The geometric moments in three dimensions (3D) are defined
The moment tensor is defined
where x1=x, x2=y and x3=z. If p indices equal 1, q indices equal 2 and r indices equal 3, then
If we need rotation invariants, we work with the moment tensors as with Cartesian tensors. Each tensor product of the moment tensors, where each index is used just twice, is then the invariant to the 3D rotation. E.g.
The division by the suitable power of m000 is normalization to scaling.
In the following appendix, there are all linerly independent invariants that have from 1 to 8 indices in the generating tensor product. The tensor products are expressed by graphs (lists of edges), where each edge corresponds to one index and each node corresponds to one moment tensor.
Method of composite complex moment forms
The 3D complex moment is defined in spherical coordinates
order s=0, 1, 2, ...
latitudinal repetition l=rem(s,2), rem(s,2)+2, ..., s-2, s
longitudinal repetition m=-l, -l+1, ..., -1, 0, 1, ..., l-1, l
rem(s,2) is reminder after division, rem(s,2)=0 for even s and 1 for odd s.
is so-called spherical harmonics. The 3D complex moment can be computed as the proper linear combination of the 3D geometric moments.
Composite complex moment form is defined
where is Clebsch-Gordan coefficient.
We can construct 3D rotation invariants as moments ,
composite complex moment form ,
tensor product of a composite complex moment form and a moment
or the tensor product of two composite complex moment forms
If both forms are the same, we note them as square .The index j must be even.
The coefficients of the invariants are square roots of rational numbers. The conversion of the coefficients from approximate floating-point format to this format is main limitation of the increasing of the order, therefore we present here the invariants to the fifth order only. We cannot guarantee the precise conversion for higher orders.
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