perform_arith_coding.sci

signal procesing toolbox

function [y,nbr_bits] = perform_arith_coding(xC, dir)

// perform_arith_coding - perform adaptive arithmetic coding
//
// 	[y,nbr_bits] = perform_arith_coding(x, dir);
//
//   dir=1 for encoding, dir=-1 for decoding.
//
// Interface to the code of (c) Karl Skretting
//
//   Copyright (c) 2008 Gabriel Peyre

//
// ------------------------------------------------------------------
// Arguments:
//  y        a column vector of non-negative integers (bytes) representing
//           the code, 0 <= y(i) <= 255.
//  Res      a matrix that sum up the results, size is (NumOfX+1)x4
//           one line for each of the input sequences, the columns are
//           Res(:,1) - number of elements in the sequence
//           Res(:,2) - unused (=0)
//           Res(:,3) - bits needed to code the sequence
//           Res(:,4) - bit rate for the sequence, Res(:,3)/Res(:,1)
//           Then the last line is total (which include bits needed to store NumOfX)
//  x       a cell array of column vectors of integers representing the
//           symbol sequences. (should not be to large integers)
//           If only one sequence is to be coded, we must make the cell array
//           like: xC=cell(2,1); xC{1}=x; // where x is the sequence
// ------------------------------------------------------------------
// Note: this routine is extremely slow since it is all Matlab code

// SOME NOTES ON THE FUNCTION
// This function is almost like Arith06, but some important changes have
// been done. Arith06 is buildt almost like Huff06, but this close connection
// is removed in Arith07. This imply that to understand the way Arith06
// works you should read the dokumentation for Huff06 and especially the
// article on Recursive Huffman Coding. To understand how Arith07 works it is
// only confusing to read about the recursive Huffman coder, Huff06.
//
// Arith07 code each of the input sequences in xC in sequence, the
// method used for each sequence depends on what kind (xType) the
// sequence is. We have
//  xType  Explanation
//   0     Empty sequence (L=0)
//   1     Only one symbol (L=1) and x>1
//   9     Only one symbol (L=1) and x=0/1
//  11     Only one symbol (L=1) and x<0
//   2     all symbols are equal, L>1, x(1)=x(2)=...=x(L)>1
//   8     all symbols are equal, L>1, x(1)=x(2)=...=x(L)=0/1
//  10     all symbols are equal, L>1, x(1)=x(2)=...=x(L)<0
//   3     non-negative integers, 0<=x(l)<=1000
//   4     not to large integers, -1000<=x(l)<=1000
//   5     large non-negative integers possible, 0<=x(l)
//   6     also possible with large negative numbers
//   7     A binary sequence, x(l)=0/1.
// Many functions are defined in this m-file:
//  PutBit, GetBit     read and write bit from/to bitstream
//  PutABit, GetABit   Arithmetic encoding of a bit, P{0}=P{1}=0.5
//  PutVLIC, GetVLIC   Variable Length Integer Code
//  PutS(x,S,C), GetS(S,C)  A symbol x, which is in S, is aritmetic coded
//                     according to the counts given by C.
//                     Ex: A binary variable with the givene probabilities
//                     P{0}=0.8 and P{1}=0.2, PutS(x,[0,1],[4,1]).
//  PutN(x,N), GetN(N) Arithmetic coding of a number x in range 0<=x<=N where
//                     we assume equal probability, P{0}=P{1}=...=P{N}=1/(N+1)


xC = uint32(xC);

if dir==1
//    xC = {xC};
end

//----------------------------------------------------------------------
// Copyright (c) 1999-2001.  Karl Skretting.  All rights reserved.
// Hogskolen in Stavanger (Stavanger University), Signal Processing Group
// Mail:  [e-mail]   Homepage:  http://www.ux.his.no/~karlsk/
//
// HISTORY:
// Ver. 1.0  19.05.2001  KS: Function made (based on Arith06 and Huff06)
//----------------------------------------------------------------------

// these global variables are used to read from or write to the compressed sequence
global y Byte BitPos
// and these are used by the subfunctions for arithmetic coding
global high low range ub hc lc sc K code

Mfile='Arith07';
K=24;  // number of bits to use in integers  (4 <= K <= 24)
Display=0;      // display progress and/or results

// check input and output arguments, and assign values to arguments
if (argn(2) < 1);
    error('function must have input arguments, see help.');
end
if (argn(1) < 1);
    error('function must have output arguments, see help.');
end

if dir==0 // (~iscell(xC))
    Encode=0;Decode=1;
    y=xC(:);            // first argument is y
    y=[y;0;0;0;0];      // add some zeros to always have bits available
else
    Encode=1;Decode=0;
    NumOfX = 1; // length(xC);
end

Byte=0;BitPos=1;         // ready to read/write into first position
low=0;high=2^K-1;ub=0;   // initialize the coder
code=0;
// we just select some probabilities which we belive will work quite well
TypeS=[  3,  4,  5,  7,  6,  1,  9, 11,  2,  8, 10,  0];
TypeC=[100, 50, 50, 50, 20, 10, 10, 10,  4,  4,  2,  1];

if Encode
    Res=zeros(NumOfX,4);
    // initalize the global variables
    y=zeros(10,1);    // put some zeros into y initially
    // start encoding, first write VLIC to give number of sequences
    PutVLIC(NumOfX);
    // now encode each sequence continuously
    Ltot=0;
    for num=1:NumOfX
        x=xC(:); // {num};
//        x=full(x(:));        // make sure x is a non-sparse column vector
        L=length(x);
        Ltot=Ltot+L;
        y=[y(1:Byte);zeros(50+2*L,1)];  // make more space available in y
        StartPos=Byte*8-BitPos+ub;      // used for counting bits
        // find what kind of sequenqe we have, save this in xType
        if (L==0)
            xType=0;
        elseif L==1
            if (x==0) | (x==1)       // and it is 0/1
                xType=9;
            elseif (x>1)             // and it is not 0/1 (and positive)
                xType=1;
            else                     // and it is not 0/1 (and negative)
                xType=11;
            end
        else
            // now find some more info about x
            maxx=max(x);
            minx=min(x);
            rangex=maxx-minx+1;
            if (rangex==1)                    // only one symbol
                if (maxx==0) | (maxx==1)       // and it is 0/1
                    xType=8;
                elseif (maxx>1)                // and it is not 0/1 (and positive)
                    xType=2;
                else                           // and it is not 0/1 (and negative)
                    xType=10;
                end
            elseif (minx == 0) & (maxx == 1)   // a binary sequence
                xType=7;
            elseif (minx >= 0) & (maxx <= 1000)
                xType=3;
            elseif (minx >= 0)
                xType=5;
            elseif (minx >= -1000) & (maxx <= 1000)
                xType=4;
            else
                xType=6;
            end
        end        // if (L==0)
        //
        if sum(xType==[4,6])        // negative values are present
            I=find(x);               // non-zero entries in x
            Sg=(sign(x(I))+1)/2;     // the signs will be needed later, 0/1
            x=abs(x);
        end
        if sum(xType==[5,6])        // we take the logarithms of the values
            I=find(x);               // non-zero entries in x
            xa=x;                    // additional bits
            x(I)=floor(log2(x(I)));
            xa(I)=xa(I)-2.^x(I);
            x(I)=x(I)+1;
        end
        //  now do the coding of this sequence
        PutS(xType,TypeS,TypeC);
        if (xType==1)       // one symbol and x=x(1)>1
            PutVLIC(x-2);
        elseif (xType==2)   // L>1 but only one symbol, x(1)=x(2)=...=x(L)>1
            PutVLIC(L-2);
            PutVLIC(x(1)-2);
        elseif sum(xType==[3,4,5,6])  // now 'normalized' sequences: 0 <= x(i) <= 1000
            PutVLIC(L-2);
            M=max(x);
            PutN(M,1000);         // some bits for M
            // initialize model
            T=[ones(M+2,1);0];
            Tu=flipud((-1:(M+1))');   // (-1) since ESC never is used in Tu context
            // and code the symbols in the sequence x
            for l=1:L
                sc=T(1);
                m=x(l);
                hc=T(m+1);lc=T(m+2);
                if hc==lc      // unused symbol, code ESC symbol first
                    hc=T(M+2);lc=T(M+3);
                    EncodeSymbol;      // code escape with T table
                    sc=Tu(1);hc=Tu(m+1);lc=Tu(m+2);  // symbol with Tu table
                    Tu(1:(m+1))=Tu(1:(m+1))-1;       // update Tu table
                end
                EncodeSymbol;  // code actual symbol with T table (or Tu table)
                // update T table, MUST be identical in Encode and Decode
                // this avoid very large values  in T even if sequence is very long
                T(1:(m+1))=T(1:(m+1))+1;
                if (rem(l,5000)==0)
                    dT=T(1:(M+2))-T(2:(M+3));
                    dT=floor(dT*7/8+1/8);
                    for m=(M+2):(-1):1; T(m)=T(m+1)+dT(m); end;
                end
            end          // for l=1:L
            // this end the "elseif sum(xType==[3,4,5,6])"-clause
        elseif (xType==7)   // L>1 and   0 <= x(i) <= 1
            PutVLIC(L-2);
            EncodeBin(x,L);    // code this sequence a special way
        elseif (xType==8)   // L>1 and   0 <= x(1)=x(2)=...=x(L) <= 1
            PutVLIC(L-2);
            PutABit(x(1));
        elseif (xType==9)   // L=1 and   0 <= x(1) <= 1
            PutABit(x(1));
        elseif (xType==10)       // L>1 and  x(1)=x(2)=...=x(L) <= -1
            PutVLIC(L-2);
            PutVLIC(-1-x(1));
        elseif (xType==11)       // L=1 and   x(1) <= -1
            PutVLIC(-1-x);
        end         // if (xType==1)
        // additional information should be coded as well
        if 0         // first the way it is not done any more
            if sum(xType==[4,6])           // sign must be stored
                for i=1:length(Sg); PutABit(Sg(i)); end;
            end
            if sum(xType==[5,6])        // additional bits must be stored
                for i=1:L
                    for ii=(x(i)-1):(-1):1
                        PutABit(bitget(xa(i),ii));
                    end
                end
            end
        else         // this is how we do it
            if sum(xType==[4,6])           // sign must be stored
                EncodeBin(Sg,length(I));    // since length(I)=length(Sg)
            end
            if sum(xType==[5,6])        // additional bits must be stored
                b=zeros(sum(x)-length(I),1);  // number of additional bits
                bi=0;
                for i=1:L
                    for ii=(x(i)-1):(-1):1
                        bi=bi+1;
                        b(bi)=bitget(xa(i),ii);
                    end
                end
                if (bi~=(sum(x)-length(I)))
                    error([Mfile,': logical error, bi~=(sum(x)-length(I)).']);
                end
                EncodeBin(b,bi);    // since bi=(sum(x)-length(I))
            end
        end
        //
        EndPos=Byte*8-BitPos+ub;    // used for counting bits
        bits=EndPos-StartPos;
        Res(num,1)=L;
        Res(num,2)=0;
        Res(num,3)=bits;
        if L>0; Res(num,4)=bits/L; else Res(num,4)=bits; end;
    end         // for num=1:NumOfX
    // flush the arithmetic coder
    PutBit(bitget(low,K-1));
    ub=ub+1;
    while ub>0
        PutBit(~bitget(low,K-1));
        ub=ub-1;
    end
    // flush is finished
    y=y(1:Byte);
    // varargout(1) = {y};
    if (nargout >= 2)
        // now calculate results for the total
        Res(NumOfX+1,1)=Ltot;
        Res(NumOfX+1,2)=0;
        Res(NumOfX+1,3)=Byte*8;
        if (Ltot>0); Res(NumOfX+1,4)=Byte*8/Ltot; else Res(NumOfX+1,4)=Byte*8; end;
        // varargout(2) = {Res};
    end
end         // if Encode

if Decode
    for k=1:K
        code=code*2;
        code=code+GetBit;   // read bits into code
    end
    NumOfX=GetVLIC;   // first read number of sequences
    xC=cell(NumOfX,1);
    for num=1:NumOfX
        // find what kind of sequenqe we have, xType, stored first in sequence
        xType=GetS(TypeS,TypeC);
        // now decode the different kind of sequences, each the way it was stored
        if (xType==0)       // empty sequence, no more symbols coded
            x=[];
        elseif (xType==1)       // one symbol and x=x(1)>1
            x=GetVLIC+2;
        elseif (xType==2)   // L>1 but only one symbol, x(1)=x(2)=...=x(L)>1
            L=GetVLIC+2;
            x=ones(L,1)*(GetVLIC+2);
        elseif sum(xType==[3,4,5,6])  // now 'normalized' sequences: 0 <= x(i) <= 1000
            L=GetVLIC+2;
            x=zeros(L,1);
            M=GetN(1000);      // M is max(x)
            // initialize model
            T=[ones(M+2,1);0];
            Tu=flipud((-1:(M+1))');   // (-1) since ESC never is used in Tu context
            // and decode the symbols in the sequence x
            for l=1:L
                sc=T(1);
                range=high-low+1;
                counts=floor(( (code-low+1)*sc-1 )/range);
                m=2; while (T(m)>counts); m=m+1; end;
                hc=T(m-1);lc=T(m);m=m-2;
                RemoveSymbol;
                if (m>M)     // decoded ESC symbol, find symbol from Tu table
                    sc=Tu(1);range=high-low+1;
                    counts=floor(( (code-low+1)*sc-1 )/range);
                    m=2; while (Tu(m)>counts); m=m+1; end;
                    hc=Tu(m-1);lc=Tu(m);m=m-2;
                    RemoveSymbol;
                    Tu(1:(m+1))=Tu(1:(m+1))-1;   // update Tu table
                end
                x(l)=m;
                // update T table, MUST be identical in Encode and Decode
                // this avoid very large values  in T even if sequence is very long
                T(1:(m+1))=T(1:(m+1))+1;
                if (rem(l,5000)==0)
                    dT=T(1:(M+2))-T(2:(M+3));
                    dT=floor(dT*7/8+1/8);
                    for m=(M+2):(-1):1; T(m)=T(m+1)+dT(m); end;
                end
            end          // for l=1:L
            // this end the "elseif sum(xType==[3,4,5,6])"-clause
        elseif (xType==7)   // L>1 and   0 <= x(i) <= 1
            L=GetVLIC+2;
            x=DecodeBin(L);    // decode this sequence a special way
        elseif (xType==8)   // L>1 and   0 <= x(1)=x(2)=...=x(L) <= 1
            L=GetVLIC+2;
            x=ones(L,1)*GetABit;
        elseif (xType==9)   // L=1 and   0 <= x(1) <= 1
            x=GetABit;
        elseif (xType==10)       // L>1 and  x(1)=x(2)=...=x(L) <= -1
            L=GetVLIC+2;
            x=ones(L,1)*(-1-GetVLIC);
        elseif (xType==11)       // L=1 and   x(1) <= -1
            x=(-1-GetVLIC);
        end         // if (xType==0)
        // additional information should be decoded as well
        L=length(x);
        I=find(x);
        if 0
            if sum(xType==[4,6])           // sign must be retrieved
                Sg=zeros(size(I));
                for i=1:length(I); Sg(i)=GetABit; end;   // and the signs   (0/1)
                Sg=Sg*2-1;                               // (-1/1)
            end
            if sum(xType==[5,6])        // additional bits must be retrieved
                xa=zeros(L,1);
                for i=1:L
                    for ii=2:x(i)
                        xa(i)=2*xa(i)+GetABit;
                    end
                end
                x(I)=2.^(x(I)-1);
                x=x+xa;
            end
        else
            if sum(xType==[4,6])           // sign must be retrieved
                Sg=DecodeBin(length(I));    // since length(I)=length(Sg)
                Sg=Sg*2-1;                  // (-1/1)
            end
            if sum(xType==[5,6])        // additional bits must be retrieved
                bi=sum(x)-length(I);     // number of additional bits
                b=DecodeBin(bi);
                bi=0;
                xa=zeros(L,1);
                for i=1:L
                    for ii=2:x(i)
                        bi=bi+1;
                        xa(i)=2*xa(i)+b(bi);
                    end
                end
                x(I)=2.^(x(I)-1);
                x=x+xa;
            end
        end
        if sum(xType==[4,6])           // sign must be used
            x(I)=x(I).*Sg;
        end
        // now x is the retrieved sequence
        xC = x; // {num}=x;
    end         // for num=1:NumOfX
    // varargout(1) = {xC};
end

if dir==1
    nbr_bits = Res(1,3);
else
    // undef for decoding
    nbr_bits = -1;
    y = xC; // {1};
end


endfunction


//----------------------------------------------------------------------
//----------------------------------------------------------------------

// --- The functions for binary sequences: EncodeBin and DecodeBin -------