Stylianos Kotsios, Evangelos Melas

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The goal of this paper is to present invariants of planar point clouds, that
is functions which take the same value before and after a linear transformation of a
planar point cloud via a 2 × 2 invertible matrix. In the approach we adopt here, these
invariants are functions of two variables derived from the least squares straight line of
the planar point cloud under consideration. A linear transformation of a point cloud
induces a nonlinear transformation of these variables. The said invariants are solutions
to certain Partial Differential Equations, which are obtained by employing Lie theory.
We find cloud invariants in the general case of a four−parameter transformation matrix, as well as, cloud invariants of various one−parameter sets of transformations which can be practically implemented. Case studies and simulations which verify our findings are also provided.


Invariants, Nonlinear Transformations, Lie Theory, Point Cloud, OCR, Image Analysis, Computational Geometry


Invariants, Nonlinear Transformations, Lie Theory, Point Cloud, OCR, Image Analysis, Computational Geometry.

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bibitem{kn:FLU} Flusser, J. Moment Invariants in Image Analysis,

it International Journal of Computer, Electrical, Automation, Control and Information Engineering, normalfont V.1, N.11, 2007.

bibitem{kn:kalouptsidis} Kalouptsidis, N. it Signal Processing Systems, Theory and Design, normalfont

John Wiley and Sons, 1997.

bibitem{kn:hu} Hu, M.K. Visual Pattern Recognition by Moment Invariants, it IRE Transactions of Information Theory., normalfont V.8, N.2, 1962, pp.179$-$187. ·

%March 1962

bibitem{kn:haim} Schweitzer, H.,

Straach, J.

Utilizing Moment Invariants and Groebner Bases to Reason About Shapes, it

Computational Intelligence, normalfont V.14, N.4, 1998, p.p. 461$-$474.

% November 1998

% Pages 461$–$474

bibitem{kn:suk} Flusser, J., Suk, T.

Degraded image analysis: An invariant approach, it

IEEE Transactions on Pattern Analysis and Machine Intelligence, normalfont V.20, N.6, 1998, p.p. 590$-$603. ·

bibitem{kn:mitra} Niloy, J. M.

Estimating surface normals in noisy point cloud data, it


Proceedings of the nineteenth annual symposium on Computational geometry, normalfont ACM New York, NY, 2003, p.p. 322$-$328.

it Advances in Pattern Recognition $—$ ICAPR 2001, normalfont

V. 2013 of the series Lecture Notes in Computer Science, p.p. 361$-$370.

bibitem{kn:neusel} Neusel, M. D. {it Invariant Theory}. Student Mathematical Library. 36. Providence, R.I.: American Mathematical Society, 2007.

bibitem{kn:gil} Gilmore, R. {it Lie Groups,Physics, and Geometry}, Cambridge University Press, New York, 2008.

bibitem{cant} Cantwell, B.J. {it Introduction to Symmetry Analysis}, Cambridge University Press, New York, 2002.

% bibitem{kn:kemper} Derksen, H., Kemper, G. %{it Computational invariant theory.} Springer, %Heidelberg, 2015.




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