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Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc. == Laplace == The Laplace–Beltrami operator is involved in many important differential equations, such as the heat equation and the wave equation. It can be defined on a Riemannian manifold as the divergence of the gradient of a real-valued function ''f'': : Its spectral components can be computed by solving the Helmholtz equation (or Laplacian eigenvalue problem): : The solutions are the eigenfunctions (modes) and corresponding eigenvalues , representing a diverging sequence of positive real numbers. The first eigenvalue is zero for closed domains or when using the Neumann boundary condition. For some shapes, the spectrum can be computed analytically (e.g. rectangle, flat torus, cylinder, disk or sphere). For the sphere, for example, the eigenfunctions are the spherical harmonics. The most important properties of the eigenvalues and eigenfunctions are that they are isometry invariants. In other words, if the shape is not stretched (e.g. a sheet of paper bent into the third dimension), the spectral values will not change. Bendable objects, like animals, plants and humans, can move into different body postures with only minimal stretching at the joints. The resulting shapes are called near-isometric and can be compared using spectral shape analysis. However, the isometric deformation of surfaces in 3D in the strict sense are rigid transformations. To characterize the actual deformation undergoing for the interest of nonrigid shape analysis, smooth deformations are introduced as an alternative family of deformation to isometry, where eigenvalues are allowed to perturb with finite error bounds.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「spectral shape analysis」の詳細全文を読む スポンサード リンク
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