Once again, the dominant cost is the factorization, which is O( n 3). (2010) also propose restricting the velocity to the normal direction, leading to a matrix in R n × n. Similarly, a vector-based vibration distance is introduced using a similar definition to the diffusion distance. (2010) introduce a vector version named the vibration signature, which uses the principal velocity eigenmodes. Analogous to the regular heat kernel signature introduced as the trace of heat kernel of Laplacian, Hildebrandt et al. The resulting Hessian is a matrix in R 3 n × 3 n for a mesh with n vertices, and its eigenfunctions in R 3 n can be thought of as principal velocity fields along which the energy varies. Analyzing eigenmodes of the Hessian is known as modal analysis, a technique that has been applied to reduced physical simulation ( Barbič and James, 2005), shape segmentation ( Huang et al., 2009), and shape analysis ( Hildebrandt et al., 2010). (2010), this observation provides an approach to systematically derive new operators by linearizing any nonlinear deformation energy. Discussed in Zorin (2005) and Hildebrandt et al. This Hessian is guaranteed to be p.s.d., since the deformation energy is minimized at the rest pose. ![]() The Hessian matrix of any nonlinear deformation energy at the rest pose can be used as an alternative to the Laplacian. Yu Wang, Justin Solomon, in Handbook of Numerical Analysis, 2019 5.8 Hessian and normal-restricted Hessian: A family of linearized energies
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