Calculating geometric properties from Fourier representation
Pattern Recognition
Regular Article: Computing Fourier Transforms and Convolutions on the 2-Sphere
Advances in Applied Mathematics
Parametrization of closed surfaces for 3-D shape description
Computer Vision and Image Understanding
Rotation invariant spherical harmonic representation of 3D shape descriptors
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
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The volume, location of the centroid, and second-order moments of a three-dimensional star-shaped object are determined in terms of the spherical harmonic coefficients of its boundary function. Bounds on the surface area of the object are derived in terms of the spherical harmonic coefficients as well. Sufficient conditions under which the moments and area computed from the truncated spherical harmonic series converge to the actual moments and area are established. The proposed method is verified using a scanned head model and by recent measurements of the 433 Eros asteroid. An extension to non-star-shaped objects of genus 0 is provided. The computational complexity of our method is shown to be equal to that of the discrete spherical harmonic transform, which is O(N^2log^2N), where N is the maximum order of coefficients retained in the expansion.