Patent Publication Number: US-7583842-B2

Title: Enhanced approach of m-array decoding and error correction

Description:
TECHNICAL FIELD 
     The present invention relates to interacting with a medium using a digital pen. More particularly, the present invention relates to determining the location of a digital pen during interaction with one or more surfaces. 
     BACKGROUND 
     Computer users are accustomed to using a mouse and keyboard as a way of interacting with a personal computer. While personal computers provide a number of advantages over written documents, most users continue to perform certain functions using printed paper. Some of these functions include reading and annotating written documents. In the case of annotations, the printed document assumes a greater significance because of the annotations placed on it by the user. One of the difficulties, however, with having a printed document with annotations is the later need to have the annotations entered back into the electronic form of the document. This requires the original user or another user to wade through the annotations and enter them into a personal computer. In some cases, a user will scan in the annotations and the original text, thereby creating a new document. These multiple steps make the interaction between the printed document and the electronic version of the document difficult to handle on a repeated basis. Further, scanned-in images are frequently non-modifiable. There may be no way to separate the annotations from the original text. This makes using the annotations difficult. Accordingly, an improved way of handling annotations is needed. 
     One technique of capturing handwritten information is by using a pen whose location may be determined during writing. One pen that provides this capability is the Anoto pen by Anoto Inc. This pen functions by using a camera to capture an image of paper encoded with a predefined pattern. An example of the image pattern is shown in  FIG. 11 . This pattern is used by the Anoto pen (by Anoto Inc.) to determine a location of a pen on a piece of paper. However, it is unclear how efficient the determination of the location is with the system used by the Anoto pen. To provide efficient determination of the location of the captured image, a system is needed that provides efficient decoding of the captured image. 
     SUMMARY 
     Aspects of the present invention provide solutions to at least one of the issues mentioned above, thereby enabling one to locate a position or positions of the captured image on a viewed document with a predefined pattern. The viewed document may be on paper, LCD screen or any other medium with the predefined pattern. Aspects of the present invention include a decoding process that permits efficient decoding of a captured image, providing for efficient determination of the location of the image. 
     With one aspect of the invention, a decoding process tactfully selects a subset of bits from bits extracted from the captured image. With another aspect of the invention, a process adjusts the number of iterations that the decoding process executes. With another aspect of the invention, a process determines the X,Y coordinates of the location of the extracted bits so that the X,Y coordinates are consistent with a local constraint such as a destination area. These and other aspects of the present invention will become known through the following drawings and associated description. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The foregoing summary of the invention, as well as the following detailed description of preferred embodiments, is better understood when read in conjunction with the accompanying drawings, which are included by way of example, and not by way of limitation with regard to the claimed invention. 
         FIG. 1  shows a general description of a computer that may be used in conjunction with embodiments of the present invention. 
         FIGS. 2A and 2B  show an image capture system and corresponding captured image in accordance with embodiments of the present invention. 
         FIGS. 3A through 3F  show various sequences and folding techniques in accordance with embodiments of the present invention. 
         FIGS. 4A through 4E  show various encoding systems in accordance with embodiments of the present invention. 
         FIGS. 5A through 5D  show four possible resultant corners associated with the encoding system according to  FIGS. 4A and 4B . 
         FIG. 6  shows rotation of a captured image portion in accordance with embodiments of the present invention. 
         FIG. 7  shows various angles of rotation used in conjunction with the coding system of  FIGS. 4A through 4E . 
         FIG. 8  shows a process for determining the location of a captured array in accordance with embodiments of the present invention. 
         FIG. 9  shows a method for determining the location of a captured image in accordance with embodiments of the present invention. 
         FIG. 10  shows another method for determining the location of captured image in accordance with embodiments of the present invention. 
         FIG. 11  shows a representation of encoding space in a document according to prior art. 
         FIG. 12  shows a flow diagram for decoding extracted bits from a captured image in accordance with embodiments of the present invention. 
         FIG. 13  shows bit selection of extracted bits from a captured image in accordance with embodiments of the present invention. 
         FIG. 14  shows an apparatus for decoding extracted bits from a captured image in accordance with embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Aspects of the present invention relate to determining the location of a captured image in relation to a larger image. The location determination method and system described herein may be used in combination with a multi-function pen. 
     The following is separated by subheadings for the benefit of the reader. The subheadings include: terms, general-purpose computer, image capturing pen, encoding of array, decoding, error correction, and location determination. 
     Terms 
     Pen—any writing implement that may or may not include the ability to store ink. In some examples, a stylus with no ink capability may be used as a pen in accordance with embodiments of the present invention. 
     Camera—an image capture system that may capture an image from paper or any other medium. 
     General Purpose Computer 
       FIG. 1  is a functional block diagram of an example of a conventional general-purpose digital computing environment that can 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 Pen 
     Aspects of the present invention include placing an encoded data stream in a displayed form that represents the encoded data stream. (For example, as will be discussed with  FIG. 4B , the encoded data stream is used to create a graphical pattern.) The displayed form may be printed paper (or other physical medium) or may be a display projecting the encoded data stream in conjunction with another image or set of images. For example, the encoded data stream may be represented as a physical graphical image on the paper or a graphical image overlying the displayed image (e.g., representing the text of a document) or may be a physical (non-modifiable) graphical image on a display screen (so any image portion captured by a pen is locatable on the display screen). 
     This determination of the location of a captured image may be used to determine the location of a user&#39;s interaction with the paper, medium, or display screen. In some aspects of the present invention, the pen may be an ink pen writing on paper. In other aspects, the pen may be a stylus with the user writing on the surface of a computer display. Any interaction may be provided back to the system with knowledge of the encoded image on the document or supporting the document displayed on the computer screen. By repeatedly capturing images with a camera in the pen or stylus as the pen or stylus traverses a document, the system can track movement of the stylus being controlled by the user. The displayed or printed image may be a watermark associated with the blank or content-rich paper or may be a watermark associated with a displayed image or a fixed coding overlying a screen or built into a screen. 
       FIGS. 2A and 2B  show an illustrative example of pen  201  with a camera  203 . Pen  201  includes a tip  202  that may or may not include an ink reservoir. Camera  203  captures an image  204  from surface  207 . 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 a personal computer (for example, via Bluetooth or other wireless protocols). 
       FIG. 2B  represents an image as viewed by camera  203 . In one illustrative example, the field of view of camera  203  (i.e., the resolution of the image sensor of the camera) is 32×32 pixels (where N=32). In the embodiment, a captured image (32 pixels by 32 pixels) corresponds to an area of approximately 5 mm by 5 mm of the surface plane captured by camera  203 . Accordingly,  FIG. 2B  shows a field of view of 32 pixels long by 32 pixels wide. The size of N is adjustable, such that a larger N corresponds to a higher image resolution. Also, while the field of view of the camera  203  is shown as a square for illustrative purposes here, the field of view may include other shapes 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 system configuration and performance requirement. The size of the captured image frame may be large or small, depending on system configuration and performance requirement. 
     The image captured by camera  203  may be used directly by the processing system or may undergo pre-filtering. This pre-filtering may occur in pen  201  or may occur outside of pen  201  (for example, in a personal computer). 
     The image size of  FIG. 2B  is 32×32 pixels. If each encoding unit size is 3×3 pixels, then the number of captured encoded units would be approximately 100 units. If the encoding unit size is 5×5 pixels, then the number of captured encoded units is approximately 36. 
       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 . Lens  208  may be a single lens or a multi-part lens system, but is represented here 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  is referred to as the virtual pen tip. It is noted that the virtual pen tip location with respect to image sensor  211  is fixed because of the constant relationship between the pen tip, the lens  208 , and the image sensor  211 . 
     The following transformation F S→P  transforms position coordinates in the image captured by camera to position coordinates in the real image on the paper:
 
 L   paper   =F   S→P  ( L   Sensor )
 
     During writing, the pen tip and the paper are on the same plane. Accordingly, the transformation from the virtual pen tip to the real pen tip is also F S→P :
 
 L   pentip   =F   s→p ( L   virtual-pentip )
 
     The transformation F S→P  may be estimated as an affine transform. This simplifies as: 
               F     S   -&gt;   P     ′     =     {                 s   x     ⁢   sin   ⁢           ⁢     θ   y           cos   ⁢           ⁢     θ   x     ⁢           ⁢   sin   ⁢           ⁢     θ   y       -     cos   ⁢           ⁢     θ   y     ⁢           ⁢   sin   ⁢           ⁢     θ   x           ,             -         s   x     ⁢   cos   ⁢           ⁢     θ   y           cos   ⁢           ⁢     θ   x     ⁢           ⁢   sin   ⁢           ⁢     θ   y       -     cos   ⁢           ⁢     θ   y     ⁢           ⁢   sin   ⁢           ⁢     θ   x             ,         0               -         s   y     ⁢   sin   ⁢           ⁢     θ   x           cos   ⁢           ⁢     θ   x     ⁢           ⁢   sin   ⁢           ⁢     θ   y       -     cos   ⁢           ⁢     θ   y     ⁢           ⁢   sin   ⁢           ⁢     θ   x             ,                 s   y     ⁢   cos   ⁢           ⁢     θ   x           cos   ⁢           ⁢     θ   x     ⁢           ⁢   sin   ⁢           ⁢     θ   y       -     cos   ⁢           ⁢     θ   y     ⁢           ⁢   sin   ⁢           ⁢     θ   x           ,         0             0   ,           0   ,         1         }           
as the estimation of F S→P , in which θ x , θ y , s x , and s y  are the rotation and scale of two orientations of the pattern captured at location  204 . Further, one can refine F S→P  by matching the captured image with the corresponding real image on paper. “Refine” means to get a more precise estimation of the transformation F S→P  by a type of optimization algorithm referred to as a recursive method. The recursive method treats the matrix F′ S→P  as the initial value. The refined estimation describes the transformation between S and P more precisely.
 
     Next, one can determine the location of virtual pen tip by calibration. 
     One places the pen tip  202  on a fixed location L pentip  on paper. Next, one tilts the pen, allowing the camera  203  to capture a series of images with different pen poses. For each image captured, one may obtain the transformation F S→P . From this transformation, one can obtain the location of the virtual pen tip L virtual-pentip :
 
 L   virtual-pentip   =F   P→S ( L   pentip )
 
where L pentip  is initialized as (0, 0) and
 
 F   S→P =( F   S→P ) −1 
 
     By averaging the L virtual-pentip  obtained from each image, a location of the virtual pen tip L virtual-pentip  may be determined. With L virtual-pentip , one can get a more accurate estimation of L pentip . After several times of iteration, an accurate location of virtual pen tip L virtual-pentip  may be determined. 
     The location of the virtual pen tip L virtual-pentip  is now known. One can also obtain the transformation F S→P  from the images captured. Finally, one can use this information to determine the location of the real pen tip L pentip :
 
 L   pentip   =F   S→P ( L   virtual-pentip )
 
Encoding of Array
 
     A two-dimensional array may be constructed by folding a one-dimensional sequence. Any portion of the two-dimensional array containing a large enough number of bits may be used to determine its location in the complete two-dimensional array. However, it may be necessary to determine the location from a captured image or a few captured images. So as to minimize the possibility of a captured image portion being associated with two or more locations in the two-dimensional array, a non-repeating sequence may be used to create the array. One property of a created sequence is that the sequence does not repeat over a length (or window) n. The following describes the creation of the one-dimensional sequence then the folding of the sequence into an array. 
     Sequence Construction 
     A sequence of numbers may be used as the starting point of the encoding system. For example, a sequence (also referred to as an m-sequence) may be represented as a q-element set in field F q . Here, q=p n  where n≧1 and p is a prime number. The sequence or m-sequence may be generated by a variety of different techniques including, but not limited to, polynomial division. Using polynomial division, the sequence may be defined as follows: 
                 R   l     ⁡     (   x   )           P   n     ⁡     (   x   )             
where P n (x) is a primitive polynomial of degree n in field F q [x] (having q n  elements). R l (x) is a nonzero polynomial of degree l (where l&lt;n) in field F q [x]. The sequence may be created using an iterative procedure with two steps: first, dividing the two polynomials (resulting in an element of field F q ) and, second, multiplying the remainder by x. The computation stops when the output begins to repeat. This process may be implemented using a linear feedback shift register as set forth in an article by Douglas W. Clark and Lih-Jyh Weng, “Maximal and Near-Maximal Shift Register Sequences: Efficient Event Counters and Easy Discrete Logarithms,” IEEE Transactions on Computers 43.5 (May 1994, pp 560-568). In this environment, a relationship is established between cyclical shifting of the sequence and polynomial R l (x): changing R l (x) only cyclically shifts the sequence and every cyclical shifting corresponds to a polynomial R l (x). One of the properties of the resulting sequence is that, the sequence has a period of q n −1 and within a period, over a width (or length) n, any portion exists once and only once in the sequence. This is called the “window property”. Period q n −1 is also referred to as the length of the sequence and n as the order of the sequence.
 
     The process described above is but one of a variety of processes that may be used to create a sequence with the window property. 
     Array Construction 
     The array (or m-array) that may be used to create the image (of which a portion may be captured by the camera) is an extension of the one-dimensional sequence or m-sequence. Let A be an array of period (m 1 , m 2 ), namely A(k+m 1 ,l)=A(k,l+m 2 )=A(k,l). When an n 1 ×n 2  window shifts through a period of A, all the nonzero n 1 ×n 2  matrices over F q  appear once and only once. This property is also referred to as a “window property” in that each window is unique. A widow may then be expressed as an array of period (m 1 , m 2 ) (with m 1  and m 2  being the horizontal and vertical number of bits present in the array) and order (n 1 , n 2 ). 
     A binary array (or m-array) may be constructed by folding the sequence. One approach is to obtain a sequence then fold it to a size of m 1 ×m 2  where the length of the array is L=m 1 ×m 2 =2 n −1. Alternatively, one may start with a predetermined size of the space that one wants to cover (for example, one sheet of paper, 30 sheets of paper or the size of a computer monitor), determine the area (m 1 ×m 2 ), then use the size to let L≧m 1 ×m 2 , where L=2 n −1. 
     A variety of different folding techniques may be used. For example,  FIGS. 3A through 3C  show three different sequences. Each of these may be folded into the array shown as  FIG. 3D . The three different folding methods are shown as the overlay in  FIG. 3D  and as the raster paths in  FIGS. 3E and 3F . We adopt the folding method shown in  FIG. 3D . 
     To create the folding method as shown in  FIG. 3D , one creates a sequence {a i } of length L and order n. Next, an array {b kl } of size m 1 ×M 2 , where gcd(m 1 , m 2 )=1 and L=m 1 ×m 2 , is created from the sequence {a i } by letting each bit of the array be calculated as shown by equation 1:
 
 b   kl   =a   i , where  k=i  mod( m   1 ),  l=i  mod( m   2 ),  i= 0 , . . . ,L− 1  (1)
 
     This folding approach may be alternatively expressed as laying the sequence on the diagonal of the array, then continuing from the opposite edge when an edge is reached. 
       FIG. 4A  shows sample encoding techniques that may be used to encode the array of  FIG. 3D . It is appreciated that other encoding techniques may be used. For example, an alternative coding technique is shown in  FIG. 11 . 
     Referring to  FIG. 4A , a first bit  401  (for example, “1”) is represented by a column of dark ink. A second bit  402  (for example, “0”) is represented by a row of dark ink. It is appreciated that any color ink may be used to represent the various bits. The only requirement in the color of the ink chosen is that it provides a significant contrast with the background of the medium to be differentiable by an image capture system. The bits in  FIG. 4A  are represented by a 3×3 matrix of cells. The size of the matrix may be modified to be any size as based on the size and resolution of an image capture system. Alternative representation of bits  0  and  1  are shown in  FIGS. 4C-4E . It is appreciated that the representation of a one or a zero for the sample encodings of  FIGS. 4A-4E  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  shows pixel representations in columns and rows in an irregular spacing format (e.g., two dark dots followed by a blank dot). 
     Referring back to  FIG. 4A , if a bit is represented by a 3×3 matrix and an imaging system detects a dark row and two white rows in the 3×3 region, then a zero is detected (or one). If an image is detected with a dark column and two white columns, then a one is detected (or a zero). 
     Here, more than one pixel or dot is used to represent a bit. Using a single pixel (or bit) to represent a bit is fragile. Dust, creases in paper, non-planar surfaces, and the like create difficulties in reading single bit representations of data units. However, it is appreciated that different approaches may be used to graphically represent the array on a surface. Some approaches are shown in  FIGS. 4C through 4E . It is appreciated that other approaches may be used as well. One approach is set forth in  FIG. 11  using only space-shifted dots. 
     A bit stream is used to create the graphical pattern  403  of  FIG. 4B . Graphical pattern  403  includes 12 rows and 18 columns. The rows and columns are formed by a bit stream that is converted into a graphical representation using bit representations  401  and  402 .  FIG. 4B  may be viewed as having the following bit representation: 
               [         0       1       0       1       0       1       1       1       0           1       1       0       1       1       0       0       1       0           0       0       1       0       1       0       0       1       1           1       0       1       1       0       1       1       0       0         ]               
Decoding
 
     When a person writes with the pen of  FIG. 2A  or moves the pen close to the encoded pattern, the camera captures an image. For example, pen  201  may utilize a pressure sensor as pen  201  is pressed against paper and pen  201  traverses a document on the paper. The image is then processed to determine the orientation of the captured image with respect to the complete representation of the encoded image and extract the bits that make up the captured image. 
     For the determination of the orientation of the captured image relative to the whole encoded area, one may notice that not all the four conceivable corners shown in  FIG. 5A-5D  can present in the graphical pattern  403 . In fact, with the correct orientation, the type of corner shown in  FIG. 5A  cannot exist in the graphical pattern  403 . Therefore, the orientation in which the type of corner shown in  FIG. 5A  is missing is the right orientation. 
     Continuing to  FIG. 6 , the image captured by a camera  601  may be analyzed and its orientation determined so as to be interpretable as to the position actually represented by the image  601 . First, image  601  is reviewed to determine the angle θ needed to rotate the image so that the pixels are horizontally and vertically aligned. It is noted that alternative grid alignments are possible including a rotation of the underlying grid to a non-horizontal and vertical arrangement (for example, 45 degrees). Using a non-horizontal and vertical arrangement may provide the probable benefit of eliminating visual distractions from the user, as users may tend to notice horizontal and vertical patterns before others. For purposes of simplicity, the orientation of the grid (horizontal and vertical and any other rotation of the underlying grid) is referred to collectively as the predefined grid orientation. 
     Next, image  601  is analyzed to determine which corner is missing. The rotation amount o needed to rotate image  601  to an image ready for decoding  603  is shown as o=(θ plus a rotation amount {defined by which corner missing}). The rotation amount is shown by the equation in  FIG. 7 . Referring back to  FIG. 6 , angle θ is first determined by the layout of the pixels to arrive at a horizontal and vertical (or other predefined grid orientation) arrangement of the pixels and the image is rotated as shown in  602 . An analysis is then conducted to determine the missing corner and the image  602  rotated to the image  603  to set up the image for decoding. Here, the image is rotated  90  degrees counterclockwise so that image  603  has the correct orientation and can be used for decoding. 
     It is appreciated that the rotation angle θ may be applied before or after rotation of the image  601  to account for the missing corner. It is also appreciated that by considering noise in the captured image, all four types of corners may be present. We may count the number of corners of each type and choose the type that has the least number as the corner type that is missing. 
     Finally, the code in image  603  is read out and correlated with the original bit stream used to create image  403 . The correlation may be performed in a number of ways. For example, it may be performed by a recursive approach in which a recovered bit stream is compared against all other bit stream fragments within the original bit stream. Second, a statistical analysis may be performed between the recovered bit stream and the original bit stream, for example, by using a Hamming distance between the two bit streams. It is appreciated that a variety of approaches may be used to determine the location of the recovered bit stream within the original bit stream. 
     Once one has the recovered bits, one needs to locate the captured image within the original array (for example, the one shown in  FIG. 4B ). The process of determining the location of a segment of bits within the entire array is complicated by a number of items. First, the actual bits to be captured may be obscured (for example, the camera may capture an image with handwriting that obscures the original code). Second, dust, creases, reflections, and the like may also create errors in the captured image. These errors make the localization process more difficult. In this regard, the image capture system may need to function with non-sequential bits extracted from the image. The following represents a method for operating with non-sequential bits from the image. 
     Let the sequence (or m-sequence) I correspond to the power series I(x)=1/P n (x), where n is the order of the m-sequence, and the captured image contains K bits of I b=(b 0  b 1  b 2  . . . b K−1 ) t , where K≧n and the superscript t represents a transpose of the matrix or vector. The location s of the K bits is just the number of cyclic shifts of I so that b 0  is shifted to the beginning of the sequence. Then this shifted sequence R corresponds to the power series x s |P n (x), or R=T s (I), where T is the cyclic shift operator. We find this s indirectly. The polynomials modulo P n (x) form a field. It is guaranteed that x s ≡r 0 +r 1 x+ . . . r n−1 x n−1 mod(P n (x)). Therefore, we may find (r 0 , r 1 , . . . ,r n−1 ) and then solve for s. 
     The relationship x s ≡r 0 +r 1 x+ . . . r n−1 x n−1 mod(P n (x)) implies that R=r 0 +r 1 T(I)+ . . . +r n−1 T n−1 (I). Written in a binary linear equation, it becomes:
 
R=r t A  (2)
 
where r=(r 0  r 1  r 2  . . . r n−1 ) t , and A=(I T(I). . . T n−1 (I)) t  which consists of the cyclic shifts of I from 0-shift to (n−1)-shift. Now only sparse K bits are available in R to solve r. Let the index differences between b i  and b 0  in R be k i , i=1,2, . . . ,k−1, then the 1 st  and (k i  +1)-th elements of R, i=1,2, . . . ,k−1, are exactly b 0 , b 1 , . . . , b k−1 . By selecting the 1 st  and (k i +1)-th columns of A, i=1,2, . . . ,k−1, the following binary linear equation is formed:
 
b t =r t M  (3)
         where M is an n×K sub-matrix of A.       

     If b is error-free, the solution of r may be expressed as:
 
 r   t   ={tilde over (b)}   t   {tilde over (M)}   −1   (4)
 
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.
 
     With known r, we may use the Pohlig-Hellman-Silver algorithm as noted by Douglas W. Clark and Lih-Jyh Weng, “Maximal and Near-Maximal Shift Register Sequences: Efficient Event Counters and Easy Discrete Logorithms,” IEEE Transactions on Computers 43.5 (May 1994, pp 560-568) to find s so that x s ≡r 0 +r 1 x+ . . . r n−1 x n−1 mod(P n (x)). 
     As matrix A (with the size of n by L, where L=2 n −1) may be huge, we should avoid storing the entire matrix A. In fact, as we have seen in the above process, given extracted bits with index difference k i , only the first and (k i +1)-th columns of A are relevant to the computation. Such choices of k i  is quite limited, given the size of the captured image. Thus, only those columns that may be involved in computation need to saved. The total number of such columns is much smaller than L (where L=2 n −1 is the length of the m-sequence). 
     Error Correction 
     If errors exist in b, then the solution of r becomes more complex. Traditional methods of decoding with error correction may not readily apply, because the matrix M associated with the captured bits may change from one captured image to another. 
     We adopt a stochastic approach. Assuming that the number of error bits in b, n e , is relatively small compared to K, then the probability of choosing correct n bits from the K bits of b and the corresponding sub-matrix {tilde over (M)} of M being non-degenerate is high. 
     When the n bits chosen are all correct, the Hamming distance between b t  and r t M, or the number of error bits associated with r, should be minimal, where r is computed via equation (4). Repeating the process for several times, it is likely that the correct r that results in the minimal error bits can be identified. 
     If there is only one r that is associated with the minimum number of error bits, then it is regarded as the correct solution. Otherwise, if there is more than one r that is associated with the minimum number of error bits, the probability that n e  exceeds the error correcting ability of the code generated by M is high and the decoding process fails. The system then may move on to process the next captured image. In another implementation, information about previous locations of the pen can be taken into consideration. That is, for each captured image, a destination area where the pen may be expected next can be identified. For example, if the user has not lifted the pen between two image captures by the camera, the location of the pen as determined by the second image capture should not be too far away from the first location. Each r that is associated with the minimum number of error bits can then be checked to see if the location s computed from r satisfies the local constraint, i.e., whether the location is within the destination area specified. 
     If the location s satisfies the local constraint, the X, Y positions of the extracted bits in the array are returned. If not, the decoding process fails. 
       FIG. 8  depicts a process that may be used to determine a location in a sequence (or m-sequence) of a captured image. First, in step  801 , a data stream relating to a captured image is received. In step  802 , corresponding columns are extracted from A and a matrix M is constructed. 
     In step  803 , n independent column vectors are randomly selected from the matrix M and vector r is determined by solving equation (4). This process is performed Q times (for example, 100 times) in step  804 . The determination of the number of loop times is discussed in the section Loop Times Calculation. 
     In step  805 , r is sorted according to its associated number of error bits. The sorting can be done using a variety of sorting algorithms as known in the art. For example, a selection sorting algorithm may be used. The selection sorting algorithm is beneficial when the number Q is not large. However, if Q becomes large, other sorting algorithms (for example, a merge sort) that handle larger numbers of items more efficiently may be used. 
     The system then determines in step  806  whether error correction was performed successfully, by checking whether multiple r&#39;s are associated with the minimum number of error bits. If yes, an error is returned in step  809 , indicating the decoding process failed. If not, the position s of the extracted bits in the sequence (or m-sequence) is calculated in step  807 , for example, by using the Pohig-Hellman-Silver algorithm. 
     Next, the (X,Y) position in the array is calculated as: x=s mod m 1  and y=s mod m 2  and the results are returned in step  808 . 
     Location Determination 
       FIG. 9  shows a process for determining the location of a pen tip. The input is an image captured by a camera and the output may be a position coordinates of the pen tip. Also, the output may include (or not) other information such as a rotation angle of the captured image. 
     In step  901 , an image is received from a camera. Next, the received image may be optionally preprocessed in step  902  (as shown by the broken outline of step  902 ) to adjust the contrast between the light and dark pixels and the like. 
     Next, in step  903 , the image is analyzed to determine the bit stream within it. 
     Next, in step  904 , n bits are randomly selected from the bit stream for multiple times and the location of the received bit stream within the original sequence (or m-sequence) is determined. 
     Finally, once the location of the captured image is determined in step  904 , the location of the pen tip may be determined in step  905 . 
       FIG. 10  gives more details about  903  and  904  and shows the approach to extract the bit stream within a captured image. First, an image is received from the camera in step  1001 . The image then may optionally undergo image preprocessing in step  1002  (as shown by the broken outline of step  1002 ). The pattern is extracted in step  1003 . Here, pixels on the various lines may be extracted to find the orientation of the pattern and the angle θ. 
     Next, the received image is analyzed in step  1004  to determine the underlying grid lines. If grid lines are found in step  1005 , then the code is extracted from the pattern in step  1006 . The code is then decoded in step  1007  and the location of the pen tip is determined in step  1008 . If no grid lines were found in step  1005 , then an error is returned in step  1009 . 
     Outline of Enhanced Decoding and Error Correction Algorithm 
     With an embodiment of the invention as shown in  FIG. 12 , given extracted bits  1201  from a captured image (corresponding to a captured array) and the destination area, a variation of an m-array decoding and error correction process decodes the X,Y position.  FIG. 12  shows a flow diagram of process  1200  of this enhanced approach. Process  1200  comprises two components  1251  and  1253 .
         Decode Once. Component  1251  includes three parts.
           random bit selection: randomly selects a subset of the extracted bits  1201  (step  1203 )   decode the subset (step  1205 )   determine X,Y position with local constraint (step  1209 )   
           Decoding with Smart Bit Selection. Component  1253  includes four parts.
           smart bit selection: selects another subset of the extracted bits (step  1217 )   decode the subset (step  1219 )   adjust the number of iterations (loop times) of step  1217  and step  1219  (step  1221 )   determine X,Y position with local constraint (step  1225 )   
               

     The embodiment of the invention utilizes a discreet strategy to select bits, adjusts the number of loop iterations, and determines the X,Y position (location coordinates) in accordance with a local constraint, which is provided to process  1200 . With both components  1251  and  1253 , steps  1205  and  1219  (“Decode Once”) utilize equation (4) to compute r. 
     Let {circumflex over (b)} be decoded bits, that is:
 
{circumflex over (b)} t =r t M  (5)
 
The difference between b and {circumflex over (b)} are the error bits associated with r.
 
       FIG. 12  shows a flow diagram of process  1200  for decoding extracted bits  1201  from a captured image in accordance with embodiments of the present invention. Process  1200  comprises components  1251  and  1253 . Component  1251  obtains extracted bits  1201  (comprising K bits) associated with a captured image (corresponding to a captured array). In step  1203 , n bits (where n is the order of the m-array) are randomly selected from extracted bits  1201 . In step  1205 , process  1200  decodes once and calculates r. In step  1207 , process  1200  determines if error bits are detected for b. If step  1207  determines that there are no error bits, X,Y coordinates of the position of the captured array are determined in step  1209 . With step  1211 , if the X,Y coordinates satisfy the local constraint, i.e., coordinates that are within the destination area, process  1200  provides the X,Y position (such as to another process or user interface) in step  1213 . Otherwise, step  1215  provides a failure indication. 
     If step  1207  detects error bits in b, component  1253  is executed in order to decode with error bits. Step  1217  selects another set of n bits (which differ by at least one bit from the n bits selected in step  1203 ) from extracted bits  1201 . Steps  1221  and  1223  determine the number of iterations (loop times) that are necessary for decoding the extracted bits. Step  1225  determines the position of the captured array by testing which candidates obtained in step  1219  satisfy the local constraint. Steps  1217 - 1225  will be discussed in more details. 
     Smart Bit Selection 
     Step  1203  randomly selects n bits from extracted bits  1201  (having K bits), and solves for r 1 . Using equation (5), decoded bits can be calculated. Let I 1 ={kε{1,2, . . . ,K}|b k ={circumflex over (b)} k  }, Ī={kε{1,2, . . . ,K}|b k ≠{circumflex over (b)} k }, where {circumflex over (b)} k  is the k th  bit of {circumflex over (b)}, B 1 ={b k |kεI 1 } and  B   1 ={b k |kεĪ 1 }, that is, B 1  are bits that the decoded results are the same as the original bits, and  B   1  are bits that the decoded results are different from the original bits, I 1  and Ī 1  are the corresponding indices of these bits. It is appreciated that the same r 1  will be obtained when any n bits are selected from B 1 . Therefore, if the next n bits are not carefully chosen, it is possible that the selected bits are a subset of B 1 , thus resulting in the same r 1 , being obtained. 
     In order to avoid such a situation, step  1217  selects the next n bits according to the following procedure:
         1. Choose at least one bit from  B   1    1303  and the rest of the bits randomly from B 1    1301  and  B   1    1303 , as shown in  FIG. 13  corresponding to bit arrangement  1351 . Process  1200  then solves r 2  and finds B 2    1305 ,  1309  and  B   2    1307 ,  1311  by computing {circumflex over (b)} 2   t =r 2   t M 2 .   2. Repeat step  1 . When selecting the next n bits, for every  B   i  (i=1, 2, 3. . . , x−1, where x is the current loop number), there is at least one bit selected from  B   i . The iteration terminates when no such subset of bits can be selected or when the loop times are reached.
 
Loop Times Calculation
       

     With the error correction component  1253 , the number of required iterations (loop times) is adjusted after each loop. The loop times is determined by the expected error rate. The expected error rate p e  in which not all the selected n bits are correct is: 
                     p   e     =         (     1   -       C     K   -     n   e       n       C   K   n         )     lt     ≈     -     ⅇ     -       lt   ⁡     (       K   -   n     K     )         n   e                       (   6   )               
where lt represents the loop times and is initialized by a constant, K is the number of extracted bits from the captured array, n e  represents the minimum number of error bits incurred during the iteration of process  1200 , n is the order of the m-array, and C K   n  is the number of combinations in which n bits are selected from K bits.
 
     In the embodiment, we want p e  to be less than e −5 =0.0067. In combination with (6), we have: 
                     lt   i     =     min   ⁡     (       lt     i   -   1       ,       5       (       K   -   n     K     )       n   e         +   1       )               (   7   )               
Adjusting the loop times may significantly reduce the number of iterations of process  1253  that are required for error correction.
 
Determine X, Y Position with Local Constraint
 
     In steps  1209  and  1225 , the decoded position should be within the destination area. The destination area is an input to the algorithm, and it may be of various sizes and places or simply the whole m-array depending on different applications. Usually it can be predicted by the application. For example, if the previous position is determined, considering the writing speed, the destination area of the current pen tip should be close to the previous position. However, if the pen is lifted, then its next position can be anywhere. Therefore, in this case, the destination area should be the whole m-array. The correct X,Y position is determined by the following steps. 
     In step  1224  process  1200  selects r i  whose corresponding number of error bits is less than: 
                     N   e     =         log   10     ⁡     (     3   lt     )             log   10     ⁡     (       K   -   n     K     )       ×       log   10     ⁡     (     10   lr     )                   (   8   )               
where lt is the actual loop times and lr represents the Local Constraint Rate calculated by:
 
                   lr   =       area   ⁢           ⁢   of   ⁢           ⁢   the   ⁢           ⁢   destination   ⁢           ⁢   area     L             (   9   )               
where L is the length of the m-array.
 
     Step  1224  sorts r i  in ascending order of the number of error bits. Steps  1225 ,  1211  and  1212  then finds the first r i  in which the corresponding X,Y position is within the destination area. Steps  1225 ,  1211  and  1212  finally returns the X,Y position as the result (through step  1213 ), or an indication that the decoding procedure failed (through step  1215 ). 
     Illustrative Example of Enhanced Decoding and Error Correction Process 
     An illustrative example demonstrates process  1200  as performed by components  1251  and  1253 . Suppose n=3, K=5, I=(I 0  I 1  . . . I 6 ) t  is the m-sequence of order n=3. Then 
                   A   =     (           I   0           I   1           I   2           I   3           I   4           I   5           I   6               I   6           I   0           I   1           I   2           I   3           I   4           I   5               I   5           I   6           I   0           I   1           I   2           I   3           I   4           )             (   10   )               
Also suppose that the extracted bits b=(b 0  b 1  b 2  b 3  b 4 ) t , where K=5, are actually the s th , (s+1) th , (s+3) th , (s+4) th , and (s+6) th  bits of the m-sequence (these numbers are actually modulus of the m-array length L=2 n −1=2 3 −1=7). Therefore
 
                   M   =     (           I   0           I   1           I   3           I   4           I   6               I   6           I   0           I   2           I   3           I   5               I   5           I   6           I   1           I   2           I   4           )             (   11   )               
which consists of the 0 th , 1 st , 3 rd , 4 th , and  6   th  columns of A. The number s, which uniquely determines the X,Y position of b 0  in the m-array, can be computed after solving r=(r 0  r 1  r 2 ) t that are expected to fulfill b t =r t M. Due to possible error bits in b, b t =r t M may not be completely fulfilled.
 
     Process  1200  utilizes the following procedure. Randomly select n=3 bits, say {tilde over (b)} 1   t =(b 0  b 1  b 2 ), from b. Solving for r 1 :
 
{tilde over (b)} 1   t =r 1   t {tilde over (M)} 1   (12)
 
where {tilde over (M)} 1  consists of the 0th, 1st, and 2nd columns of M. (Note that {tilde over (M)} 1  is an n×n matrix and r 1   t  is a 1×n vector so that {tilde over (b)} 1   t  is a 1×n vector of selected bits.)
 
     Next, decoded bits are computed:
 
{circumflex over (b)} 1   t =r 1   t M  (13)
 
where M is an n×K matrix and r 1   t  is a 1×n vector so that {circumflex over (b)} 1   t  is a 1×K vector. If {circumflex over (b)} 1  is identical to b, i.e., no error bits are detected, then step  1209  determines the X,Y position and step  1211  determines whether the decoded position is inside the destination area. If so, the decoding is successful, and step  1213  is performed. Otherwise, the decoding fails as indicated by step  1215 . If {circumflex over (b)} 1  is different from b, then error bits in b are detected and component  1253  is performed. Step  1217  determines the set  B   1 , say {b 0  b 1  b 2  b 3 }, where the decoded bits are the same as the original bits. Thus,  B   1 ={b 4 } (corresponding to bit arrangement  1351  in  FIG. 13 ). Loop times (lt) is initialized to a constant, e.g., 100, which may be variable depending on the application. Note that the number of error bits corresponding to r 1  is equal to 1. Then step  1221  updates the loop time (lt) according to equation (7), lt 1 =min(lt,13)=13.
 
     Step  1217  next chooses another n=3 bits from b. If the bits all belong to B 1 , say {b 0  b 2  b 3 }, then step  1219  will determine r 1  again. In order to avoid such repetition, step  1217  may select, for example, one bit {b 4 } from  B   1 , and the remaining two bits {b 0  b 1 } from B 1 . 
     The selected three bits form {tilde over (b)} 2   t =(b 0  b 1  b 4 ). Step  1219  solves for r 2 :
 
{tilde over (b)} 2   t =r 2   t {tilde over (M)} 2   (14)
 
where {tilde over (M)} 2  consists of the  0   th , 1 st , and 4 th  columns of M.
 
     Step  1219  computes {circumflex over (b)} 2   t =r 2   t M. Find the set B 2 , e.g., {b 0  b 1  b 4 }, such that {circumflex over (b)} 2  and b are the same. Then  B   2 ={b 2  b 3 } (corresponding to bit arrangement  1353  in  FIG. 13 ). Step  1221  updates the loop times (lt ) according to equation (7). Note that the number of error bits associated with r 2  is equal to 2. Substituting into (7), lt 2 =min(lt 1 ,32)=13. 
     Because another iteration needs to be performed, step  1217  chooses another n=3 bits from b. The selected bits shall not all belong to either B 1  or B 2 . So step  1217  may select, for example, one bit {b 4 } from  B   1 , one bit {b 2 } from  B   2 , and the remaining one bit {b 0 }. 
     The solution of r, bit selection, and loop times adjustment continues until we cannot select any new n=3 bits such that they do not all belong to any previous B i &#39;s, or the maximum loop times lt is reached. 
     Suppose that process  1200  calculates five r i (i=1,2,3,4,5), with the number of error bits corresponding to 1, 2, 4, 3, 2, respectively. (Actually, for this example, the number of error bits cannot exceed 2, but the illustrative example shows a larger number of error bits to illustrate the algorithm.) Step  1224  selects r i &#39;s, for example, r 1 ,r 2 ,r 4 ,r 5 , whose corresponding numbers of error bits are less than N e  shown in (8). 
     Step  1224  sorts the selected vectors r 1 ,r 2 ,r 4 ,r 5  in ascending order of their error bit numbers: r 1 ,r 2 ,r 5 ,r 4 . From the sorted candidate list, steps  1225 ,  1211  and  1212  find the first vector r, for example, r 5 , whose corresponding position is within the destination area. Step  1213  then outputs the corresponding position. If none of the positions is within the destination area, the decoding process fails as indicated by step  1215 . 
     Apparatus 
       FIG. 14  shows an apparatus  1400  for decoding extracted bits  1201  from a captured array in accordance with embodiments of the present invention. Apparatus  1400  comprises bit selection module  1401 , decoding module  1403 , position determination module  1405 , input interface  1407 , and output interface  1409 . In the embodiment, interface  1407  may receive extracted bits  1201  from different sources, including a module that supports camera  203  (as shown in  FIG. 2A ). Bit selection module  1401  selects n bits from extracted bits  1201  in accordance with steps  1203  and  1217 . Decoding module  1403  decodes the selected bits (n bits selected from the K extracted bits as selected by bit selection module  1401 ) to determine detected bit errors and corresponding vectors r i  in accordance with steps  1205  and  1219 . Decoding module  1403  presents the determined vectors r i  to position determination module  1405 . Position determination module  1405  determines the X,Y coordinates of the captured array in accordance with steps  1209  and  1225 . Position determination module  1405  presents the results, which includes the X,Y coordinates if successful and an error indication if not successful, to output interface  1409 . Output interface  1409  may present the results to another module that may perform further processing or that may display the results. 
     Apparatus  1400  may assume different forms of implementation, including modules utilizing computer-readable media and modules utilizing specialized hardware such as an application specific integrated circuit (ASIC). 
     As can be appreciated by one skilled in the art, a computer system with an associated computer-readable medium containing instructions for controlling the computer system can be utilized to implement the exemplary embodiments that are disclosed herein. The computer system may include at least one computer such as a microprocessor, digital signal processor, and associated peripheral electronic circuitry. 
     Although the invention has been defined using the appended claims, these claims are illustrative in that the invention is intended to include the elements and steps described herein in any combination or sub combination. Accordingly, there are any number of alternative combinations for defining the invention, which incorporate one or more elements from the specification, including the description, claims, and drawings, in various combinations or sub combinations. It will be apparent to those skilled in the relevant technology, in light of the present specification, that alternate combinations of aspects of the invention, either alone or in combination with one or more elements or steps defined herein, may be utilized as modifications or alterations of the invention or as part of the invention. It may be intended that the written description of the invention contained herein covers all such modifications and alterations.