Multi-Scale Feature Spaces for Shape Processing and Analysis Giuseppe Patan` e and Bianca Falcidieno Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche Genova, Italy {patane,falcidieno}@ge.imati.cnr.it Abstract—In digital geometry processing and shape modeling, the Laplace-Beltrami and the heat diffusion operator, together with the corresponding Laplacian eigen- maps, harmonic and geometry-aware functions, have been used in several applications, which range from surface parameterization, deformation, and compression to seg- mentation, clustering, and comparison. Using the linear FEM approximation of the Laplace-Beltrami operator, we derive a discrete heat kernel that is linear, stable to an irregular sampling density of the input surface, and scale covariant. With respect to previous work, this last property makes the kernel particularly suitable for shape analysis and comparison; in fact, local and global changes of the surface correspond to a re-scaling of the time parameter without affecting its spectral component. Finally, we study the scale spaces that are induced by the proposed heat kernel and exploited to provide a multi-scale approximation of scalar functions defined on 3D shapes. Keywords-scale-space methods; heat kernel; Laplacian matrix; spectral analysis; signal and function smoothing; critical points; shape analysis 1. I NTRODUCTION In digital geometry processing and shape modeling, the Laplace-Beltrami and the heat diffusion operator, together with the corresponding Laplacian eigenmaps, harmonic and geometry-aware functions, have been used in several contexts. For instance, the eigenvectors of the graph Laplacian are exploited to project the input signal into the frequency [23], [50] or a lower dimensional space [2], [22]; to smooth surfaces [8], [19], [20], [27], [29], [48] and signals [33], [45]; to compress 3D shapes [18], [42]; to process meshes [31], [32] and graphs [9], [21]. The Laplacian eigenvectors [34], [35], [37], [38], [46] are also used for parameterizing surfaces homeomorphic to the sphere [15] or with an arbitrary genus [51], [52]. In the frequency space, mesh segmentation [24], [49], shape correspondence [16] and comparison [34], [37], [38], [39], [40] have been suc- cessfully addressed. Finally, mesh Laplacian operators, whose stability and convergence have been studied in [12], [17], [47] and [34], [36], [37], play a central role in the definition of differential coordinates for surface deformation [41], [43] and quadrilateral remeshing [2], [10], [11], [30]. The Laplace-Beltrami operator is strictly related to the heat diffusion equation, which provides an embedding of a given function in a hierarchy of smoothed approxima- tions. The heat kernel and the associated diffusion metric have been exploited to approximate the gradient of maps defined on triangulated surfaces or point sets [25]. Finally, the heat kernel has important applications such as shape segmentation [14], [7] and matching [3], [28] with diffusion distances [6], [22], multi-scale [44] and isometry-invariant [3], [28] signatures. The main limitation of the current discretizations of the heat kernel k t proposed by previous work is its dependence on the scale of M; i.e., rescaling M to αM, α ∈ R + , k t becomes ˜ k t = α 2 k α 2 t . This means that global and local changes to the surface provide different kernels, which must be “normalized” before their use as shape signatures. To partially overcome this drawback, in [26] the heat kernel signature is sampled logarithmically in time, scaled, and derived; then, the descriptor is defined by the magnitude of the Fourier Transform coefficients. In this case, the normalization steps are neither unique nor intrinsically defined by the input shape. Alternatively, in [14] the eigenvalues are normalized by the first non-null eigenvalue, thus guaranteeing that ˜ k t = α 2 k t but without removing the scale term α 2 . Overview and contributions Combining the weak formulation of the heat equation with the linear FEM approximation of the Laplace- Beltrami operator, we derive a discretization of the heat kernel that is linear, intrinsically invariant to surface scalings, and stable to irregular sampling densities. As main feature with respect to previous work, the proposed kernel is scale covariant; i.e., local and global changes of the surface correspond to a re-scaling of the time pa- rameter without affecting the spectral term of the kernel. Then, the scale invariance is achieved by normalizing the eigenvalues by the first non-null eigenvalue, thus avoiding a-posteriori changes to the kernel itself or to the surface. Due to these two properties, the kernel is particularly suitable for shape comparison and the proposed approach improves the invariance and stability of the corresponding shape signatures. To assess these aims, the underlying idea is to compute the solution of the heat equation using the inner product induced by the mass matrix of the linear FEM discretization. In fact, this scalar product is adapted to the sampling density of M through the distribution of the areas of its triangles.