A Concise Parametrization of Affine Transformation

Abstract

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.Throughout this paper all vectors should be considered as real column vectors, and hence, matrices act on them by the multiplication from the left.2010 Mathematics Subject Classification. 68U05,65D18,65F60,15A16.

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

“…It is now worth commenting on the connection between our approach and the work of Kaji and Ochiai [47]. As a particular case, their results provide a parametrization of the group SE(3) in terms of the algebra se(3).…”

“…For the special case of a Kirchhoff rod, this discretization scheme reduces to the one used by Bertails, Audoly, Cani, Querleux, Leroy and Lévêque [11]. We moreover discuss the connection between our approach and the interpolation of affine transformations introduced very recently by Kaji and Ochiai [47] in the context of computer graphics applications.…”

Tetrisation of Triangular Meshes and Its Application in Shape Blending

“…where log c is the "continuous" logarithm such that it chooses the nearest branch of logarithm to the adjacent tetrahedra when i varies (see [8] for details). The indeterminacy of log for SO(3) is in the rotation angle and log c chooses the angle continuously for adjacent tetrahedra.…”

Multi‐modal robust inverse‐consistent linear registration

“…In this study, we work with rigid and affine transformations. While a standard parameterization for the rigid case exists [Zefran et al, ], different parameterizations were proposed for affine transformations [Arsigny et al, ; Kaji et al, ; Kaji and Ochiai, ]. For the parameterization in [Arsigny et al, ], no closed form of the exponential map exists.…”

“…For the parameterization in [Arsigny et al, ], no closed form of the exponential map exists. The parameterization in [Kaji et al, ; Kaji and Ochiai, ] describes the transformation matrix as a product of two matrix exponentials, which complicates the computation of the half transform. A viable alternative in practice is to directly update the parameters of the transformation matrix, as gradient descent optimizations are unlikely to produce negative determinants when started from identity.…”