3D Spatial Autocorrelation Function for Anisotropy

[Michael Doube]

Hi all

I've been working on a plugin that calculates anisotropy in 3D using the
  mean intercept length, which seems to work OK.  I've come across
another method that uses spatial autocorrelation, which has the
advantage that segmentation is not required.

http://dx.doi.org/10.1118/1.2437281

I remember there was chat earlier this month about 1D autocorrelation,
which might work (i.e. work out the autocorrelation function of line
probes at a range of angles through the stack) but the paper I've read
states that anisotropy can be worked out faster using k-space data from
magnetic resonance imaging.  I'm a bit confused as I though that k-space
data were essentially Fourier transformations of 2D images, so you get
your 2D cross-sectional image by doing an inverse Fourier transform on
the k-space data.  Additionally, I don't have MRI k-space images, I have
microCT images...

So does the method I'm reading propose that I work out the 3D Fourier
transform of a stack, then work out the ACF at each 3D angle?

Any insight here would be appreciated,

Mike

[Gluender-3]

Dear Mike,

most probably I'm lacking information on what you have in mind and
what you've already considered. However, here are some comments that
may channel further discussions:

1.
What exactly is meant by anisotropy and how can autocorrelation
provide a measure for it?
Does anisotropy in your case mean the uneven distribution of
intensities (grey-values) in space or does it mean an uneven extent
object extent in space?

In both cases I doubt that autocorrelation is a promising starting
point for analyses but the calculation of moments may be considered.

2.
The 1D Fourier transform of a projection of an object slice with a
parallel beam  is the spectral profile along a straight line through
the origin of the 2D Fourier spectrum of the object slice.

Commonly in todays CT fan-beam X-ray illumination is used which means
that you first have to re-arrange the values form several projections
in order to obtain the data for the 1D Fourier transform.

3.
An established feature in image analysis is the integral of the grey
values on straight lines either through origin of the 2D
power-spectrum or of the 2D autocorrelation function, both as a
function of the lines' angle. Back in 1986 I managed to prove that
both are mathematically identical:

<http://www.gluender.de/Writings/WritingsTexts/HardText.html#Gl-1986-2>

Maybe this feature or the orthogonal one, i.e. the integrals on
circles as a function of the diameter, may provide a sufficient
measure of anisotropy.

>Hi all
>
>I've been working on a plugin that calculates anisotropy in 3D using
>the  mean intercept length, which seems to work OK.  I've come
>across another method that uses spatial autocorrelation, which has
>the advantage that segmentation is not required.
>
>http://dx.doi.org/10.1118/1.2437281
>
>I remember there was chat earlier this month about 1D
>autocorrelation, which might work (i.e. work out the autocorrelation
>function of line probes at a range of angles through the stack) but
>the paper I've read states that anisotropy can be worked out faster
>using k-space data from magnetic resonance imaging.  I'm a bit
>confused as I though that k-space data were essentially Fourier
>transformations of 2D images, so you get your 2D cross-sectional
>image by doing an inverse Fourier transform on the k-space data.
>Additionally, I don't have MRI k-space images, I have microCT
>images...
>
>So does the method I'm reading propose that I work out the 3D
>Fourier transform of a stack, then work out the ACF at each 3D angle?
>
>Any insight here would be appreciated,
>
>Mike

HTH
--

                   Herbie

          ------------------------
          <http://www.gluender.de>

[dscho]

Hi,

On Tue, 28 Apr 2009, Michael Doube wrote:

> I've been working on a plugin that calculates anisotropy in 3D using the
> mean intercept length, which seems to work OK.  I've come across another
> method that uses spatial autocorrelation, which has the advantage that
> segmentation is not required.
>
> http://dx.doi.org/10.1118/1.2437281
>
> I remember there was chat earlier this month about 1D autocorrelation,
> which might work (i.e. work out the autocorrelation function of line
> probes at a range of angles through the stack) but the paper I've read
> states that anisotropy can be worked out faster using k-space data from
> magnetic resonance imaging.  I'm a bit confused as I though that k-space
> data were essentially Fourier transformations of 2D images, so you get
> your 2D cross-sectional image by doing an inverse Fourier transform on
> the k-space data.  Additionally, I don't have MRI k-space images, I have
> microCT images...
>
> So does the method I'm reading propose that I work out the 3D Fourier
> transform of a stack, then work out the ACF at each 3D angle?

I looked at the paper only briefly, but maybe I can help with the ACF <->
FFT thingie.

The autocorrelation of an image with itself can be described as a
convolution of the image with the flipped version of itself (where
"flipped" means "flipped in all available axis").

The advantage of looking at it that way is that convolution can be
described as a point-wise multiplication in Fourier space, reducing the
computational complexity rather dramatically.  You have to extend the
dimensions, though, as Fourier wraps around at the borders.  For example,
you have to embed a cube into a cube that is 2x2x2 times as large, filling
the rest with zeroes.

Now, I can only _guess_ that the result (basically, the autocorrelation
with respect to all possible offsets) is what they refer to as the
"anisotropy tensor", and I would expect that the analysis boils down to
determining the PCA of the offset vectors (weighted by the corresponding
autocorrelation value).

Well, I hope that I could help you at least a little bit...

Ciao,
Dscho