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Copy pathTVRegSpikeRemoveMultiL1BiCont2.m
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TVRegSpikeRemoveMultiL1BiCont2.m
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function [fest,spike,imu,festfine,sigmaminus]= TVRegSpikeRemove( data, iter, u0, scale, ep, dx,nanti, plotflag, diagflag )
% u = TVRegDiff( data, iter, alph, u0, scale, ep, dx, plotflag, diagflag );
% u = tvdiff( e, dx, iter, ep, alph );
% Rick Chartrand ([email protected]), Apr. 10, 2011
% Please cite Rick Chartrand, "Numerical differentiation of noisy,
% nonsmooth data," ISRN Applied Mathematics, Vol. 2011, Article ID 164564,
% 2011.
%
% Inputs: (First three required; omitting the final N parameters for N < 7
% or passing in [] results in default values being used.)
% data Vector of data to be differentiated.
%
% iter Number of iterations to run the main loop. A stopping
% condition based on the norm of the gradient vector g
% below would be an easy modification. No default value.
%
% alph Regularization parameter. This is the main parameter
% to fiddle with. Start by varying by orders of
% magnitude until reasonable results are obtained. A
% value to the nearest power of 10 is usally adequate.
% No default value. Higher values increase
% regularization strenght and improve conditioning.
%
% u0 Initialization of the iteration. Default value is the
% naive derivative (without scaling), of appropriate
% length (this being different for the two methods).
% Although the solution is theoretically independent of
% the intialization, a poor choice can exacerbate
% conditioning issues when the linear system is solved.
%
% scale 'large' or 'small' (case insensitive). Default is
% 'small'. 'small' has somewhat better boundary
% behavior, but becomes unwieldly for data larger than
% 1000 entries or so. 'large' has simpler numerics but
% is more efficient for large-scale problems. 'large' is
% more readily modified for higher-order derivatives,
% since the implicit differentiation matrix is square.
%
% ep Parameter for avoiding division by zero. Default value
% is 1e-6. Results should not be very sensitive to the
% value. Larger values improve conditioning and
% therefore speed, while smaller values give more
% accurate results with sharper jumps.
%
% dx Grid spacing, used in the definition of the derivative
% operators. Default is the reciprocal of the data size.
%
% plotflag Flag whether to display plot at each iteration.
% Default is 1 (yes). Useful, but adds significant
% running time.
%
% diagflag Flag whether to display diagnostics at each
% iteration. Default is 1 (yes). Useful for diagnosing
% preconditioning problems. When tolerance is not met,
% an early iterate being best is more worrying than a
% large relative residual.
%
% Output:
%
% u Estimate of the regularized derivative of data. Due to
% different grid assumptions, length( u ) =
% length( data ) + 1 if scale = 'small', otherwise
% length( u ) = length( data ).
% Copyright notice:
% Copyright 2010. Los Alamos National Security, LLC. This material
% was produced under U.S. Government contract DE-AC52-06NA25396 for
% Los Alamos National Laboratory, which is operated by Los Alamos
% National Security, LLC, for the U.S. Department of Energy. The
% Government is granted for, itself and others acting on its
% behalf, a paid-up, nonexclusive, irrevocable worldwide license in
% this material to reproduce, prepare derivative works, and perform
% publicly and display publicly. Beginning five (5) years after
% (March 31, 2011) permission to assert copyright was obtained,
% subject to additional five-year worldwide renewals, the
% Government is granted for itself and others acting on its behalf
% a paid-up, nonexclusive, irrevocable worldwide license in this
% material to reproduce, prepare derivative works, distribute
% copies to the public, perform publicly and display publicly, and
% to permit others to do so. NEITHER THE UNITED STATES NOR THE
% UNITED STATES DEPARTMENT OF ENERGY, NOR LOS ALAMOS NATIONAL
% SECURITY, LLC, NOR ANY OF THEIR EMPLOYEES, MAKES ANY WARRANTY,
% EXPRESS OR IMPLIED, OR ASSUMES ANY LEGAL LIABILITY OR
% RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR USEFULNESS OF
% ANY INFORMATION, APPARATUS, PRODUCT, OR PROCESS DISCLOSED, OR
% REPRESENTS THAT ITS USE WOULD NOT INFRINGE PRIVATELY OWNED
% RIGHTS.
% BSD License notice:
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions
% are met:
%
% Redistributions of source code must rsigmaminusin the above
% copyright notice, this list of conditions and the following
% disclaimer.
% Redistributions in binary form must reproduce the above
% copyright notice, this list of conditions and the following
% disclaimer in the documentation and/or other materials
% provided with the distribution.
% Neither the name of Los Alamos National Security nor the names of its
% contributors may be used to endorse or promote products
% derived from this software without specific prior written
% permission.
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
% CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,
% INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
% MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
% DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR
% CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
% SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
% LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
% USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED
% AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
% LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
% ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
% POSSIBILITY OF SUCH DAMAGE.
% code starts here
% Make sure we have a column vector.
data = data( : );
% Get the data size.
n = length( data );
% Default checking. (u0 is done separately within each method.)
if nargin < 9 || isempty( diagflag )
diagflag = 1;
end
if nargin < 8 || isempty( plotflag )
plotflag = 1;
end
if nargin < 7 || isempty( dx )
dx = 1 / n;
end
if nargin < 6 || isempty( ep )
ep = 1e-6;
end
if nargin < 5 || isempty( scale )
scale = 'small';
end
% nanti=1;
expvar=1;
dd=abs(diff(data));
sigmaplus=sum(dd.*(1-dd/max(dd)))/sum(1-abs(dd)/max(abs(dd)));
alph=((sigmaplus*sigmaplus)^expvar)*dx^(nanti-1);
falph=ones(length(data),1);
falph(:)=alph;
pinf=0;
thresh=1;
threshrm=1;
taumin=1;
% Construct antidifferentiation operator and its adjoint.
str='v';
for ii=1:nanti
str=['cumsum(' str ')*dx'] ;
end
AN=str2func(['@(v,dx)' str]);
str='w';
for ii=1:nanti
str=['AT(' str ')*dx'];
end
if nargin < 4 || isempty( u0 )
% u0 = [ 0; diff( data +0.1*rand(size(data)))/dx ];
u0 = [ 0; diff( data )/dx ];
i=1;
while i<nanti
u0=[0; diff( u0 )/dx;];
i=i+1;
end
if plotflag
subplot(2,1,1);
plot(1:n,AN(u0,dx),'-r',1:n,data,'-b');
subplot(2,1,2);
plot( u0, 'ok' ), drawnow;
end
end
f=AN(u0,dx);
u = u0;
% res=zeros(size(u));
eta=exp(-(sum(diff(data))^2)/(2*sigmaplus^2));% + 0.5;
% eta=(eta-0.5)*2;
tau=taumin+eta*(thresh-taumin);
% tau=taumin+1/(1-eta);
phi=1;
chi=1;
beta=0.2;
sigmaminus=sigmaplus;
function calcDen8()
nsr=sigmaplus/(max(AN(u,dx))-min(AN(u,dx)));
wdu=(1-((abs(D*u))/max(abs(D*u))));
wu=(1-((abs(u))/max(abs(u))));
% ddp=[0;abs(diff(data)).*(1-abs(u(2:end))/max(abs(u(2:end)))).*wdu(2:end)]...
% +[abs(diff(data)).*(1-abs(u(1:end-1))/max(abs(u(1:end-1)))).*wdu(1:end-1);0];
% ddp=[sigmaminus;abs(diff(data)).*(1-abs(u(2:end))/max(abs(u(2:end)))+nsr)]/2+...
% [abs(diff(data)).*(1-abs(u(1:end-1))/max(abs(u(1:end-1)))+nsr);sigmaminus]/2;
ddp=[sigmaminus;abs(diff(data)).*(wdu(1:end-1)+nsr)]/2+...
[abs(diff(data)).*(wdu(2:end)+nsr);sigmaminus]/2;
% den8 = (abs((res)/(sigmaplus)).*(1-wdu))/(sum(1-wdu));
dd2=[abs(diff(data));abs(diff(data(end-1:end)))];
if exist('res')
rsmax=(sum(abs((res)/(sigmaminus)).*(1-wdu))/(sum(1-wdu)));
rsm=(sum(abs((res(~spike))/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
else
rsm=(sum(abs((dd2)/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
rsp=(mean(abs((dd2)/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
rsmax=rsm;
end
if exist('res')
% if ~isempty(newnoise)
ares=abs(res);
p=sum(ares<sigmaminus)/n;
if sum(spike)~=0
threshc=mean(ares(~spike)/sigmaplus)+std(ares(spike)/sigmaplus);
threshrm=mean(ares(spike)/sigmaplus)-3*std(ares(spike)/sigmaplus);
else
threshc=1;
end
% plotDensity(ares(spike)/sigmaplus);
thresh=max(thresh,threshc);
threshrm=max(threshrm,threshc);
% threshrm=min(abs(res(newnoise)/sigmaplus))/2
% end
else
ares=sigmaplus;
p=sum(ddp<sigmaplus)/n;
threshc=1;
thresh=2;
threshrm=1;
end
tau=(1-eta)*taumin+(eta)*(thresh);
sigmamax=min(sigmaplus,sigmaminus);
neff=sum(~spike);
eta=exp(-((sigmaminus-sigmamax)^2)/(2*sigmamax^2))*exp(-(abs(rsp)^2)*neff);
qq=(rsm/rsmax)^(1/1);
e8=((1-nsr^(tau/thresh))/(sqrt(1+eta)*(rsmax*(1-qq)+qq*rsm)));
e8=(1-nsr^(1/rsm));
% e8=(1+nsr^(tau/thresh))/(max(rsm,1));
% e8=-1+1/eta;
% e8=1;
%
% den8=ep^(threshrm)+(exp(-(abs(ddp-max(ddp)))/std(ddp))).^e8;
% % den8(den8<ep^(e8))=ep^(e8);
if exist('res')
sigmalocal=filter(b,a,ddp);
else
sigmalocal=ddp;
end
%
% pp=exp(-((abs(sigmalocal-max(sigmalocal)).^2)/var(sigmalocal))-(abs(u)/var(abs(u))));
% pp=exp(-(abs(u.^2)/var(abs(u))));
% ms=max(ddp(ddp<(mean(ddp)+3*std(ddp))));
% jj=(1-abs(u)/max(abs(u)));
%
% kk=(abs(dd2)/max(abs(dd2)));
% pp=exp(-((abs(sigmalocal-max(sigmalocal)).^2)/(ms)^2)-((u.^2)/var(u))).*kk.*jj;
% pp=pp/max(pp);
% % plot(pp);
% ind=sigmalocal>sigmaplus&abs(u)<std(u);
%
% % den8(ind)=den8(ind)+sigmalocal(ind);
% % den8=(1-pp).*den8+pp.*(sigmalocal+sigmaplus);
% den8=abs(u)/max(abs(u))+ep+sigmalocal/max(AN(u,dx));
ee=mean(ddp(ddp<=(1-nsr^(1/rsmax))*(mean(ddp(~spike))-std(ddp(ddp<mean(ddp(~spike)))))));
mean(ddp<mean(ddp));
mmu=Inf;
while std(ddp(ddp<mmu))>mean(ddp(ddp<mmu))/3
mmu=mean(ddp(ddp<mmu));
end
ee = mean(ddp<mmu);
% ee=mean(ddp(ddp<=-log((1-(p^(1+rsmax))))/mean(ddp)));
% plotDensity(ddp);hold on;y=ylim();h=plot([ee ee],y,'-g');set(h,'Linewidth',1);hold off;
den8=nsr.^((threshc*sigmaplus/rsmax)./ddp + std(u.^2)./(max(abs(u))-abs(u)).^2)+ee;%+(exp(-(abs(ddp-max(ddp)))/std(ddp))).^e8;
ddp2=ddp;
ind=ddp2<sigmalocal;
ddp2(ind)=sigmalocal(ind);
ff=std(u*dx)*sum(u>std(u)/2)/(max(AN(u,dx))-min(AN(u,dx)));
uu=[u(2:end);u(end)]+(([wu(2:end);wu(end)]).^(2)).*[u(1);u(1:end-1)];
% AAU = AN(AN(u,dx)-min(AN(u,dx)),1);
% AAU(end)/(2*sqrt(2)*sigmaminus*n);
ef=max((rsmax+1)*abs(diff(data))/(2*sigmaminus))-std((rsmax+1)*abs(diff(data))/(2*sigmaminus));
ef=exp(-abs(ef).^(sqrt(rsmax)/2));
den8=nsr.^((((0.5+(1/threshc))*sigmaplus/(1+rsmax))^1/2)./((sigmalocal+ddp).^1/2)+ mean(abs(u))./(max(abs(uu))-abs(uu)+nsr))...
+exp(sqrt(2+rsmax)*log(nsr)*(abs([u(2:end);u(end)])/min(abs(u(u~=0)))).^(1/2))*(1/sigmaplus^(1+sqrt(nsr)))...
+ ef/sigmaplus + ee;%+(exp(-(abs(ddp-max(ddp)))/std(ddp))).^e8;
% den8=1;
%
% den8=nsr.^((threshc*sigmaplus)./sigmalocal + mean((u))./(max(abs(u))-abs(u)+sigmaplus));%+(exp(-(abs(ddp-max(ddp)))/std(ddp))).^e8;
% plot(1:n,log(den8),1:n,sigmalocal );
if plotflag
plot(1:n,log(den8),1:n,ddp)
1;
end
end
rs=0;
rsp=1;
rsm=1;
rsmax=1;
% Different methods for small- and large-scale problems.
spike=false(size(data));
ATN=str2func(['@(w,dx)' str]);
%AT2 = @(w) AT(AT(w));
% Construct differentiation matrix.
c = ones( n, 1 );
D = spdiags( [ -c c ], [ 0 1 ], n, n ) / dx;
D( n, n ) = 0;
Db = spdiags( [ -c c ], [ 1 0 ], n, n ) / dx;
Db( n, n ) = 0;
clear c
DT = D';
% Since Au( 0 ) = 0, we need to adjust.
ofst=data( 1 );
data = data - ofst;
datap=data;
% Default initialization is naive derivative.
u = u0;
uprev=u0;
% Precompute.
ATNd = ATN( data ,dx);
weight=ones(size(data));
% thresh=6;
% weight=linspace(1,0,n)';
NRG=@(z) alph*sum(abs(diff(z)*dx))+0.5*sum((weight.*(AN( z,dx )-data)*dx).^2);
NRGP=@(z) alph*weight.*[0; abs(diff(z)*dx)]+0.5*(weight.*(AN( z,dx )-data)*dx).^2;
E0=NRG(u);
E=E0;
% Main loop.
ii=1;
iii=1;
tic;
[mu,imu]=max(u);
su=std(u);
detect=true;
gamma=0;
abort=false;
while ii<=iter && ~abort
if toc>30
break;
end
% Diagonal matrix of weights, for linearizing E-L equation.
windowSize = 3;
ep=sigmaplus/(max(AN(u,dx))-min(AN(u,dx)));
ep=ep^2;
b = (1/windowSize)*ones(1,windowSize);
a = 1;
% epu=filter(b,a,abs(D * u))+(ep*max(abs(u)))^2;
aDu=abs(D*u);
epu=filter(b,a,abs(D * u))+(ep*max(abs(u)))^2;
epu=min(aDu(aDu~=0));
epu=std(aDu(aDu<=sqrt(ep)*max(aDu)));
% Q = spdiags( 1./ (sqrt( (D * u) .^2 ) + epu), 0, n, n );
if iii>1
p=(1-eta^2)+(eta^2)*pinf;
p=1;
% peff=(1-peff)*p+peff;
% cf(99);plot(peff);
else
p=1;
peff=1;
end
Q = spdiags( (abs( (D * u) ) + epu ).^(p-2), 0, n, n );
% Linearized diffusion matrix, also approximation of Hessian.
try
L = dx * DT * Q * D;
catch e
disp(e);
end
% Gradient of functional.
%
% gamma=((1-alph^iii)*exp((1/iter)^2))*exp(-(iii/iter)^2)+alph^iii
dd=abs(diff(data(~spike)));
sigmaplus=sum(dd.*(1-dd/max(dd)))/sum(1-abs(dd)/max(abs(dd)));
% res=(datap-AN(u,dx));
if iii>1
falph=abs(res);
windowSize = 3;
b = (1/windowSize)*ones(1,windowSize);
a = 1;
falph = filter(b,a,falph);
% sigmaminus=mean(falph)
falph=falph;
% falph=((falph.*falph).^expvar)*dx^(nanti-1);
falph=alph*exp(-((max(abs(res))-abs(res)).^2)/var(abs(res)));
% falph=(falph+alph);
mfalph=mean(falph);
falph(:)=alph;
if plotflag
subplot(2,1,1);
h=plot(1:n,data,'-b',1:n,data,'.b',1:n,AN(u,dx),'-r',find(spike),data(spike),'og',imu,data(imu),'vb',1:n,min(data)+falph*(max(data)-min(data))/max(falph));
set(h(3),'linewidth',2);
legend('Regularized Estimate','Data','Data','Spikes');
subplot(2,1,2);
plot(1:n, u, 'ok',imu,mu,'vb'), drawnow;
end
% falph(:)=alph;
% del0=(1-(abs(sigmaminus-sigmaplus)/max(sigmaminus,sigmaplus)))^2;
% del0=((1-eta^2)+(1-chi))/2 +eta^2;
else
del0=1;
end
% del=min(del*del,1)
if iii>1
gamma=sigmaplus;
% del=1-del
% gamma=sigmaplus*(sigmaplus*(1-phi)*(1-eta))+(sigmaplus*(eta^2));
gamma=sigmaplus*((1-eta)^2)+sigmaminus*sigmaminus*eta;
else
gamma=sigmaplus*((1-eta)^2)+sigmaminus*sigmaminus*eta;
end
gamma=(sigmaplus^1.5)*eta+sigmaplus*sigmaplus*(1-eta);
gamma = sigmaminus*sigmaminus*p;
% del=del0
gamma=0;
del=del0;
del=1;
% gamma=0
% if gamma==0
% disp('its over!!!')
% end
% % del=del
% % gamma=0;
% % gamma=alph^(iii/iter)
% delta=0;
if iii>1
res=AN( u ,dx)-data;
ep1=std(abs(res(~spike)));
b = (1/windowSize)*ones(1,windowSize);
a = 1;
sigmalocal=filter(b,a,abs(res));
sigmaplocal=filter(b,a,dd.*(1-dd/max(dd)));
ep2=filter(b,a,abs(res));
ep2=ep*min(ep2(ep2~=0));
% ep2(ep2>ep1)=ep1;
else
ep2=ep;
end
den3=1;
epd=min(abs(u(u~=0)))/max(abs(u));
abu=abs(filter(b,a, u));
mxu=max(abu);
abDu=abs(filter(b,a,D * u));
mxDu=max(abDu);
epDu=min(abDu(abDu~=0));
den4=(abu/mxu);
den5=exp(-(((abDu).^2))/(2*var(abDu)));
den6=1-exp(-(((mxu-abu).^2))/(2*var(mxu-abu)));
% delta=abs[data(1:end)-[data(2:end); data(end)] +]
% den5=den5/max(den5);
den2=(den4.*den5.*den6.*abs( (AN( u ,dx)-data)));
den2=den2+min(dd(dd~=0));
den2=abs( (AN( u ,dx)-data))+min(dd(dd~=0));
% min(dd(dd~=0))
if iii>1
% den8=sigmalocal0/sigmaplus+ep;
% plot(den8);
else
calcDen8()
end
if iii>1
% denu=filter(b,a,abs(res));
% denu=denu/sigmaplus;
% denu=denu/max(denu);
% denu=1-denu+max(min(abs(res(res~=0))),ep*max(AN(u,dx)));
else
denu=1;
end
% cf(777);plot(1:n,den4,'-g',1:n,den5,'-r',1:n,den6,'-c');legend('den4','den5','den6');
%%mulitply den2 by something which is 1 most of the time and
%%very small at the maximum of u or possibly Du
% den2=(abs( weight.*(AN( u ,dx)-data))+ep2);
% weight=weight;
% % weight(weight==0)=0;
% wdu=(exp(-((mean(abs(D*u))-abs(D*u)).^2)/var(abs(D*u))));
% den2=(wdu+ep).*(abs( weight.*(AN( u ,dx)-data))+ep2);
% rs=(sum(res(~spike).*wdu(~spike))/(sum(wdu(~spike))/sum(~spike)));
if diagflag
fprintf('sigmaplus= %f, sigmaminus = %f, phi = %f, chi = %f , rsp = %f, rsm = %f\n',[sigmaplus sigmaminus phi chi rsp rsm])
fprintf('alph = %f, gamma = %f, eta = %f tau = %f \n',[alph gamma eta tau])
end
w2=weight;
g = gamma*(ATN( w2.*(AN( u ,dx)-data)./den2,dx ))+del.*(ATN( w2.*(AN( u ,dx)-data)./(den8),dx ));
g= g + falph .* (L * u).*denu;
% Build preconditioner.
c = cumsum( n : -1 : 1 ).';
B = alph * L + spdiags( c( end : -1 : 1 ), 0, n, n );
%droptol = 1.0e-2;
try
R = ichol( B, struct('droptol',1.0e-3) );
catch
alpha = max(sum(abs(B),2)./diag(B))-2;
R = ichol( B, struct('droptol',1.0e-3,'diagcomp',alpha) );
end
% Prepare to solve linear equation.
tol = nsr;
maxit = int16(sqrt(n))*n;
hs = @(x) ( alph *( L * x).*denu + gamma*(ATN( w2.*(AN( x ,dx))./den2,dx )) + del.*ATN( w2.*AN( x,dx )./den8,dx ) );
if diagflag
[s,flag,relres,ip] = pcg( hs , -g, tol, maxit, R', R );
if flag==0
disp(['pcg converged at iteration ' int2str(ip) ' to a solution with relative residual ' num2str(relres) '.']);
else
disp(flag);
abort=true;
s = pcg( hs , -g, tol, maxit, R', R );
end
fprintf( 'iteration %2d: relative change = %.3e, gradient norm = %.3e, gradient elm max = %.3e, dF = %.3e\n', ii, norm( s ) / norm( u ), norm( g ),max(abs(g)),NRG(u+s)-NRG(u));
else
[s,flag,relres,ip] = pcg( hs , -g, tol, maxit, R', R );
end
if norm( s ) / norm( u ) < (nsr)^(3/4)
abort=true;
end
% Update current solution
u = u + s;
% uprev=u;
if ~diagflag
hess = ( alph * L * (u) + ATN( AN( u,dx ),dx ) );
end
E=NRG(u);
res=(data-AN(u,dx));
% Display plot.
% exp(-(res(res>0)/(thresh*sigma)));
% weight(res>thresh*sigma)=0;
if plotflag
subplot(2,1,1);
h=plot(1:n,data,'-b',1:n,data,'.b',1:n,AN(u,dx),'-r',find(spike),data(spike),'og',imu,data(imu),'vb',1:n,min(data)+falph*(max(data)-min(data))/max(falph));
set(h(3),'linewidth',2);
legend('Regularized Estimate','Data','Data','Spikes');
subplot(2,1,2);
plot(1:n, u, 'ok',imu,mu,'vb'), drawnow;
end
wdu=(1-((abs(D*u))/max(abs(D*u))));
rsp=(mean(abs(res(~spike))/(sigmaplus).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
sigmamax=min(sigmaplus,sigmaminus);
neff=sum(~spike);
eta=exp(-((sigmaminus-sigmamax)^2)/(2*sigmamax^2))*exp(-(abs(rsp)^2)*neff);
nsr=(1/(((max(AN(u,dx))-min(AN(u,dx)))/sigmaminus)));
pinf= (1-nsr)^rsm;
% eta=(eta-0.5)*2;
tau=(1-eta)*taumin+(eta)*(thresh);
ares=abs(res);
ares(spike)=mean(ares(~spike));
sigmalocal=filter(b,a,ares);
sigmalocal0=sigmalocal;
peff=exp(-(((sigmalocal-sigmaplus).^2)/(sigmaplus^2)));
sigmalocal(sigmalocal>(mean(sigmalocal)+3*std(sigmalocal)))=(mean(sigmalocal)+3*std(sigmalocal));
sigmalocal(sigmalocal<sigmaplus)=sigmaplus;
wdu=(1-((abs(D*u))/max(abs(D*u))));
rsmax=(sum(abs((res)/(sigmaminus)).*(1-wdu))/(sum(1-wdu)));
% thresh=max(thresh,2*rsmax);
rsm=(sum(abs((res(~spike))/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
neff=sum(~spike);
eta=exp(-((sigmaminus-sigmamax)^2)/(2*sigmamax^2))*exp(-(abs(rsp)^2)/n);
tau=(1-eta)*taumin+(eta)*(thresh);
sigmamax=min(sigmaplus,sigmaminus);
% weight(spike)=sqrt(exp(-((abs(res(spike))-sigmaplus).^2)./(2*(tau*sigmalocal(spike)).^2)));
% cf(88);plot(weight)
% sigmalocal(spike)=mean(sigmalocal(~spike));
% cf(888);plot(1:n,tau*sigmalocal,1:n,abs(res))
% tau=taumin+1/(1-eta);
% indnew=~spike&abs(res)>tau*sigmaplus;
% tau=3;
% indnew=~spike&(res>tau*sigmalocal|-res>max(1,tau^2)*sigmalocal);
indnew=~spike&(res>tau*sigmaplus|-res>max(1,tau^2)*sigmaplus);
indrm=spike&(res<threshrm*sigmaminus|-res>max(1,threshrm^2)*sigmaminus);
indnem(indrm)=false;
rem = sum(indrm);
if sum(spike)~=0 && detect
newold=indnew|spike;
if spike(end) || spike(1)
if ~spike(1)
uclean=interp1([find(~spike);n],[u(~spike);u(end-1)],1:n);
elseif ~spike(end)
uclean=interp1([1;find(~spike)],[u(2);u(~spike)],1:n);
else
uclean=interp1([1;find(~spike);n],[u(2);u(~spike);u(end)],1:n);
end
else
uclean=interp1(find(~spike),u(~spike),1:n);
end
% uclean(newold)=0;
try
[mu,imu]=findpeaks(uclean);
catch e
disp(e)
end
% if ~isempty(find(imu==24))
% disp('24 is in the race');
% end
ind=mu>std(uclean(abs(uclean)<1.3236*std(uclean)/sqrt(2)));%μ%>=beta*max(uclean)&(data(imu)-(data(imu-1)+u(imu-1)*dx)<tau*sigmaplus | data(imu)-(data(imu+1)-u(imu)*dx)<tau*sigmaplus)'&(~spike(imu+1) | ~spike(imu-1))';
mu=mu(ind);
imu=imu(ind);
else
try
[mu,imu]=findpeaks(u);
catch e
disp(e)
end
ind=mu>std(u(abs(u)<1.3236*std(u)/sqrt(2)));%&mu>=beta*max(u)&(data(imu)-data(imu-1)<tau*sigmaplus | data(imu)-data(imu+1)<tau*sigmaplus);
mu=mu(ind);
imu=imu(ind);
newold=spike;
end
jj=1;
while jj<=length(imu)
% if indnew(imu(jj))
% indnew(imu(jj))=false;
% end
% if imu(jj)<n-1 && newold(imu(jj)+1) && ~indnew(imu(jj)+2)
% kk=imu(jj)+1;
% while kk<n
% if newold(kk)
% % if data(kk)<data(kk+1) || u(kk+1)*(dx^nanti)>-tau*sigmaplus
% if u(kk+1)*(dx^nanti)>-tau*sigmaplus
% spike(kk)=false;
% indnew(kk)=false;
% weight(kk)=1;
% end
% else
% kk=n;
% end
% kk=kk+1;
% end
% end
% if imu(jj)>2 && newold(imu(jj)-1) && ~indnew(imu(jj)-2)
% kk=imu(jj)-1;
% while kk>1
% if kk==24
% disp('this better not pass');
% end
% if newold(kk)
% % if data(kk)<data(kk-1) || u(kk-1)*(dx^nanti)<tau*sigmaplus
% if u(kk-1)*(dx^nanti)<tau*sigmaplus
% spike(kk)=false;
% indnew(kk)=false;
% weight(kk)=1;
% end
% else
% kk=1;
% end
% kk=kk-1;
% end
%
% end
jj=jj+1;
end
newnoise=find(indnew);
pred=AN(u,dx);
if (~isempty(newnoise)||sum(indrm)~=0) && detect
ii=0;
% if ~isempty(find(newnoise==24))
% disp('not today!')
% end
spike(newnoise)=true;
weight(newnoise)=0;
spike(indrm)=false;
weight(indrm)=1;
if plotflag
subplot(2,1,1);
h=plot(1:n,data,'-b',1:n,data,'.b',1:n,AN(u,dx),'-r',find(spike),data(spike),'og',imu,data(imu),'vb',1:n,min(data)+falph*(max(data)-min(data))/max(falph));
set(h(3),'linewidth',2);
legend('Regularized Estimate','Data','Data','Spikes');
subplot(2,1,2);
plot(1:n, u, 'ok',imu,mu,'vb'), drawnow;
end
datap=AN(u,dx);
for jj=1:n
if spike(jj) && jj~=1 && jj~=n
st=jj-1;
jj=jj+1;
while jj<n && spike(jj)
jj=jj+1;
end
en=jj;
datap(st:en)=linspace(datap(st),datap(en),en-st+1);
end
end
% datap(~noise)=pred(~noise);
u0p = [ u(1); diff( datap )/dx ];
% i = 1;
% while i<nanti
%
% u0p=[0; diff( u0p )/dx;];
% i=i+1;
% end
% alph=((sigmaplus*sigmaplus)^expvar)*dx^(nanti-1)*((abs(sigmaminus-sigmaplus)/max(sigmaminus,sigmaplus))^2)*(max(abs(u*dx))/max(abs(diff(data(~spike)))))^2;
if plotflag
subplot(2,1,1);
h=plot(1:n,data,'-b',1:n,data,'.b',1:n,AN(u0p,dx),'-r',find(spike),data(spike),'og',imu,data(imu),'vb',1:n,min(data)+falph*(max(data)-min(data))/max(falph));
set(h(3),'LineWidth',2);
legend('Regularized Estimate','Data','Spikes');
subplot(2,1,2);
plot(1:n, u, 'ok',imu,mu,'vb',1:n,u0p,'ob' ), drawnow;
end
u=u0p;
else
if diagflag
disp('no new noise');
end
% expvar=1;
if ii==iter && detect
ii=0;
expvar=0.75;
detect=false;
ufine=u;
end
% Display plot.
if plotflag
subplot(2,1,1);
h=plot(1:n,data,'-b',1:n,data,'.b',1:n,AN(u,dx),'-r',find(spike),data(spike),'og',imu,data(imu),'vb',1:n,min(data)+falph*(max(data)-min(data))/max(falph));
set(h(3),'linewidth',2);
subplot(2,1,2);
plot( 1:n,u, 'ok',imu,mu,'vb' ), drawnow;
end
end
sigmaminus=mean(abs(res(~spike)));
dd=abs(diff(data(~spike)));
sigmaplus=sum(dd.*(1-dd/max(dd)))/sum(1-abs(dd)/max(abs(dd)));
% ddp2=;
phi=(max(abs(u*dx))/max(abs(diff(data(~spike)))));
chi=(abs(sigmaminus-sigmaplus)/max(sigmaminus,sigmaplus));
% wdu=(1-((abs(D*u))/max(abs(D*u))));
%
%
%
% % ddp=[0;abs(diff(data)).*(1-abs(u(2:end))/max(abs(u(2:end)))).*wdu(2:end)]...
% % +[abs(diff(data)).*(1-abs(u(1:end-1))/max(abs(u(1:end-1)))).*wdu(1:end-1);0];
%
% ddp=[0;abs(diff(data)).*(1-abs(u(2:end))/max(abs(u(2:end))))]...
% +[abs(diff(data)).*(1-abs(u(1:end-1))/max(abs(u(1:end-1))));0];
%
%
% % den8 = (abs((res)/(sigmaplus)).*(1-wdu))/(sum(1-wdu));
% % den8(den8<sqrt(ep))=sqrt(ep);
% den8=((((abs(u))/max(abs(u)))).^2);
% % den8(den8<ep)=ep;
%
%
% % den8(den8==0)=min(den8(den8~=0));
%
%
% %
% % if (sum(abs((res(~spike))/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))))>rsm
% % break;
% % else
% % rsm=(sum(abs((res(~spike))/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
% % end
% rsm=(sum(abs((res(~spike))/(sigmaminus)).*(1-wdu(~spike)))/(sum(1-wdu(~spike))));
% thresh=max(thresh,2*rsmax);
% tau=(1-eta)*taumin+(eta)*(thresh);
% sigmamax=min(sigmaplus,sigmaminus);
% eta=exp(-((sigmaminus-sigmamax)^2)/(2*sigmamax^2))*exp(-abs(rsp-1)^2);
%
% qq=(rsm/rsmax)^(1/1);
% e8=((1+nsr^((1-eta)*tau/thresh))/(sqrt(1+eta)*(rsmax*(1-qq)+qq*rsm)));
% % e8=-1+1/eta;
% % e8=1;
% sigmalocal=filter(b,a,abs(res));
%
% den8=(exp(-(abs(ddp-max(ddp)))/std(ddp))+ep).^(e8);
% % den8(den8<ep^(e8))=ep^(e8);
%
% pp=exp(-((abs(sigmalocal-max(sigmalocal)).^2)/var(sigmalocal))-(abs(u)/var(abs(u))));
% pp=exp(-(abs(u.^2)/var(abs(u))));
% ms=max(ddp(ddp<(mean(ddp)+3*std(ddp))));
% jj=(1-abs(u)/max(abs(u)));
% dd2=[abs(diff(data));abs(diff(data(end-1:end)))];
% kk=(abs(dd2)/max(abs(dd2)));
% pp=exp(-((abs(sigmalocal-max(sigmalocal)).^2)/(ms)^2)-((u.^2)/var(u))).*kk.*jj;
% pp=pp/max(pp);
% plot(pp);
% ind=sigmalocal>sigmaplus&abs(u)<std(u);
%
% % den8(ind)=den8(ind)+sigmalocal(ind);
% den8=(1-pp).*den8+pp.*(sigmalocal+sigmaplus);
%
% % den8=(1-(abs(ddp-max(ddp)))/std(ddp).^(e8));
% % den8=(abs(ddp)/max(ddp)).^(e8);
%
%
% plot(1:n,log(den8));
% legend('den8','pp','sigmalocal')
% rs2=(mean(abs(res(~spike))));
% alph=sigmaplus*((1-eta)^2)*phi + sigmaplus*sigmaplus*phi*chi+sigmaminus*sigmaminus*eta;
calcDen8();
alph=sigmaplus*sigmaplus;
sigmaminus=mean(abs(res(~spike)));
% tau=taumin+1/(1-eta);
% sigma=sigmaminus
% sigma=mean(abs(res(~noise)));
% alph=(mean(abs(res(~spike)))^expvar)*dx^(nanti-1);
ii=ii+1;
iii=iii+1;
end
fest=zeros(length(data),1+nanti);
fest(:,1+nanti)=u;
ii=nanti;
while ii>0;
fest(:,ii)=cumsum(fest(:,ii+1))*dx;
ii=ii-1;
end
fest(:,1)=fest(:,1)+ofst;
festfine=zeros(length(data),1+nanti);
if exist('ufine','var')==0;
ufine=u;
end
festfine(:,1+nanti)=ufine;
ii=nanti;
while ii>0;
festfine(:,ii)=cumsum(festfine(:,ii+1))*dx;
ii=ii-1;
end
festfine(:,1)=festfine(:,1)+ofst;
if plotflag
subplot(2,1,1);
h=plot(1:n,data,'-b',1:n,data,'.b',1:n,AN(u,dx),'-r',find(spike),data(spike),'og');
set(h(3),'linewidth',2);
subplot(2,1,2);