I'm wondering how does ndimage.measurements.center_of_mass calculate the center of mass values for a two dimensional numpy array.

For example let's say we have an array m:

m = numpy.array([[1,2,3,1],[4,4,4,1],[5,6,7,2]])

Then one would get the following for the coordinates of the center of mass (x, y):

x, y = ndimage.measurements.center_of_mass(m) 

x = 1.325  and  y = 1.3

So, how are those values of x and y mathematically derived (actual derivation)?

When I follow the steps in the function definition center_of_mass, I don't get that result.

  • $\begingroup$ Can you show us your attempts where you implement their method? $\endgroup$
    – WBM
    Mar 4, 2021 at 13:52

1 Answer 1


If you go to the source page for this method you can find the function:

def center_of_mass(input, labels=None, index=None):
    Calculate the center of mass of the values of an array at labels.
    input : ndarray
        Data from which to calculate center-of-mass.
    labels : ndarray, optional
        Labels for objects in `input`, as generated by `ndimage.label`.
        Only used with `index`.  Dimensions must be the same as `input`.
    index : int or sequence of ints, optional
        Labels for which to calculate centers-of-mass. If not specified,
        all labels greater than zero are used.  Only used with `labels`.
    center_of_mass : tuple, or list of tuples
        Coordinates of centers-of-mass.
    normalizer = sum(input, labels, index)
    grids = numpy.ogrid[[slice(0, i) for i in input.shape]]

    results = [sum(input * grids[dir].astype(float), labels, index) / normalizer
               for dir in range(input.ndim)]

    if numpy.isscalar(results[0]):
        return tuple(results)

    return [tuple(v) for v in numpy.array(results).T]

Whereby sum is (sum - centre)^2 of the input labels

  • $\begingroup$ I hadn't seen that they define a function "sum". The first element of grids[0] or grids [1] is "0". Does that imply that in calculating "input * grids[dir]" the first element in the array has no effect in calculating the center of mass? $\endgroup$ Mar 4, 2021 at 20:35
  • $\begingroup$ Not quite following. What do you mean grids[:2] = 0? Have you implemented sum yourself? Is this for your input data? $\endgroup$
    – WBM
    Mar 4, 2021 at 20:59
  • $\begingroup$ Let me know if you have any questions otherwise please accept the answer $\endgroup$
    – WBM
    Mar 14, 2021 at 18:02

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