Someone solve this problem for me

[Pingu]

This is the kind of thing I had in mind:



I wanted to know the series of symmetrical spheres you'd get, starting with the smallest, and how many voxels each would have.

[Jet Black]

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

The emphasis of the question didn't come across right. Why Voxels?

[Jet Black]

I've seen those before.

finally, a jew skin lampshade.

[Brother Daniel]

OK so assuming that the centre of the sphere is the centre of a voxel, and that we include any voxel whose centre is within the sphere, we get (for voxels of size 1x1x1):

r2	number of voxels
0	1
1	7
2	19
3	27
4	33
5	57
6	81
8	93
9	123
10	147
11	171
12	179
13	203
14	251
16	257
17	305
18	341
19	365
20	389
21	437
22	461
24	485
25	515
26	587
27	619
29	691
30	739
32	751
33	799
34	847
35	895
36	925

[Brother Daniel]

Blue: number of voxels as a function of radius
Red: (4/3) p r³

[Pingu]

The emphasis of the question didn't come across right. Why Voxels?

Because voxels are how we plot data in brains.

[Pingu]

THANK YOU BD!!!!

I was beginning to think my question was incomprehensible.

I got as far as the 7 voxel one, then I got stuck.

[buttershug]

OK so assuming that the centre of the sphere is the centre of a voxel, and that we include any voxel whose centre is within the sphere, we get (for voxels of size 1x1x1):

You have voxels partly out of the sphere?
I'm picturing your second one as a cube with a cube on each face.
That would give a radius of 1.5 for sphere that encloses all of each voxel.

If you remove the requirement for the centre of the sphere being the centre of a voxel the second one would be 6 where you have four cubes forming a square with a voxel on top and bottom of the square.

Something else to consider is what volume of the sphere that completely contains all voxels is filled by voxel?

Otherwise you can get a really bad sphere with two side by side voxels.

[Brother Daniel]

You have voxels partly out of the sphere?

Yes, I was including any voxel whose centre is on or within the sphere.

If we specify that the voxel has to be completely within the sphere, then you get something like

r2	number of voxels
0	0
0.75	1
2.75	7
4.75	19
6.75	33
8.75	57

and so on.

I'm picturing your second one as a cube with a cube on each face.

yes.

That would give a radius of 1.5 for sphere that encloses all of each voxel.

More like 1.658 or so, to get all the corners in.

If you remove the requirement for the centre of the sphere being the centre of a voxel the second one would be 6 where you have four cubes forming a square with a voxel on top and bottom of the square.

Hmmm. I was assuming that the voxels all fit into a cubic grid. Here you seem to be thinking of the voxels as "loose". That's a harder problem.

But even with the grid, we'll get different answers if we (e.g.) put the centre of the sphere on one of the corners instead of at the centre of a voxel. We'd jump from 0 voxels to 8 voxels, for example.

[Brother Daniel]

THANK YOU BD!!!!

I was beginning to think my question was incomprehensible.

I got as far as the 7 voxel one, then I got stuck.

yw

I haven't found a tidy formula. I had to do a bit of coding.

Note that if you have a value of r² that's in between two of the numbers in the left-hand column of my table, then you round down.

So your picture in #26, for example, looks like the case of r = 5, i.e. r² = 25, in my table. (Just staring at it and counting, it looks like you have 515 voxels there.) And so it's applicable to 25 ≤ r² < 26.

[Brother Daniel]

Something else to consider is what volume of the sphere that completely contains all voxels is filled by voxel?

Here's the fraction of the sphere's volume that is accounted for by the voxels that are entirely within the sphere. (Given as a function of the radius.)

[Pingu]

yw

I haven't found a tidy formula. I had to do a bit of coding.

Note that if you have a value of r² that's in between two of the numbers in the left-hand column of my table, then you round down.

So your picture in #26, for example, looks like the case of r = 5, i.e. r² = 25, in my table. (Just staring at it and counting, it looks like you have 515 voxels there.) And so it's applicable to 25 ≤ r² < 26.

Yes, I was trying to code something. Then I decided that messy Duplo nuggets were better anyway.

But one day this will come in handy I think, for instance if we start with coordinates from an 8mm grid (which we may well do, once we leave fMRI behind).

[noAnthro]

You have voxels partly out of the sphere?
I'm picturing your second one as a cube with a cube on each face.
That would give a radius of 1.5 for sphere that encloses all of each voxel.

If you remove the requirement for the centre of the sphere being the centre of a voxel the second one would be 6 where you have four cubes forming a square with a voxel on top and bottom of the square.

Something else to consider is what volume of the sphere that completely contains all voxels is filled by voxel?

Otherwise you can get a really bad sphere with two side by side voxels.

Butters, but now you have the top and bottom voxel centered at the intersection of the other four voxels? That is aren't they offset on the horizontal plane by one half voxel from the other four voxels? Assuming that the other four voxels are oriented in the horizontal plane.

[Pingu]

Yes, the centre of the ROI needs to be the centre of a voxel, for my purposes anyway. For playing-with-math purposes you can do what you like

[Brother Daniel]

For playing-with-math purposes you can do what you like

Right. So if we change from Pingu's problem to butters's problem for a moment (where we're just trying to fit as many equal-sized cubes as possible within a sphere, allowing the cubes to fall loosely instead of conforming to a cubic grid):

radius	number of cubes
0.866	1
1.225	2
1.382	3
1.471	4
1.546	5
1.631	6
1.658	7
1.732	8

Note that butters's suggestion for 6 cubes requires a sphere big enough to hold 7 cubes. A tighter arrangement of 6 cubes has 2 on top of 4.

[PerseusOmega9]

For playing-with-math purposes you can do what you like

oh god, don't encourage them

[Brother Daniel]

lol

[ffejrxx]

instead of making the voxels smaller can you make the world bigger?

[Brother Daniel]

[Heinz Hershold]

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

And that is why there is a problem. Like fitting a square peg into a round hole.

It is easy enough to convert from rectangular coordinates to polar or spherical. The question is, can your software work with different types of coordinate systems?

It seems to me, if you want to graph a sphere, you want to use spherical coordinates, but then again, what do I know?

[kowalskil]

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?

Redefine foxel.

[Pingu]

Heinz, I think you are missing the point.

[Heinz Hershold]

Heinz, I think you are missing the point.

Probably true. Anyway, this is what I was talking about. I don't know if that is any use to you or not.

[Pingu]

Yes, that's essentially what I do.

[ffejrxx]

Voxelized
http://store.steampowered.com/app/34..._7_7_204_150_1

on sale for 9¢