Affine interpolation in a lie group framework

Abstract

Affine transformations are of vital importance in many tasks pertaining to motion design and animation. Interpolation of affine transformations is non-trivial. Typically, the given affine transformation is decomposed into simpler components which are easier to interpolate. This may lead to unintuitive results, while in some cases, such solutions may not work. In this work, we propose an interpolation framework which is based on a Lie group representation of the affine transformation. The Lie group representation decomposes the given transformation into simpler and meaningful components, on which computational tools like the exponential and logarithm maps are available in closed form. Interpolation exists for all affine transformations while preserving a few characteristics of the original transformation. A detailed analysis and several experiments of the proposed framework are included.

References

1. Marc Alexa. 2002. Linear combination of transformations. In ACM Transactions on Graphics (TOG), Vol. 21. ACM, 380--387.
2. Marc Alexa, Daniel Cohen-Or, and David Levin. 2000. As-rigid-as-possible shape interpolation. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques. ACM Press/Addison-Wesley Publishing Co., 157--164.
3. Kevin G Der, Robert W Sumner, and Jovan Popović. 2006. Inverse kinematics for reduced deformable models. In ACM Transactions on Graphics (TOG), Vol. 25. ACM, 1174--1179.
4. Oren Freifeld and Michael J. Black. 2012. Lie Bodies: A Manifold Representation of 3D Human Shape. In European Conf. on Computer Vision (ECCV) (Part I, LNCS 7572). Springer-Verlag, 1--14.
5. Jean Gallier and Jocelyn Quaintance. 2017. Notes on Differential Geometry and Lie Groups. (2017). Available online at www.cis.upenn.edu/jean/gbooks/manif.html.
6. Alec Jacobson, Zhigang Deng, Ladislav Kavan, and JP Lewis. 2014. Skinning: Real-time Shape Deformation. In ACM SIGGRAPH 2014 Courses.
7. S. Kaji, S. Hirose, S. Sakata, Y. Mizoguchi, and K. Anjyo. 2012. Mathematical Analysis on Affine Maps for 2D Shape Interpolation. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA '12). Eurographics Association, Goslar Germany, Germany, 71--76. http://dl.acm.org/citation.cfm?id=2422356.2422368
8. Ladislav Kavan, Steven Collins, Jiří Žára, and Carol O'Sullivan. 2008. Geometric skinning with approximate dual quaternion blending. ACM Transactions on Graphics (TOG) 27, 4 (2008), 105.
9. Ladislav Kavan and Jiří Žára. 2005. Spherical blend skinning: a real-time deformation of articulated models. In Proceedings of the 2005 symposium on Interactive 3D graphics and games. ACM, 9--16.
10. Jun Li and Peng-wei Hao. 2006. Smooth interpolation on homogeneous matrix groups for computer animation. Journal of Zhejiang University-SCIENCE A 7, 7 (2006), 1168--1177.
11. Nadia Magnenat-Thalmann, Richard Laperrire, and Daniel Thalmann. 1988. Joint-dependent local deformations for hand animation and object grasping. In Proceedings on Graphics interface.
12. Richard M. Murray, S. Shankar Sastry, and Li Zexiang. 1994. A Mathematical Introduction to Robotic Manipulation (1st ed.). CRC Press, Inc., Boca Raton, FL, USA.
13. Hiroyuki Ochiai and Ken Anjyo. 2013. Mathematical description of motion and deformation: from basics to graphics applications. In SIGGRAPH Asia 2013, Hong Kong, China, November 19--22, 2013, Courses. 2:1--2:47.
14. M. M. Postnikov. 2001. Geometry VI: Riemannian Geometry. Springer-Verlag Berlin Heidelberg.
15. Jarek Rossignac and Álvar Vinacua. 2011. Steady affine motions and morphs. ACM Transactions on Graphics (TOG) 30, 5 (2011), 116.
16. J. M. Selig. 2010. Geometric Fundamentals of Robotics (2nd ed.). Springer Publishing Company, Incorporated.
17. Ken Shoemake. 1985. Animating Rotation with Quaternion Curves. SIGGRAPH Comput. Graph. 19, 3 (July 1985), 245--254.
18. Ken. Shoemake. 1992. Matrix animation and polar decomposition. Graphics Interface'92, 1992 (1992).
19. Robert W. Sumner, Matthias Zwicker, Craig Gotsman, and Jovan Popović. 2005. Mesh-based Inverse Kinematics. ACM Trans. Graph. 24, 3 (July 2005), 488--495.
20. B. Whited, G. Noris, M. Simmons, R. Sumner, M. Gross, and J. Rossignac. 2010. BetweenIT: An Interactive Tool for Tight Inbetweening. Comput. Graphics Forum (Proc. Eurographics) 29, 2 (2010), 605--614.
21. Milos Zefran and Vijay Kumar. 1998. Interpolation schemes for rigid body motions. Computer-Aided Design 30, 3 (1998), 179 -- 189. Motion Design and Kinematics.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation