[  b     (0) ] t )=[ r   (1) ] t ( M   (0)   ,  M     (0) )+( E   J   ,[e   (0) ] t   +E   J   P   (0) )
 
and
 
[ r   (1) ] t   =[r   (0) ] t   +E   J   [M   (0) ] −1  
 
   where 
               E   J     =       ∑     j   =   1     J     ⁢     e     k   j     t         ,       e   i   t     =       (         0       …       0         1   i         0       …       0         )       1   ×   n         ,         
P (0) =[M (0) ] −1   M   (0) . Now, D (1) =(E J ,[e (0) ] t +E J P (0) ), and the number of different bits d (1)  is: d (1) =HammingWeight(D (1) )=HammingWeight([e (0) ] t +E J P (0) )+J. If d (1) &lt;d (0) , then r (1)  is a better solution of r than r (0) .
 
   The vector r is referred to as a location vector. Since division x s /P n (x) and division r(x)/P n (x) generates the same m-sequence R, once r, i.e. the coefficients of r(x), is solved, s can be obtained by using a discrete logarithm. Therefore, s′, the location of R in the original m-sequence m, can be obtained. Methods for solving a discrete logarithm are well known in the art. For example, one technique for solving a discrete logarithm is described in “Maximal and Near-Maximal Shift Register Sequences: Efficient Event Counters and Easy Discrete Logarithms,” Clark, D. W. and Weng, L-J.,  IEEE Transactions on Computers,  43(5), (1994), pp. 560-568, which is incorporated entirely herein by reference. 
   Thus, this simplified decoding process can be summarized by the following steps. First, n independent bits b (0)  are randomly selected from the total set of bits b captured in an image of a document. The bits n may be randomly selected using, for example, Gaussian elimination. Once the bits n are selected, then the relationship ([b (0) ] t ,[  b   (0) ] t )=[r (0) ] t (M (0) ,  M   (0) )+(0 n   t ,[e (0) ] t ) is solved to determine r. If the HammingWeight value d (0)  is 0, then the value of r is output and used to determine s′ as described above, giving the position of this bit in the document. 
   If the value d (0)  is not 0, then J bits of the chosen n bits are flipped, where 1≦J&lt;n, and the number of different bits using the equation d (1) =HammingWeight([e (0) ] t +E J P (0) )+J is computed. Next, another set of n independent bits is selected, and the process is repeated. The new b (0)  is different from all previous sets. Finally, the value of r is output that corresponds to the smallest d, i.e. the least number of different bits. In various implementations of the invention, up to two bits are flipped, and b (0)  is only selected once. 
   Tool for Decoding an M-Array 
     FIG. 7  illustrates an example of a decoding tool  701  that may be implemented according to various examples of the invention. As seen in this figure, the tool  401  receives image information from a pen camera device  201 , and provides a bit position in a pattern. The decoding tool  701  includes a coefficient matrix M preparation module  703  and a bM matrix preparation module  705 . It also includes a stochastic decoder module  707  and a discrete logarithm determination module  709 . With various examples of the invention, one or more of these modules may be implemented using analog circuitry. More typically, however, one or more of these modules will be implemented by software instruction executing on a programmable computer, such as the programmable computer shown in  FIG. 1 . Each of these modules  703 - 709  will be discussed in more detail below. 
   Coefficient Matrix M Preparation 
   In order to solve for r as discussed above, the arrays b and M are configured. First, all of the bits extracted for one dimension are stored in a matrix called Extracted_Bits_Array. For dimension b, where b=0, 1, . . . , 7, the Extracted_Bits_Array (m, n)=B b   m,n . As illustrated in  FIG. 8 , the bits extracted for one dimension are stored in Extracted_Bits_Array. In this figure, the null values are shown as “FF”.  FIG. 8  also indicates the position that will be determined by the decoding process. The decoded position is the position of the first element of the m-array stored in the Extracted_Bits_Array. In the case of the m-array representing positions using (x,y) Cartesian coordinates, the decoded position will be the coordinates of point C X′Y′  in the pattern array. 
   Once an Extracted_Bits_Array is created for a dimension, the total number of non-FF bits is counted. If the number is fewer than n, where n is the order of the m-array (in the illustrated example, n=28), then too few bits have been obtained to decode the array, and the decoding fails for this dimension. If the number is more than 2n, up to the 2n bits that have the highest recognition confidence values are kept, and “FF” is assigned to all other elements in the Extracted_Bits_Array. 
   In the illustrated example, it should be noted that the size of Extracted_Bits_Array is 20×20. This size is considered large enough to account for all possible positions of the extracted bits for a pattern encoded using an 8-a-16 symbol. That is, given the 128×100 pixel image sensor and the size of the symbol 8-a-16, a size 20×20 matrix is considered large enough to hold the bits in the image, regardless of how the image is rotated. 
   To obtain M, the coefficient matrix M preparation module  703  creates a matrix called M_Const_Matrix as a constant table. The size of M_Const_Matrix is the same as the size of Extracted_Bits_Array, i.e. 20×20 in the illustrated implementation. The M_Const_Matrix table is constructed in the following manner. For every i and j, where 1≦i≦20, 1≦j≦20,
 
 M ( i,j ) T =( A ( i,j ),  A ( i+ 1 ,j+ 1), . . . ,  A ( i+ 26 ,j+ 26),  A ( i+ 27 ,j+ 27)) T  
 
   where A(i,j) is element (i,j) of the m-array based on the m-sequence m.  FIG. 9  shows an illustration of how M_Const_Matrix is constructed. 
   Next, the bM matrix preparation module  705  constructs matrix bm_Matrix to contain b and M. For every non-FF bit in the Extracted_Bits_Array, the bM matrix preparation module  705  places the bit in the last column of bM_Matrix. Next, the corresponding element in M_Const_Matrix is retrieved (which is a vector), and that element is placed in the first n columns of the same row of bM_Matrix. With various examples of the invention, the bM matrix preparation module  705  may reorder th rows of bM_Matrix according to the recognition confidence of the corresponding bits, from highest to lowest.  FIG. 9  for an illustration of how bM_Matrix is constructed. As a result, the first n columns of bM_Matrix is M (transposed). The last column of bM_Matrix is b. bM_Matrix has n+1 columns and up to 2n rows. For calculation purposes, another matrix, bM_Copy may be created, which is exactly the same as bM_Matrix. 
   Stochastic Decoding 
   Next, the stochastic decoder module  707  obtains a solution for r. More particularly, a first solution for r may be obtained with Gaussian elimination. In the bM_Matrix, through Gaussian elimination, n linearly independent bits are selected to solve for r. The process proceeds as follows. In bM_Matrix, starting from the first row down, a row is located that has a “1” in the first column. If it is not the first row of bM_Matrix, the row is switched with the first row of bM_Matrix. Next, in the bM_Matrix, the new first row (with a “1” in the first column) is used to perform a XOR operation with all the remaining rows that have a “1” in the first column and the result of the operation replaces the value of the original row. Now, all of the rows in bM_Matrix have a “0” in the first column except the first row, which has a “1” in the first column. 
   Next, starting from the second row down in the bM_Matrix, a row is identified that has a “1” in the second column. If it is not the second row of the bM_Matrix, this row is switched with the second row of bM_Matrix. In bM_Matrix, the new second row (with a “1” in the second column) to perform an XOR operation with all the remaining rows (including the first row of bM_Matrix) that have a “1” in the second column, letting the result replace the original value for the row. Now, all the rows in bM_Matrix have a “0” in the second column except the second row which has a “1” in the second column. This process continues until there is a “1” along the diagonal of the first n rows of bM_Matrix, as shown in  FIG. 10 . 
   The first n rows of bM_Matrix correspond to the n bits selected for solving r, i.e. b (0)  as described above. The rest of the rows of bM_Matrix correspond to the rest of the bits, i.e.  b   (0)  also described above. Further, the last column of the first n rows of the bM_Matrix is the solution for r (0)  noted above, which will be referred to as r_Vector here. The last column of the rest of the rows is e (0)  noted above, which will be referred to as e_Vector here. Letting d be the number of 1&#39;s in e_Vector, d is the number of different bits, d (0) , described above. If d=0, it means there are no error bits. The process is stopped, and r_Vector is output as the as the solution of r. If d&gt;0, however, then there are error bits, and the process is continued. 
   In bM_Copy, the same row switching is done as in bM_Matrix, but no XOR operation is performed. The first n rows and n columns of bM_Copy is M (0)  (transposed) as described above, which will be referred to as M_Matrix here. The rest of the rows and the first n columns of bM_Copy is the  M   (0)  (transposed) described above, which will be referred to as MB_Matrix here. From M_Matrix and MB_Matrix, MR_Matrix is obtained, which is [M (0) ] −1  (transposed), and P_Matrix, which is P (0)  described above:
 
MR_Matrix=M_Matrix −1  
 
 P _Matrix= MB _Matrix· MR _Matrix
 
   Because there may be error bits in b, it can be assumed that each of the n bits selected for solving r may be wrong, and its value “flipped” (i.e., the value changed from 0 to 1 or from 1 to 0) to solve for r again. If the new r results in a smaller d, the new r is a better solution for r, and d min  is initialized as d. 
   For every flipped bit, to calculate the new d, it is not necessary to repeat the process of Gaussian elimination. As previously discussed, d (1) =HammingWeight([e (0) ] t +E J P (0) )+J, therefore if [e (0) ] t +E J P (0)  can be obtained, then a new d is obtained. 
   Accordingly, each of the n bits selected is flipped. For every column of P_Matrix, the column, the XOR operating is performed with e_Vector. The result is e_Vector_Flip. As illustrated in  FIG. 11 , e_Vector_Flip=[e (0) ] t +E J P (0) , where J=1. 
   Letting d=HammingWeight(e_Vector_Flip)+1, where d is the new count of different bits. If d&lt;d min , then let d min =d, and i 1 =index of the corresponding column in P_Matrix. This process continues until all columns in P_Matrix have been processed. If d min =1, the process is stopped, as the error bit has been located. As discussed in detail above, [r (1) ] t =[r (0) ] t +E J [M (0) ] −1 , where J=1. Therefore, the new r_Vector is calculated by performing the XOR operation on the i 1 -th row of MR_Matrix and the original r_Vector (the one from Gaussian elimination), as shown in  FIG. 12 . 
   If d min ≠1, it means that there are more than 1 error bits. Accordingly, two of the n selected bits are flipped to determine if a smaller d can be obtained. For every pair of columns of P_Matrix, the two columns are obtained and the XOR operation is performed with e_Vector. As shown in  FIG. 13 , the result is e_Vector_Flip. Letting d=HammingWeight(e_Vector_Flip)+2, d is the new count of different bits. If d&lt;d min , then d min =d, and i 1 =index of the first corresponding column, and i 2 =index of the second corresponding column in P_Matrix. This process continues for all pairs of columns in P_Matrix. 
   If d min =2, then the process is stopped, as it indicates that the two error bits have been identified. As discussed above, [r (1) ] t =[r (0) ] t +E J [M (0) ] −1 , where J=2. Therefore, the new r_Vector is calculated by performing the XOR operation on the i 1 -th and i 2 -th row of MR_Matrix and the original r_Vector (the one from Gaussian elimination). As shown in  FIG. 14 , the new r_Vector is output as the solution of r. If d min ≠2, the process continues to the next step. 
   Thus, if d min  is the d obtained with no bit flipping, the original r_Vector (the one from Gaussian elimination) is output as the solution to r. If d min  is the d obtained with one bit flipping, the new r_Vector is calculated by performing the XOR operation on the i 1 -th row of MR_Matrix and the original r_Vector. The new r_Vector is output as the solution to r. If d min  is the d obtained with two bit flipping, the new r_Vector by is calculated by performing the XOR operating with the i 1 -th and i 2 -th row of MR_Matrix and the original r_Vector. The new r_Vector is output as the solution to r. Thuse, the output of the stochastic decoding process is the location vector r. 
   Calculation of L by Discrete Logarithm 
   Given location vector r, the discrete logarithm determination module  709  can obtain L (referred to as the bit “s” above in paragraphs 42 and 43) by a discrete logarithm determination technique. L is the location of the first element in the Extracted_Bits_Array of the m-sequence, and L∈{0, 1, . . . , 2 n −2}, where n is the order of the m-sequence. r can be viewed as an element of the finite field F 2     n   . It can be proven that:
 
r=α L  
 
   where α is a primitive element of the finite field F 2     n    and is known from the division of polynomials that generates the m-sequence. Therefore, given r, L can be solved from the above equation. 
   Letting n be the order of the m-sequence, m be the period of the m-sequence, i.e. m=2 n 1, m i  be the prime factors of m=2 n −1, and w be the number of m i &#39;s. For each m i , ν i  is chosen such that 
   
     
       
         
           
             
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   where i=1, . . . , w. 
   In the illustrated implementation, n=28, so α=(1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1) t  (correspondingly, the primitive polynomial in division 
           1       P   n     ⁡     (   x   )             
that generates the m-sequence is P n (x)=1+x 3 +x 28 ), m=2 28 −1. There are 6 prime factors of m, i.e., w=6, and the prime factors are: 3, 43, 127, 5, 29, 113. Correspondingly, ν i  are: 2, 25, 32, 1, 1, 30. All these are stored in constant tables.
 
   For each m i , q∈{0, 1, 2, . . . m i −1} is found such that 
               (     α     m     m   i         )     q     =       r     m     m   i         .           
Note that again, these are multiplications over the finite field F 2     n   . Letting p i =q, then,
 
   
     
       
         
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   Localization in the M-Array 
   Based on the method used in generating the m-array from the m-sequence, the position of the first element in Extracted_Bits_Array in m-array can be obtained:
 
 x =mod( L,m   1 )
 
 y =mod( L,m   2 )
 
   where m 1  is the width of the m-array, and m 2  is the height of the m-array. When the order of the m-sequence is n, 
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   For each dimension, the decoding process described above outputs position (x,y). Letting (x p ,y p ) be the output of the dimension representing the X, Y position in Cartesian coordinates, as illustrated above, (x p ,y p ) are the coordinates of point C X′Y′  in the symbol pattern array. 
   Solving Multiple Dimensions of m-Arrays Simultaneously 
   As discussed in detail above, a document may have multiple (e.g., 8) dimensions of m-arrays. Supposing that the dimensions are b i , i=1, 2, . . . , C, and the metadata are encoded by the relative shift d j  between b j  and b 1 , where b 1  is the position dimension and j=2, 3, . . . , C. The metadata are the same no matter where the image is obtained. Therefore, the metadata can be extracted sometime before the error-correcting decoding starts. When d j , j=2, 3, . . . , C, are known, b i , i=1, 2, . . . , C, can be jointly used for the decoding of position. The process is as follows. 
   Supposing b i   t =[r b     i   ] t M b     i   , i=1, 2, . . . , C, then the relationship between r b     j    and r b     1    is [r b     j   ] t =[r b     1   ] t Q −d     j   , where Q −d     j   ={circumflex over (M)} 0˜(n−1) A d     j   [{circumflex over (M)} 0˜(n−1) ] −1 , {circumflex over (M)} 0˜(n−1)  are the sub-matrices of {circumflex over (M)}, consisting of the first n columns of {circumflex over (M)}, and A d     j   =(a d     j    a d     j     +1  . . . a d     j     +n−1 ), where a d     j     +k  is the coefficients when α d     j     +k  is expressed as the linear combination of 1, α, . . . , α n−1  where α is a primitive element of F 2     n    and the root of x n P n (x −1 ). Therefore the location of vector r b     1    may be solved via:
 
( b   1   t    b   2   t    . . . b   C   t )=[ r   b     1   ] t ( M   b     1      M   b     2      . . . M   b     C   ),
 
   The procedure to solve this equation is exactly the same as solving b i   t =[r b     i   ] t M b     i   , i=1, 2, . . . , C, separately. However, solving them jointly is more efficient in two ways. First, the speed can be nearly C times faster because only one linear system is solved instead (but with some overhead to compute Q −d     j    and more XOR operations to solve a larger system). Second, the probability of obtaining the correct solution is also greatly increased, especially when none of the dimensions has enough bits for computing the solution. 
   CONCLUSION 
   While the invention has been described with respect to specific examples including presently preferred modes of carrying out the invention, those skilled in the art will appreciate that there are numerous variations and permutations of the above described systems and techniques that fall within the spirit and scope of the invention as set forth in the appended claims.