I'm dealing with physical simulation (position based dynamic). Now I'm trying to realize tracking for two meshes. In order to explain what does it mean, let's assume that we have two similar meshes: one with fine resolution and other with coarse. A need to make in the way that mesh with fine resolution followed the coarse mesh moving. But it should not be exact following, instead it should follow only in general. Let's consider this task in terms of mathematics: First we need to build Laplace-Beltrami discrete operator. I use next definition $$L=V^{-1/2}CV^{-1/2},$$ where $V$ is a diagonal matrix of Voronoi areas, $C$ is a sparse symmetric matrix of cotangent weights. After that I perform an eigenvalue decomposition $$L=QAQ^{T}$$ The next step is to build the matrix of harmonic functions $T$;

On Internet I have found that it could be built in the next way $$T=(V^{-1/2}Q)^{T}$$ So now I have $n*n$ matrix T, where n - is a number of vertices's in the fine mesh. Now, if have T matrix I can build the next constraint $$T*p_f==TBp_c$$

$p_f$ is a vector of vertices's in the fine mesh

$p_c$ is a vector of vertices in the coarse mesh

$B$ such matrix that $Bp_c=p_f$.

If this constraint is satisfied fine mesh exactly follows for the coarse mesh. Now, if we discard from matrix $T$ those eigenvectors, which correspond to the biggest eigenvalues we obtain the $k*n$ matrix $T$, and it means that we discarded information about high frequent harmonics in the mesh spectral representation. As result mesh with fine resolution will follow for the coarse only in general but such small details like wrinkle will be different.

In my implementation it works only if I use full $T$ matrix, but when I discard at least one eigenvector, behavior becomes unpredictable.

Question: Is there any fundamental errors in this approach? I'm confident in all, except of definition for $T$ matrix.

  • $\begingroup$ Writing down the equations, the matrix $T$ satisfying $C T = T A$ (eigenvectors of $C$) is defined by $T = (V^{-1/2}Q)$, without the transpose. I dont know if it is the matrix or the harmonic functions you are looking for. Anyway, $k$ is the remaining number of rows? are you solving $Tp_f = TBp_c$ to obtain $p_f$ by a method that minimizes the residual (The system is rectangular after removing rows)? Why are you playing with $T$ and not with $B$ (svd of $B$ and keep only a few singular values)? $\endgroup$ – sebas Mar 3 '14 at 21:38

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