Posted by
dscho on
Apr 29, 2009; 3:50pm
URL: http://imagej.273.s1.nabble.com/3D-Spatial-Autocorrelation-Function-for-Anisotropy-tp3692740p3692741.html
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