Graphics Reference
In-Depth Information
C H A P T E R
5
Spectral Methods for Shape
Analysis
e decomposition of a shape into simpler pieces is the ultimate goal of most mathematical meth-
ods that aim at combining synthesis with saliency. is is the peculiar characteristic of spectral
methods: the decomposition and expression of a complex function into a set of simpler ones.
Probably, the most popular example is Fourier analysis, which studies the way general functions
may be represented or approximated by sums of simpler trigonometric functions. One best-known
application of Fourier analysis is signal processing: in this case a signal is decomposed into a lin-
ear combination of the eigenvectors of the Laplace operator applied to the signal. e Laplace
operator is a popular tool for modeling physical phenomena such as the diffusion equation for
heat and fluid flow and wave propagation.
e first application of Fourier analysis to 3D shapes was proposed by Taubin in 1995,
[ 193 ], to show how shape compression and smoothing can take advantage of the spectral analy-
sis of the mesh geometry. is first example of mesh Laplace operator originated much further
development, suggesting methods not only for analyzing but also for editing and interpolating
meshes, using for instance the differential coordinates.
In general, the popularity of spectral methods derives from the flexibility and the gener-
ality of the paradigm behind Laplace operators: indeed, they project the analysis of a shape to
the study of the model eigenstructures (eigenvalues, eigenvectors, or eigenspaces) derived from
appropriately defined mesh operators. e recent literature is rich with successful applications of
spectral analysis to graph theory, computer vision, machine learning, graph drawing, etc., and
several surveys have been proposed [ 187 , 215 ]; in this chapter we mainly focus on the extension
of the Laplace operator and the heat equation to 3D shape analysis.
5.1
LAPLACE OPERATORS
e Laplace operator, or Laplacian, is a differential operator given by the divergence of the gra-
dient of a function on the Euclidean space E n , formally:
X
@ 2
@x 2 i
f WD div . grad f /DrrD
;
i
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