octave-communications-common 1.2.0-2 / usr / share / octave / packages / communications-1.2.0 / lloyds.m

## Copyright (C) 2003 David Bateman
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn  {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @var{init_codes})
## @deftypefnx {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @var{len})
## @deftypefnx {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @dots{}, @var{tol})
## @deftypefnx {Function File} {[@var{table}, @var{codes}] =} lloyds (@var{sig}, @dots{}, @var{tol}, @var{type})
## @deftypefnx {Function File} {[@var{table}, @var{codes}, @var{dist}] =} lloyds (@dots{})
## @deftypefnx {Function File} {[@var{table}, @var{codes}, @var{dist}, @var{reldist}] =} lloyds (@dots{})
##
## Optimize the quantization table and codes to reduce distortion. This is
## based on the article by Lloyd
##
##  S. Lloyd @emph{Least squared quantization in PCM}, IEEE Trans Inform
##  Theory, Mar 1982, no 2, p129-137
##
## which describes an iterative technique to reduce the quantization error
## by making the intervals of the table such that each interval has the same
## area under the PDF of the training signal @var{sig}. The initial codes to
## try can either be given in the vector @var{init_codes} or as scalar
## @var{len}. In the case of a scalar the initial codes will be an equi-spaced
## vector of length @var{len} between the minimum and maximum value of the
## training signal.
##
## The stopping criteria of the iterative algorithm is given by
##
## @example
## abs(@var{dist}(n) - @var{dist}(n-1)) < max(@var{tol}, abs(@var{eps}*max(@var{sig}))
## @end example
##
## By default @var{tol} is 1.e-7. The final input argument determines how the
## updated table is created. By default the centroid of the values of the
## training signal that fall within the interval described by @var{codes}
## are used to update @var{table}. If @var{type} is any other string than
## "centroid", this behavior is overridden and @var{table} is updated as
## follows.
##
## @example
## @var{table} = (@var{code}(2:length(@var{code})) + @var{code}(1:length(@var{code}-1))) / 2
## @end example
##
## The optimized values are returned as @var{table} and @var{code}. In
## addition the distortion of the optimized codes representing the training
## signal is returned as @var{dist}. The relative distortion in the final
## iteration is also returned as @var{reldist}.
##
## @seealso{quantiz}
## @end deftypefn

function [table, code, dist, reldist] = lloyds (sig, init, tol, type)

  if (nargin < 2 || nargin > 4)
    print_usage ();
  endif

  if (min (size (sig)) != 1)
    error ("lloyds: SIG must be a vector");
  endif

  sig = sig(:);
  sigmin = min (sig);
  sigmax = max (sig);

  if (length (init) == 1)
    len = init;
    init = [0:len-1]' / (len - 1) * (sigmax - sigmin) + sigmin;
  elseif (min (size (init)))
    len = length (init);
  else
    error ("lloyds: invalid initial codebook");
  endif
  lcode = length (init);

  if (any (init != sort (init)))
    ## Must be monotonically increasing
    error ("lloyds: INIT_CODES must be monotonically increasing");
  endif

  if (nargin < 3)
    tol = 1e-7;
  elseif (isempty (tol))
    tol = 1e-7;
  endif
  stop_criteria = max (eps * abs (sigmax), abs (tol));

  if (nargin < 4)
    type = "centroid";
  elseif (!ischar (type))
    error ("lloyds: TYPE must be a string");
  endif

  ## Setup initial codebook, table and distortion
  code = init(:);
  table = (code(2:lcode) + code(1:lcode-1))/2;
  [indx, ignore, dist] = quantiz (sig, table, code);
  reldist = abs (dist);

  while (reldist > stop_criteria)
    ## The formula of the code at the new iteration is
    ##
    ##  code = Int_{table_{i-1}}^{table_i} x PSD(sig(x)) dx / ..
    ##          Int_{table_{i-1}}^{table_i} PSD(sig(x)) dx
    ##
    ## As sig is a discrete signal, this comes down to counting the number
    ## of times "sig" has values in each interval of the table, and taking
    ## the mean of these values. If no value of the signals in interval, take
    ## the middle of the interval. That is calculate the centroid of the data
    ## of the training signal falling in the interval. We can reuse the index
    ## from the previous call to quantiz to define the values in the interval.
    for i = 1:lcode
      psd_in_interval = find (indx == i-1);
      if (!isempty (psd_in_interval))
        code(i) = mean (sig(psd_in_interval));
      elseif (i == 1)
        code(i) = (table(i) + sigmin) / 2;
      elseif (i == lcode)
        code(i) = (sigmax + table(i-1)) / 2;
      else
        code(i) = (table(i) + table(i-1)) / 2;
      endif
    endfor

    ## Now update the table. There is a problem here, in that I believe
    ## the elements of the new table should be given by b(i)=(c(i+1)+c(i))/2,
    ## but Matlab doesn't seem to do this. Matlab seems to also take the
    ## centroid of the code for the table (this was a real pain to find
    ## why my results and Matlab's disagreed). For this reason, I have a
    ## default behavior the same as Matlab, and allow a flag to overide
    ## it to be the behavior I expect. If any one wants to tell me what
    ## the CORRECT behavior is, then I'll get rid of of the irrelevant
    ## case below.
    if (strcmp (type, "centroid"))
      for i = 1:lcode-1;
        sig_in_code = sig(find (sig >= code(i)));
        sig_in_code = sig_in_code(find (sig_in_code < code(i+1)));
        if (!isempty (sig_in_code))
          table(i) = mean (sig_in_code);
        else
          table(i) = (code(i+1) + code(i)) / 2;
        endif
      endfor
    else
      table = (code(2:lcode) + code(1:lcode-1))/2;
    endif

    ## Update the distortion levels
    reldist = dist;
    [indx, ignore, dist] = quantiz (sig, table, code);
    reldist = abs (reldist - dist);
  endwhile

  if (size (init, 1) == 1)
    code = code';
    table = table';
  endif

endfunction

%% Test input validation
%!error lloyds ()
%!error lloyds (1)
%!error lloyds (1, 2, 3, 4, 5)
%!error lloyds (1, [3 2 1])
%!error lloyds (1, 2, 3, 4)