perform_arith_coding.sci.svn-base

signal procesing toolbox

// These function may call themselves recursively
function EncodeBin(x,L)
global y Byte BitPos
global high low range ub hc lc sc K code

Display=0;
x=x(:);
if (length(x)~=L); error('EncodeBin: length(x) not equal L.'); end;

// first we try some different coding methods to find out which one
// that might do best. Many more methods could have been tried, for
// example methods to check if x is a 'byte' sequence, or if the bits
// are grouped in other ways. The calling application is the best place
// to detect such dependencies, since it will (might) know the process and
// possible also its statistical (or deterministic) properties.
// Here we just check some few coding methods: direct, split, diff, diff+split
// The main variables used for the different methods are:
//  direct: x, I, J, L11, b0
//  split: x is split into x1 and x2, L1, L2, b1 is bits needed
//  diff: x3 is generated from x, I3, J3 L31, b2
//  diff+split: x3 is split into x4 and x5, L4, L5, b3
//
MetS=[0,1,2,3];       // the different methods, direct, split, diff, diff+split
MetC=[9,3,3,1];       // and the counts (which gives the probabilities)
// first set how many bits needed to code the method
b0=log2(sum(MetC))-log2(MetC(1));
b1=log2(sum(MetC))-log2(MetC(2));
b2=log2(sum(MetC))-log2(MetC(3));
b3=log2(sum(MetC))-log2(MetC(4));

I=find(x(1:(L-1))==1);      // except last element in x
J=find(x(1:(L-1))==0);
L11=length(I);
// perhaps is 'entropy' interesting
// N1=L11+x(L);
// N0=L-N1;
// b=N1*log2(L/N1)+N0*log2(L/N0);
// disp(['L=',int2str(L),',  N1=',int2str(N1),',  (e)bits=',num2str(b)]);
b0=b0+BitEst(L,L11+x(L));   // bits needed for the sequence without splitting
if Display
    disp(['EncodeBin: x of length ',int2str(L),' and ',int2str(L11+x(L)),...
        ' ones (p=',num2str((L11+x(L))/L,'//1.3f'),...
        ') can be coded by ',num2str(b0,'//6.0f'),' bits.']);
end
// diff with a binary sequence is to indicate wether or not a symbol is
// the same as preceeding symbol or not, x(0) is assumed to be '0' (zero).
// This is the DPCM coding scheme on a binary sequence
x3=abs(x-[0;x(1:(L-1))]);
I3=find(x3(1:(L-1))==1);      // except last element in x3
J3=find(x3(1:(L-1))==0);
L31=length(I3);
b2=b2+BitEst(L,L31+x3(L));  // bits needed for the sequence without splitting
if Display
    disp(['EncodeBin: diff x, L=',int2str(L),' gives ',int2str(L31+x3(L)),...
        ' ones (p=',num2str((L31+x3(L))/L,'//1.3f'),...
        ') can be coded by ',num2str(b2,'//6.0f'),' bits.']);
end
//
if (L>40)
    // only now we try to split the sequences x and x3
    if (L11>3) & ((L-L11)>3)
        // try to split x into x1 and x2, depending on previous symbol
        if L11<(L/2)
            x1=x(I+1);
            x2=[x(1);x(J+1)];
            L1=L11;L2=L-L11;
        else
            x1=[x(1);x(I+1)];
            x2=x(J+1);
            L1=L11+1;L2=L-L11-1;
        end
        b11=BitEst(L1,length(find(x1)));  // bits needed for x1
        b12=BitEst(L2,length(find(x2)));  // bits needed for x2
        // b1 is bits to code: Method, L11, x1 and x2
        b1=b1+log2(L)+b11+b12;
        if Display
            disp(['EncodeBin, x -> x1+x2:      lengths are ',int2str(L1),'+',int2str(L2),...
                '  bits are ',num2str(b11,'//6.0f'),'+',num2str(b12,'//6.0f'),...
                '.  Total is ',num2str(b1,'//6.0f'),' bits.']);
        end
    else
        b1=b0+1;      // just to make this larger
    end
    if (L31>3) & ((L-L31)>3)
        // try to split x3 into x4 and x5, depending on previous symbol
        if L31<(L/2)
            x4=x3(I3+1);
            x5=[x3(1);x3(J3+1)];
            L4=L31;L5=L-L31;
        else
            x4=[x3(1);x3(I3+1)];
            x5=x3(J3+1);
            L4=L31+1;L5=L-L31-1;
        end
        b31=BitEst(L4,length(find(x4)));  // bits needed for x4
        b32=BitEst(L5,length(find(x5)));  // bits needed for x5
        // b3 is bits to code: Method, L31, x4 and x5
        b3=b3+log2(L)+b31+b32;
        if Display
            disp(['EncodeBin, diff x -> x4+x5: lengths are ',int2str(L4),'+',int2str(L5),...
                '  bits are ',num2str(b31,'//6.0f'),'+',num2str(b32,'//6.0f'),...
                '.  Total is ',num2str(b3,'//6.0f'),' bits.']);
        end
    else
        b3=b2+1;      // just to make this larger
    end
else
    b1=b0+1;      // just to make this larger
    b3=b2+1;      // just to make this larger
end
// now code x by the best method of those investigated
[b,MetI]=min([b0,b1,b2,b3]);
MetI=MetI-1;
PutS(MetI,MetS,MetC);   // code which method to use
//
if MetI==0
    // code the sequence x
    N1=L11+x(L);
    N0=L-N1;
    PutN(N1,L);            // code N1, 0<=N1<=L
    for n=1:L
        if ~(N0*N1); break; end;
        PutS(x(n),[0,1],[N0,N1]);   // code x(n)
        N0=N0-1+x(n);N1=N1-x(n);    // update model (of rest of the sequence)
    end
elseif MetI==1
    // code x1 and x2
    clear x4 x5 x3 x I3 J3 I J
    PutN(L11,L-1);         // 0<=L11<=(L-1)
    EncodeBin(x1,L1);
    EncodeBin(x2,L2);
elseif MetI==2
    // code the sequence x3
    N1=L31+x3(L);
    N0=L-N1;
    PutN(N1,L);            // code N1, 0<=N1<=L
    for n=1:L
        if ~(N0*N1); break; end;
        PutS(x3(n),[0,1],[N0,N1]);   // code x3(n)
        N0=N0-1+x3(n);N1=N1-x3(n);    // update model (of rest of the sequence)
    end
elseif MetI==3
    // code x4 and x5
    clear x1 x2 x3 x I3 J3 I J
    PutN(L31,L-1);         // 0<=L31<=(L-1)
    EncodeBin(x4,L4);
    EncodeBin(x5,L5);
end
return    // end of EncodeBin

function x = DecodeBin(L)
global y Byte BitPos
global high low range ub hc lc sc K code

// these must be as in EncodeBin
MetS=[0,1,2,3];       // the different methods, direct, split, diff, diff+split
MetC=[9,3,3,1];       // and the counts (which gives the probabilities)
MetI=GetS(MetS,MetC);    // encode which method to use

if (MetI==1) | (MetI==3)   // a split was done
    L11=GetN(L-1);         // 0<=L11<=(L-1)
    if L11<(L/2)
        L1=L11;L2=L-L11;
    else
        L1=L11+1;L2=L-L11-1;
    end
    x1=DecodeBin(L1);
    x2=DecodeBin(L2);
    // build sequence x from x1 and x2
    x=zeros(L,1);
    if L11<(L/2)
        x(1)=x2(1);
        n1=0;n2=1;   // index for the last in x1 and x2
    else
        x(1)=x1(1);
        n1=1;n2=0;   // index for the last in x1 and x2
    end
    for n=2:L
        if (x(n-1))
            n1=n1+1;
            x(n)=x1(n1);
        else
            n2=n2+1;
            x(n)=x2(n2);
        end
    end
else                      // no split
    N1=GetN(L);
    N0=L-N1;
    x=zeros(L,1);
    for n=1:L
        if (N0==0); x(n:L)=1; break; end;
        if (N1==0); break; end;
        x(n)=GetS([0,1],[N0,N1]);   // decode x(n)
        N0=N0-1+x(n);N1=N1-x(n);    // update model (of rest of the sequence)
    end
end

if (MetI==2) | (MetI==3)   // x is diff coded
    for n=2:L
        x(n)=x(n-1)+x(n);
    end
    x=rem(x,2);
end

return    // end of DecodeBin


// ------- Other subroutines ------------------------------------------------

// Functions to write and read a Variable Length Integer Code word
// This is a way of coding non-negative integers that uses fewer
// bits for small integers than for large ones. The scheme is:
//   '00'   +  4 bit  - integers from 0 to 15
//   '01'   +  8 bit  - integers from 16 to 271
//   '10'   + 12 bit  - integers from 272 to 4367
//   '110'  + 16 bit  - integers from 4368 to 69903
//   '1110' + 20 bit  - integers from 69940 to 1118479
//   '1111' + 24 bit  - integers from 1118480 to 17895695
//   not supported  - integers >= 17895696 (=2^4+2^8+2^12+2^16+2^20+2^24)
function PutVLIC(N)
global y Byte BitPos
global high low range ub hc lc sc K code
if (N<0)
    error('Arith07-PutVLIC: Number is negative.');
elseif (N<16)
    PutABit(0);PutABit(0);
    for (i=4:-1:1); PutABit(bitget(N,i)); end;
elseif (N<272)
    PutABit(0);PutABit(1);
    N=N-16;
    for (i=8:-1:1); PutABit(bitget(N,i)); end;
elseif (N<4368)
    PutABit(1);PutABit(0);
    N=N-272;
    for (i=12:-1:1); PutABit(bitget(N,i)); end;
elseif (N<69940)
    PutABit(1);PutABit(1);PutABit(0);
    N=N-4368;
    for (i=16:-1:1); PutABit(bitget(N,i)); end;
elseif (N<1118480)
    PutABit(1);PutABit(1);PutABit(1);PutABit(0);
    N=N-69940;
    for (i=20:-1:1); PutABit(bitget(N,i)); end;
elseif (N<17895696)
    PutABit(1);PutABit(1);PutABit(1);PutABit(1);
    N=N-1118480;
    for (i=24:-1:1); PutABit(bitget(N,i)); end;
else
    error('Arith07-PutVLIC: Number is too large.');
end

endfunction




// Aritmetic coding of a number or symbol x, the different symbols (numbers)
// are given in the array S, and the counts (which gives the probabilities)
// are given in C, an array of same length as S.
// x must be a number in S.
// example with symbols 0 and 1, where probabilites are P{0}=0.8, P{1}=0.2
// PutS(x,[0,1],[4,1]) and x=GetS([0,1],[4,1])
// An idea: perhaps the array S should be only in the calling function, and
// that it do not need to be passed as an argument at all.
// Hint: it may be best to to the most likely symbols (highest counts) in
// the beginning of the tables S and C.
function PutS(x,S,C)
global y Byte BitPos
global high low range ub hc lc sc K code
N=length(S);     // also length(C)
m=find(S==x);    // m is a single value, index in S, 1 <= m <= N
sc=sum(C);
lc=sc-sum(C(1:m));
hc=lc+C(m);
// disp(['PutS: lc=',int2str(lc),' hc=',int2str(hc),' sc=',int2str(sc),' m=',int2str(m)]);
EncodeSymbol;  // code the bit
return

function x=GetS(S,C)
global y Byte BitPos
global high low range ub hc lc sc K code
range=high-low+1;
sc=sum(C);
counts=floor(( (code-low+1)*sc-1 )/range);
m=1;
lc=sc-C(1);
while (lc>counts); m=m+1; lc=lc-C(m); end;
hc=lc+C(m);
x=S(m);
// disp(['GetS: lc=',int2str(lc),' hc=',int2str(hc),' sc=',int2str(sc),' m=',int2str(m)]);
RemoveSymbol;
return

// Aritmetic coding of a number x, 0<=x<=N, P{0}=P{1}=...=P{N}=1/(N+1)
function PutN(x,N)     // 0<=x<=N
global y Byte BitPos
global high low range ub hc lc sc K code
sc=N+1;
lc=x;
hc=x+1;
EncodeSymbol;  // code the bit
return

function x=GetN(N)
global y Byte BitPos
global high low range ub hc lc sc K code
range=high-low+1;
sc=N+1;
x=floor(( (code-low+1)*sc-1 )/range);
hc=x+1;lc=x;
RemoveSymbol;
return

// Aritmetic coding of a bit, probability is 0.5 for both 1 and 0
function PutABit(Bit)
global y Byte BitPos
global high low range ub hc lc sc K code
sc=2;
if Bit
    hc=1;lc=0;
else
    hc=2;lc=1;
end
EncodeSymbol;  // code the bit
return

function Bit=GetABit
global y Byte BitPos
global high low range ub hc lc sc K code
range=high-low+1;
sc=2;
counts=floor(( (code-low+1)*sc-1 )/range);
if (1>counts)
    Bit=1;hc=1;lc=0;
else
    Bit=0;hc=2;lc=1;
end
RemoveSymbol;
return;

// The EncodeSymbol function encode a symbol, (correspond to encode_symbol page 149)
function EncodeSymbol
global y Byte BitPos
global high low range ub hc lc sc K code
range=high-low+1;
high=low+floor(((range*hc)/sc)-1);
low=low+floor((range*lc)/sc);
while 1          // for loop on page 149
    if bitget(high,K)==bitget(low,K)
        PutBit(bitget(high,K));
        while ub > 0
            PutBit(~bitget(high,K));
            ub=ub-1;
        end
    elseif (bitget(low,K-1) & (~bitget(high,K-1)))
        ub=ub+1;
        low=bitset(low,K-1,0);
        high=bitset(high,K-1,1);
    else
        break
    end
    low=bitset(low*2,K+1,0);
    high=bitset(high*2+1,K+1,0);
end
return

// The RemoveSymbol function removes (and fill in new) bits from
// file, y, to code
function RemoveSymbol
global y Byte BitPos
global high low range ub hc lc sc K code
range=high-low+1;
high=low+floor(((range*hc)/sc)-1);
low=low+floor((range*lc)/sc);
while 1
    if bitget(high,K)==bitget(low,K)
        // do nothing (shift bits out)
    elseif (bitget(low,K-1) & (~bitget(high,K-1)))
        code=bitset(code,K-1,~bitget(code,K-1));     // switch bit K-1
        low=bitset(low,K-1,0);
        high=bitset(high,K-1,1);
    else
        break
    end
    low=bitset(low*2,K+1,0);
    high=bitset(high*2+1,K+1,0);
    code=bitset(code*2+GetBit,K+1,0);
end
if (low > high); error('low > high'); end;
return

// Functions to write and read a Bit
function PutBit(Bit)
global y Byte BitPos
BitPos=BitPos-1;
if (~BitPos); Byte=Byte+1; BitPos=8; end;
y(Byte) = bitset(y(Byte),BitPos,Bit);
return

function Bit=GetBit
global y Byte BitPos
BitPos=BitPos-1;
if (~BitPos); Byte=Byte+1; BitPos=8; end;
Bit=bitget(y(Byte),BitPos);
return

function b=BitEst(N,N1);
if (N1>(N/2)); N1=N-N1; end;
N0=N-N1;
if (N>1000)
    b=(N+3/2)*log2(N)-(N0+1/2)*log2(N0)-(N1+1/2)*log2(N1)-1.3256;
elseif (N1>20)
    b=(N+3/2)*log2(N)-(N0+1/2)*log2(N0)-(N1+1/2)*log2(N1)-0.020984*log2(log2(N))-1.25708;
else
    b=log2(N+1)+sum(log2(N-(0:(N1-1))))-sum(log2(N1-(0:(N1-1))));
end
return