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

Abstract:

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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