Re: Hyperstack with nz=1 problem

Posted by Fred Damen on
URL: http://imagej.273.s1.nabble.com/Hyperstack-with-nz-1-problem-tp5019980p5020075.html

Greetings,

Most MRI datasets, e.g., DICOM, files contain the slice thickness and in some
ways of viewing MRI images it is indeed used.  Voxels have dimensions.  Almost
all the time, the voxels within the slice plane, (i.e., xy) is acquired as a
square, thus rarely does anyone try to display the voxels with rectangles
instead of squares when displaying the x,y plane.  The z direction, or third
spatial dimension, which is commonly a different dimension than in the x,y
directions, is displayed including the third dimension, i.e., xz or yz planes,
the voxel is displayed as rectangles.  Also if there are inter-slice gaps,
then black space should be displayed in between voxels in the z direction as
in this case the voxels are not contiguous. If one is trying to compare an
isometric dataset with one that has a different z dimension, this is a very
important issue.

Since, in MRI, the data within a 2D slice or a 3D-dataset is acquired in the
frequency domain, it is correct to say that the space between the voxels in
the xy or xy/xz/yz, respectively, can be interpolated from the acquired data.
The Fourier components, i.e., acquired samples, are discrete, yet represent a
continuous spatial domain.  In Fourier acquired data this is done by
zero-padding. Interpolating in the z direction of a stack of slices is
complicated as the data is not spatially continuous and may not be contiguous.
 In MRI, being able to interpolate in the spatial domain in no way implies the
ability to interpolate in the frequency domain, this would be considered
inventing information.  For example, try acquiring every other line of the
Fourier data and then try to invent the unacquired, i.e., skipped, lines.  In
fact, to be able to extract the interpolated acquired samples within x,y plane
using data acquired from many parallel coils is hot topic in the MRI world,
and, to be able to do it in the z direction is a very hot topic.

Without the philosophical aspect this is just a bunch of numbers...

Thanks,

Fred

On Sat, February 10, 2018 5:07 am, Herbie wrote:

> Dear Christophe,
>
> to avoid a possible misunderstanding, I don't think that there is
> anything wrong with the very article and of course it is in no way
> contradictory to signal / systems theory. Personally, I find the
> article's style a bit one-sided and in any case it doesn't deal with the
> whole story, i.e. "signal discretization and reconstruction of the
> continuous signal".
>
> "[...] in any case it's always interesting to learn more about the
> "philosophical" aspect of what we're dealing with every day."
>
> I would prefer to call it mathematical not philosophical but you've put
> the latter in inverted commas, so it is perfectly ok.
>
> The advantage of mathematical conclusions is that they are generally not
> a matter of taste.
>
> For instance and with respect to Fred's tomographic slice question:
> Although the slice thickness may be known from the process of data
> acquisition, it is not contained in the slice data and therefore can't
> be reconstructed from it. However, if one has sufficient discrete
> neighboring slices and if one knows that the sampling not only in x and
> y but also in z was made according to the sampling theorem, then one can
> reconstruct the void volume between the slices (and of course between
> the samples of each slice). If one calls this reconstruction "thickness
> of the slice" is indeed rather a philosophical question than a
> mathematical one...
>
> Best greetings
>
> Herbie
>
> ::::::::::::::::::::::::::::::::::::::::::::::::::
> Am 10.02.18 um 11:30 schrieb Christophe Leterrier:
>> Hi Herbie,
>>
>> I didn't dare to send this often-posted link for you, Kenneth or the other
>> knowledgable people already discussing in this thread - I figured it could
>> be a nice introduction to the discussed problem for the whole list. The
>> argument made is along the line of your earlier statements about a pixel
>> not having a surface or a slice not having a thickness. Sadly I don't read
>> German to really understand if your underlying reasons or explanation
>> differ from Alvy Ray Smith - in any case it's always interesting to learn
>> more about the "philosophical" aspect of what we're dealing with every day.
>>
>> Best Regards,
>>
>> Christophe
>>
>> 2018-02-10 11:09 GMT+01:00 Herbie <[hidden email]>:
>>
>>> Bonjour Christophe,
>>>
>>> thanks for chiming in and of course I know this paper. However and
>>> although it is to the point, a more thorough view would be desirable.
>>>
>>> For those who read German, here is a link to an article written for those
>>> who prefer thorough scientific explanations that require only moderate
>>> mathematical knowledge:
>>> <www.gluender.de/Writings/WritingsTexts/WritingsDownloads/20
>>> 16_Diskretisierung.zip>
>>>
>>> Best
>>>
>>> Herbie
>>>
>>> ::::::::::::::::::::::::::::::::::::::::::::::::::
>>> Am 10.02.18 um 10:47 schrieb Christophe Leterrier:
>>>
>>> Hi everyone,
>>>>
>>>> I think the discussion has reached the "post the Alvy Ray Smith paper"
>>>> point:
>>>> https://news.ycombinator.com/item?id=8614159
>>>> "A Pixel Is Not A Little Square, A Pixel Is Not A Little Square, A Pixel
>>>> Is
>>>> Not A Little Square! (And a Voxel is Not a Little Cube)"
>>>>
>>>> As a biologist-microscopist I can't say I understand everything in it, but
>>>> I got the part about pixels not being little squares :)
>>>>
>>>> Christophe
>>>>
>>>>
>>>>
>>>> 2018-02-10 10:11 GMT+01:00 Herbie <[hidden email]>:
>>>>
>>>> Good day Fred,
>>>>>
>>>>> no problem with your statements. They are compatible with signal theory
>>>>> (see my answer to Kenneth).
>>>>>
>>>>> Mathematical fact is that samples are numbers (or in RGB number triplets)
>>>>> that have no spatial or temporal extension.
>>>>>
>>>>> If you consider the physical process of sampling, the question may arise
>>>>> of how an extended little area, or in your tomographic case, an extended
>>>>> little volume eventually leads to a number. The answer is easy:
>>>>>
>>>>>           By spatial integration.
>>>>>
>>>>> Consequently, it is not the little integration area or the little
>>>>> integration volume that is later (after pictorial or tomographic
>>>>> reconstruction) displayed as little area or little volume. It is the
>>>>> number
>>>>> (gray value) that resulted from integration during signal acquisition
>>>>> that
>>>>> is smeared out or interpolated in some fashion on a display or by a
>>>>> projector and presented as a light intensity.
>>>>>
>>>>> Of course the integration area or volume during signal acquisition may be
>>>>> much larger than the sampling distance. In tomography (CT and MRI) this
>>>>> is
>>>>> not only the case regarding the z-direction but also, usually not as
>>>>> pronounced, in the xy-directions. The classic example however, is the
>>>>> flying spot scanner in which the spot represents the integration area and
>>>>> the sampling distance may be chosen independently from the spot size.
>>>>>
>>>>> Hopefully I could clarify the topic a bit.
>>>>>
>>>>> Regards
>>>>>
>>>>> Herbie
>>>>>
>>>>> ::::::::::::::::::::::::::::::::::::::::
>>>>> Am 10.02.18 um 02:54 schrieb Fred Damen:
>>>>>
>>>>> Greetings,
>>>>>
>>>>>>
>>>>>> Beauty is in the eyes of the beholder...  and my beauty is MRI.
>>>>>>
>>>>>> In MRI a slice is a 3D entity with length, width and depth -- which we
>>>>>> call
>>>>>> slice thickness.  A voxel, i.e., volume element, represents a single
>>>>>> value for
>>>>>> a location in 3D space. Voxels are contiguous within the slice and
>>>>>> depending
>>>>>> on how data was collected may be contiguous in z also -- you can have
>>>>>> what we
>>>>>> call an interslice gap.  In MRI there is no way to acquire a slice with
>>>>>> infinitesimally thin slice thickness.  Usually the slice thickness is
>>>>>> more
>>>>>> than twice that of the in-slice voxel size.
>>>>>>
>>>>>> Thanks for the info,
>>>>>>
>>>>>> Fred
>>>>>>
>>>>>> On Fri, February 9, 2018 11:03 am, Herbie wrote:
>>>>>>
>>>>>> Good day!
>>>>>>>
>>>>>>> "[...] so that ImageJ treats a single slice as a volume?"
>>>>>>>
>>>>>>> A slice is an image!
>>>>>>>
>>>>>>> A slice has no extension orthogonal to itself.
>>>>>>> A pixel also has no extension in any direction because it is a
>>>>>>> mathematical
>>>>>>> point in 2D, i.e. a number or sample value.
>>>>>>> A voxel also has no extension in any direction because it is a
>>>>>>> mathematical
>>>>>>> point in 3D, i.e. a number or sample value.
>>>>>>>
>>>>>>> Pixels, i.e. values at points in 2D, are arranged in a 2D grid and the
>>>>>>> sometimes equidistant *spacing* of the grid points is often confused
>>>>>>> with
>>>>>>> the pixel size, that actually doesn't exist.
>>>>>>> (A pixel doesn't have a size.)
>>>>>>>
>>>>>>> Voxels, i.e. values at points in 3D, are arranged in a 3D grid and the
>>>>>>> sometimes equidistant *spacing* of the grid points is often confused
>>>>>>> with
>>>>>>> the voxel size, that actually doesn't exist.
>>>>>>> (A voxel doesn't have a size.)
>>>>>>>
>>>>>>> In short:
>>>>>>> A slice has no neighbors orthogonal to itself, i.e. there is no
>>>>>>> (defined)
>>>>>>> spacing in the third dimension.
>>>>>>>
>>>>>>> That said, you may indeed use dummy slices to define the missing
>>>>>>> spacing!
>>>>>>>
>>>>>>> HTH
>>>>>>>
>>>>>>> Herbie
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> --
>>>>>>> Sent from: http://imagej.1557.x6.nabble.com/
>>>>>>>
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>>>>>>>
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