ACCV 2016 paper. It includes complementary evaluations to Sec. 4.1 for general geometric configurations as well as some degenerate configura- tions such as ...
Unifying Algebraic Solvers for Scaled Euclidean Registration from Point, Line and Plane Constraints Supplemental Material Document
Folker Wientapper1 , Arjan Kuijper2 1 2
VRAR, Fraunhofer IGD, Darmstadt, Germany MAVC, TU Darmstadt, Darmstadt, Germany
Abstract. The present document provides supplemental material to our ACCV 2016 paper. It includes complementary evaluations to Sec. 4.1 for general geometric configurations as well as some degenerate configurations such as the central PnP problem and the case when the lines and planes are all parallel to some common direction.
1 1.1
Evaluations on General Geometric Configurations Registration Accuracy Versus Noise
We made a complementary evaluation to Sec. 4.1, but instead of varying the number of correspondences we varied the noise level on the reference points xk . The number of correspondences was kept fixed to 50 throughout all evaluations. For the ’relative noise level’ we compute the mean of the sigular values of the covariance matrix of the points xk and the covariance of the noise. Then we compute the square-root of their ratios. Again, we evluated general geometric configurations of the lines, planes, and points. Fig. 1 shows the results. Again, it can be seen that DLS/gDLS and UPnP do not succeed in correctly determining the registration parameters from point-topoint correspondences whereas our solver does. For the other cases we see that gDLS has slightly higher mean errors for very low noise levels. 1.2
Fixed Scale and the Inhomogeneous Case
We also analyzed the benefits of the UPnP-rotation-solver, since this is the only one that accepts the inhomogeneous matrix Mi ∈ R10×10 as in Eq. 15. The other algorithms can only handle 9 × 9 input matrices. Again we performed the evaluation on general geometric configurations by varying the number of input correspondences. The relative noise ratio was set to 0.0001. For our own solver and the DLS/gDLS-solver we computed the homogeneous matrix Mh as in Sec. 2.2. Note that this corresponds to leaving the scale as free parameter. Fig. 2 shows the results. Using the UPnP solver on Mi leads to more accurate results as expected, because the problem is modeled more appropriately. A rather
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Fig. 1. Mean errors of the estimated rotation, translation, and scale for scaled Euclidean registration with 50 correspondences in general geometric configurations with varying noise. The relative noise level refers to the ratio between the standard deviations of the noise and the centered points xk .
surprising result is that this only affects the translation, whereas the rotation estimates are equally accurate for all solvers. Note also, that UPnP is the only algorithm that can solve the minimal cases of six point-to-plane or three point-to-line correspondences.
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Evaluations on Degenerate Geometric Configurations
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Central Case for Lines and the PnP Problem
As we mentioned in the paper in Sec. 2.4, the central case, i.e. when all 3d lines have a common point of intersection, can be considered as a degenerate situation because the scale cannot be computed. This special case has a direct connection to the PnP-problem, where all lines intersect at the camera center. For this reason - and also because the PnP-problem has a very high relevance in computer vision - we use the PnP evaluation framework of Zheng et al. [1]3 and 3
The evaluation framework of Zheng and a collection of the most prominent PnPsolvers are available at https://sites.google.com/site/yinqiangzheng/.
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Fig. 2. Mean errors of the estimated rotation and translation for Euclidean registration with known and fixed scale (inhomogeneous case) versus number of used correspondences. The relative noise level was set to 0.0001. Only UPnP can handle the minimal cases, i.e. 3 line-correspondences or 6 plane correspondences.
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Fig. 3. Accuracy for the central PnP-problem. Left: error of rotation and translation as a function of the number of input correspondences (pixel noise is set to σ = 1.0). Right: error with respect to pixel noise using 10 input correspondences.
included UPnP, gDLS/DLS and our own solver into it. Again we replaced the linear parameter elimination of the algorithms by our orthogonal complement based version (marked by ’(OC)’). Newton (OC) refers to the direct second order (Newton) based minimization of Eq. 10 in Sec. 2.2 of the paper. For comparison we also include the well-known algorithms LHM [2] and EPnP [3], as well as OPnP [1], which probably is the most robust and accurate PnP-algorithm currently available. LM refers to the solution obtained by minimizing the reprojection error in the image plane with Levenberg-Marquardt iterations which represents the (gold-standard) baseline for PnP-problems. 0 The evaluation setup is as follows. Random points xk in the camera coordinate frame are created inside the cube [(−2, −2, 4) × (2, 2, 8)] and corresponding 2D image measurements qk ∈ R2 are generated by perspective division and by adding pixel noise with varying variance. Next, with a random ground-truth
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rotation RGT and translation tGT the corresponding reference points xk are obtained. The image points, qk , and the reference points, xk , are used as input for all PnP-algorithms. Fig. 3 shows the results. It can be seen that the modified algorithms as considered in the paper compare favourably with the state-of-the-art even for the minimal cases of three correspondences. Perhaps more important is the fact that the PnP-problem - as being a degenerate case in the scaled Euclidean registration context - is handled properly by the SVD-based pseudo-inverse computation from Sec. 2.2, as all algorithms (UPnP, DLS/gDLS and our solver) are supplied with the homogeneous matrix Mh from Eq. 9. We note again that linear parameter elimination inside the original gDLS-algorithm [4] cannot be used here, because it will attempt to invert a singular matrix to obtain the pseudo-inverse ˜ † . Likewise the linear parameter elimination technique from the original DLSA algorithm [5] is not applicable whenever the scale needs to be estimated in general geometric configurations. Parallel Lines and Planes
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Fig. 4. Mean errors of the estimated rotation, translation, and scale for scaled Euclidean registration with the lines and planes being parallel to one common direction.
We evaluated another degenerate case, i.e. when all planes and lines are parallel in one common direction v0 . Then, the translation is ambiguous up to this common vector, so any t = t0 + γv0 represents a valid solution for arbitrary γ. Again, the linear parameter elimination techniques inside the original algo˜ TA ˜ rithms [5, 4, 6] will fail, because they attempt to invert a singular matrix A (see Sec. 2.3 in the paper). By constrast, our SVD-based computation of the pseudo-inverse will handle this degeneracy correctly.
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The evaluation was similar to Sec. 4.1 except that the lines and planes were constrained to be parallel to some random direction v0 . For the translation error we projected the error vector between the translation offsets t0 returned by the algorithms and the ground-truth translation tGT onto the orthogonal complement of v0 , i.e. errt = k(I − v0 vT0 )(tGT − t0 )k. In Fig. 4 the results are depicted. It can be seen that all algorithms succeed in estimating correct results with similar accuracy. We also note that this is possible even for three point-to-line and six point-to-plane correspondences. This is, because again only six DoF in total need to be estimated.
References 1. Zheng, Y., Kuang, Y., Sugimoto, S., strm, K., Okutomi, M.: Revisiting the pnp problem: A fast, general and optimal solution. In: IEEE Proc. Int’l Conf. on Computer Vision (ICCV). (2013) 2344–2351 2. Lu, C.P., Hager, G.D., Mjolsness, E.: Fast and globally convergent pose estimation from video images. IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI) 22 (2000) 610–622 3. Lepetit, V., Moreno-Noguer, F., Fua, P.: Epnp: An accurate o(n) solution to the pnp problem. LNCS Int’l J. of Computer Vision (IJCV) 81 (2008) 155–166 4. Sweeney, C., Fragoso, V., H¨ ollerer, T., Turk, M.: gdls: A scalable solution to the generalized pose and scale problem. In Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T., eds.: LNCS Proc. European Conf. on Computer Vision (ECCV), Cham, Springer International Publishing (2014) 16–31 5. Hesch, J.A., Roumeliotis, S.I.: A direct least-squares (dls) method for pnp. Volume 0., Los Alamitos, CA, USA, IEEE Computer Society (2011) 383–390 6. Kneip, L., Li, H., Seo, Y.: Upnp: An optimal o(n) solution to the absolute pose problem with universal applicability. In Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T., eds.: LNCS Proc. European Conf. on Computer Vision (ECCV), Cham, Springer International Publishing (2014) 127–142
