Adjacency matrix for image pixel graphs

One of my favorite NumPy functions is roll(). Here's an example of using this function to get neighborhood indices to create adjacency matrices for images.

An adjacency matrix is a way of representing a graph and shows which nodes of the graph are adjacent to which other nodes. For n nodes is is an n*n matrix A where a_ij is the number of edges from vertex i to vertex j. The number of edges can also be replaced with edge weights depending on the application.

For images, a pixel neighborhood defines a graph which is sparse since each pixel is only connected to its neighbors (usually 4 or 8). A sparse representation, for example using dictionaries, is preferable if only the edges are needed. For clustering using spectral graph theory, a full matrix is needed. The following function creates an adjacency matrix in sparse or full matrix form given an image:

from numpy import *

def adjacency_matrix(im,connectivity=4,sparse=False):
 """ Create a pixel connectivity matrix with
  4 or 8 neighborhood. If sparse then a dict is
  returned, otherwise a full array. """

 m,n = im.shape[:2]

 # center pixel index
 c = arange(m*n)
 ndx = c.reshape(m,n)

 if sparse:
  A = {}
 else:
  A = zeros((m*n,m*n))

 if connectivity==4: # 4 neighborhood
  hor = roll(ndx,1,axis=1).flatten() # horizontal
  ver = roll(ndx,1,axis=0).flatten() # vertical
  for i in range(m*n):
   A[c[i],hor[i]] = A[hor[i],c[i]] = 1
   A[c[i],ver[i]] = A[ver[i],c[i]] = 1

 elif connectivity==8: # 8 neighborhood
  hor = roll(ndx,1,axis=1).flatten() # horizontal
  ver = roll(ndx,1,axis=0).flatten() # vertical
  diag1 = roll(roll(ndx,1,axis=0),1,axis=1).flatten() # diagonal
  diag2 = roll(roll(ndx,1,axis=0),1,axis=0).flatten() # diagonal
  for i in range(m*n):
   A[c[i],hor[i]] = A[hor[i],c[i]] = 1
   A[c[i],ver[i]] = A[ver[i],c[i]] = 1
   A[c[i],diag1[i]] = A[diag1[i],c[i]] = 1
   A[c[i],diag2[i]] = A[diag2[i],c[i]] = 1

 return A

Use it like this:

from pylab import *

im = random.random((10,10))
A = adjacency_matrix(im,8,False)

figure()
imshow(1-A)
gray()
show()

Which should give you an image like the one below.