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I've struggled to find any literature explaining this, so here's hoping someone can help out.

I work in a subfield of MRI where the data that we collect is 4D (3D spatial and 1D temporal). Due to certain hardware (and participant) constraints, it is frequently the case that the data we collect is spatially anisotropic. For example, a signal intensity at some point in space reflects the measurement of some physical phenomenon emanating from a 0.5 mm x 0.5 mm x 1.0 mm volume in space.

So for a given time point, the raw data we have in our possession is a 3D matrix where each cell represents the measured signal coming from an anisotropic volume of space.

At any rate, one of the customary preprocessing methods that takes place prior to any sort of statistical analysis is to resample these data sets into spatially isotropic forms.

No where can I find any justification for why this is done. If anyone could shed some light onto some underlying theory that motivates this decision, it would be greatly appreciated. Cheers~


Edit: To offer some additional details...

Consider a 2D space in "real-world" that spans 5mm x 5mm. Suppose this space takes on discrete measurements representing 1.25 mm x 1.00 mm (an anisotropic area). This means that we will have an 4x5 matrix of measurements representing our real-world space. This is the data that the MRI outputs.

Now, suppose I want to "resample" this anisotropic data into an isotropic format using a grid of 1.00 mm x 1.00 mm. For this procedure, lay down the anisotropic matrix onto the real-world coordinate system, and, on top of this, superimpose the isotropic grid. Then, depending on your spatial resampling method (e.g. nearest neighbor) the isotropic grid will be populated with values to form a 5x5 matrix (where these values are dictated by what values were initially present in the acquired anisotropic dataset) This would look something like this:

Picture 1

The above example is only a 2D spatial data set...but, hopefully, it is clear how to generalize this procedure to the higher 3D case.

To add further context, these datasets are typically subjected to spatial transformations that can be either linear or non-linear. Further, the data sets are smoothed (using a whole range of different approaches...some more 'standard' than others) for noise minimization. Once some other preprocessing steps take place, statistical analysis finally occurs.

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    $\begingroup$ Welcome to SE.SP! I'm not sure we'll be able to get an answer, but it's an interesting question. Would it be possible for you to edit your question and add some more detail (perhaps just appropriate links) to explain what you mean (mathematically) by resample these data sets into spatially isotropic forms ? $\endgroup$
    – Peter K.
    Feb 4 at 0:21
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    $\begingroup$ @PeterK. certainly - I'll add as much detail as I can (may take a few...to make sure I have accurately portrayed the methodology). $\endgroup$
    – S.Cramer
    Feb 4 at 0:22
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    $\begingroup$ @PeterK. I added some additional commentary, which hopefully clarifies what I mean. $\endgroup$
    – S.Cramer
    Feb 4 at 1:49
  • $\begingroup$ Excellent. Thank-you. $\endgroup$
    – Peter K.
    Feb 4 at 2:05
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    $\begingroup$ @Royi the way in which I used the terms reflects how the field uses the terms. I am quite certain of this. If you prefer to think of this as a question of "why prefer square grids over rectangular grids", then the theme of the question does not change. $\endgroup$
    – S.Cramer
    Feb 4 at 12:42

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I think this is simply done to get "uniform" sampling and the same sample rate in all spatial coordinates.

If you want to represent a continuous quantity that is a function of time and space $v_C(t,x,y,z)$ inside a computer, you need to sample it, i.e. turn it into numbers. $v_C$ can be voltage, pressure, field strength, etc.

Sampling means you have a discrete sequence $v_D$ that samples $v_C$ at discrete instances of $t$, $x$ ..., i.e.

$$v_D[k,l,m,n] = v_C(t_k,x_l,y_m,z_n) \; \; \; k,l,m,n \in \mathbb{Z}$$

So in order for the sampling to be uniform you want the sampling points to be equidistant , for example

$$x_1-x_0 = x_1-x_2 = x_3-x_2 = .... = \Delta_x$$

And if you want your spatial sample rate to be the same in all dimensions you need to make sure that

$$\Delta_x = \Delta_y = \Delta_z$$

If your hardware doesn't do this by default, you need to resample, which I believe is what you are describing.

The reason for this is pretty straight forward: the vast majority of existing signal processing algorithms assuming a uniform grid and for multi-dimensional data, they also assume that the sample interval in all dimensions is the same. You can work around it but almost everything will get a heck of a lot easier if you use an equally spaced uniform gird, provided you can do the re-sampling without introducing too much error or noise.

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  • $\begingroup$ Yes, I think the uniform sampling in each dimension is the reason, so that other algorithms can be applied as-is. $\endgroup$
    – Peter K.
    Feb 4 at 13:25
  • $\begingroup$ @PeterK. and Hilmar. Thank you for the answer, but I find this sort of discomforting hah. When we resample data, we necessarily introduce fictitious and eliminate real measurements. Assuming I am reading your final paragraph correctly, your claim is that the field is willing to sacrifice the fidelity of the data sets for mere convenience? Is that correct? $\endgroup$
    – S.Cramer
    Feb 4 at 14:42
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    $\begingroup$ @S.Cramer: You need to ask the "field" what they are willing to do and way. Resampling is a well understood (if somewhat complicated) algorithm. You can do a lot to minimize damage and in some cases avoid any damage at all. It depends on your data and the algorithm you use. $\endgroup$
    – Hilmar
    Feb 4 at 22:38

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