Scikit-image and central moments: what is the meaning?
Looking for examples of how to use image processing tools to "describe" images and shapes of any sort, I have stumbled upon the Scikit-image skimage.measure.moments_central(image, cr, cc, order=3) function.
They give an example of how to use this function:
from skimage import measure #Package name in Enthought Canopy
import numpy as np
image = np.zeros((20, 20), dtype=np.double) #Square image of zeros
image[13:17, 13:17] = 1 #Adding a square of 1s
m = moments(image)
cr = m[0, 1] / m[0, 0] #Row of the centroid (x coordinate)
cc = m[1, 0] / m[0, 0] #Column of the centroid (y coordinate)
In[1]: moments_central(image, cr, cc)
Out[1]:
array([[ 16., 0., 20., 0.],
[ 0., 0., 0., 0.],
[ 20., 0., 25., 0.],
[ 0., 0., 0., 0.]])
1) What do each of the values represent? Since the (0,0) element is 16, I get this number corresponds to the area of the square of 1s, and therefore it is mu zero-zero. But how about the others?
2) Is this always a symmetric matrix?
3) What are the values associated with the famous second central moments?
The array returned by measure.moments_central correspond to the formula of https://en.wikipedia.org/wiki/Image_moment (section central moment). mu_00 corresponds indeed to the area of the object.
The inertia matrix is not always symmetric, as shown by this example where the object is a rectangle instead of a square.
>>> image = np.zeros((20, 20), dtype=np.double) #Square image of zeros
>>> image[14:16, 13:17] = 1
>>> m = measure.moments(image)
>>> cr = m[0, 1] / m[0, 0]
>>> cc = m[1, 0] / m[0, 0]
>>> measure.moments_central(image, cr, cc)
array([[ 8. , 0. , 2. , 0. ],
[ 0. , 0. , 0. , 0. ],
[ 10. , 0. , 2.5, 0. ],
[ 0. , 0. , 0. , 0. ]])
As for second-order moments, they are mu_02, mu_11, and mu_20 (coefficients on the diagonal i + j = 1). The same Wikipedia page https://en.wikipedia.org/wiki/Image_moment explains how to use second-order moments for computing the orientation of objects.