A Hu moment invariant as a shape circularity measure

In this paper we propose a new circularity measure which defines the degree to which a shape differs from a perfect circle. The new measure is easy to compute and, being area based, is robust—e.g., with respect to noise or narrow intrusions. Also, it satisfies the following desirable properties:
• it ranges over (0,1](0,1] and gives the measured circularity equal to 1 if and only if the measured shape is a circle;
• it is invariant with respect to translations, rotations and scaling.Compared with the most standard circularity measure, which considers the relation between the shape area and the shape perimeter, the new measure performs better in the case of shapes with boundary defects (which lead to a large increase in perimeter) and in the case of compound shapes. In contrast to the standard circularity measure, the new measure depends on the mutual position of the components inside a compound shape.Also, the new measure performs consistently in the case of shapes with very small (i.e., close to zero) measured circularity. It turns out that such a property enables the new measure to measure the linearity of shapes.In addition, we propose a generalisation of the new measure so that shape circularity can be computed while controlling the impact of the relative position of points inside the shape. An additional advantage of the generalised measure is that it can be used for detecting small irregularities in nearly circular shapes damaged by noise or during an extraction process in a particular image processing task.