Journal of Mathematical Imaging and Vision

, Volume 56, Issue 1, pp 125–136 | Cite as

Shape Interpretation of Second-Order Moment Invariants

  • Joviša Žunić
  • Dragiša Žunić


This paper deals with the following problem: What can be said about the shape of an object if a certain invariant of it is known? Such a, herein called, shape invariant interpretation problem has not been studied/solved for the most of invariants, but also it is not known to which extent the shape interpretation of certain invariants does exist. In this paper, we consider a well-known second-order affine moment invariant. This invariant has been expressed recently Xu and Li (2008) as the average square area of triangles whose one vertex is the shape centroid while the remaining two vertices vary through the shape considered. The main results of the paper are (i) the ellipses are shapes which minimize such an average square triangle area, i.e., which minimize the affine invariant considered; (ii) this minimum is \(1/(16\pi ^2)\) and is reached by the ellipses only. As by-products, we obtain several results including the expression of the second Hu moment invariant in terms of one shape compactness measure and one shape ellipticity measure. This expression further leads to the shape interpretation of the second Hu moment invariant, which is also given in the paper.


Moments Moment invariants Shape Pattern recognition Image processing 



This work is partially supported by the Serbian Ministry of Science—projects OI174008/OI174026.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical InstituteSerbian Academy of Sciences and ArtsBelgradeSerbia
  2. 2.Faculty of Computer ScienceBelgradeSerbia

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