boundary_conditions_emd.m

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%BOUNDARY_CONDITIONS_EMD  extends an extrema set to limit edge effects on the interpolations
%
% 
% [TMIN,TMAX,ZMIN,ZMAX] = BOUNDARY_CONDITIONS_EMD(INDMIN,INDMAX,T,X,Z,NBSYM)
%
% inputs:
%   - INDMIN, INDMAX: indices of minima and maxima in the real signal X
%   - T: sampling times
%   - X: real signal in which INDMIN and INDMAX are the indices of extrema
%   - Z: signal which values are interpolated in the final envelope
%   - NBSYM: number of points added to each end
%
% outputs:
%   - TMIN, TMAX: extended sampling times
%   - ZMIN, ZMAX: extended "extrema" set
%
% use:
%   - for a real signal X:
%     [TMIN,TMAX,ZMIN,ZMAX] = BOUNDARY_CONDITIONS_EMD(INDMIN,INDMAX,T,X,X,NBSYM)
%   - for a complex signal Z and a direction PHI:
%     X = exp(-i*PHI)*Z;
%     [TMIN,TMAX,ZMIN,ZMAX] = BOUNDARY_CONDITIONS_EMD(INDMIN,INDMAX,T,X,Z,NBSYM)
%
% rem: it has to be noted that this function was originally written for the 
% classical EMD and adapted to the bivariate case without a proper study of its
% effects. The edge effects problem for the bivariate EMD has not been studied yet. 
%
% See also
%  extr
%
% G. Rilling, last modification 3.2007
% [e-mail]
function [tmin,tmax,zmin,zmax] = boundary_conditions(indmin,indmax,t,x,z,nbsym)
	
	lx = length(x);
	
	if (length(indmin) + length(indmax) < 3)
		error('not enough extrema')
	end

    % boundary conditions for interpolations :

	if indmax(1) < indmin(1)
    	if x(1) > x(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 x(1) < x(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 x(end) < x(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 x(end) > x(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 
          
	zlmax =z(lmax); 
	zlmin =z(lmin);
	zrmax =z(rmax); 
	zrmin =z(rmin);
     
	tmin = [tlmin t(indmin) trmin];
	tmax = [tlmax t(indmax) trmax];
	zmin = [zlmin z(indmin) zrmin];
	zmax = [zlmax z(indmax) zrmax];
end