Abstract
Verified distance computation is an important task in various application domains. In some domains a proof of correctness is crucial. In this paper, we show how we can apply the methods provided by our uniform framework for verified geometric computations to derive verified bounds on the distance between non-convex objects. The framework features a layered structure enabling the algorithm to run independently whether the objects are described by implicit functions or parametric ones or by polyhedrons. The approach is based on the use of adaptively constructed hierarchical decompositions of the models. As a practical example we use various scenarios occurring in automatic surgery assistance systems for total hip replacement (THR). To ensure that an implant selected by the system fits into the patient’s femoral shaft, we have to derive verified bounds on the distance between them. In this case, the models are either superquadrics or polyhedrons, both of which can be non-convex We first show how to increase the enclosure quality of implicit objects by incorporating interval contractors into the hierarchical space decomposition. Next, we describe the construction of a decomposition structure for parametric objects. After that, we present an improvement of the case selector for computing the distance between interval tree nodes, yielding tighter results. We also show how to integrate surface normals into the algorithm if first-order information is available and how to accelerate the solving process by incorporating information gained by non-verified floating-point solvers. Finally, we provide numerical results for all distance query types occurring during the THR procedure and examine whether it is advisable to perform the computation on the implicit model or on the parametric one if both are available. Further numerical results are presented for test cases involving contractors in the decomposition structures.
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Notes
f(x, y, z) = x 500 + y 500 + z 500 − 1.
f(x, y, z) = x 4 + y 4 + z 4 − x 2 − y 2 − z 2 + 0.5.
f(x, y, z) = (x 2 + y 2 + z 2 + 2y − 1) ((x 2 + y 2 + z 2 − 2y − 1)2 − 8z 2) + 16xz(x 2 + y 2 + z 2 − 2y − 1)).
\(f(x,y,z) = 2-\cos(x+\alpha y)+\cos(x-\alpha y)+\cos(y+\alpha z)+\cos(y- \alpha z)+\cos(z+\alpha x)+\cos(z- \alpha x) \mathrm{\ with\ } \alpha = 1.61803.\)
References
Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, New York
Araya I, Trombettoni G, Neveu B (2010) Exploiting monotonicity in interval constraint propagation. In: AAAI
Auer E, Chuev A, Cuypers R, Kiel S, Luther W (2011) Relevance of accurate and verified numerical algorithms for verification and validation in biomechanics. In: EUROMECH Colloquium 511, Ponta Delgada, Azores, Portugal
Auer E, Cuypers R, Luther W (2012) Process-oriented approach to verification in engineering. In: ICINCO (2), pp 513–518
Barr A (1981) Superquadrics and angle-preserving transformations. Comput Graph Appl IEEE 1(1):11–23
Bastl B, Jezek F (2005) Comparison of implicitization methods. J Geom Graph 9(1):11–29
Brunet P, Navazo I (1990) Solid representation and operation using extended octrees. ACM Trans Graph (TOG) 9(2):170–197
Bühler K (2002) Implicit linear interval estimations. In: Proceedings of the 18th spring conference on Computer graphics. ACM, New York, p 132
Bühler K, Dyllong E, Luther W (2004) Reliable distance and intersection computation using finite precision geometry. In: Alt R, Frommer A, Kearfott R, Luther W (eds) Numerical software with result verification. Lecture notes in computer science, vol 2991. Springer Berlin, pp 579–600
Carlbom I, Chakravarty I, Vanderschel D (1985) A hierarchical data structure for representing the spatial decomposition of 3-d objects. IEEE Comput Graph Appl 5(4):24–31
Chakraborty N, Peng J, Akella S, Mitchell JE (2008) Proximity queries between convex objects: an interior point approach for implicit surfaces. IEEE Trans Rob 24(1):211–220
Cuypers R (2011) Geometrische Modellierung mit Superquadriken zur Optimierung skelettaler Diagnosesysteme. Logos
Du K, Kearfott RB (1994) The cluster problem in multivariate global optimization. J Global Optim 5:253–265
Dyllong E, Grimm C (2007) Verified adaptive octree representations of constructive solid geometry objects. In: Simulation und Visualisierung, pp 223–235
Dyllong E, Kiel S (2010) Verified distance computation between convex hulls of octrees using interval optimization techniques. PAMM 10(1):651–652
Dyllong E, Kiel S (2012) A comparison of verified distance computation between implicit objects using different arithmetics for range enclosure. Computing 94:281–296
Edelsbrunner H (1995) Algebraic decomposition of non-convex polyhedra. In: Proceedings. 36th annual symposium on foundations of computer science, pp 248–257
de Figueiredo L, Stolfi J (1997) Self-validated numerical methods and applications. IMPA, Rio de Janeiro
Gilbert E, Johnson D, Keerthi S (1988) A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE J Robot Autom 4(2):193–203
Hansen E, Walster GW (2004) Global optimization using interval analysis. Marcel Dekker, New York
Hofschuster W, Krämer W (2004) C-XSC 2.0 A C++ library for extended scientific computing. In: Alt R, Frommer A, Kearfott R, Luther W (eds) Numerical software with result verification. Lecture notes in computer science, vol 2991. Springer, Berlin, pp 259–276
Jaklič A, Leonardis A, Solina F (2000) Segmentation and recovery of superquadrics, vol 20. Springer, Amsterdam
Kiel S (2010) Verified spatial subdivision of implicit objects using implicit linear interval estimations. In: Curves and surfaces, pp 402–415
Kiel S (2012) YalAA: yet another library for affine arithmetic. Reliable Comput 16:114–129
Kockara S, Halic T, Bayrak C, Iqbal K, Rowe R (2009) Contact detection algorithms. J Comput 4(10):1053
Lennerz C, Schomer E (2002) Efficient distance computation for quadratic curves and surfaces. In: Geometric modeling and processing, 2002. Proceedings, pp 60–69
Makino K, Berz M (2003) Taylor models and other validated functional inclusion methods. Int J Pure Appl Math 4(4):379–456
Mirtich B (1998) V-Clip: fast and robust polyhedral collision detection. ACM Trans Graph 17:177–208
Quinlan S (1994) Efficient distance computation between non-convex objects. In, Robot Autom
Samet H (2006) Foundations of multidimensional and metric data structures. Morgan Kaufmann, San Francisco
Shapiro V (2007) Semi-analytic geometry with R-functions. ACTA Numer 16
Snyder JM, Woodbury AR, Fleischer K, Currin B, Barr AH (1993) Interval methods for multi-point collisions between time-dependent curved surfaces. In: Proceedings of the 20th annual conference on Computer graphics and interactive techniques, SIGGRAPH ’93. ACM, New York, pp 321–334
Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57
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We would like to thank all referees for their suggestions which helped us to improve the paper and to significantly reduce the computation times.
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Communicated by V. Kreinovich.
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Kiel, S., Luther, W. & Dyllong, E. Verified distance computation between non-convex superquadrics using hierarchical space decomposition structures. Soft Comput 17, 1367–1378 (2013). https://doi.org/10.1007/s00500-013-1005-y
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DOI: https://doi.org/10.1007/s00500-013-1005-y