Thank you Bruce, 
I think, your explanation is quite sensible. Obviously, Gaussian filtering in the magnitude domain is not equivalent to the frequency-space multiplication as half of the information (phase) is lost. Not to mention that the fourier transform of the inverse-gaussian (Wald) function is not analytically defined. 
pk

________________________________

From: FSL - FMRIB's Software Library on behalf of Bruce Fischl
Sent: Sat 3/7/2009 10:20 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 Peter,

I'm pretty sure it is. You can think about Gaussian convolution as moving
intensities around the image since the diffusion equation obeys an
underlying conservation law. So any one voxel has its intensity "come from"
other voxels in the image. Unfortunately there is nothing unique about it,
so you don't know for example if the intensity came from a high value voxel
far away or a moderately valued voxel nearby. The inverse diffusion
equation is fundamentally ill-conditioned. The k-space thing is probably
only true in the limit of infinite support, etc..., but I haven't thought
about it.

cheers,
Bruce


On Sat, 7 Mar 2009, Kochunov, Peter wrote:

> 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
>>
>>
>>
>
>
>