Abstract
This paper discusses the efficient optimization of iterative closest points (ICP) algorithms. While many algorithms formulate the optimization problem in terms of quadratic error functionals, the discontinuities introduced by varying changing correspondences usually motivate the optimization by quasi-Newton or Gauss-Newton methods. These disregard the fact that the Hessian matrix in these cases is constant, and can thus be precomputed analytically and inverted a-priori. We demonstrate on the example of Allen et al.’s seminal paper “The space of human body shapes”, that all relevant quantities for a full Newton method can be derived easily, and lead to an optimization process that reduces computation time by around 98% while achieving results of almost equal quality (about 1% difference). Along the way, the paper proposes minor improvements to the original problem formulation by Allen et al., aimed at making the results more reproducible.
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Notes
- 1.
- 2.
It should be noted that “smoothness” here refers to the matrix elements, not the mesh: Each matrix element should change smoothly between neighboring points. The resulting mesh need not be smooth in shape at all, but neighboring points should be transformed similarly.
- 3.
In the present example, there are \(\mathcal O(|M|^2)\) entries in \(\mathbf H\), which can be considerable.
- 4.
In the theoretical limit of a continuous surface, this happens almost permanently. In this case, \(\mathbf g\) is almost always continuous and \(\mathbf H\) cannot be given as shown here (i.e. in terms of a polygon topology). In general, \(\mathbf H\) is not constant then. As initially stated, this case is not considered.
- 5.
The aforementioned overhead in parameters (Sect. 2.1) is mirrored here in the fact that the gradient is of the form \(\mathbf u \mathbf v^{\mathsf T}\), which means it has rank 1 (dimension 3), corresponding to the expected \(3^{\circ }\) of freedom out of 12 parameters.
- 6.
It should be noted at this point that here the issues discussed in Sect. 2 become apparent, where a lack of specified units will lead to different optimization goals for models in meters, feet or inches, for example. In this case, the unit is considered to be meters.
- 7.
The approximative algorithm outperforming the exact one in terms of result quality may seem counter-intuitive, but is immanent to the problem structure: Newton’s method is exact at finding the closest local optimum where point associations do not change. L-BFGS instead is more likely to miss this “direct” minimum. As its approximation combines information across different point associations, it can thereby “learn” the overall shape of the surface, which is, for high-density surface points, more accurate than the purely local quadratic view. In turn, for sparse surface models, Newtons method is more accurate, as it better captures the dominant structure.
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German, L., Ziehn, J.R., Rosenhahn, B. (2017). Rapid Analytic Optimization of Quadratic ICP Algorithms. In: Chen, CS., Lu, J., Ma, KK. (eds) Computer Vision – ACCV 2016 Workshops. ACCV 2016. Lecture Notes in Computer Science(), vol 10118. Springer, Cham. https://doi.org/10.1007/978-3-319-54526-4_5
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