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Subject:

Re: Actual implementation? [Re: Q: How to de-smooth BOLD images, previously smoothed with a known kernel-width?]

From:

Mark Jenkinson <[log in to unmask]>

Reply-To:

FSL - FMRIB's Software Library <[log in to unmask]>

Date:

Sat, 7 Mar 2009 18:08:49 +0000

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (177 lines)

Hi,

My understanding of this is that you are essentially working with a  
discrete
signal and applying a discrete operation to it.  You can either  
consider this
as a discrete convolution operation in image space, or as a  
multiplication in
the Fourier space, but not the continuous k-space, the space you get  
to with
the discrete FFT.  This does contain phase and magnitude information,
but not necessarily the same as the original k-space, as they are not  
the
same due to the magnitude operation and the discrete operations.
However, it is perfectly equivalent to perform the convolution by  
using the
FFT, multiplying and then doing the inverse FFT.  Any description in  
terms
of diffusion, or k-space are actually continuous analogues of this  
process
and not quite what you are really doing by blurring the discrete,  
magnitude
reconstructed signal.

As for inverting the operation, the problem is simply one of machine
precision.  Since we are talking about an already acquired signal that
has been convolved with a (discrete) Gaussian, then even noise is not
an issue.  So what really counts is the ability to represent the values
sufficiently accurately, and the precision to which the FFT can be
calculated.  Because the FFT involves a large summation, much of
which is partially canceling, it is sensitive enough that the  
inversion is
usually not sufficiently accurate since the suppression and then
enhancement of the high-frequency components in the image
typically have rounding errors which affect the image sufficiently to
cause problems.

So I agree that in practice it probably will not give good enough  
results,
but in theory there isn't really a loss of information in the discrete  
case
except with respect to the machine precision.

All the best,
	Mark



On 7 Mar 2009, at 16:43, Kochunov, Peter wrote:

> 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
>>>
>>>
>>>
>>
>>
>>
>

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