Patent Publication Number: US-7710297-B2

Title: Method and apparatus for entropy coding and decoding

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This is a Continuation Application of PCT Application No. PCT/JP2007/058993, filed Apr. 19, 2007, which was published under PCT Article 21(2) in Japanese. 
    
    
     This application is based upon and claims the benefit of priority from prior Japanese Patent Application No. 2006-118169, filed Apr. 21, 2006, the entire contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to method and apparatus for entropy coding and decoding. 
     2. Description of the Related Art 
     In entropy coding, it is necessary to obtain the coding probability of symbols to be coded. As a method of calculating a coding probability, there has been proposed a method of using a full tree (also called a context tree) which associates one-dimensional data sequences output in the past with paths from the leaf nodes to the root node. The method disclosed in F. M. J. Willems, Y. M. Shtarkov, and T. J. Tjalkens, “The Context Tree Weighting Method: Basic Properties”, IEEE Trans. Inform. Theory, vol. 41, no. 3, pp. 653-664, May 1995 is called a context-tree weighting (CTW) algorithm. It is known that using the coding probability calculation method disclosed in this reference and T. Matsushima and S. Hirasawa, “A Bayes Coding Algorithm for FSM Sources”, Proc. Int. Symp. on Information Theory pp. 388, September 1995 makes it possible to obtain a high coding efficiency with respect to a one-dimensional data sequence output of a Markov process. 
     Assume that two matrices, namely an information matrix as symbols to be coded and a reference matrix are given. In this case, as an application of entropy coding, it is conceivable to use a technique of coding or decoding components of an information matrix by using components of a reference matrix. When using a coding probability calculation method using a context tree for coding/decoding of matrix components in this manner, it is necessary to set a path from a leaf node of a context tree to a root node in accordance with the correlation between matrices. For example, regarding the pixel levels of two temporarily consecutive frames in a video sequence as an information matrix and a reference matrix, the correlation between a component of the information matrix and a component of the reference matrix generally increases with a decrease in the distance between the two matrix components. In this case, performing coding/decoding while associating a one-dimensional data sequence, in which a reference matrix is simply arranged in the raster scan order, i.e., the horizontal direction, with a path from a leaf node to a root node of a context tree amounts to performing coding/decoding without any consideration to the correlation between a component of an information matrix of interest and a component of a reference matrix arranged in the vertical direction. This is not preferable in terms of coding efficiency. 
     BRIEF SUMMARY OF THE INVENTION 
     It is an object of the present invention to form a context tree suitable for the correlation between an information matrix and a reference matrix and allow high-efficiency entropy coding and decoding by using the context tree. 
     According to a first aspect of the present invention, there is provided an entropy coding method comprising: generating a reference matrix containing matrix components from an information matrix, the reference matrix being coded so as to have a correlation with the information matrix; classifying the matrix components within a range around an i-th row/j-th column component (where  i  and  j  are arbitrary integers) of the reference matrix to component sets with reference to distances from the i-th row/j-th column component, the component sets each containing matrix components equal in distance from the i-th row/j-th column component of the reference matrix; converting the matrix components belonging to the each component set into symbols; forming a context tree including a root node, a plurality of internal nodes corresponding to the component sets, and a plurality of branches which have one-to-one correspondence with the symbols and a plurality of leaf nodes; associating a sequence obtained by arranging the symbols in descending order of distance between the i-th row/j-th column component of the reference matrix and the matrix components with a path extending from the leaf node to the root node of the context tree, and calculating a coding probability of the i-th row/j-th column component of the information matrix as a weighting sum of probabilities which the respective nodes hold; and generating a codeword by arithmetically coding the i-th row/j-th column component of the information matrix in accordance with the coding probability. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
         FIG. 1  is a block diagram showing an entropy coding apparatus according to an embodiment; 
         FIG. 2  is a view showing an example of an information matrix; 
         FIG. 3  is a view showing an example of a reference matrix; 
         FIG. 4  is a view showing an example of classification in an information classifier; 
         FIG. 5  is a flowchart showing a context tree forming procedure; 
         FIG. 6  is a view showing an example of a context tree; 
         FIG. 7  is a view showing an example of how symbol sequences are associated with paths from the leaf nodes to the root node of the context tree in  FIG. 6 ; 
         FIG. 8  is a flowchart showing a coding probability calculation procedure; 
         FIG. 9  is a flowchart showing an entropy coding procedure in a coder; 
         FIG. 10  is a block diagram of an entropy decoding apparatus according to an embodiment; and 
         FIG. 11  is a flowchart showing an entropy decoding procedure in a decoder. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The embodiments of the present invention will be described below. 
     (Entropy Coding Apparatus) 
     As shown in  FIG. 1 , an entropy coding apparatus according to an embodiment includes a reference sequence generator  105 , information classifier  103 , information converter  104 , coding probability calculator  101 , and coder  102 . The reference sequence generator  105 , the coding probability calculator  101  and the coder  102  receive an information matrix  11  to be coded. 
     The reference sequence generator  105  is a memory which stores, for example, the coded components of an information matrix, and outputs already coded components of an information matrix as a reference matrix  14 . The information classifier  103  forms a set  15  by classifying components within a predetermined range around the ith row/jth column component (where  i  and  j  are arbitrary integers) of the reference matrix  14  in accordance with a predetermined rule. In this specification, components of the information matrix  11  and reference matrix  14  are also called matrix components. The information converter  104  converts each matrix component of the set  15  into a symbol  16  in accordance with a predetermined operation. The coding probability calculator  101  calculates a coding probability  12  by using the ith row/jth column component of the information matrix  11  and a sequence  17  in which the symbol  16  output from the information converter  104  are arranged in descending order of the distance between the ith row/jth column component of the reference matrix and a corresponding component of the reference matrix. The coder  102  generates a codeword  13  of the ith row/jth column component of the information matrix  11  by arithmetic coding in accordance with the coding probability  12  which is calculated by, for example, the CTW algorithm. 
     Each component of the entropy coding apparatus in  FIG. 1  will be described in detail below. 
     Assume that both the information matrix  11  and the reference matrix  14  are V rows×H columns matrices, and the distance between the i m th row/j m th column component and the i s th row/j s th column component is given by
 
√{square root over ((i m −i s ) 2 +(j m −j s ) 2 )}{square root over ((i m −i s ) 2 +(j m −j s ) 2 )}  (11)
 
Letting Ξ and Ψ respectively be sets of values which the information matrix  11  and the reference matrix  14  can take, |Ξ| and |Ψ| respectively be the numbers of elements of Ξ and Ψ, and N be a total set of natural numbers equal to or greater than 1, the distance between the ith row/jth column component of the information matrix and a matrix component of the set  15  of the reference matrix  14  is given by
 
√{square root over ((i−i s ) 2 +(j−j s ) 2 )}{square root over ((i−i s ) 2 +(j−j s ) 2 )}  (12)
 
where i s  and j s  represent the row number and column number of the above matrix component, respectively.
 
     The reference sequence generator  105  generates an already coded component of the information matrix  11  as a component of the reference matrix  14 . Assume that the components of the information matrix  11  are coded in the raster scan order. In this case, the reference sequence generator  105  generates a component at a position corresponding to the already coded component of the reference matrix  14  by using an already coded component of the information matrix  11  in accordance with a predetermined rule. For example, as shown in  FIG. 2 , when coding an ith row/jth column component x i,j  of the information matrix  11  as shown in  FIG. 2 , the reference sequence generator  105  generates a component of the reference matrix  14  as shown in  FIG. 3 . 
     The information classifier  103  classifies components within a predetermined range around the ith row/jth column component of the reference matrix  14  with reference to the distances from the ith row/jth column component, and forms the set  15  having components equal in distance from the ith row/jth column component. For example, as shown in  FIG. 4 , the information classifier  103  classifies the components within the range of the ith row/(j−1)th column, (i−1)th row/jth column, (i−1)th row/(j+1)th column, (I−1)th row/(j−1)th column, ith row/(j−2)th column, and (i−2)th row/jth column of the reference matrix  14  according to distances of 1, √{square root over (2)}, and 2 from the ith row/jth column component to form three sets, i.e., a combination of the ith row/(j−1)th column component and the (i−1)th row/jth column component, a combination of the (i−1)th row/(j+1)th column component and the (i−1)th row/(j−1)th column component, and a combination of the ith row/(j−2)th column component and the (i−2)th row/jth column component. 
     This apparatus uses predetermined constants for components, of the ith row/(j−1)th column, (i−1)th row/jth column, (i−1)th row/(j+1)th column, (i−1)th row/(j−1)th column, ith row/(j−2)th column, and (i−2)th row/jth column components, which fall outside the range of the reference matrix  14 . If, for example, h, vεN, h&lt;0 or h&gt;H, and v&lt;0 or v&gt;V, then y v,h =0. 
     The information converter  104  converts each matrix component of the set  15  into the symbol  16  in accordance with a predetermined operation. Let z r  be the symbol  16  obtained by converting a matrix component of a set, of the set  15  formed by the information classifier  103 , which includes the rth nearest component to the ith row/jth column component, Z r  be a set of values which z r  can take, and |Z r | be the number of elements of Z r . Consider, for example, a case wherein the information converter  104  converts a matrix component of the following set:
 
 G={y   g   εΨ|gεZ,  0≦ g≦|G|− 1}  (13)
 
which is classified by the information classifier  103  into
 
                     ∑     g   =   0            G        -   1       ⁢     y   g             (   14   )               
In this case, the information converter  104  generates symbols z 1 , z 2  and z 3  by performing the following operation for a combination of an ith row/(j−1)th column component y i,j−1  and an (i−1)th row/jth column component y i−1,j , a combination of an (i−1)th row/(j+1)th column component y i−1,j+1  and an (i−1)th row/(j−1)th column component y i−1,j−1 , and a combination of an ith row/(j−2)th column component y i,j−2  and an (i−2)th row/jth column component  v .
   z   1   =y   i,j−1   +y   i−1,j   (15.1)   z   2   =y   i−1,j+1   +y   i−1,j−1   (15.2)   z   3   =y   i,j−2   +y   i−2,j   (15.3) 
     When converting a matrix component of a set G into 
                     ∑     g   =   0            G        -   1       ⁢       y   g     ⁢          Ψ        g               (   16   )               
the information converter  104  generates symbols z 1 , z 2 , and z 3  by performing the operation represented by the following equations:
   z   1   =y   i,j−1   +y   i−1,j ×|Ψ|   z   2   =y   i−1,j+1   +y   i−1,j−1 ×|Ψ|   z   3   =y   i,j−2   +y   i−2,j ×|Ψ|  (17) 
     The coding probability calculator  101  calculates a coding probability by using the ith row/jth column component of the information matrix  11 , the sequence  17  of the symbols  16  output from the information converter  104 , and a context tree formed in advance. That is, the coding probability calculator  101  associates a sequence in which the symbols  16  are arranged in descending order of the distance between the ith row/jth column component of the reference matrix  14  and the matrix components of the set G with paths from the leaf nodes to the root node of a context tree, and calculates the coding probability of the ith row/jth column component of the information matrix  11  as the weighting sum of probabilities which the respective nodes hold. 
     A context tree forming method will be described below with reference to  FIGS. 5 ,  6 , and  7 . The flowchart of  FIG. 5  shows a context tree forming procedure. 
     First of all, in step S 101 , this apparatus sets a maximum depth D of a context tree. Specifically, the maximum depth D is set to be equal to the number of sets  15  formed by the information classifier  103 . 
     In step S 102 , the apparatus generates a root node of the content tree. In this case, letting  d  be a counter representing a depth and = be an assignment operator, d=0. 
     In step S 103 , the apparatus determines whether  d  is smaller than the maximum depth D. If  d  is smaller than D, the apparatus performs the processing in step S 104 . Otherwise, the apparatus terminates the formation of the context tree. 
     In step S 104 , the apparatus associates a node at the depth  d  with a set, of the sets  15  formed by the information classifier  103 , which includes the (d+1)th nearest component to the ith row/jth column component. 
     In step S 105 , the apparatus generates branches extending from the node at the depth  d  and child nodes. In this case, the number of branches is set to be equal to a number |Z d | of values which symbols obtained by converting components of a set corresponding to the node at the depth  d  can take. 
     In step S 106 , the branches generated in step S 105  are associated in one-to-one correspondence with the symbols obtained by converting the components of the set corresponding to the node at the depth  d . 
     In step S 107 , d=d+1 is set. The process then returns to step S 103 . 
     The above processing makes it possible to associate the sequences  17  of the symbols  16  output from the information converter  104  in one-to-one correspondence with paths from the leaf nodes of the context tree to the root node. For example, with regard to a component y v,h  (h, vεN, 1≦h≦H, 1≦v≦V) of a reference matrix, y v,h ε{0, 1} is set. The information classifier  103  forms sets by classifying the components within the range of the ith row/(j−1)th column, (i−1)th row/jth column, (i−1)th row/(j+1)th column, (i−1)th row/(j−1)th column, ith row/(j−2)th column, and (i−2)th row/jth column of the reference matrix. 
     When converting the components of the respective sets according to equations (15.1), (15.2), and (15.3), the coding probability calculator  101  obtains a context tree like that shown in  FIG. 6 . If, for example, the symbols  16  output from an information converter  104  are z 1 =0, z 2 =1, and z 3 =2, z 3  z 2  z 1 =210 which is the sequence  17  of the symbols  16  as shown in  FIG. 7  corresponds to a path from a leaf node to the node root of the context tree, which is drawn with the thick line in  FIG. 7 . As is understood from this example, in the sequence  17 , the symbols  16  are arranged in descending order of the distance between the ith row/jth column component of the reference matrix  14  and a corresponding matrix component of the set  15 . That is, the symbols  16  are sequentially arranged starting from a symbol corresponding to a set including a matrix component at the most distant position from the ith row/jth column component of the reference matrix  14 . 
     A procedure of calculating a coding probability by using a context tree formed in this manner according to the CTW algorithm will be described next. When obtaining the coding probability of the ith row/jth column component of an information matrix, this apparatus can perform coding probability calculation by using T. Matsushima and S. Hirasawa, “A Bayes Coding Algorithm for FSM Sources”, Proc. Int. Symp. on Information Theory pp. 388, September 1995, the entire contents of which are incorporated herein by reference. A procedure of performing coding probability calculation in the raster scan order will be described below with reference to  FIG. 8 . 
     Let x t  be the tth component of an information matrix in the raster scan order. With regard to a context tree, q t (s), p t,s (x|s), p t,m (x|s), n t (x|s), α(x|s)(xεΞ) is made to correspond to an arbitrary node  s  if it is an internal node. p t,s (x|s), p t,m (x|s), n t (x|s), α(x|s)(xεΞ) is made to correspond to the node  s  if it is a node. 
     In step S 201 , this apparatus associates a sequence z D z D−1  . . . z 1  comprising symbols z d  (1≦d≦D) output from the information converter  104  with a path from a node on the context tree to the root node. 
     In step S 202 , the apparatus initializes the counter  d  representing a depth by the following equation.
 
d=D  (*)
 
     In step S 203 , it is determined whether  d  is equal to or greater than zero. If YES in step S 203 , the apparatus performs the processing in step S 204 . If NO in step S 203 , the apparatus performs the processing in step S 208 . 
     In step S 204 , it is determined whether  d  is smaller than the maximum depth D. If YES in step S 204 , the apparatus performs the processing in step S 205 . If NO in step S 204 , the apparatus performs the processing in step S 206 . 
     In step S 205 , the apparatus performs the following calculation with respect to a node s d  at the depth d on the path set in step S 201 . 
     
       
         
           
             
               
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     In step S 206 , the apparatus calculates the following calculation with respect to the node s d  at the depth  d  on the path set in step S 201 . 
     
       
         
           
             
               
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α( x|s   d )=½( x εΞ)  (19)
 
     In step S 207 , d=d−1 is set. 
     In step S 208 , the apparatus initializes the counter  d  representing a depth by the equation (*) again. 
     In step S 209 , it is determined whether  d  is equal to or greater than zero. If YES in step S 209 , the apparatus performs the processing in step S 210 . If NO in step S 209 , the apparatus terminates the CTW algorithm. 
     In step S 210 , it is determined whether d is smaller than D. If YES in step S 210 , the apparatus performs the processing in step S 211 . If NO in step S 210 , the apparatus performs the processing in step S 212 . 
     In step S 211 , the apparatus performs the following calculation with respect to the node s d  at the depth  d  on the path set in step S 201 . 
     
       
         
           
             
               
                 
                   
                     
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     The apparatus performs the calculation of equation (20) with respect to all xεΞ. 
     In step S 212 , the apparatus performs the following calculation with respect to the node s d  at the depth  d  on the path set in step S 201 . 
     
       
         
           
             
               
                 
                   
                     
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                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     The apparatus performs the calculation of equation (21) with respect to all xεΞ. 
     In step S 213 , d=d−1 is set. 
     The apparatus performs the above processing and regards the following value corresponding to a root node s 0  as a coding probability.
 
P t,m (x|s 0 )  (22)
 
     The coder  102  generates a codeword by performing arithmetic coding, i.e., entropy coding (variable-length coding), with respect to the ith row/jth column component of the information matrix  11  in accordance with the coding probability calculated in this manner. That is, the coder  102  generates a codeword corresponding to the ith row/jth column component of the information matrix  11  by an entropy coding technique of controlling the codeword length in accordance with the coding probability. In other words, the coder  102  generates a codeword corresponding to the ith row/jth column component of the information matrix  11  by performing arithmetic computation based on addition, subtraction, multiplication, division, or bit processing computation in accordance with the coding probability. In addition, the coder  102  fragments an interval on a number line into small sections each of which has a width corresponding to the coding probability of a sequence and is associated with the sequence, and expresses one point in a small section corresponding to a sequence to be coded as a codeword, thereby generating a codeword corresponding to the ith row/jth column component of the information matrix  11 . 
     A case wherein the codes of the information matrix  11  are coded in the raster scan order to generate binary codewords (variable-length codes) will be described below with reference to  FIG. 9 . 
     Let  n  be the number of components of the information matrix  11  and reference matrix  14 , x t  be the first to tth sequences in the raster scan order, and x 0  be a null sequence. The number of components of a matrix is given by n=H×V. 
     In step S 301 , the value of the counter is set to t=1. 
     In step S 302 , it is determined whether  t  is equal to or smaller than  n . If YES in step S 302 , the apparatus performs the processing in step S 303 . If NO in step S 302 , the apparatus performs the processing in step S 305 . 
     In step S 303 , the apparatus performs the following calculation in binary notation. 
                       T   ⁡     (       x     t   -   1       ⁢   x     )       =       ⌊       T   ⁡     (     x     t   -   1       )       ⁢       P     t   ,   m       ⁡     (     x   ❘     s   0       )         ⌋     ω       ⁢     
     ⁢     (     x   ∈   Ξ     )     ⁢     
     ⁢       F   ⁡     (     x   t     )       =       F   ⁡     (     x     t   -   1       )       +       ∑       x   ′     &lt;   x       ⁢     T   ⁡     (       x     t   -   1       ⁢     x   ′       )                     (   23   )               
where └ ┘ ω  indicates that binary floating-point representation is truncated to ω digits.
 
     Assume that F(x 0 ) and T(x 0 ) are set to F(x 0 )=0 and T(x 0 )=1. 
     In step S 304 , t=t+1 is set. 
     In step S 305 , the apparatus generates a codeword by sequentially arranging values from the first decimal place of F(x n ) calculated in the preceding processing to the least significant digit upon addition based on mathematical expression (23). 
     As described above, the entropy coding apparatus can form a context tree suitable for the correlation between an information sequence and a reference matrix, and hence improves coding efficiency as compared with the case of simply associating a one-dimensional data sequence in which a reference matrix is simply arranged in the raster scan order with a path from a leaf node of a context tree to the root node. 
     (Entropy Decoding Apparatus) 
     An entropy decoding apparatus corresponding to the above entropy coding apparatus will be described next. As shown in  FIG. 10 , the entropy decoding apparatus according to an embodiment includes a decoder  201 , information classifier  202 , information converter  203 , coding probability calculator  204 , and reference sequence generator  205 . The decoder  201  receives a codeword (variable-length code)  21  to be decoded. The decoder  201  generates an information matrix  22  by entropy-decoding the codeword  21 . 
     The reference sequence generator  205  is a memory which stores, for example, a decoded component of an information matrix, and outputs an already decoded component of an information matrix as a reference matrix  23 . The information classifier  202  forms a set  24  by classifying components within a predetermined range around the ith row/jth column component (where  i  and  j  are arbitrary integers) of the reference matrix  23  in accordance with a predetermined rule. In this specification, components of the information matrix  22  and reference matrix  23  are also called matrix components. The information converter  203  converts each matrix component of the set  24  into a symbol  25  in accordance with a predetermined operation. The coding probability calculator  204  calculates a coding probability  27  by using the ith row/jth column component of the information matrix  11  and a sequence  26  in which the symbol  25  output from the information converter  203  are arranged in descending order of the distance between the ith row/jth column component of the reference matrix and a component of the reference matrix. The decoder  201  generates the information matrix  22  by entropy-decoding the codeword  21  in accordance with the coding probability  27  calculated by, for example, the CTW algorithm. The reference sequence generator  205  and the coding probability calculator  204  receive the ith row/jth column component of the information matrix  22 . 
     Each component of the entropy decoding apparatus in  FIG. 10  will be described in detail below. 
     A case wherein the entropy coding apparatus shown in  FIG. 1  codes the components of an information matrix in the raster scan order will be described. The reference sequence generator  205  generates a component of the reference matrix  23  from a component of the information matrix  22 , which is obtained by decoding the codeword  21 , in accordance with a predetermined rule. The rule based on which the reference matrix  23  is generated from components of the information matrix  22  is the same as the rule based on which the reference sequence generator  105  in the entropy coding apparatus shown in  FIG. 1  generates the reference matrix  14  from components of the information matrix  11 . 
     The information classifier  202 , information converter  203 , and coding probability calculator  204  each perform the same processing as that performed by a corresponding one of the information classifier  103 , information converter  104 , and coding probability calculator  101  in the entropy coding apparatus shown in  FIG. 1 . The information classifier  202  classifies components within a predetermined range around the ith row/jth column component of the reference matrix  23 , e.g., the components within the range of the ith row/(j−1)th column, (i−1)th row/jth column, (i−1)th row/(j+1)th column, (i−1)th row/(j−1)th column, ith row/(j−2)th column, and (i−2)th row/jth column of the reference matrix  22  shown in  FIG. 4 , with reference to the distances from the ith row/jth column component, and forms the set  24  having components equal in distance from the ith row/jth column component. Assume that a predetermined range around the ith row/jth column component of the reference matrix  23  is set in advance, which is common to the entropy coding apparatus. In addition, with regard to components, of the ith row/(j−1)th column, (i−1)th row/jth column, (i−1)th row/(j+1)th column, (i−1)th row/(j−1)th column, ith row/(j−2)th column, and (i−2)th row/jth column components, which fall outside the range of the reference matrix  23 , constants to be applied are set in advance so as to be common to those in the entropy coding apparatus shown in  FIG. 1 . 
     The information converter  203  converts each matrix component of the set  24  into the symbol  25  in accordance with the same operation as that performed by the information converter  104  in the entropy coding apparatus shown in  FIG. 1 , e.g., the operation based on mathematical expressions (12) to (17), and outputs the sequence  26  of the symbols  25 . The coding probability calculator  204  calculates a coding probability by using the sequence  26  of the symbols  25  from the information converter  203  and a context tree formed in advance. The context tree forming method and the coding probability calculation method based on it which are used in this case are the same as those used by the coding probability calculator  101  in the entropy coding apparatus shown in  FIG. 1 . 
     The decoder  201  entropy-decodes (variable-length decodes) the codeword  21  in accordance with the coding probability  27  to reconstruct the ith row/jth column component of the information matrix  22 . The procedure executed by the decoder  201  will be described with reference to  FIG. 11  by exemplifying the case of decoding the codeword generated by the entropy coding apparatus shown in  FIG. 1 . 
     In step S 401 , the decoder  201  sets a counter value  t  to t=1. 
     In step S 402 , the decoder  201  determines whether the counter value  t  is equal to or smaller than  n . If YES in step S 402 , the decoder  201  performs the processing in step S 403 . If NO in step S 402 , the decoder  201  terminates the decoding operation. 
     In step S 403 , the decoder  201  decodes a component x t  of an information matrix which corresponds to the tth place in the raster scan order according to the following equation. Assume that the decoder  201  is to perform calculation in binary notation. 
                       T   ⁡     (       x     t   -   1       ⁢   x     )       =       ⌊       T   ⁡     (     x     t   -   1       )       ⁢       P     t   ,   m       ⁡     (     x   ❘     s   0       )         ⌋     ω       ⁢     
     ⁢     (     x   ∈   Ξ     )     ⁢     
     ⁢     x   t     =     max   ⁢     {       x   ∈   Ξ     |         ∑       x   ′     &lt;   x       ⁢     T   ⁡     (     xt   -     1   ⁢     x   ′         )         ≤     W     t   -   1           }               (   24   )               
where W 0 =F(x n ).
 
     In step S 404 , the decoder  201  sets 
     
       
         
           
             
               
                 
                   
                     W 
                     t 
                   
                   = 
                   
                     
                       W 
                       
                         t 
                         - 
                         1 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           
                             x 
                             ′ 
                           
                           &lt; 
                           x 
                         
                       
                       ⁢ 
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               
                                 t 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               x 
                               ′ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     In step S 405 , the decoder  201  sets t=t+1. The process returns to step S 402 . 
     Another embodiment of the present invention will be described next. The above embodiment uses the memory storing a coded information sequence or a decoded information sequence as a reference sequence generator. The above description has exemplified the case wherein an information sequence to be coded or an information sequence obtained by decoding a codeword and a reference sequence are the same sequence, i.e., the reference sequence is an information sequence which has already been coded or decoded. However, a reference sequence need not be the same sequence as an information sequence, and may be a sequence having a correlation with an information sequence. 
     Application examples of the entropy coding apparatus and entropy decoding apparatus according to the embodiment will be described next. For example, a video coding/decoding apparatus uses entropy coding/decoding. The video coding apparatus divides one frame of a video sequence into blocks, and orthogonally transforms the pixel levels of an input video sequence or prediction residue for each block by discrete cosine transform (DCT) or the like, thereby generating orthogonal transformation coefficients. This apparatus quantizes orthogonal transformation coefficients and entropy-codes the quantized orthogonal transformation coefficients. 
     Such a video coding apparatus may use a method of sending, as information for determining whether all the quantized orthogonal transformation coefficients contained in each block are zero, a value of 0 when all the coefficients are zero, and a value of 1 when all the coefficients are not zero, and sending the values of the quantized orthogonal transformation coefficients only when the value of the flag is 1. In this case, the apparatus generates an information sequence by arranging pieces of information, each indicating whether all the quantized orthogonal transformation coefficients contained in each block are zero, in accordance with the positions of the blocks of one frame. The entropy coding apparatus according to an embodiment is suitable for entropy-coding such an information sequence. In addition, the entropy decoding apparatus according to an embodiment is suitable for decoding the codeword generated by the above operation to reproduce the original information sequence. 
     Additional advantages and modifications will readily occur to those skilled in the art. Therefore, the invention in its broader aspects is not limited to the specific details and representative embodiments shown and described herein. Accordingly, various modifications may be made without departing from the spirit or scope of the general inventive concept as defined by the appended claims and their equivalents.