Re: maximum entropy method for deconvolution

*To*: mathgroup at smc.vnet.net*Subject*: [mg75469] Re: maximum entropy method for deconvolution*From*: dantimatter <dantimatter at gmail.com>*Date*: Wed, 2 May 2007 03:49:06 -0400 (EDT)*References*: <200704281000.GAA09123@smc.vnet.net><f146p6$mcq$1@smc.vnet.net>

Thanks to all for the help, and a special thanks to Guillermo Sanchez for the Modeling and Simulation notebook. I'm finding that my answer is still 'unpleasant'. Could it be that the convolution of a function with a step-function is something that simply cannot be deconvolved? That perhaps there's information lost in the convolution and it could never be recovered? Dan On Apr 30, 2:44 am, "turnback" <turnb... at bluebottle.com> wrote: > Hi, > > Your problem is a deconvolution problem. It is difficult in general. But > since your function p is simply a rectangle function, things may not be that > bad. > > Let me define t0:=33.6, g(t):=G, f(t):=F > > Then according to the definition of convolution, we can get: > > g(t)=Integrate[f(s), {s, t-t0, t}] > > and for any small dt, we can get: > > ==> g(t+dt)=Integrate[f(s), {s, t+dt-t0, t+dt}] > > ==> g(t+dt)-g(t) > =Integrate[f(s), {s, t+dt-t0, t+dt}]-Integrate[f(s), {s, t-t0, t}] > =Integrate[f(s), {s, t, t+dt}]-Integrate[f(s), {s, t-t0, t-t0+dt}] > =Integrate[f(s)-f(s-t0), {s, t, t+dt}] > > define l(t):=f(t)-f(t-t0), then > > g(t+dt)-g(t)=Integrate[l(s), {s,t,t+dt}] > ==> Limit[(g(t+dt)-g(t))/dt,dt->0] = Limit[(Integrate[l(s), > {s,t,t+dt}])/dt,dt->0] > > Suppose g(t) is differentiable, then > > ==> g'(t)=l(t)=f(t)-f(t-t0) > > then performing Fourier transform: > > ==> I*w*G(w)=(1-Exp[-I*w*t0])*F(w) > > ==> F(w)=G(w)*I*w/(1-Exp[-I*w*t0]) > > With inverse transform, you get f(t). > > By the way, According to my knowledge, p's Fourier transform is a Sinc > function. When you claim function p has a lot of zeros in the frequency > domain, you may implement the transform by using Discrete Fourier Transform > and you pick up a specific sampling rate. If real time performance is not > your concern, try to increase your sampling rate a lot. You should get > correct result with your original approach. > > After all, I just realise that instead of writing > p=UnitStep[t]*UnitStep[33.6-t], you can write it as: > p[t]=UnitStep[t]-UnitStep[t-33.6]. You may find much easier proof. > > hui. > > > > ----- Original Message ----- > From: "dantimatter" <dantimat... at gmail.com> > To: <mathgr... at smc.vnet.net> > Sent: Saturday, April 28, 2007 6:00 AM > Subject: maximum entropy method for deconvolution > > > hello all, > > > first off, many thanks to 'Roman' et al for all the previous help with > > my inversion problem. > > > i have a convolution function G which is the convolution of F and p (G > > = F**p). i know G and i know p, and i'd like to extract F. i can do > > this by taking the Fourier transform of G, dividing by the Fourier > > transform of p, and inverting the result to get F. the problem is > > that p is a step function (p = UnitStep[t]*UnitStep[33.6-t]) which has > > a lot of zeros in frequency space, and thus it is difficult to get at > > F via inversion. Mathematica is happily doing the inversion but the > > results are very noticeably wrong. > > > i understand from my conversations with some of you and much time > > spent in the library that this is in general a difficult problem, but > > there are some methods that are known to make this type of problem > > tractable, such as the maximum entropy method (MEM) for inversion. is > > anyone aware of an implementation of a MEM algorithm in Mathematica? > > i have read Numerical Recipes a couple of times and i am unable to get > > my head around the relevant chapter. If there isn't a Mathematica > > implementation, perhaps someone could offer some advice on where else > > to look? if it exists, a "for dummies" type book with step-by-step > > instructions would be the best resource for me... > > > cheers, > > dan > > ---------------------------------------------------------------------- > Find out how you can get spam free email.http://www.bluebottle.com

**Follow-Ups**:**Re: Re: maximum entropy method for deconvolution***From:*Sseziwa Mukasa <mukasa@jeol.com>

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