Someone solve this problem for me

[Pingu]

I want to approximate a sphere with voxels.

My voxels are 8mm x 8mm x 8mm.

If I assume that my centre is in the centre of a voxel, what size symmetrical "spheres" can I get, at different radii?

For instance, if I go to a max radius of 12 voxels, I could presumably get a seven voxel cluster. How many voxels in the next size up and so on?

[Mike PSS]

Is this your starting point Vauxhall?

[Linus]

For instance, if I go to a max radius of 12 voxels, I could presumably get a seven voxel cluster. How many voxels in the next size up and so on?

I don't understand the numerical example, but if a voxel is a cube of size L, then a sphere of radius R can fit at most

4 pi/3 * R^3 / L^3

voxels. Intuitively, I think the following is a lower bound (reduce the radius by the length of the diagonal in a voxel):

4 pi/3 * (R - sqrt(3) L)^3 / L^3.

[praying]

what the hell is a voxel?

[Linus]

A pixel, but 3D.

[Pingu]

It's OK, I solved it. Well, I didn't exactly solve it, but I figured that I should drop the requirement that the centre be in the center of the central voxel.

I've ended up with some weird "spheres" though. The voxels are just a bit big for the diameter "sphere" we want. Like trying to build golf balls out of lego.

[praying]

A pixel, but 3D.

thanks!

[Pingu]

what the hell is a voxel?

Pixel = little pic tures
Voxel = little vo lumes

Just as you represent a 2D picture in little coloured pixels, you can represent a 3D brain, say, as a set of little coloured (or not) voxels.

[Pingu]

Is this your starting point Vauxhall?

A pixellated Voxel!

[Mike PSS]

A pixellated Voxel!

I deserve an award for that word/picture/meaning association.

[Pingu]

You do

[Jet Black]

What are you using voxels for?

[gandalf]

Im guessing 3d giant penis0rz

[Heinz Hershold]

Pixel = little pic tures
Voxel = little vo lumes

Just as you represent a 2D picture in little coloured pixels, you can represent a 3D brain, say, as a set of little coloured (or not) voxels.

That will happen if all of the voxels are cubes. Have you tried different shapes?

Do you use mango?

[Pingu]

What are you using voxels for?

I want to specify a small spherical region of brain centered on a set (xyz) of coordinates in millimeters.

That will happen if all of the voxels are cubes. Have you tried different shapes?

Well, no, because I'm using cartesian coordinates.

The issue is that I have a location-of-interest derived from fMRI, on a series of scan using a 3mm grid (i.e. 3 x 3 x 3 mm voxels). Usually, if you want to define a spherical region (a "Region Of Interest" ROI) with radius, say 10mm, of a specific voxel, you just include all voxels within 10mm (Euclidian distance) of the coordinates and you get a set of sphericalish looking things of approximately the same size i.e. same number of voxels.

But I am going to an image that is much more grainy - the voxels are 8mm. So I wanted to standardise my "sphere" so I have the same shape and size ROI for each coordinate. But for a small "sphere" there's a lot of variance in shape and size.

It's a bit like trying to copy a Lego design in Duplo.

Interestingly, for a 10mm radius, most come out as 8 voxels, one as 12 - because the location I'm interested in is not dead centre of any 8mm voxel, so instead of finding a nice symmetrical 7 voxel object, it finds a lumpy thing.

But they are OK. And probably better than finding symmetrical "spheres" with a centre in the in the wrong place.

Do you use mango?

No, we use SPM mostly.

[ffejrxx]

modeling a sphere out of voxels is a challenge

making the voxels as small as possible would be best
at a 10mm rad
5mm voxels = 33
will be rounder

+/-10mm 1 voxel
+/-5mm = 9 voxels 3x3
+0mm = 13 voxels 3x3+4, 1 in the center of each side
total = 33

[Pingu]

Yabbut the size of the voxels is a given.

If it wasn't, I wouldn't have a problem. They are quite big.

[Mike PSS]

I want to specify a small spherical region of brain centered on a set (xyz) of coordinates in millimeters.

The issue is that I have a location-of-interest derived from fMRI, on a series of scan using a 3mm grid (i.e. 3 x 3 x 3 mm voxels). Usually, if you want to define a spherical region (a "Region Of Interest" ROI) with radius, say 10mm, of a specific voxel, you just include all voxels within 10mm (Euclidian distance) of the coordinates and you get a set of sphericalish looking things of approximately the same size i.e. same number of voxels.

But I am going to an image that is much more grainy - the voxels are 8mm. So I wanted to standardise my "sphere" so I have the same shape and size ROI for each coordinate. But for a small "sphere" there's a lot of variance in shape and size.

It's a bit like trying to copy a Lego design in Duplo.

Interestingly, for a 10mm radius, most come out as 8 voxels, one as 12 - because the location I'm interested in is not dead centre of any 8mm voxel, so instead of finding a nice symmetrical 7 voxel object, it finds a lumpy thing.

But they are OK. And probably better than finding symmetrical "spheres" with a centre in the in the wrong place.

No, we use SPM mostly.

I wonder if some crystal packing rules can help.



Although you say that the cartesian grid is "fixed" (i.e. the voxel construct of the brain in question) and the coordinate of interest is variable?

[Pingu]

Yes. I have a set of coordinates referring to voxels on a 3mm grid. I want spheres around them, but for an 8mm grid, same origin.

[Mike PSS]

Yes. I have a set of coordinates referring to voxels on a 3mm grid. I want spheres around them, but for an 8mm grid, same origin.

What's your radius of interest of each coordinate? Fixed? Variable?

[Pingu]

Well, it's kind of suck it and see. I wanted it small enough so that close ones don't overlap, but big enough so that you've a chance of hitting the signal of interest.

We'll probably take the first principle component from the voxels of each sphere, and you need a decent number of spheres to be reasonably confident that the first principle component is capturing the relevant variance.

[Pingu]

The data is from a magnetoencephalography (MEG) scan.

[Mike PSS]

The data is from a magnetencephalography (MEG) scan.

I've seen those before.

[Mike PSS]

Yes. I have a set of coordinates referring to voxels on a 3mm grid. I want spheres around them, but for an 8mm grid, same origin.

If each 3x3x3 voxel is a unit cube, then each 8x8x8 voxel is 2-2/3 rds x 2-2/3 rds x 2-2/3 rds unit cubes. That's almost 19 unit cubes (18.96) per 8x8x8 voxel. If your coordinate space is less than, say 500 units then your error in estimating 19 unit cubes per 8x8x8 voxel will not be very far off.

Transpose your cartesian coordinates by 2-2/3rds in each direction to hit the boundries of your new voxel frames, then use a radius to define your new sphere and you should be able to define the boundry edges of each new 8x8x8 block from the new coordinate boundries radiating from the sample center ± radius on each axis.

Since your estimating a cube instead of a sphere you'll end up with almost double the number of unit cells gathered (volume of sphere vs. volume of cube). But the transpose math should be easier.

Do you need to transpose each 3x3x3 voxel into some 8x8x8 sphere?

[Pingu]

No, I have 13 locations on a 3x3x3 grid, i.e. 13 sets of 3D coordinates.

I just want spheres round them, which wouldn't normally be a problem (and I only want spheres because I want a uniformish distance from the point-of-interest), because I'm usually dealing with the same grain of grid.

But the MEG spatial resolution is quite coarse, and a bit dodgy so I want a reasonable stab at finding a reproducible signal, across lots of subjects.

I think I'm going to have to go with my lumpy duplo "spheres".