Modal Shape Analysis
beyond Laplacian
(CAGP 2012)
Klaus Hildebrandt, Christian Schulz,
Christoph von Tycowicz, Konrad Polthier
(brief)
Presenter: ShiHao.Wu
Outline
• 1, Quick review of Spectral
• 2, Main contributions of the paper
• 3, Results
Chladni plates
• Let us “see” the vibration of the metal plate
from the white sands.
Chladni plates VS spetral
Left: In the late 1700's, the physicist Ernst Chladni was amazed
by the patterns formed by sand on vibrating metal plates. Right: numerical
simulations obtained with a discretized Laplacian.
How to transform to spetral space?
• In order to build eigenstructure of the surface.
• Discrete view:
– Define matrices on the graphs of surface, then
find eigenvectors and eigenvalues.
• Continuous view:
– Define operators on the manifolds, then find
eigenfunctions(eigenmodes) and eigenvalues.
Laplace operator(Laplacian)
Laplace operator(Laplacian)
• Advantage:
– eigenfunctions of the Laplacian form
an orthogonal basis of the space of
the surface(like Fourier basis)
– eigenvalues and eigenfunctions are
invariant under isometric
deformations of the surface
• Disadvantage:
– insensitivity to extrinsic features of
the surface, like sharp bends
Motivation and Contributions
• New operators derive from surface energy as
alternatives to Lapacian, which is sensitive to
features.
• Application to global mesh shape analysis
– vibration signature(descriptor): measures the
similarity of points on a surface
– feature signature: identify features of surfaces
What’s surface energy?
• A twice continuously differentiable function of
the surface.
Surface
Energy
Hessian on
minima
Eigenvalues
Eigenmodes
• (skip)At a minimum of E, the eigenmodes
associated to the low eigenvalues of the
Hessian of E, point into the direction that
locally cause the least increase of energy.
Example1 of surface energy
(foundation of this paper)
• Dirichlet energy:
– measure of how variable of σ , is intimately
connected to Laplace basically.
– its Hessian equals the Laplace operator if the
surface is a part of the Euclidean plane
Example2 of surface energy(skip)
(foundation of this paper)
• Deformation energy:
– x is a surface mesh (the positions of the vertices)
– ƒ and ω measure angles, length of edges, or area
of triangles. Example, thin shell simulation:
Key observation
• Laplacian eigenvalue problem appears as the
eigenvalue problem of the Hessian of the
Dirichlet energy.
So, By modifying the Dirichlet energy, we
could get new operators better than Laplacian,
using normal information
Modified Dirichlet energy
Discretization of Dirichlet energy(skip)
Comparison and possible Explanation
• Laplacian
Comparison and possible Explanation
• The energy
has its minimum at the origin of
the space of normal vector fields.
• Therefore, eigenmodes of Hessian of
that
correspond to small eigenvalues have small
function values in areas of high curvature,
because then a variation in this direction causes
less increase of energy
Modes of Deformation energy(skip)
• Deformation energy:
Eigen vibration modes
from shell deformation energy(skip)
Eigen vibration modes
from shell deformation energy(skip)
Vibration signature(skip)
• Based on the vibration modes of the surface
with respect to the Discrete Shells energy
–
denote the eigenvalues and vibration
eigenmodes of a mesh x.
– The eigenmodes with smaller eigenvalue receive
higher weights and t is a parameter to control that
Vibration signature distance
• measures the similarity of points on a surface
• Different distance threshold
Feature signature
• Identify features of surfaces by normal information
Vibration signature
VS Heat kernel signature
• VS
• HKS
Heat kernel signature
VS Vibration signature
• HKS
• VS
different t
Feature signature
VS Heat kernel signature
• FS
• HKS
Feature signature
VS Heat kernel signature
• FS
• HKS
Efficiency
Future work
• Spectral quadrangulations sensitive to feature
• Create a subspace of the shape space of a
mesh by vibration modes.(SIG 2012)
Application: use of eigenvectors
• Whenever clustering is applicable, e.g.,
– Mesh segmentation in spectral domain [Liu & Zhang 04, 05, 07]
– Surface reconstruction: grouping “inside” and “outside” tetrahedra
[Kullori et al. 05]
– Shape correspondence: finding clusters of consistent or agreeable
pair-wise matching [Leordeanu & Hebert 06]
29
Favorite spectral project
• Music Visualization and editing
Thank you !