emd_local.m

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function [imf,ort,nbits] = emd_local(x,t,stop,tst);

%EMD_LOCAL  ("Local" Empirical Mode Decomposition) computes a local version of EMD
%
% stopping criterion for sifting : 
%   at each point : mean amplitude < threshold2*envelope amplitude
%   &
%   mean of boolean array ((mean amplitude)/(envelope amplitude) > threshold) < tolerance
%   &
%   |#zeros-#extrema|<=1
%
%
% [imf,ort,nbits] = EMD_LOCAL(x,t,stop,tst)
% inputs: 
%         - x : analyzed signal (line vector)
%         - t (optional) : sampling times (line vector) (default : 1:length(x))
%         - stop (optional) : threshold, threshold2 and tolerance (optional)
%                             for sifting stopping criterion 
%                             default : [0.05,0.5,0.05]
%         - tst (optional) : if equals to 1 shows sifting steps with pause
%                            if equals to 2 no pause
%
% outputs:
%         - imf : intrinsic mode functions (last line = residual)
%         - ort : index of orthogonality
%         - nbits : number of iterations for each mode
%
% calls:
%         - extr (finds extrema and zero-crossings)
%         - io : computes the index of orthogonality
% 
% G. Rilling, July 2002
% [e-mail]

defstop = [0.05,0.5,0.05];

if(nargin==1)
  t = 1:length(x);
  stop = defstop;
  tst = 0;
end

if(nargin==2)
  stop = defstop;
  tst = 0;
end

if (nargin==3)
  tst=0;
end

S = size(x);
if ((S(1) > 1) & (S(2) > 1)) | (length(S) > 2)
  error('x must have only one row or one column')
end

if S(1) > 1
  x = x';

end

S = size(t);
if ((S(1) > 1) & (S(2) > 1)) | (length(S) > 2)
  error('t must have only one row or one column')
end

if S(1) > 1
  t = t';
end

if (length(t)~=length(x))
  error('x and t must have the same length')
end

S = size(stop);
if ((S(1) > 1) & (S(2) > 1)) | (S(1) > 3) | (S(2) > 3) | (length(S) > 2)
  error('stop must have only one row or one column of max three elements')

end

if S(1) > 1
  stop = stop';
  S = size(stop);
end

if S(2) <2
  stop(2)=defstop(2);
end

if S(2) < 3
  stop(3)=defstop(3);
end

sd = stop(1);
sd2 = stop(2);
tol = stop(3);

if tst
  figure
end

% maximum number of iterations
MAXITERATIONS=2000;

% minimum length of the constant part of curve appended to zones badsx==1
LARGMIN = 5;

% maximum number of points symmmetrized for interpolations
NBSYM = 5;

lx = length(x);

sdt(lx) = 0;
sdt = sdt+sd;
sd2t(lx) = 0;
sd2t = sd2t+sd2;

% number of extrema and zero-crossings in residual
ner = lx;
nzr = lx;

r = x;
imf = [];
k = 1;

% iterations counter for the extraction of 1 mode
nbit=0;

% total iterations counter
NbIt=0;

% loop start : modes extracted while at least 3 extrema
while ner > 2
        
  % current mode
  m = r;
  
  % mode at previous iteration
  mp = m;

  sx = sd + 1;
  badsx = 1;

  % tests if enough extrema to proceed
  test = 0;
  
  [indmin,indmax,indzer] = extr(m);
  lm=length(indmin);
  lM=length(indmax);
  nem=lm + lM;
  nzm=length(indzer);
  
  j=1;
  
  % sifting loop
  while (mean(badsx) > tol | (abs(nzm-nem)>1))&(test == 0)&nbit<MAXITERATIONS

     
   if(nbit>MAXITERATIONS/5 & mod(nbit,floor(MAXITERATIONS/10))==0)
      disp(['mode ',int2str(k),' nombre d iterations : ',int2str(nbit)])
      disp(['stop parameter mean value : ',num2str(s)])
   end


    % boundary conditions for interpolations :
    
    if indmax(1) < indmin(1)
      if m(1) > m(indmin(1))
        lmax = fliplr(indmax(2:min(end,NBSYM+1)));
        lmin = fliplr(indmin(1:min(end,NBSYM)));
        lsym = indmax(1);
      else
        lmax = fliplr(indmax(1:min(end,NBSYM)));
        lmin = [fliplr(indmin(1:min(end,NBSYM-1))),1];
        lsym = 1;
      end
    else
      if m(1) < m(indmax(1))
        lmax = fliplr(indmax(1:min(end,NBSYM)));
        lmin = fliplr(indmin(2:min(end,NBSYM+1)));
        lsym = indmin(1);
      else
        lmax = [fliplr(indmax(1:min(end,NBSYM-1))),1];
        lmin = fliplr(indmin(1:min(end,NBSYM)));
        lsym = 1;
      end
    end
    
    if indmax(end) < indmin(end)
      if m(end) < m(indmax(end))
        rmax = fliplr(indmax(max(end-NBSYM+1,1):end));
        rmin = fliplr(indmin(max(end-NBSYM,1):end-1));
        rsym = indmin(end);
      else
        rmax = [lx,fliplr(indmax(max(end-NBSYM+2,1):end))];
        rmin = fliplr(indmin(max(end-NBSYM+1,1):end));
        rsym = lx;
      end
    else
      if m(end) > m(indmin(end))
        rmax = fliplr(indmax(max(end-NBSYM,1):end-1));
        rmin = fliplr(indmin(max(end-NBSYM+1,1):end));
        rsym = indmax(end);
      else
        rmax = fliplr(indmax(max(end-NBSYM+1,1):end));
        rmin = [lx,fliplr(indmin(max(end-NBSYM+2,1):end))];
        rsym = lx;
      end
    end
    

    tlmin = 2*t(lsym)-t(lmin);
    tlmax = 2*t(lsym)-t(lmax);
    trmin = 2*t(rsym)-t(rmin);
    trmax = 2*t(rsym)-t(rmax);
    
    % in case symmetrized parts do not extend enough
    if tlmin(1) > t(1) | tlmax(1) > t(1)
      if lsym == indmax(1)

        lmax = fliplr(indmax(1:min(end,NBSYM)));
      else
        lmin = fliplr(indmin(1:min(end,NBSYM)));
      end
      if lsym == 1
        error('bug')
      end
      lsym = 1;
      tlmin = 2*t(lsym)-t(lmin);
      tlmax = 2*t(lsym)-t(lmax);
    end
    
    
    if trmin(end) < t(lx) | trmax(end) < t(lx)
      if rsym == indmax(end)
        rmax = fliplr(indmax(max(end-NBSYM+1,1):end));
      else
        rmin = fliplr(indmin(max(end-NBSYM+1,1):end));
      end
      if rsym == lx
        error('bug')
      end
      rsym = lx;
      trmin = 2*t(rsym)-t(rmin);
      trmax = 2*t(rsym)-t(rmax);
    end
    
          
    mlmax =m(lmax); 
    mlmin =m(lmin);
    mrmax =m(rmax); 
    mrmin =m(rmin);
    
    % definition of enveloppes from interpolation
        
    envmax = interp1([tlmax t(indmax) trmax],[mlmax m(indmax) mrmax],t,'spline');
    envmin = interp1([tlmin t(indmin) trmin],[mlmin m(indmin) mrmin],t,'spline');
    
    envmoy = (envmax + envmin)/2;

    % estimation of mode amplitude    
    amp = abs(envmax-envmin)/2;
    if any(amp == 0)
      amp = amp+1e-300;   % circumvents divide by zero
    end
    mamp = max(amp);
    sx=(abs(envmoy))./amp;
    s = mean(sx);
    
    % definition of f equal to 1 where sifting is needed and 
    % dcays fast to 0 in the neighbourhood

    badsx = (sx > sd & amp > mamp/20) | (sx > sd2 & amp > mamp/100);
    d = diff([0 badsx 0]);
    debs = find(d==1);
    fins = find(d==-1)-1;
    if length(debs)~=length(fins)
      error('pb avec le regroupement en composantes connexes')

    end
    indextr = sort([indmin indmax]);
    d = diff(indextr);
    f = [];
    connexe = [];
    lc = length(debs);
    if lc > 0
      connexe(lc,lx) = 0;
      for i = 1:lc
        connexe(i,debs(i):fins(i)) = 1;
    % indices of previous and next extrema

    indp = min([nem-1,length(find(indextr < debs(i)))]);
        inds = max([2,length(find(indextr <= fins(i)))+1]);
		
    % evaluation of extrema densities left and right
        llarg = mean(d(min([nem-1,max([1,indp])]):max([1,min([nem-1,indp + 2])])));
        rlarg = mean(d(min([nem-1,max([1,inds - 1])]):max([1,min([nem-1,inds + 1])])));
    larg = mean([rlarg,llarg]);
	
    % special cases...
    if indp == 0
          if inds ~= (nem+1)
            larg = (indextr(inds)-indextr(inds-1));
          else
            larg = round(lx/2);
          end
        else
          if  inds ~= (nem+1)
            larg = max([(indextr(inds)-indextr(inds-1)),(indextr(indp+1)-indextr(indp))]);
          else
            larg =(indextr(indp+1)-indextr(indp));
          end
        end
        larg = 2*max([round((fins(i)-debs(i))/4),larg,LARGMIN]);

    w(1:round(larg/2)) = [1:round(larg/2)]/round(larg/2);
        w((2*larg+2-round(larg/2)):(2*larg+1)) = fliplr(w(1:round(larg/2)));

        w(round(larg/2):(2*larg+2-round(larg/2))) = 1;
        indd = max(1,debs(i)-larg);
        indf = min(lx,fins(i)+larg);
        connexe(i,indd:debs(i)) = w((larg+1-debs(i)+indd):(larg+1));
        connexe(i,fins(i):indf) = w((larg+1):(larg+1+indf-fins(i)));
      end

      f = max(connexe,[],1);
    
    else
      f(lx) = 0;
    end
    
    m = m - f.*envmoy;
    
    [indmin,indmax,indzer] = extr(m);
    lm=length(indmin);
    lM=length(indmax);
    nem = lm + lM;
    nzm = length(indzer);
    
    % display
    
     if tst
      subplot(5,1,1)
      plot(t,mp);hold on;
      plot(t,envmax,'--k');plot(t,envmin,'--k');plot(t,envmoy,'r');
      title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' before sifting']);
      set(gca,'XTick',[])
      hold  off

      subplot(5,1,2)
      plot(t,sx)
      hold on
      plot(t,sdt,'--r')
      plot(t,sd2t,':k')
      title('stop parameter')
      set(gca,'XTick',[])
      hold off

      subplot(5,1,3)
      plot(f)
      title('mean ponderation')
      set(gca,'XTick',[])
      
      subplot(5,1,4)
      plot(t,m)
      title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' after sifting']);
      set(gca,'XTick',[])

      subplot(5,1,5);
      plot(t,r-m)
      title('residue');
      disp(['stop parameter mean value : ',num2str(s)])
      if tst == 2
        pause(0.01)
      else
        pause
      end
    end
    
    
    % end loop :
    
    if nem < 3
      test = 1;
    end        
    
    mp = m;
    nbit=nbit+1;
    NbIt=NbIt+1;

    if(nbit==(MAXITERATIONS-1))
      warning(['forced stop of sifting : too many iterations... mode ',int2str(k),'. stop parameter mean value : ',num2str(s)])
    end
  
  end
  imf(k,:) = m;
  nbits(k) = nbit;
  k = k+1;
  r = r - m;
  [indmin,indmax,indzer] = extr(r);
  ner = length(indmin) + length(indmax);
  nzr = length(indzer);
  nbit=1;
  
  if (max(r) - min(r)) < (1e-10)*(max(x) - min(x))
    if ner > 2
      warning('forced stop of EMD : too small amplitude')
    else
      disp('forced stop of EMD : too small amplitude')
    end
    break
  end
  
end


imf(k,:) = r;


ort = io(x,imf);

if tst
  close
end