# Catalogue of Artificial Intelligence Techniques

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## Binary and Grey Scale Moments

Keywords: Grey Scale Moments, eccentricity

### Categories: Pattern Recognition and Image Processing

Author(s): H.W. Hughes

In computer vision, moments are scalar values that encode some property of the shape or distribution of an object. They are often used as a compact representation of simple binary shapes, because: (i) they are easy to compute (i.e., real-time hardware to compute moments exists), and (ii) a few different moments are often sufficient to characterise uniquely a limited set of shapes. It is also possible to calculate functions of the moments that are invariant to translation, rotation and scale changes. The $i{j}_{}^{th}$ moment (${m}_{ij}^{}$ ) of a binary image about the centre of mass $\left({x}_{0}^{},{y}_{0}^{}\right)$ is given by:

${m}_{ij}^{}=\sum \left({x}_{0}^{}-x{\right)}_{}^{i}\left({y}_{0}^{}-y{\right)}_{}^{j}$

By varying the parameters $i$ and $j$ the resulting moment can yield a number of useful results such as the area and the eccentricityeccentricity. For a grey scale image whose intensity can be expressed as a function,$f\left(x,y\right)$ of$x$ and $y$ , the moment ${m}_{ij}^{}$ is given by:

${m}_{ij}^{}=\sum f\left(x,y\right)\left({x}_{0}^{}-x{\right)}_{}^{i}\left({y}_{0}^{}-y{\right)}_{}^{j}$

### References:

• Gonzalez, R.C. and Wintz, P., Digital Image Processing , Addison-Wesley, London, 1979 (second edition ).