|Stochastic Systems Group|
Lie Algebraic Representation of Geometric Transforms
In this talk, I will present a new algebraic formulation to describe the affine transformations of 2D shapes and their dynamics in a continuous-time process. In traditional computer vision applications, transformations are typically given in their natural form that lie on a manifold with group structure (Lie group), which leads to difficulties in applying statistical modeling approaches that are mostly based on vector space representations. In this work, a vector space based on Lie algebra is explored to describe and manipulate geometric transformations. Compared to traditional methods, the new representation has three advantages: (1) decomposition and combination of transformations can be done by linear operations, which makes the migration of vector space based models much easier; (2) the structure of the new representation conveys significant geometric information; (3) geometric constraints that are difficult to enforce in traditional representations can be converted into linear constraints. The Lie algebraic representation is further extended to model the deformation of a triangle mesh, in which the consistency at vertices are preserved by keeping the changes on a particular subspace. The characteristics of the extended formulation are also discussed.
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