# Image Analysis: Identification of Objects via Polynomial Systems

• Robert H. Lewis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)

## Abstract

The problem is to identify a movable object that is in some sense known, if it is encountered later. Suppose we have a sensor, on a fixed radar station or a moving platform. We have an object, say object A, previously measured, with certain distinct identifiable points $$p_i.$$ We know the distances between these points. We later encounter a similar object B and want to know if it is A. We have a sensor that sends and receives electronic signals, and so we measure the distances $$t_i$$ from the sensor to the distinguished points on B.

We first consider the two-dimensional case. Assume there are three distinct points on A. We have our measured distances $$t_1, t_2, t_3$$ and previously known distances between the points on A, $$d_1, d_2, d_3$$. We derive a polynomial system relating these quantities and show that it is easy to solve yielding a resultant that is the “signature” for A. Its use will eliminate B if B is not A.

The generalization to three dimensions is immediate. We need a fourth point. The polynomial system contains many parameters, but we solve it symbolically. We then discuss generalizations involving flexibility. In those cases we need five points and the systems are much more complex.

We compare solutions on Magma, Maple, and Fermat computer algebra systems.

## Keywords

Image analysis Polynomial system Resultant Parameters Dixon Gröbner basis

## References

1. 1.
Buse, L., Elkadi, M., Mourrain, B.: Generalized resultants over unirational algebraic varieties. J. Symbolic Comp. 29, 515–526 (2000)
2. 2.
Cox, D., Little, J., O’Shea, D.: Using algebraic geometry. In: Graduate Texts in Mathematics, vol. 185. Springer, New York (1998).
3. 3.
Dixon, A.L.: The eliminant of three quantics in two independent variables. Proc. London Math. Soc. 6, 468–478 (1908)
4. 4.
Faugere, J.-C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139, 61–88 (1999)
5. 5.
Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using Dixon resultants. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation. ACM Press (1994)Google Scholar
6. 6.
Lewis, R.H.: Dixon-EDF: The Premier Method for Solution of Parametric Polynomial Systems. In: Kotsireas, I.S., Martinez-Moro, E. (eds.) Applications of Computer Algebra, Proceedings in Mathematics & Statistics, Kalamata, Greece, vol. 198. Springer, Cham (2017).
7. 7.
Lewis, R.H.: Heuristics to accelerate the Dixon resultant. Math. Comput. Simul. 77(4), 400–407 (2008)
8. 8.
Lewis, R.H., Stiller, P.: Solving the recognition problem for six lines using the Dixon resultant. Math. Comput. Simul. 49, 203–219 (1999)
9. 9.
Lewis, R.H.; Computer algebra system Fermat. http://home.bway.net/lewis/
10. 10.
Lewis, R.H.; Fermat code for Dixon-EDF. http://home.bway.net/lewis/dixon
11. 11.
Stiller, P.; Symbolic computation of object/image equations. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. pp. 359–364. ACM Press, New York (1997)Google Scholar
12. 12.
Sturmfels, B.: Solving systems of polynomial equations. In: CBMS Regional Conference Series in Mathematics, vol. 97. American Mathematical Society (2003)Google Scholar