Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or reach. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C². In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^∞ manifold without significantly reducing the reach. More precisely, given a manifold with reach R, we construct a manifold that is ε-close to it in the C¹ sense (both the manifold and its tangent spaces are close), and has reach at least R-ε. The proof employs techniques from differential topology - partitions of unity and smoothing using convolution kernels. This result implies that nearly all theorems established for C² or manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C², for free!

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $τ$. Moreover, let $μ$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$, $j=1,2,\dots,N$ be $N$ independent random samples from distribution $μ$. Also, let $ξ_j$, $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $σ^2I$, where $σ$ is less than a certain specified function of $d, V$ and $τ$. We assume that we are given the data points $y_j=x_j+ξ_j,$ $j=1,2,\dots,N$, modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cdσ^2/τ$ and whose reach is greater than $cτ/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $σ^{-1}$ and exponentially on $d$.

Given a finite set of points P sampling an unknown smooth surface ℳ ⊆ ℝ³, our goal is to triangulate ℳ based solely on P. Assuming ℳ is a smooth orientable submanifold of codimension 1 in ℝ^d, we introduce a simple algorithm, Naive Squash, which simplifies the α-complex of P by repeatedly applying a new type of collapse called vertical relative to ℳ. Naive Squash also has a practical version that does not require knowledge of ℳ. We establish conditions under which both the naive and practical Squash algorithms output a triangulation of ℳ. We provide a bound on the angle formed by triangles in the α-complex with ℳ, yielding sampling conditions on P that are competitive with existing literature for smooth surfaces embedded in ℝ³, while offering a more compartmentalized proof. As a by-product, we obtain that the restricted Delaunay complex of P triangulates ℳ when ℳ is a smooth surface in ℝ³ under weaker conditions than existing ones.

Conference version, full version is given in hal-03721463

In our companion paper "Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting.

We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.

In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(K′)) ≲ dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.

In this paper, we focus on one particular instance of the shape reconstruction problem, in which the shape we wish to reconstruct is an orientable smooth submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the submanifold (such as the Čech complex or the Rips complex), we recast the reconstruction problem as a 𝓁₁-norm minimization problem in which the optimization variable is a chain of K. Providing that K satisfies certain reasonable conditions, we prove that the considered minimization problem has a unique solution which triangulates the submanifold and coincides with the flat Delaunay complex introduced and studied in a companion paper [D. Attali and A. Lieutier, 2022]. Since the objective is a weighted 𝓁₁-norm and the contraints are linear, the triangulation process can thus be implemented by linear programming.

Abstract We address the problem of simplifying two‐dimensional polygonal partitions that exhibit strong regularities. Such partitions are relevant for reconstructing urban scenes in a concise way. Preserving long linear structures spanning several partition cells motivates a point‐line projective duality approach in which points represent line intersections, and lines possibly carry multiple points. We propose a simplification algorithm that seeks a balance between the fidelity to the input partition, the enforcement of canonical relationships between lines (orthogonality or parallelism) and a low complexity output. Our methodology alternates continuous optimization by Riemannian gradient descent with combinatorial reduction, resulting in a progressive simplification scheme. Our experiments show that preserving canonical relationships helps gracefully degrade partitions of urban scenes, and yields more concise and regularity‐preserving meshes than common mesh‐based simplification approaches.

Shapes are often designed to satisfy structural properties and serve a particular functionality in the physical world. Unfortunately, most existing generative models focus primarily on the geometric or visual plausibility, ignoring the physical or structural constraints. To remedy this, we present a novel method aimed to endow deep generative models with physical reasoning. In particular, we introduce a loss and a learning framework that promote two key characteristics of the generated shapes: their connectivity and physical stability. The former ensures that each generated shape consists of a single connected component, while the latter promotes the stability of that shape when subjected to gravity. Our proposed physical losses are fully differentiable and we demonstrate their use in end-to-end learning. Crucially we demonstrate that such physical objectives can be achieved without sacrificing the expressive power of the model and variability of the generated results. We demonstrate through extensive comparisons with the state-of-the-art deep generative models, the utility and efficiency of our proposed approach, while avoiding the potentially costly differentiable physical simulation at training time.

Abstract We consider the following setting: suppose that we are given a manifold M in $${\mathbb {R}}^d$$ R d with positive reach. Moreover assume that we have an embedded simplical complex $${\mathcal {A}}$$ A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in $${\mathcal {A}}$$ A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then $${\mathcal {A}}$$ A is a triangulation of the manifold, that is, they are homeomorphic.

This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP) for integer modulo 2 coefficients, where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation when the simplicial complex is a pseudomanifold. We then show how this particular instance of the problem is relevant, by providing an application in the context of point cloud triangulation.

This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We first consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a finite point set is collapsible. We then consider a convex domain defined as the convex hull of a finite point set. We show that if the point set samples sufficiently densely the domain, then both the Cech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a filtration by sweeping space with a growing sphere whose center has been fixed and studying events occurring through the filtration. Since the filtration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory.

We consider the domain of non-empty convex and compact subsets of a finite dimensional Euclidean space to represent partial or imprecise points in computational geometry. The convex hull map on such imprecise points is given domain-theoretically by an inner and an outer convex hull. We provide a practical algorithm to compute the inner convex hull when there are a finite number of convex polytopes as partial points. A notion of pre-inner support function is introduced, whose convex hull gives the support function of the inner convex hull in a general setting. We then show that the convex hull map is Scott continuous and can be extended to finitely generable subsets, represented by the Plotkin power domain of the underlying domain. This in particular allows us to compute, for the first time, the convex hull of attractors of iterated function systems in fractal geometry. Finally, we derive a program logic for the convex hull map in the sense of the weakest pre-condition for a given post-condition and show that the convex hull predicate transformer is computable.

In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach.

When learning 3D shapes we are usually interested in their intrinsic geometry rather than in their orientation. To deal with the orientation variations the usual trick consists in augmenting the data to exhibit all possible variability, and thus let the model learn both the geometry as well as the rotations. In this paper we introduce a new auto-encoder model for encoding and synthesis of 3D shapes. To get rid of undesirable input variability our model learns a manifold in a quotient space of the input space. Typically, we propose to quotient the space of 3D models by the action of rotations. Thus, our quotient auto-encoder allows to directly learn in the space of interest, ignoring side information. This is reflected in better performances on reconstruction and interpolation tasks, as our experiments show that our model outperforms a vanilla auto-encoder on the well-known Shapenet dataset. Moreover, our model learns a rotation-invariant representation, leading to interesting results in shapes co-alignment. Finally, we extend our quotient auto-encoder to quotient by non-rigid transformations.

International audience

We associate with each compact set X of Rn two real-valued functions cX and hX defined on R+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on cP(t) entails that the Rips complex of P at scale r collapses to the Cech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on hX over some interval guarantees a topologically correct reconstruction of the shape X either with a Cech complex of P or with a Rips complex of P. Regarding the reconstruction with Cech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive μ-reach. Most importantly, our work shows that Rips complexes can also be used to provide topologically correct reconstruction of shapes. This may be of some computational interest in high dimensions.

We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference G - K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.

Sampling conditions for recovering the homology of a set using topological persistence are much weaker than sampling conditions required by any known polynomial time algorithm for producing a topologically correct reconstruction. Under the former sampling conditions which we call weak sampling conditions, we give an algorithm that outputs a topologically correct reconstruction. Unfortunately, even though the algorithm terminates, its time complexity is unbounded. Motivated by the question of knowing if a polynomial time algorithm for reconstruction exists under the weak sampling conditions, we identify at the heart of our algorithm a test which requires answering the following question: given two 2-dimensional simplicial complexes L ⊂ K, does there exist a simplicial complex containing L and contained in K which realizes the persistent homology of L into K? We call this problem the homological simplification of the pair (K, L) and prove that this problem is NP-complete, using a reduction from 3SAT.

A simple way to reconstruct a shape A from a sample P is to output an r-offset P + r B, where B = {x ∈ RN x ≤ 1} designates the unit Euclidean ball centered at the origin. Recently, it has been proved that the output P + r B is homotopy equivalent to the shape A, for a dense enough sample P of A and for a suitable value of the parameter r. In this paper, we extend this result and find convex sets C ⊂ RN, besides the unit Euclidean ball B, for which P + rC reconstructs the topology of A. This class of convex sets includes in particular N-dimensional cubes in RN. We proceed in two steps. First, we establish the result when P is an ε-offset of A. Building on this first result, we then consider the case when P is a finite noisy sample of A.

Abstract We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the Gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ‐reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive μ‐reach can thus be approximated by the curvature measures of the offset of a point‐cloud sample.

Abstract Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this paper, using the notion of polyhedral chains introduced by Whitney, we introduce a new general framework to define an analogous of the set of critical points of piecewise linear maps defined over discrete objects that can be easily computed. We illustrate our results by showing how they can be used to compute Minkowski sums of polyhedra and volumes swept by moving polyhedra.

It has been observed for a long time that the operation consisting of offsetting a solid by a quantity r and then offsetting its complement by d < r produces, in some cases, a new solid with the same topology but with a smooth boundary. While this fact has been widely used in Computer Aided Geometric Design or in the field of image processing, we provide here for the first time a tight and robust condition that guarantees the smoothness of the new solid and gives a lower bound on its reach (distance to the medial axis). This condition is based on the general properties of the distance function to a compact set and relies on the recently introduced critical function and μ-reach.

We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size, and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points and critical values of distance functions.

Given a smooth compact codimension one submanifold S of Rk and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisy-approximation that generalize sampling conditions introduced by Amenta & al. and Dey & al. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. Our results allow to consider balls of different radii. For the first approximation condition, we also prove that a subset (known as the λ medial axis) of the medial axis of Rk\K is homotopy equivalent to the medial axis of S. We obtain similar results for smooth compact submanifolds S of Rk of any codimension.

In this work, one proves that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it. For this, one introduces the weak feature size (wfs) that generalizes the notion of local feature size. Our results apply to open sets with positive wfs, which include many sets whose boundaries are not smooth and even nowhere smooth. This class includes also the piecewise analytic open sets which cover many cases encountered in practical applications. The proofs are based on the study of distance functions to closed sets and their critical points. As an application, one gives an algorithmic way, thanks to persistent homology techniques, to compute the homology groups of open sets from noisy samples of points on their boundary.

Consider two (n - 1)-dimensional manifolds, S and S' in Rn. We say that they are projection-homeomorphic when the closest projection of each one onto the other is a homeomorphism. We give tight conditions under which S and S' are projection-homeomorphic. These conditions involve the local feature size for S and for S' and the Hausdorff distance between them. Our results hold for arbitrary n.

We introduce a domain-theoretic framework for differential calculus. We define the set of primitive maps as well as the derivative of an interval-valued Scott continuous function on the domain of intervals, and show that they are dually related, providing an extension of the classical duality of differentiation and integration as in the fundamental theorem of calculus. It is shown that, for locally Lipschitz functions of a real variable, the domain-theoretic derivative coincides with the Clarke's derivative. We then construct a domain for differentiable real-valued functions of a real variable by pairing consistent information about the function and information about its derivative. The set of classical $C^1$ functions, equipped with its $C^1$ norm, is embedded into the set of maximal elements of this countably based, bounded complete continuous domain. This domain also provides a model for the differential properties of piecewise $C^1$ functions, locally Lipschitz functions and more generally of all continuous functions. We prove that consistency of function information and derivative information is decidable on rational step functions, which shows that our domain can be given an effective structure. We thus obtain a data type for differential calculus. As an immediate application, we present a domain-theoretic formulation of Picard's theorem, which provides a data type for solving differential equations.

It is well known that the complexity of the Delaunay triangulation of N points in R 3, i.e. the number of its faces, can be O (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface.In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of R 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N).

We provide an algorithmic formalization of ordinary differential equations in the framework of domain theory. Given a Scott continuous, interval-valued and time-dependent scalar field and a Scott continuous initial function consistent with the scalar field, the domain-theoretic analogue of the classical Picard operator, whose fix-points give the solutions of the differential equation, acts on the domain of continuously differentiable functions by successively updating the information about the solution and the information about its derivative. We present a linear and a quadratic algorithm respectively for updating the function information and the derivative information on the basis elements of the domain. In the generic case of a classical initial value problem with a continuous scalar field, which is Lipschitz in the space component, this provides a novel technique for computing the unique solution of the differential equation up to any desired accuracy, such that at each stage of computation one obtains two continuous piecewise linear maps which bound the solution from below and above, thus giving the precise error. When the scalar field is continuous and computable but not Lipschitz, it is known that no computable classical solution may exist. We show that in this case the interval-valued domain-theoretic solution is computable and contains all classical solutions. This framework also allows us to compute an interval-valued solution to a differential equation when the initial value and/or the scalar field are interval-valued, i.e. imprecise.

A data-type for differential calculus is introduced, which is based on domain theory. We define the integral and also the derivative of a Scott continuous function on the domain of intervals, and present a domain-theoretic generalization of the fundamental theorem of calculus. We then construct a domain for differentiable real valued functions of a real variable. The set of classical C/sup 1/ functions, equipped with its C/sup 1/ norm, is embedded into the set of maximal elements of this domain, which is a countably based bounded complete continuous domain. This gives a data type for differential calculus. The construction can be generalized to C/sup k/ and C/sup /spl infin// functions. As an immediate application, we present a domain-theoretic generalization of Picard's theorem, which provides a data type for solving differential equations.