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Bruce, Is that really the case? I mean, the k-space-domain operations equivalent to convolution/deconvolution with the Gaussian function are inversable? pk -----Original Message----- From: FSL - FMRIB's Software Library on behalf of Bruce Fischl Sent: Sat 3/7/2009 8:43 AM To: [log in to unmask] Subject: Re: [FSL] Actual implementation? [Re: Q: How to de-smooth BOLD images, previously smoothed with a known kernel-width?] Hi Raj, Gaussian blurring is the equivalent of running the diffusion equation for time proportional to sigma^2 (since the Gaussian is the Green's Function of it), which is not time-reversible. Information is irretrievably lost in diffusion, so I'm afraid the inversion isn't possible. sorry :< Bruce On Fri, 6 Mar 2009, Rajeev Raizada wrote: > On Fri, 6 Mar 2009 09:27:24 -0800, Michael T Rubens > <[log in to unmask]> wrote: > >> take FFT of smoothed image, divided by FFT of gaussian. the inverse FFT >> should be your unsmoothed data. > > Thanks... > But please see below... :-) > >> On Fri, Mar 6, 2009 at 5:12 AM, Rajeev Raizada <[log in to unmask] wrote: > [...] >>> Non-specific high-level exhortations to recast the smoothing >>> as a 3D Fourier filter and then to apply the inverse filter >>> are also welcome, but probably won't be quite as useful :-) > > I believe that the application of an inverse filter > may be easier said than done. > It appears that for Gaussian deblurring, the inverse is "ill-conditioned", > e.g. http://ieeexplore.ieee.org/iel5/5992/26914/01196312.pdf > > Two additional complications: > 1. Apparently there are some analytical results for deblurring of 2D discrete Gaussians, > but I don't know enough to know whether these hold in 3D as well. > 2. I believe that the 3D smoothing is actually done by a Gaussian convolved > by a sinc function, not just a plain vanilla Gaussian. > > Does anyone have an actual implementation of such "de-smoothing", > as opposed to an "in principle" description of what it ought to involve? > Googling for gaussian deblurring turns up a lot of hits for blind deconvolution > and methods of counteracting noise. > However, in this case the deconvolution is not blind at all, > as we know that it was a gaussian kernel of FWHM 6mm, > and also there wasn't any noise in the blurring process. > So, in principle those two facts ought to make things easier, I think? > > Any help greatly appreciated. > The more specific the better. :-) > > Raj > > >