The geometric moments in three dimensions (3D) are defined:
Tensor method
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.
Mii :
MijMij :
MijMjkMki :
The division by the suitable power of m000 is normalization to scaling.
In the following appendix, all independent invariants have from 1 to 10 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.
independent 3D rotation invariants up to the 10 indices in the tensor product.
Method of composite complex moment forms
The 3D complex moment is defined in spherical coordinates
where:
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 the main limitation of increasing the order. Therefore, we present here the invariants to the fifth order only. We cannot guarantee the precise conversion for higher orders.
3D complex moment rotation invariants up to the fifth order
Publications:
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Tensor Method for Constructing 3D Moment Invariants, Computer Analysis of Images and Patterns. Proceedings, Eds: A. Berciano, D. Díaz-Pernil, W.G. Kropatsch, H. Molina-Abril, P. Real, Springer, (Berlin 2011), pp. 213–219. CAIP'11 /14./, (Sevilla, ESP, 29.08.2011-31.08.2011). Download
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Moments and Moment Invariants in Pattern Recognition, Wiley & Sons Ltd., 2009 .
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: "Moment forms invariant to rotation and blur in arbitrary number of dimensions", IEEE Transactions on Pattern Analysis and Machine Intelligence vol.25, 2, pp. 234-246, 2003
Relevant publications by other authors:
- D. Cyganski, J.A. Orr, "Object Recognition and Orientation Determination by Tensor Methods," In: T.S. Huang (ed.) Advances in Computer Vision and Image Processing, pp. 101–144. JAI Press 1988.
- T.H. Reiss, P.J.W. Rayner, "On Generating Features Invariant to Linear Transformations in Two and Three Dimensions," Technical report CUED/F-INFENG/TR.87, Cambridge University Engineering Department 1991.
- C.H. Lo, H.S. Don, "3D moment forms: Their construction and application to object identification and positioning, " IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1053-1064, vol.11, 10, 1989.
