MODWT HAAR LOOSES SUMMATION RECONSTRUCTION PROPERTY AFTER COEFFICIENTS THRESHOLDING

Modwt haar is known to be one side causal and shift invariant.

It also owns the summation reconstruction property.

This means that summing the coefficients gives the same result of using IMODWT reconstruction

except that IMODWT is not causal and so it has border distorsion whereas summation has not that problem.

The problem is that this only works if the coefficients are not modified in any way.

If instead I carry out the thresolding all the properties are lost.

x=rand(1,1000);
swc=modwt(x,'haar',4);
xrec= imodwt(swc,'haar');
xsum=sum(swc);
mean(abs(xrec-xsum))
figure(1);scatter(xrec,xsum)

ans =

   1.5834e-16

for i=1:5
thswc(i,:)= wthresh(swc(i,:),'s',0.08);  % thresholding
end

thxrec= imodwt(thswc,'haar');
thxsum=sum(thswc);
mean(abs(thxrec-thxsum))
figure(2);scatter(thxrec,thxsum)


ans =

    0.0453

MODWT HAAR & SOFT THRESHOLDING ARE BOTH ORTHOGONAL SO THE SUMMATION AND THE IMODWT RECONSTRUCTION WOULD MUST BE EQUAL !!!

Does anyone know the reason for this and how to fix it?

Thanks!