Geometric moment computation for polygonal shapes and objects with a spline boundary
Prof. Dr. A. Tuzikov
National Academy of Sciences of Belarus, Minsk


It is of interest for different applications to compute geometric moments of plane or volumetric objects. Moments of zero order define the object area and volume, respectively. The object centroid is computed using first order moments and the orientation (we mean axes of inertia) --- from second order moments. It is well known that geometric moments and moment invariants are very useful for recognition of objects and images. We present explicit formulae and efficient algorithms for geometric moment calculation of polygonal shapes. The algorithms take advantages of polygonal shape representation and use only coordinates of shape vertices and face orientation. We present also an approach for computation of area and geometric moments for a plane object with a spline curve boundary. The explicit formulae are obtained for area and low order moment calculation. The complexity of calculation depends on the moment order, spline degree and the number of control points used in spline representation. The formulae proposed use the advantage that the sequence of spline control points is cyclic. It allowed to reduce substantially the number of summands in them. The formulae might be useful in different applications where it is necessary to perform measurements for shapes with a smooth boundary. We consider also an algorithm for volume evaluation of a 3D object from area measurements done in non-parallel cross-sections. The algorithm is based on Watanabe formula for volume computation and uses an interpolation by cubic splines. The same splines are applied also for computation of object area and centroid in every cross-section. It allowed us to derive an explicit formula for volume computation with application for 3D freehand ultrasound imaging.

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