A Concise Parametrization of Affine Transformation

Good parametrisations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, and animation. It has been one of the central research topics in computer graphics. However, there is no single perfect method and each one has both advantages and disadvantages. In this paper, we propose a novel parametrisation of affine transformations, which is a generalisation to or an improvement of existing methods. Our method adds yet another choice to the existing toolbox and shows better performance in some applications. A C++ implementation is available to make our framework ready to use in various applications.

[1]  Jonathan H. Manton,et al.  A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups , 2004, ICARCV 2004 8th Control, Automation, Robotics and Vision Conference, 2004..

[2]  Leo Dorst,et al.  Geometric Algebra , 2019, Computer Vision, A Reference Guide.

[3]  Hiroyuki Ochiai,et al.  Anti-commutative Dual Complex Numbers and 2D Rigid Transformation , 2016, ArXiv.

[4]  Tom Duff,et al.  Matrix animation and polar decomposition , 1992 .

[5]  Myung-Soo Kim,et al.  Interpolating solid orientations with circular blending quaternion curves , 1995, Comput. Aided Des..

[6]  Sung Yong Shin,et al.  General Construction of Time-Domain Filters for Orientation Data , 2002, IEEE Trans. Vis. Comput. Graph..

[7]  W. Culver.  On the existence and uniqueness of the real logarithm of a matrix , 1966 .

[8]  Nicolas Courty,et al.  Motion Compression using Principal Geodesics Analysis , 2009, Comput. Graph. Forum.

[9]  Jovan Popović,et al.  Mesh-based inverse kinematics , 2005, SIGGRAPH 2005.

[10]  G. Nielson.  Smooth Interpolation of Orientations , 1993 .

[11]  F. Sebastian Grassia,et al.  Practical Parameterization of Rotations Using the Exponential Map , 1998, J. Graphics, GPU, & Game Tools.

[12]  Samuel R. Buss,et al.  Spherical averages and applications to spherical splines and interpolation , 2001, TOGS.

[13]  Frank Chongwoo Park,et al.  Smooth invariant interpolation of rotations , 1997, TOGS.

[14]  Ken Shoemake,et al.  Animating rotation with quaternion curves , 1985, SIGGRAPH.

[15]  Ken-ichi Anjyo,et al.  Mathematical description of motion and deformation: from basics to graphics applications , 2013, SA '13.

[16]  John A. Vince,et al.  Geometric algebra for computer graphics , 2008 .

[17]  Roger W. Brockett,et al.  Robotic manipulators and the product of exponentials formula , 1984 .

[18]  Ken-ichi Anjyo,et al.  Mathematical analysis on affine maps for 2D shape interpolation , 2012, SCA '12.

[19]  Jirí Zára,et al.  Geometric skinning with approximate dual quaternion blending , 2008, TOGS.

[20]  Ignacio Llamas,et al.  Twister: a space-warp operator for the two-handed editing of 3D shapes , 2003, ACM Trans. Graph..

[21]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[22]  Marc Alexa,et al.  As-rigid-as-possible shape interpolation , 2000, SIGGRAPH.

[23]  安生 健一,et al.  A Lie Theoretic Parameterization of Affine Transformation , 2013 .

[24]  Ravi Ramamoorthi,et al.  Fast construction of accurate quaternion splines , 1997, SIGGRAPH.

[25]  Sung Yong Shin,et al.  A general construction scheme for unit quaternion curves with simple high order derivatives , 1995, SIGGRAPH.

[26]  D. Hestenes,et al.  Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics , 1984 .

[27]  Max K. Agoston,et al.  Computer Graphics and Geometric Modelling - Mathematics , 2005 .

[28]  John F. Hughes,et al.  Smooth interpolation of orientations with angular velocity constraints using quaternions , 1992, SIGGRAPH.

[29]  N. Higham.  The Scaling and Squaring Method for the Matrix Exponential Revisited , 2005, SIAM J. Matrix Anal. Appl..

[30]  Jarek Rossignac,et al.  Steady affine motions and morphs , 2011, TOGS.

[31]  Joan Lasenby,et al.  Mesh Vertex Pose and Position Interpolation Using Geometric Algebra , 2008, AMDO.

[32]  Joan Lasenby,et al.  Applications of Conformal Geometric Algebra in Computer Vision and Graphics , 2004, IWMM/GIAE.

[33]  Shizuo Kaji,et al.  Tetrisation of triangular meshes and its application in shape blending , 2016, ArXiv.

[34]  A. W. Knapp.  Lie groups beyond an introduction , 1988 .

[35]  Ken Shoemake.  Polar Matrix Decomposition , 1994, Graphics Gems.

[36]  Marc Alexa,et al.  Linear combination of transformations , 2002, ACM Trans. Graph..

[37]  Ken Shoemake.  QUATERNIONS AND 4 × 4 MATRICES , 1991 .

[38]  Ken-ichi Anjyo,et al.  Mathematical basics of motion and deformation in computer graphics , 2014, SIGGRAPH '14.

[39]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[40]  Stephen Mann,et al.  Geometric algebra for computer science - an object-oriented approach to geometry , 2007, The Morgan Kaufmann series in computer graphics.

[41]  Gengdai Liu,et al.  Probe-Type Deformers , 2015 .