| 作成者 |
|
|
|
|
|
|
|
| 本文言語 |
|
| 出版者 |
|
|
|
| 発行日 |
|
| 収録物名 |
|
| 巻 |
|
| 開始ページ |
|
| 終了ページ |
|
| 出版タイプ |
|
| アクセス権 |
|
| 関連DOI |
|
|
|
|
|
| 関連URI |
|
|
|
|
|
| 関連情報 |
|
|
|
|
|
| 概要 |
Affine transformation (or geometric transformation) provides a mathematical foundation for shape manipulation and motion analysis in computer graphics. In particular, the set Aff^+(3) of positive (or,... reflection free) affine transformations is important since it consists of rotation, shear, translation, and their compositions. The elements in Aff^+(3) are usually represented by 4 × 4-homogeneous matrices with algebraic operations such as addition, scalar product, and product. While the product corresponds to the composition of the transformations, geometric meaning of addition and scalar product are not clear. There are many situations where we want to have geometrically meaningful weighted sum (linear combination) of transformations, for example, for skinning [10], and for motion analysis and compression [1]. To mention a few parameterization developed previously: Euler angle, and Quaternion parameterizes the rotation. Dual quaternion, and axis-angle presentation parameterizes the rigid transformation (rotation and translation altogether). The above parameterizations are all partial; they deal only subsets of Aff^+(3), and cannot handle shear and scale. On the other hand, Alexa [1] introduced a Euclidean parameterization of Aff^+(3) using the Lie correspondence. The idea is that Aff^+(3) forms a Lie group and it corresponds to a linear space called Lie algebra through the matrix exponential and logarithm. However, this method yet fails to give a parameterization for the whole transformations; it is limited for transformations without negative eigenvalues. The limitation is due to mathematical nature of the Lie correspondence, which guarantees only local bijectivity. Here we introduce a novel parameterization of Aff^+(3) based on Lie theory. Our general framework can also be found in [12], which includes more precise definitions of Lie group and Lie algebra. It has several advantages over previous ones; ? No limitation; it parameterizes the whole transformations. ? Smooth and having the same degree of freedom; ordinary variational techniques can be applied. ? With geometrically meaningful operations; for example, the sum of two rigid transformations is again rigid. ? Low computational cost; fast enough for real-time applications. In the following sections, we will give the new parameterization method, a computation algorithm, and applications to shape deformation.続きを見る
|
Mendeley出力
