In recent years a considerable amount of work in graphics and geometric optimization used tools based on the Laplace-Beltrami operator on a surface. The applications of the Laplacian include mesh editing, surface smoothing, and shape interpolations among others. However, it has been shown [12,23,25] that the popular cotangent approximation schemes do not provide convergent point-wise (or even L 2 ) estimates, while many applications rely on point-wise estimation. Existence of such schemes has been an open question [12].In this paper we propose the first algorithm for approximating the Laplace operator of a surface from a mesh with point-wise convergence guarantees applicable to arbitrary meshed surfaces. We show that for a sufficiently fine mesh over an arbitrary surface, our mesh Laplacian is close to the Laplace-Beltrami operator on the surface at every point of the surface.Moreover, the proposed algorithm is simple and easily implementable. Experimental evidence shows that our algorithm exhibits convergence empirically and outperforms cotangent-based methods in providing accurate approximation of the Laplace operator for various meshes. *

We present an algorithm for approximating the LaplaceBeltrami operator from an arbitrary point cloud obtained from a k-dimensional manifold embedded in the ddimensional space. We show that this PCD Laplace (PointCloud Data Laplace) operator converges to the LaplaceBeltrami operator on the underlying manifold as the point cloud becomes denser. Unlike the previous work, we do not assume that the data samples are independent identically distributed from a probability distribution and do not require a global mesh. The resulting algorithm is easy to implement. We present experimental results indicating that even for point sets sampled from a uniform distribution, PCD Laplace converges faster than the weighted graph Laplacian. We also show that using our PCD Laplacian we can directly estimate certain geometric invariants, such as manifold area.

Next-generation structural materials are expected to be lightweight, high strength, and tough composites with embedded functionalities to sense, adapt, self-repair, morph, and restore. This review highlights recent developments and concepts in bioinspired nanocomposites, emphasizing tailoring the architecture, interphases, and confinement to achieve dynamic and synergetic responses. We highlight cornerstone examples from natural materials with unique mechanical property combinations based on relatively simple building blocks produced in aqueous environments at ambient conditions. A particular focus is on structural hierarchies across multiple length scales to achieve multifunctionality and robustness. We further discuss recent advances, trends, and emerging opportunities in combining biological and synthetic components, state-ofthe-art characterization, and modelling approaches to assess the physical principles underlying nature design and mechanical responses at multiple length scales. These multidisciplinary approaches promote the synergetic enhancement of individual materials properties and an improved predictive and prescriptive design of the next era of structural materials at multi-length scales for a wide range of applications.

We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions.As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs.

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under Z 2 coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis.First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps.Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses-two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally.Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.

We revisit the problem of computing Fréchet distance between polygonal curves under L 1 , L 2 , and L ∞ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of n vertices each, a ε-approximation of their discrete Fréchet distance can be computed in roughly O(nκ 3 log n/ε 3 ) time in three dimensions, if one of the curves is κ-bounded. Previously, only a κ-approximation algorithm was known. If both curves are the socalled backbone curves, which are widely used to model protein backbones in molecular biology, we can ε-approximate their Fréchet distance in near linear time in two dimensions, and in roughly O(n 4/3 log nm) time in three dimensions. In the second part, we propose a pseudo-output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the number of switching cells, which is quadratic in the worst case, but tends to be much smaller in practice.

Given a smoothly embedded 2-manifold in Ê ¿ , we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.

Curve matching is a fundamental problem that occurs in many applications. In this paper, we study the problem of measuring partial similarity between curves. Specifically, given two curves, we wish to maximize the total length of subcurves that are close to each other, where closeness is measured by the Fréchet distance, a common distance measure for curves. The resulting maximal length is called the partial Fréchet similarity between the two input curves. Given two polygonal curves P and Q in IR d of size m and n, respectively, we present the first exact algorithm that runs in polynomial time to compute F δ (P, Q), the partial Fréchet similarity between P and Q, under the L1 and L∞ norms. Specifically, we formulate the problem of computing F δ (P, Q) as a longest path problem, and solve it in O(mn(m + n) log(mn)) time, under the L1 or L∞ norm, using a "shortest-path map" type decomposition. To the best of our knowledge, this is the first paper to study this natural definition of partial curve similarity in the continuous setting (with all points in the curve considered), and present a polynomial-time exact algorithm for it.