index to subscript, subscript to index

Alec Jacobson

October 28, 2010

weblog/

Many dimensional grids always give me a headache when trying to convert indices to subscripts. Subscripts to indices is more or less straightforward. If you're dealing with a 3D array with lengths: nx,ny,nz corresponding to subscripts: i,j,k, then converting a subscript to an index is as simple as:

index = i + nx*(j+ny*k)

The other way around is a little trickier, but has a simple derivation if you forget it. Start with the formula above:

i + nx*(j+ny*k) = index

Now notice that nx*(j+ny*k) is of course divisible by nx so moding both sides by nx gives you:

i = index % nx

Now you know index and i, so move those to the right hand side and begin again.

nx*(j+ny*k) = index - i

Divide through by nx:

             
j+ny*k = (index - i)/nx

Now you can apply the same idea to j and ny. So moding both sides by ny gives you:

             
j = (index - i)/nx % ny

Finally you know everything but k so the last equation is:

             
ny*k = (index - i)/nx - j

Divide through by ny:

             
k = ((index - i)/nx-j)/ny

In summary to convert index to i,j,k of a 3d array of size nx by ny by nz:

i = index % nx
j = (index - i)/nx % ny
k = ((index - i)/nx-j)/ny

The pattern here is pretty clear. And so it should be easy to generalize to whatever dimension array you are using 2D, 4D, N-dimensional. Note:This assumes you are using Column major ordering as in matlab (ind2sub, sub2ind). If you're using Row order just do every step in Spanish.