A Hybrid Algorithm for Solving 7 Parameters Transformation

The 7 parameters transformation is a well-known problem in engineering sciences such as Computer Vision, Survey Engineering and Photogrammetry. To solve this problem, we use an algebraic solver in which we need to transform the input system into an equivalent system, but better adapted such as a Gro¿bner basis. For operational applications of this problem, we need to compute this basis as fast as possible. In this paper, we describe an efficient hybrid algorithm which employs both Buchberger algorithm and Faugere\´s F₅ algorithm to compute this basis. Using this algorithm, we record the "trace" of the useful calculations for computing the Gro¿bner basis of the polynomial ideal generated by the equations of the system. For any coordinates of input points, this trace can provide directly the Gro¿bner basis of the ideal generated by the system without useless computations.