# Computational Analysis of PDE-Based Shape Analysis Models by Exploring the Damped Wave Equation
## 1 Introduction  
The computation of shape correspondences is a fundamental task in computer vision with many potential applications [1, 2]. In the setting of three-dimensional shape analysis, the underlying problem amounts to identifying an explicit relation between the surface elements of two or more shapes. The variety of possible shape correspondence mappings that is of interest in applications includes non-rigid transformations where shapes are just almost isometric, allowing, e.g., to match different poses of human or animal shapes. An important solution strategy is to achieve a pointwise shape correspondence using so-called descriptor-based methods. For this, a feature descriptor has to be computed that characterises each point on a shape by describing the surrounding shape surface geometry. A mathematically sound approach to compute such shape signatures is to make use of the spectral decomposition of the Laplace-Beltrami operator; see, e.g., [2, 3, 4]. To this end, the Laplace-Beltrami operator may be incorporated in a certain variety of partial differential equations (PDEs) that are potentially useful as models for feature computation. The arguably most important classic signature that can be obtained in this way is the heat kernel signature (HKS) [4], which relies on the heat equation; however, this also includes versions of the Schrodinger equation leading to the wave kernel signature (WKS) [5]. Additionally, the hyperbolic wave equation [6] has already been proposed. For the computation of such PDE-based signatures, the basic task amounts to resolving the underlying PDEs on a manifold representing a shape's boundary. The HKS and WKS areboth kernel-based methods that rely on the eigenfunction expansion of the Laplace-Beltrami operator to tackle this task and to achieve efficient algorithms. An alternative to the spectral approach is to consider the numerical integration of the underlying PDEs as proposed in [6; 7; 8]. As has been shown in the mentioned works, the shape-matching accuracy of the numerical descriptors constructed in the latter works may be slightly higher than by the kernel-based approaches, such as HKS. However, when following the path towards numerical feature construction, it is highly advocated to employ very efficient computational means such as the model order reduction framework presented in [8; 9] in order to avoid high computational times. In previous works based on numerical integration, the first-order implicit time integration has been studied in detail [7]. For the wave equation model, it has been shown that backward differencing in time may yield favourable results over simple central differences [6]. In particular, as has been illustrated in [6], the classic wave equation may yield in some experiments results of higher corresponding quality compared to the other mentioned models. Let us note that this holds again in particular when assessing the models using numerical integration with implicit first-order time stepping. **Our Contributions**: In this paper, we give an account of efforts on analysing important numerical aspects of PDE-based shape descriptors. According to the previous results described above, the question arises if some implicit schemes offer particularly useful key properties in the shape analysis context. To answer this question, we present a detailed study of three implicit time stepping schemes for the wave and damped wave equation on manifolds. We show that standard finite differences may not be appropriate since the typical initial condition used for the construction of shape signatures, which is a discrete Dirac delta function, may yield oscillatory artefacts that may spoil correspondence quality. In order to resolve this issue, we show that \(l_{0}\)-stable methods should be employed, extending the first results presented in [10] in several ways by a much more detailed exposition. We explore in detail that, in addition, a certain amount of numerical or explicit diffusion is beneficial to achieve results of best matching quality.