The perspective problem (also called pose estimation) is defined as follows: given n points in space, and the values of the angles subtended by each pair of points at an unknown “point of view,“ determine the coordinates of the point of view.

The equations stating this problem mathematically are known. The problem is in solving them, and also in determining the number of physically realizable solutions.

The authors have approached the problem by concentrating on the special case of n=3. They have set up the quadratic equations involved (I believe that the equations are quadratic in the general case also, the difference being in the number of variables and the number of equations involved). Since the elimination of variables between these gives rise to higher degree equations, general solutions are hard to find. Moreover, there is the problem of determining the number of feasible solutions as a function of the case-dependent coefficients. Then, there is the problem of the stability of the solutions, namely, whether the dependence of the number of solutions on the coefficients is continuous everywhere, and where the points of discontinuity are. Thus, one cannot depend on numerical solution methods alone before getting some analytic properties of the solutions.

The authors use a recently available method for these purposes: the Wu-Ritt’s zero-decomposition method (I recommend reference 16 in Algorithmic algebra [1] as the most accessible reference to the method). Using Wu-Ritt, the solution set of these equations is obtained by a set-theoretic union of a set of reals obtained from another set of nonlinear equations. In the three-case, the two original equations give rise to a set of 21 equations in “triangular form,” (a set of equations where the first members of the set are in one variable, and the next are in two variables). Some of the equations of the sets are of a degree as high as four.

By a painstaking analysis of these equations, and the qualitative properties of their solutions, the authors come to a major number of important conclusions regarding the properties of the physically realizable solutions. The number of solutions can be four, three, two, and one. The authors have also developed a numerical algorithm for finding all of the stable, physically realizable solutions when the parameters are away from the points of discontinuity of the step function involved. The algorithm is found to be efficient, both according to a theoretical estimation, and experimental results.

The authors also have done a geometrical analysis of the three-case of the problem. Here, the solution set is found to be on the intersection of three toroidal surfaces. This yields a visual picture of the way the number of solutions depends on the parameters.

The appendices of the paper exhibit all of the equations involved in the Wu-Ritt method in the three-case.