Review and Analysis of Solutions of the Three Point Perspective Pose Estimation Problem¶

Motivation(s)¶

Given the perspective projection of three points constituting the vertices of a known triangle in \(\mathbb{R}^3\), it is possible to determine the position of each of the vertices as well as the projectivity (a.k.a. absolute orientation). The generalization of this problem, Perspective-n-Point (PnP), is important in both photogrammetry and computer vision. However, photogrammetrists no longer consider direct solutions important with the advent of iterative solutions because in most photogrammetry situations, scale and distances are known to within 10% and angle is known to within \(15^\circ\). In contrast, the direct solution method is more important computer vision because approximate solutions are not known.

Proposed Solution(s)¶

The authors present all of the major direction solutions under a unified P3P algebraic framework (i.e. law of cosines). They also propose a linear solution for the absolute orientation problem: given three points in the 3D camera coordinate system and their corresponding three points in 3D world coordinate system, find a Euclidean transformation mapping camera coordinates to world coordinates.

Evaluation(s)¶

Figure 2 shows the differences of algebraic derivations among the six solution techniques. Excluding the techniques of Fischler and Linnainmaa, the remaining solutions suffer from algebraic singularity (Table I). This issue manifests itself in the danger cylinder, specifically when the center of perspectivity and the vertices of a triangle are concylic.

To characterize the numerical sensitivity of the solutions, the authors tested those techniques on randomly generated 3D triangles. The results of random permutation of six solutions in double precision and single precision indicate that Finsterwalder’s solution gives the best accuracy. However, the best and worst mean absolute distance error in single precision show that Grunert’s and Fischler’s solution give similar accuracy. Furthermore, the histogram of absolute distance error reveals these three techniques exhibit uniformly good behavior over many trials.

Future Direction(s)¶

How to modify Fischler’s solution to achieve higher accuracy in double precision?

Question(s)¶

Does algebraic singularity matter in practice compared to accuracy?

Analysis¶

Given the camera intrinsics, the pose of the camera can be recovered under the P3P framework.

Note that Finsterwalder’s solution in the paper contains some typos [Och]. Furthermore, the benefits of the proposed absolute orientation linear solution is doubtful when [Ume91] had already proposed an optimal solution.

One surprising experimental result is that the accuracy of each solution when switching to double precision varies by several orders of magnitude. However, the relative accuracy ordering did not change.

The section on numerical stability and rounding errors contains techniques that could be useful in analyzing other PnP solutions.

References

HLONolle94: Bert M Haralick, Chung-Nan Lee, Karsten Ottenberg, and Michael Nölle. Review and analysis of solutions of the three point perspective pose estimation problem. International journal of computer vision, 13(3):331–356, 1994.
Och: Benjamin Ochoa. Corrections to “review and analysis of solutions of the three point perspective pose estimation problem”. https://cseweb.ucsd.edu/classes/wi15/cse252B-a/haralick94corrections.pdf. Accessed on 2017-09-14.