Interpretation of spatial moments

Spatial moments are a very simple and powerful way to describe the spatial distribution of values, provided they have a sufficiently strong central tendency, that is, a tendency to cluster around some particular value. This implies that “background” pixel values (e.g. zones where the quantity of interest, such as intensity or concentration, is zero), are small.

Interpretation of spatial moments

• order 0 = TOTAL MASS [units: concentration, density, etc.]
• order 1 = location of CENTRE OF MASS in x and y from 0,0 [units: L]
• order 2 = VARIANCE around centroid in x and y [units: L^2]
• order 3 = coeff. of SKEWNESS (symmetry) in x and y [units: n/a] –> =0 : SYMMETRIC distribution –> <0 : Distribution asymmetric to the LEFT (tail extends left of centre of mass) –> >0 : Distribution asymmetric to the RIGHT (tail extends right of centre of mass)
• order 4 = KURTOSIS (flatness) in x and y [units: n/a] –> =0 : Gaussian (NORMAL) distribution –> <0 : Distribution FLATTER than normal –> >0 : Distribution MORE PEAKED than normal –> <-1.2: BIMODAL (or multimodal) distribution

Parameters derived from 2nd moments (from Applied Image Processing Awcock (1995))

• ELONGATION (ECCENTRICITY) = Ratio of longest to shortest distance vectors from the object’s centroid to its boundaries
• ORIENTATION = For elongated objects, describes the orientation (in degrees) of the “long” direction with respect to horizontal (x axis)