Abstract Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features. Although one can locate a particular feature at any index in the filtration using existing algorithms, the resulting representatives may not be compatible with the zigzag: a representative cycle at one index may not map into a representative cycle at its neighbor. For this, one needs to compute compatible representative cycles along each bar in the barcode. It is known that the barcode for a zigzag filtration with m insertions and deletions can be computed in $$O(m^\omega )$$ O ( m ω ) time, where $$\omega < 2.373$$ ω < 2.373 is the matrix multiplication exponent. However, it is not known how to compute the compatible representatives so efficiently. For a non-zigzag filtration, the classical matrix-based algorithm provides representatives in $$O(m^3)$$ O ( m 3 ) time, which can be improved to $$O(m^\omega )$$ O ( m ω ) . However, no known algorithm for zigzag filtrations computes the representatives with the $$O(m^3)$$ O ( m 3 ) time bound. We present an $$O(m^2n)$$ O ( m 2 n ) time algorithm for this problem, where $$n\le m$$ n ≤ m is the size of the largest complex in the filtration.

Given a zigzag filtration, we want to find its barcode representatives, i.e., a compatible choice of bases for the homology groups that diagonalize the linear maps in the zigzag. To achieve this, we convert the input zigzag to a levelset zigzag of a real-valued function. This function generates a Mayer-Vietoris pyramid of spaces, which generates an infinite strip of homology groups. We call the origins of indecomposable (diamond) summands of this strip their apexes and give an algorithm to find representative cycles in these apexes from ordinary persistence computation. The resulting representatives map back to the levelset zigzag and thus yield barcode representatives for the input zigzag. Our algorithm for lifting a p-dimensional cycle from ordinary persistence to an apex representative takes O(p ⋅ m log m) time. From this we can recover zigzag representatives in time O(log m + C), where C is the size of the output.

Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features. Although one can locate a particular feature at any index in the filtration using existing algorithms, the resulting representatives may not be compatible with the zigzag: a representative cycle at one index may not map into a representative cycle at its neighbor. For this, one needs to compute compatible representative cycles along each bar in the barcode. Even though it is known that the barcode for a zigzag filtration with m insertions and deletions can be computed in O (mω) time, it is not known how to compute the compatible representatives so efficiently. For a non-zigzag filtration, the classical matrix-based algorithm provides representatives in O (m3) time, which can be improved to O (mω ). However, no known algorithm for zigzag filtrations computes the representatives with the O (m3) time bound. We present an O (m2n ) time algorithm for this problem, where n ≤ m is the size of the largest complex in the filtration.

It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an $O(d n^4)$ algorithm for computing the persistent $k$-cup modules for all $k \in \{2, \dots, d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for $d>3$. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent $k$-cup modules and devise an $O(c(d)n^4)$ algorithm for computing it, where $c(d)$ is subexponential in $d$.

We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.

Abstract The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a $$\textbf{Z}^2$$ Z 2 -indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank of M over I by computing the barcode of the zigzag module obtained by restricting to that path. If M is the homology of a bifiltration F of $$t$$ t simplices (while accounting for multi-criticality) and I consists of $$t$$ t points, this computation takes $$O(t^\omega )$$ O ( t ω ) time where $$\omega \in [2,2.373)$$ ω ∈ [ 2 , 2.373 ) is the exponent of matrix multiplication. We apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M , determine whether M is interval decomposable and, if so, compute all intervals supporting its indecomposable summands.

We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module $M$ indexed by a bifiltration of $n$ simplices in $O(n^3)$ time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has $O(n^4)$ runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of $M$ from $O(n^4)$ to $O(n^3)$. In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.

Persistence diagrams (PD)s play a central role in topological data analysis. This analysis requires computing distances among such diagrams such as the 1-Wasserstein distance. Accurate computation of these PD distances for large data sets that render large diagrams may not scale appropriately with the existing methods. The main source of difficulty ensues from the size of the bipartite graph on which a matching needs to be computed for determining these PD distances. We address this problem by making several algorithmic and computational observations in order to obtain an approximation. First, taking advantage of the proximity of PD points, we condense them thereby decreasing the number of nodes in the graph for computation. The increase in point multiplicities is addressed by reducing the matching problem to a min-cost flow problem on a transshipment network. Second, we use Well Separated Pair Decomposition to sparsify the graph to a size that is linear in the number of points. Both node and arc sparsifications contribute to the approximation factor where we leverage a lower bound given by the Relaxed Word Mover's distance. Third, we eliminate bottlenecks during the sparsification procedure by introducing parallelism. Fourth, we develop an open source software called1 PDoptFlow based on our algorithm, exploiting parallelism by GPU and multicore. We perform extensive experiments and show that the actual empirical error is very low. We also show that we can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.

The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval $I$ of a $\mathbb{Z}^2$-indexed persistence module $M$ is equal to the generalized rank of the zigzag module that is induced on a certain path in $I$ tracing mostly its boundary. Hence, we can compute the generalized rank over $I$ by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing $M$ to that path. If the bifiltration and $I$ have at most $t$ simplices and points respectively, this computation takes $O(t^ω)$ time where $ω\in[2,2.373)$ is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module $M$, determine whether $M$ is interval decomposable and, if so, compute all intervals supporting its summands.

Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set - a salient feature of a combinatorial dynamical system - across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.

Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes beyond the capabilities of existing software for topological data analysis. Furthermore, this framework offers multiple opportunities for parallelization which have not been previously exploited. We provide several examples of the utility of this framework, demonstrate parallel speedups, and report on significant improvements in comparison to existing software.

Cytometry experiments yield high-dimensional point cloud data that is difficult to interpret manually. Boolean gating techniques coupled with comparisons of relative abundances of cellular subsets is the current standard for cytometry data analysis. However, this approach is unable to capture more subtle topological features hidden in data, especially if those features are further masked by data transforms or significant batch effects or donor-to-donor variations in clinical data. We present that persistent homology, a mathematical structure that summarizes the topological features, can distinguish different sources of data, such as from groups of healthy donors or patients, effectively. Analysis of publicly available cytometry data describing non-naïve CD8+ T cells in COVID-19 patients and healthy controls shows that systematic structural differences exist between single cell protein expressions in COVID-19 patients and healthy controls. We identify proteins of interest by a decision-tree based classifier, sample points randomly and compute persistence diagrams from these sampled points. The resulting persistence diagrams identify regions in cytometry datasets of varying density and identify protruded structures such as 'elbows'. We compute Wasserstein distances between these persistence diagrams for random pairs of healthy controls and COVID-19 patients and find that systematic structural differences exist between COVID-19 patients and healthy controls in the expression data for T-bet, Eomes, and Ki-67. Further analysis shows that expression of T-bet and Eomes are significantly downregulated in COVID-19 patient non-naïve CD8+ T cells compared to healthy controls. This counter-intuitive finding may indicate that canonical effector CD8+ T cells are less prevalent in COVID-19 patients than healthy controls. This method is applicable to any cytometry dataset for discovering novel insights through topological data analysis which may be difficult to ascertain otherwise with a standard gating strategy or existing bioinformatic tools.

Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general $O(m^ω)$ time complexity are not known, where $ω< 2.37286$ is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with $m$ additions and deletions on a graph with $n$ vertices and edges, the algorithm for $0$-dimension runs in $O(m\log^2 n+m\log m)$ time and the algorithm for 1-dimension runs in $O(m\log^4 n)$ time. The algorithm for $0$-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on $2$-manifolds. The algorithm for $1$-dimension pairs a negative edge with the earliest positive edge so that a $1$-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for $0$-dimension to compute the $(p-1)$-dimensional zigzag persistence for $\mathbb{R}^p$-embedded complexes in $O(m\log^2 n+m\log m+n\log n)$ time.

BACKGROUND: Interpretation of high-throughput gene expression data continues to require mathematical tools in data analysis that recognizes the shape of the data in high dimensions. Topological data analysis (TDA) has recently been successful in extracting robust features in several applications dealing with high dimensional constructs. In this work, we utilize some recent developments in TDA to curate gene expression data. Our work differs from the predecessors in two aspects: (1) Traditional TDA pipelines use topological signatures called barcodes to enhance feature vectors which are used for classification. In contrast, this work involves curating relevant features to obtain somewhat better representatives with the help of TDA. This representatives of the entire data facilitates better comprehension of the phenotype labels. (2) Most of the earlier works employ barcodes obtained using topological summaries as fingerprints for the data. Even though they are stable signatures, there exists no direct mapping between the data and said barcodes. RESULTS: The topology relevant curated data that we obtain provides an improvement in shallow learning as well as deep learning based supervised classifications. We further show that the representative cycles we compute have an unsupervised inclination towards phenotype labels. This work thus shows that topological signatures are able to comprehend gene expression levels and classify cohorts accordingly. CONCLUSIONS: In this work, we engender representative persistent cycles to discern the gene expression data. These cycles allow us to directly procure genes entailed in similar processes.

Abstract Curve reconstruction from unstructured points in a plane is a fundamental problem with many applications that has generated research interest for decades. Involved aspects like handling open, sharp, multiple and non‐manifold outlines, run‐time and provability as well as potential extension to 3D for surface reconstruction have led to many different algorithms. We survey the literature on 2D curve reconstruction and then present an open‐sourced benchmark for the experimental study. Our unprecedented evaluation of a selected set of planar curve reconstruction algorithms aims to give an overview of both quantitative analysis and qualitative aspects for helping users to select the right algorithm for specific problems in the field. Our benchmark framework is available online to permit reproducing the results and easy integration of new algorithms.

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their persistence. These concepts are well suited for spaces that are not directed. As a result, one needs a concept of homology that accommodates orientations in input space. Path-homology developed for directed graphs by Grigor'yan, Lin, Muranov and Yau has been effectively adapted for this purpose recently by Chowdhury and Mémoli. They also give an algorithm to compute this path-homology. Our main contribution in this paper is an algorithm that computes this path-homology and its persistence more efficiently for the $1$-dimensional ($H_1$) case. In developing such an algorithm, we discover various structures and their efficient computations that aid computing the $1$-dimensional path-homnology. We implement our algorithm and present some preliminary experimental results.

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to track features in a changing multivector field.

Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with $\mathbb{Z}_2$ coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in $\mathbb{R}^{d+1}$ so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data.

Automatic Extraction of road network from satellite images is a goal that can benefit and even enable new technologies. Methods that combine machine learning (ML) and computer vision have been proposed in recent years which make the task semi-automatic by requiring the user to provide curated training samples. The process can be fully automatized if training samples can be produced algorithmically. In this work, we develop such a technique by infusing a persistence-guided discrete Morse based graph reconstruction algorithm into ML framework. We elucidate our contributions in two phases. First, in a semi-automatic framework, we combine a discrete-Morse based graph reconstruction algorithm with an existing CNN framework to segment input satellite images. We show that this leads to reconstructions with better connectivity and less noise. Next, in a fully automatic framework, we leverage the power of the discrete-Morse based graph reconstruction algorithm to train a CNN from a collection of images without labelled data and use the same algorithm to produce the final output from the segmented images created by the trained CNN. We apply the discrete-Morse based graph reconstruction algorithm iteratively to improve the accuracy of the CNN. We show experimental results on datasets from SpaceNet Challenge. Full version of the paper appears in [8].

Recently it has been shown that computing the dimension of the first homology group H 1 (K) of a simplicial 2-complex K embedded linearly in R 4 is as hard as computing the rank of a sparse 0 -1 matrix.This puts a major roadblock to computing persistence and a homology basis (generators) for complexes embedded in R 4 and beyond in less than quadratic or even near-quadratic time.But, what about dimension three?It is known that when K is a graph or a surface with n simplices linearly embedded in R 3 , the persistence for piecewise linear functions on K can be computed in O(n log n) time and a set of generators of total size k can be computed in O(n + k) time .However, the question for general simplicial complexes K linearly embedded in R 3 is not completely settled.No algorithm with a complexity better than that of the matrix multiplication is known for this important case.We show that the persistence for height functions on such complexes, hence called height persistence, can be computed in O(n log n) time.This allows us to compute a basis (generators) of H i (K), i = 1, 2, in O(n log n + k) time where k is the size of the output.This improves significantly the current best bound of O(n ω ), ω being the exponent of matrix multiplication.We achieve these improved bounds by leveraging recent results on zigzag persistence in computational topology, new observations about Reeb graphs, and some efficient geometric data structures.

Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.

Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For 1-D persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most n-D persistence modules, n>1, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called 2-D interval decomposable modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called dimension distance that bounds it from below.

Automated annotation and analysis of protein molecules have long been a topic of interest due to immediate applications in medicine and drug design. In this work, we propose a topology based, fast, scalable, and parameter-free technique to generate protein signatures. We build an initial simplicial complex using information about the protein's constituent atoms, including its radius and existing chemical bonds, to model the hierarchical structure of the molecule. Simplicial collapse is used to construct a filtration which we use to compute persistent homology. This information constitutes our signature for the protein. In addition, we demonstrate that this technique scales well to large proteins. Our method shows sizable time and memory improvements compared to other topology based approaches. We use the signature to train a protein domain classifier. Finally, we compare this classifier against models built from state-of-the-art structure-based protein signatures on standard datasets to achieve a substantial improvement in accuracy.

In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice.

Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called nerve of a cover of the domain. In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage. More specifically, we show that the one-dimensional homology of the nerve complex N(U) of a path-connected cover U of a domain X cannot be richer than that of the domain X itself. Intuitively, this result means that no new H_1-homology class can be "created" under a natural map from X to the nerve complex N(U). Equipping X with a pseudometric d, we further refine this result and characterize the classes of H_1(X) that may survive in the nerve complex using the notion of size of the covering elements in U. These fundamental results about nerve complexes then lead to an analysis of the H_1-homology of Reeb spaces, mappers and multiscale mappers. The analysis of H_1-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.

We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological surveys and contagious disease surveillance. In this more general setting existing algorithms for classical (i.e.~static) clustering problems are not applicable anymore. We propose a set of optimization problems which we collectively refer to as 'temporal clustering'. The quality of a solution to a temporal clustering instance can be quantified using three parameters: the number of clusters $k$, the spatial clustering cost $r$, and the maximum cluster displacement $δ$ between consecutive time steps. We consider spatial clustering costs which generalize the well-studied $k$-center, discrete $k$-median, and discrete $k$-means objectives of classical clustering problems. We develop new algorithms that achieve trade-offs between the three objectives $k$, $r$, and $δ$. Our upper bounds are complemented by inapproximability results.

With the rapid growth of publicly available GPS traces, robust and efficient automatic road network reconstruction has become a crucial task in GIS data analysis and applications. In [20], an effective and robust road network reconstruction algorithm was developed based on the discrete Morse theory, which has the state-of-the-art performance in automatic road-network reconstruction. Based on a discrete Morse-based graph reconstruction framework, we provide two improvements of the previous algorithm [20]: (1) we further simplify it and obtain a better empirical time performance; and (2) we develop a simple but effective editing strategy that helps adding missing road segments in the output reconstruction.

We study hierarchical clusterings of metric spaces that change over time. This is a natural geo- metric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent sequence of hierarchical clusterings from a sequence of unlabeled point sets. We encode the clustering objective by embedding each point set into an ultrametric space, which naturally induces a hierarchical clustering of the set of points. We enforce temporal coherence among the embeddings by finding correspondences between successive pairs of ultrametric spaces which exhibit small distortion in the Gromov-Hausdorff sense. We present both upper and lower bounds on the approximability of the resulting optimization problems.

In topology inference from data, current approaches face two major problems. One concerns the selection of a correct parameter to build an appropriate complex on top of the data points; the other involves with the typical ‘large’ size of this complex. We address these two issues in the context of inferring homology from sample points of a smooth manifold of known dimension sitting in an Euclidean space ℝk. We show that, for a sample size of n points, we can identify a set of O(n2) points (as opposed to Voronoi vertices) approximating a subset of the medial axis that suffices to compute a distance sandwiched between the well known local feature size and the local weak feature size (in fact, the approximating set can be further reduced in size to O(n)). This distance, called the lean feature size, helps pruning the input set at least to the level of local feature size while making the data locally uniform. The local uniformity in turn helps in building a complex for homology inference on top of the sparsified data without requiring any user-supplied distance threshold. Unlike most topology inference results, ours does not require that the input is dense relative to a global feature such as reach or weak feature size; instead it can be adaptive with respect to the local feature size. We present some empirical evidence in support of our theoretical claims.

In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on P indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short of scaling well, especially for high dimensional data. In this paper, we investigate the advantages and limitations of existing approaches. Based on insights gained from the experiments, we propose an efficient new algorithm, called SimBa, for approximating the persistent homology of Rips filtrations with quality guarantees. Our new algorithm leverages a batch collapse strategy as well as a new sparse Rips-like filtration. We experiment on a variety of low and high dimensional data sets. We show that our strategy presents a significant size reduction, and our algorithm for approximating Rips filtration persistence is order of magnitude faster than existing methods in practice.

Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. Mapper, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points. The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scale at which the cover is constructed. Inspired by the concept, we explore a notion of a tower of covers which induces a tower of simplicial complexes connected by simplicial maps, which we call multiscale mapper. We study the resulting structure, and design practical algorithms to compute its persistence diagrams efficiently. Specifically, when the domain is a simplicial complex and the map is a real-valued piecewise-linear function, the algorithm can compute the exact persistence diagram only from the 1-skeleton of the input complex. For general maps, we present a combinatorial version of the algorithm that acts only on vertex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that we show to hold.

Given a real-valued function f defined over a manifold M embedded in R^d, we are interested in recovering structural information about f from the sole information of its values on a finite sample P. Existing methods provide approximation to the persistence diagram of f when geometric noise and functional noise are bounded. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice. We propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees and experimental results on the quality of our approximation of the sampled scalar field.

Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the non-linear structure hidden behind the data. In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea. This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. Our persistence-distortion distance has two properties not shared by previous methods: First, it is stable against the perturbations of the input graph metrics. Second, it is a continuous distance measure, in the sense that it is defined on an alignment of the underlying spaces of input graphs, instead of merely their nodes. This makes our persistence-distortion distance robust against, for example, different discretizations of the same underlying graph. Despite considering the input graphs as continuous spaces, that is, taking all points into account, we show that we can compute the persistence-distortion distance in polynomial time. The time complexity for the discrete case where only graph nodes are considered is much faster.

Persistent homology with coefficients in a field $$\mathbb {F}$$ coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substantially the amount of matrix operations. In addition, we propose a heuristic to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology.

Algorithms for persistent homology are well-studied where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under Z2 coefficients for a (monotone) sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis.

The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the ietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size is huge--particularly in high dimensions. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures. We investigate a complex called the graph induced complex that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one-dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.

Delaunay refinement, a versatile method of mesh generation, is plagued by memory thrashing when required to generate large output meshes. To address this space issue, a localized version of Delaunay refinement was proposed for generating meshes for smooth surfaces and volumes bounded by them. The method embodies a divide-and-conquer paradigm in that it maintains the growing set of sample points with an octree and produces a local mesh within each individual node, and stitches these local meshes seamlessly. The proofs of termination and global consistency for localized methods exploit recently developed sampling theory for smooth surfaces. Unfortunately, these proofs break down for a larger class called piecewise smooth complexes (PSCs) that allow smooth surface patches that are joined along ridges and corners. In this work, we adapt a recently developed sampling and meshing algorithm for PSCs into the localization framework. This requires revisiting the original algorithm, and more importantly re-establishing the correctness proofs to accommodate the localization framework. Our implementation of the algorithm exhibits that it can indeed generate large meshes with significantly less time and memory than the original algorithm without localization. In fact, it beats a state-of-the-art meshing tool of CGAL for generating large meshes.

Recently, various applications have motivated the study of spectral structures (eigenvalues and eigenfunctions) of the so-called Laplace-Beltrami operator of a manifold and their discrete versions. A popular choice for the discrete version is the so-called Gaussian weighted graph Laplacian which can be applied to point cloud data that samples a manifold. Naturally, the question of stability of the spectrum of this discrete Laplacian under the perturbation of the sampled manifold becomes important for its practical usage. Previous results showed that the spectra of both the manifold Laplacian and discrete Laplacian are stable when the perturbation is “nice” in the sense that it is restricted to a diffeomorphism with minor area distortion. However, this forbids, for example, small topological changes.

The fractures of thin plates often exhibit complex physical behaviors in the real world. In particular, fractures caused by tearing are different from fractures caused by in-plane motions. In this paper, we study how to make thin-plate fracture animations more realistic from three perspectives. We propose a stress relaxation method, which is applied to avoid shattering artifacts after generating each fracture cut. We formulate a fracture-aware remeshing scheme based on constrained Delaunay triangulation, to adaptively provide more fracture details. Finally, we use our multi-layered model to simulate complex fracture behaviors across thin layers. Our experiment shows that the system can efficiently and realistically simulate the fractures of multi-layered thin plates.

A special family of non-trivial loops on a surface called handle and tunnel loops associates closely to geometric features of "handles" and "tunnels" respectively in a 3D model. The identification of these handle and tunnel loops can benefit a broad range of applications from topology simplification/repair, and surface parameterization, to feature and shape recognition. Many of the existing efficient algorithms for computing non-trivial loops cannot be used to compute these special type of loops. The two algorithms known for computing handle and tunnel loops provably have a serious drawback that they both require a tessellation of the interior and exterior spaces bounded by the surface. Computing such a tessellation of three dimensional space around the surface is a non-trivial task and can be quite expensive. Furthermore, such a tessellation may need to refine the surface mesh, thus causing the undesirable side-effect of outputting the loops on an altered surface mesh. In this paper, we present an efficient algorithm to compute a basis for handle and tunnel loops without requiring any 3D tessellation. This saves time considerably for large meshes making the algorithm scalable while computing the loops on the original input mesh and not on some refined version of it. We use the concept of the Reeb graph which together with several key theoretical insights on linking number provide an initial set of loops that provably constitute a handle and a tunnel basis. We further develop a novel strategy to tighten these handle and tunnel basis loops to make them geometrically relevant. We demonstrate the efficiency and effectiveness of our algorithm as well as show its robustness against noise, and other anomalies in the input.

Abstract Reconstructing a surface mesh from a set of discrete point samples is a fundamental problem in geometric modeling. It becomes challenging in presence of ‘singularities’ such as boundaries, sharp features, and non‐manifolds. A few of the current research in reconstruction have addressed handling some of these singularities, but a unified approach to handle them all is missing. In this paper we allow the presence of various singularities by requiring that the sampled object is a collection of smooth surface patches with boundaries that can meet or intersect. Our algorithm first identifies and reconstructs the features where singularities occur. Next, it reconstructs the surface patches containing these feature curves. The identification and reconstruction of feature curves are achieved by a novel combination of the Gaussian weighted graph Laplacian and the Reeb graphs. The global reconstruction is achieved by a method akin to the well known Cocone reconstruction, but with weighted Delaunay triangulation that allows protecting the feature samples with balls. We provide various experimental results to demonstrate the effectiveness of our feature‐preserving singular surface reconstruction algorithm.

Bubbles and foams are important features of liquid surface phenomena, but they are difficult to animate due to their thin films and complex interactions in the real world. In particular, small bubbles (having diameter <1cm) in a dense foam are highly affected by surface tension, so their shapes are much less deformable compared with larger bubbles. Under this small bubble assumption, we propose a more accurate and efficient particle-based algorithm to simulate bubble dynamics and interactions. The key component of this algorithm is an approximation of foam geometry, by treating bubble particles as the sites of a weighted Voronoi diagram. The connectivity information provided by the Voronoi diagram allows us to accurately model various interaction effects among bubbles. Using Voronoi cells and weights, we can also explicitly address the volume loss issue in foam simulation, which is a common problem in previous approaches. Under this framework, we present a set of bubble interaction forces to handle miscellaneous foam behaviors, including foam structure under Plateau's laws, clusters formed by liquid surface bubbles, bubble-liquid and bubble-solid coupling, bursting and coalescing. Our experiment shows that this method can be straightforwardly incorporated into existing liquid simulators, and it can efficiently generate realistic foam animations, some of which have never been produced in graphics before.

Given a continuous function f:X -> S on a topological space X, its level set f-1(a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph which can facilitate its broader applications in shape and data analysis. The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P, we compute the H1-homology of the Reeb graph from P. It takes O(n log n) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm to compute the rank of H1(MM) from point data. The best known previous algorithm for this problem takes O(n3) time for point data.

Abstract Delaunay refinement, recognized as a versatile tool for meshing a variety of geometries, has the deficiency that it does not scale well with increasing mesh size. The bottleneck can be traced down to the memory usage of 3D Delaunay triangulations. Recently an approach has been suggested to tackle this problem for the specific case of smooth surfaces by subdividing the sample set in an octree and then refining each subset individually while ensuring termination and consistency. We extend this to localized refinement of volumes, which brings about some new challenges. We show how these challenges can be met with simple steps while retaining provable guarantees, and that our algorithm scales many folds better than a state‐of‐the‐art meshing tool provided by CGAL.

Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ($\mathbb{Z}$) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex K of dimension greater than p, the boundary matrix $[\partial_{p+1}]$ is totally unimodular if and only if $H_p(L, L_0)$ is torsion-free for all pure subcomplexes $L_0, L$ in K of dimensions p and $p+1$, respectively, where $L_0 \subsetL$. Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under $\mathbb{Z}_2$ coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal $(d-1)$-cycle in a given homology class for any triangulation of an orientable compact d-manifold or for any finite simplicial complex embedded in $\mathbb{R}^d$. Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity.

Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold M ⊂ Rd. These loops approximate a shortest basis of the one dimensional homology group H1(M) over coefficients in finite field Z2. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of H1 (Κ) for any finite simplicial complex Κ whose edges have non-negative weights.

Spectral methods have been widely used in a broad range of applications fields. One important object involved in such methods is the Laplace-Beltrami operator of a manifold. Indeed, a variety of work in graphics and geometric optimization uses the eigen-structures (i.e, the eigenvalues and eigenfunctions) of the Laplace operator. Applications include mesh smoothing, compression, editing, shape segmentation, matching, parameterization, and so on. While the Laplace operator is defined (mathematically) for a smooth domain, these applications often approximate a smooth manifold by a discrete mesh. The spectral structure of the manifold Laplacian is estimated from some discrete Laplace operator constructed from this mesh. In this paper, we study the important question of how well the spectrum computed from the discrete mesh approximates the true spectrum of the manifold Laplacian. We exploit a recent result on mesh Laplacian and provide the first convergence result to relate the spectrum constructed from a general mesh (approximating an m-manifold embedded in ℝd) with the true spectrum. We also study how stable these eigenvalues and their discrete approximations are when the underlying manifold is perturbed, and provide explicit bounds for the Laplacian spectra of two “close” manifolds, as well as a convergence result for their discrete approximations. Finally, we present various experimental results to demonstrate that these discrete spectra are both accurate and robust in practice.

Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer (Z) coefficients, we show the following: For a finite simplicial complex K of dimension greater than p, the boundary matrix [partialp+1] is totally unimodular if and only if Hp(L, L0) is torsion-free, for all pure subcomplexes L0, L in K of dimensions p and p+1 respectively, where L0 ⊂ L.

Abstract Although understanding of shape features in the context of shape matching and retrieval has made considerable progress in recent years, the case for partial and incomplete models in presence of pose variations still begs a robust and efficient solution. A signature that encodes features at multi‐scales in a pose invariant manner is more appropriate for this case. The Heat Kernel Signature function from spectral theory exhibits this multi‐scale property. We show how this concept can be merged with the persistent homology to design a novel efficient pose‐oblivious matching algorithm for all models, be they partial, incomplete, or complete. We make the algorithm scalable so that it can handle large data sets. Several test results show the robustness of our approach.

Abstract The technique of Delaunay refinement has been recognized as a versatile tool to generate Delaunay meshes of a variety of geometries. Despite its usefulness, it suffers from one lacuna that limits its application. It does not scale well with the mesh size. As the sample point set grows, the Delaunay triangulation starts stressing the available memory space which ultimately stalls any effective progress. A natural solution to the problem is to maintain the point set in clusters and run the refinement on each individual cluster. However, this needs a careful point insertion strategy and a balanced coordination among the neighboring clusters to ensure consistency across individual meshes. We design an octtree based localized Delaunay refinement method for meshing surfaces in three dimensions which meets these goals. We prove that the algorithm terminates and provide guarantees about structural properties of the output mesh. Experimental results show that the method can avoid memory thrashing while computing large meshes and thus scales much better than the standard Delaunay refinement method.

A cut locus of a point p in a compact Riemannian manifold M is defined as the set of points where minimizing geodesics issued from p stop being minimizing. It is known that a cut locus contains most of the topological information of M. Our goal is to utilize this property of cut loci to decipher the topology of M from a point sample. Recently it has been shown that Rips complexes can be built from a point sample P of M systematically to compute the Betti numbers, the rank of the homology groups of M. Rips complexes can be computed easily and therefore are favored over others such as restricted Delaunay, alpha, Cech, and witness complex. However, the sizes of the Rips complexes tend to be large. Since the dimension of a cut locus is lower than that of the manifold M, a subsample of P approximating the cut locus is usually much smaller in size and hence admits a relatively smaller Rips complex. In this paper we explore the above approach for point data sampled from surfaces embedded in any high dimensional Euclidean space. We present an algorithm that computes a subsample P' of a sample P of a 2-manifold where P' approximates a cut locus. Empirical results show that the first Betti number of M can be computed from the Rips complexes built on these subsamples. The sizes of these Rips complexes are much smaller than the one built on the original sample of M.

As a direct consequence of software quirks, designer errors, and representation flaws, often three-dimensional shapes are stored in formats that introduce inconsistencies such as small gaps and overlaps between surface patches. We present a new algorithm that simultaneously repairs imperfect geometry and topology while generating Delaunay meshes of these shapes. At the core of this approach is a meshing algorithm for input shapes that are piecewise smooth complexes (PSCs), a collection of smooth surface patches meeting at curves non-smoothly or in non-manifold configurations. Guided by a user tolerance parameter, we automatically merge nearby components while building a Delaunay mesh that has many of these errors fixed. Experimental evidence is provided to show the results of our algorithm on common computer-aided design (CAD) formats. Our algorithm may also be used to simplify shapes by removing small features which would require an excessive number of elements to preserve them in the output mesh.

Abstract We present an algorithm for the reconstruction of a surface with boundaries (including a non‐orientable one) in three dimensions from a sufficiently dense sample. It is guaranteed that the output is isotopic to the unknown sampled surface. No previously known algorithm guarantees isotopic or homeomorphic reconstruction of surfaces with boundaries. Our algorithm is surprisingly simple. It ‘peels’ slivers greedily from an α‐complex of a sample of the surface. No other post‐processing is necessary. We provide several experimental results from an implementation of our basic algorithm and also a modified version of it.

This video presents the working of a new algorithm/software called DelPSC that meshes piecewise smooth complexes in three dimensions with Delaunay simplices. Piecewise smooth complexes admit a large class of geometric domains including polyhedra, smooth and piecewise smooth surfaces with or without boundary, and non-manifolds. The algorithm and its proof of correctness are described in the paper [7].

Many applications such as topology repair, model editing, surface parameterization, and feature recognition benefit from computing loops on surfaces that wrap around their 'handles' and 'tunnels'. Computing such loops while optimizing their geometric lengths is difficult. On the other hand, computing such loops without considering geometry is easy but may not be very useful. In this paper we strike a balance by computing topologically correct loops that are also geometrically relevant. Our algorithm is a novel application of the concepts from topological persistence introduced recently in computational topology. The usability of the computed loops is demonstrated with some examples in feature identification and topology simplification.

Many applications seek to identify features like 'handles' and 'tunnels' in a shape bordered by a surface embedded in three dimensions. To this end we define handle and tunnel loops on surfaces which can help identifying these features. We show that a closed surface of genus g always has g handle and g tunnel loops induced by the embedding. For a class of shapes that retract to graphs, we characterize these loops by a linking condition with these graphs. These characterizations lead to algorithms for detection and generation of these loops. We provide an implementation with applications to feature detection and topology simplification to show the effectiveness of the method.

We present a Delaunay based algorithm for simplifying vector field datasets. Our aim is to reduce the size of the mesh on which the vector field is defined while preserving topological features of the original vector field. We leverage a simple paradigm, vertex deletion in Delaunay triangulations, to achieve this goal. This technique is effective for two reasons. First, we guide deletions by a local error metric that bounds the change of the vectors at the affected simplices and maintains regions near critical points to prevent topological changes. Second, piecewise-linear interpolation over Delaunay triangulations is known to give good approximations of scalar fields. Since a vector field can be regarded as a collection of component scalar fields, a Delaunay triangulation can preserve each component and thus the structure of the vector field as a whole. We provide experimental evidence showing the effectiveness of our technique and its ability to preserve features of both two and three dimensional vector fields.

We present an isosurface meshing algorithm, DelIso, based on the Delaunay refinement paradigm. This paradigm has been successfully applied to mesh a variety of domains with guarantees for topology, geometry, mesh gradedness, and triangle shape. A restricted Delaunay tri- angulation, dual of the intersection between the surface and the three dimensional Voronoi diagram, is often the main ingredient in Delaunay refinement. Computing and storing three dimensional Voronoi/Delaunay diagrams become bottlenecks for Delaunay refinement techniques since isosurface computations generally have large input datasets and output meshes. A highlight of our algorithm is that we find a simple way to recover the restricted Delaunay triangulation of the surface without computing the full 3D structure. We employ techniques for efficient ray tracing of isosurfaces to generate surface sample points, and demonstrate the effectiveness of our implementation using a variety of volume datasets.

This paper presents an algorithm for sampling and triangulating a generic $C^2$-smooth surface $\Sigma\subset \mathbb{R}^3$ that is input with an implicit equation. The output triangulation is guaranteed to be homeomorphic to $\Sigma$. We also prove that the triangulation has well-shaped triangles, large dihedral angles, and a small size. The only assumption we make is that the input surface representation is amenable to certain types of computations, namely, computations of the intersection points of a line and $\Sigma$, computations of the critical points in a given direction, and computations of certain silhouette points.

We present an algorithm to identify the flat and tubular regions of a three dimensional shape from its point sample. We consider the distance function to the input point cloud and the Morse structure induced by it on R3. Specifically we focus on the index 1 and index 2 saddle points and their unstable manifolds. The unstable manifolds of index 2 saddles are one dimensional whereas those of index 1 saddles are two dimensional. Mapping these unstable manifolds back onto the surface, we get the tubular and flat regions. The computations are carried out on the Voronoi diagram of the input points by approximating the unstable manifolds with Voronoi faces. We demonstrate the performance of our algorithm on several point sampled objects.

The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations, feature extractions and others. In most cases, the distance function induced by the surface is approximated by a discrete distance function induced by a discrete sample of the surface. The critical points of the distance function determine the topology of the set inducing the function. However, no earlier theoretical result has linked the critical points of the distance to a sampling of geometric structures to their topological properties. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface either lie very close to the surface or near its medial axis and this closeness is quantified with the sampling density. Based on this result, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight ε-sampling of a surface, approximates the surface geometrically, both in Hausdorff distance and normals, and captures its topology.

This paper presents an algorithm for sampling and triangulatinga smooth surface Σ ⊂ ℝ3 where the triangulation is homeomorphic to Σ. The only assumption we make is that the input surface representation is amenable to certain types of computations, namely computations of the intersection points of a line with the surface, computations of the critical points of some height functions defined on the surface and its restriction to a plane, and computations of some silhouette points. The algorithm ensures bounded aspect ratio, size optimality, and smoothness of the output triangulation. Unlike previous algorithms, this algorithm does not need to compute the local feature size for generating the sample points which was a major bottleneck. Experiments show the usefulness of the algorithm in remeshing and meshing CAD surfaces that are piecewise smooth.

We present an algorithm to compute a Delaunay mesh conforming to a polyhedron possibly with small input angles. The radius-edge ratio ofmost output tetrahedra are bounded by a constant, except possibly those that are provably close to small angles. Further, the mesh is graded, that is, edge lengths are at least a constant fraction of the local feature sizes at the edge endpoints. Unlike a previous algorithm, this algorithm is simple to implement as it avoids computing local feature sizes and protective zones explicitly. Our experimental results confirm our claims and show that few skinny tetrahedra remain.

We present an algorithm for surface reconstruction in presence of noise. We show that, under a reasonable noise model, the algorithm has theoretical guarantees. Actual performance of the algorithm is illustrated by our experimental results.

In this paper we establish a topological similarity between two apparently different shape constructors from a set of points. Shape constructors are geometric structures that transform finite point sets into continuous shapes. Due to their immense practical importance in geometric modeling various shape constructors have been proposed recently. Understanding the relations among them often leads to new insights that are potentially helpful in applications. Here we discover a topological equivalence among two such geometric structures, namely α shapes and flow shapes. Both shapes found applications in surface reconstruction and molecular modelin.

Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra, called slivers, that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic point sets, but not with boundaries. Recently a randomized point-placement strategy has been proposed to remove slivers while conforming to a given boundary. In this paper we present a deterministic algorithm for generating a weighted Delaunay mesh which respects the input boundary and has no poor quality tetrahedron including slivers. As in previous work, we assume that no input angle is acute. Our result is achieved by combining the weight pumping method for sliver exudation and the Delaunay refinement method for boundary conformation.

Within the field of computer graphics and visualization, it is often necessary to visualize polygonal models with large number of polygons. Display quality is mandatory, but it is also desirable to have the ability to rapidly update the display in order to facilitate interactive use. Point based rendering methods have been shown effective for this task. Building on this paradigm we introduce the PMR system which uses a hierarchy both in points and triangles for rendering. This hierarchy is fundamentally different from the ones used in existing methods. It is based on the feature geometry in the object space rather than its projection in the screen space. This provides certain advantages over the existing methods.

Current surface reconstruction algorithms perform satisfactorily on we ll-sampled, smooth surfaces without boundaries. However, these algorithms face difficulty with undersampling. Cases of undersampling are prevalent in real data since often they sample a part of the boundary of an object, or are derived from a surface with high curvature or nonsmoothness. In this paper we present an algorithm to detect the boundaries where dense sampling stops and undersampling begins. This information can be used to reconstruct surfaces with boundaries, and also to localize small and sharp features where usually undersampling happens. We report the effectiveness of the algorithm with a number of experimental results. Theoretically, we justify the algorithm with some mild assumptions that are valid for most practical data.

Shape modeling is an integral part of many visualization problems. Recent advances in scanning technology and a number of surface reconstruction algorithms have opened up a new paradigm for modeling shapes from samples. Many of the problems currently faced in this modeling paradigm can be traced back to two anomalies in sampling, namely undersampling and oversampling. Boundaries, non-smoothness and small features create undersampling problems, whereas oversampling leads to too many triangles. We use Voronoi cell geometry as a unified guide to detect undersampling and oversampling. We apply these detections in surface reconstruction and model simplification. Guarantees of the algorithms can be proved. The authors show the success of the algorithms empirically on a number of interesting data sets.

The problem of computing a piecewise linear approximation to a surface from a set of sample points is important in solid modeling, computer graphics and computer vision. A recent algorithm [1] using the Voronoi diagram of the sample points gave a guarantee on the distance of the output surface from the original sampled surface assuming the sample was `suciently dense'. We give a similar algorithm, simplifying the computation and the proof of the geometric guarantee. In addition, we guarantee that our output surface is homeomorphic to the original surface; to our knowledge this is the rst such topological guarantee for this problem. 1 Introduction A number of applications in CAD, computer graphics, computer vision and mathematical modeling involve the computation of a piecewise lin- Dept. of Computer Science, U. of Texas, Austin TX 78712. e-mail: [email protected], supported by NSF grant CCR-9731977 y Dept. of Computer Science, U. of Texas, Austin, TX 78712. e-mail: sunghe...

Article Reconstruction curves with sharp corners Share on Authors: Tamal K. Dey View Profile , Rephael Wenger Dept. of CIS, Ohio State University, Columbus, Ohio Dept. of CIS, Ohio State University, Columbus, OhioView Profile Authors Info & Claims SCG '00: Proceedings of the sixteenth annual symposium on Computational geometryMay 2000 Pages 233–241https://doi.org/10.1145/336154.336209Online:01 May 2000Publication History 13citation470DownloadsMetricsTotal Citations13Total Downloads470Last 12 Months7Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

A sliver is a tetrahedon whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that, if the Delaunay triangulation has the ratio property introduced in Miller et al. [1995], then there is an assignment of weights so the weighted Delaunay traingulation contains no slivers. We also give an algorithm to compute such a weight assignment.

Consider an arrangement of n hyperplanes in R d . Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no non-trivial bound was known for the general case where the polytopes may have overlapping interiors, for d > 2. We analyze families of polytopes that do not share vertices. In R 3 we show an O(k 1=3 n 2 ) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in R d is k 1=2 n d=2 ) which is a factor of p n away from the best known upper bound in the range n d 2 ...

A sliver is a tetrahedron whose four vertices lie close to a plane and whose projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

Article Curve reconstruction: connecting dots with good reason Share on Authors: Tamal K. Dey Department of CSE, IIT Kharagpur, India 721302 Department of CSE, IIT Kharagpur, India 721302View Profile , Kurt Mehlhorn Max-Planck-Institut für Informatik, D-66123 Saarbrücken, Germany Max-Planck-Institut für Informatik, D-66123 Saarbrücken, GermanyView Profile , Edgar A. Ramos Max-Planck-Institut für Informatik, D-66123 Saarbrücken, Germany Max-Planck-Institut für Informatik, D-66123 Saarbrücken, GermanyView Profile Authors Info & Claims SCG '99: Proceedings of the fifteenth annual symposium on Computational geometryJune 1999 Pages 197–206https://doi.org/10.1145/304893.304972Online:13 June 1999Publication History 33citation434DownloadsMetricsTotal Citations33Total Downloads434Last 12 Months1Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Recent developments in analyzing molecular structures and representing solid models using simplicial complexes have further enhanced the need for computing structural information about simplicial complexes in R 3 . This paper develops basic techniques required to manipulate and analyze structures of complexes in R 3 . A new approach to analyze simplicial complexes in Euclidean 3-space R 3 is described. First, methods from topology are used to analyze triangulated 3-manifolds in R 3 . Then, it is shown that these methods can, in fact, be applied to arbitrary simplicial complexes in R 3 after (simulating) the process of thickening a complex to a 3-manifold homotopic to it. As a consequence considerable structural information about the complex can be determined and certain discrete problems solved as well. For example, it is shown how to determine homology groups, as well as concrete representations of their generators, for a given complex in R 3

We prove an O(nk/sup 1/3/) upper bound for planar k-sets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k-levels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general.

Article Free Access Share on Visibility with reflection Authors: Boris Aronov Computer Science Department, Polytechnic University, Brooklyn, NY Computer Science Department, Polytechnic University, Brooklyn, NYView Profile , Alan R. Davis Div. of Computer Science, Math. and Science, St. Johns University, Jamaica, NY Div. of Computer Science, Math. and Science, St. Johns University, Jamaica, NYView Profile , Tamal K. Dey Dept. of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, India Dept. of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, IndiaView Profile , Sudebkumar P. Pal Dept. of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, India Dept. of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, IndiaView Profile , D. Chithra Prasad Dept. of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, India Dept. of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, Kharagpur 721302, IndiaView Profile Authors Info & Claims SCG '95: Proceedings of the eleventh annual symposium on Computational geometrySeptember 1995 Pages 316–325https://doi.org/10.1145/220279.220313Published:01 September 1995Publication History 3citation320DownloadsMetricsTotal Citations3Total Downloads320Last 12 Months7Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We describe an optimal algorithm to decide if one closed curve on a triangulated 2-manifold can be continuously transformed to another, i.e., if they are homotopic. Our algorithm runs in O(n+k/sub 1/+k/sub 2/) time and space, where closed curves C/sub 1/ and C/sub 2/ of lengths k/sub 1/ and k/sub 2/, resp., on a genus g surface M (g/spl ne/2 if M orientable, and g/spl ne/3,4 if M is non-orientable) are presented as edge-vertex sequences in a triangulation T of size n of M. This also implies an optimal algorithm to decide if a closed curve on a surface can be continuously contracted to a point. Except for three low genus cases, our algorithm completes an investigation into the computational complexity of the two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. However, we make novel applications of methods from modern combinatorial group theory for an approach entirely different from previous ones, and much simpler to implement.

We provide a new technique for deriving optimal sized polygonal schema for triangulated compact 2-manifolds without boundary in O(n) time, where n is the size of the given triangulation T. We first derive a polygonal schema P embedded in T using Seifert-Van Kampen's theorem. A reduced polygonal schema Q of optimal size is computed from P, where a surjective mapping from the vertices of P is retained to the vertices of Q. This helps detecting null-homotopic (contractable to a point) cycles. Given a cycle of length k we determine if it is null-homotopic in O(n+gk) time where g is the genus of the given 2-manifold. The actual contraction for a null-homotopic cycle can be computed in O(gkn) time and space. This is an improvement of a factor of g over the previous best-known algorithms for these problems.

Every collection of t ≥ 2n2 triangles with a total of n vertices in R3 has Ω(t4/n6) crossing pairs. This implies that one of their edges meets Ω(t3/n6) of the triangles. From this it follows that n points in R3 have only O(8/3) halving planes.

This paper presents a simple algorithm to compute a convex decomposition of a nonconvex polyhedron of arbitrary genus (handles) and shells (internal voids). For such a polyhedron S with n edges and rnotches (features causing nonconvexity in polyhedra), the algorithm produces a worst-case optimal $O(r^2 )$ number of convex polyhedra $S_i $, with $U_{i = 1}^k S_i = S$, in $O(nr^2 + r^{7/2} )$ time and $O(nr + r^{5/2} )$ space. Recently, Chazelle and Palios have given a fast $O((n + r^2 )\log r$) time and $O(n + r^2 )$ space algorithm to tetrahedralize a nonconvex polyhedron. Their algorithm, however, works for a simple polyhedron of genus zero and with no shells (internal voids). The algorithm, presented here, is based on the simple cut and split paradigm of Chazelle. With the help of zone theorems on arrangements, it is shown that this cut and split method is quite efficient. The algorithm is extended to work for a certain class of nonmanifold polyhedra. Also presented is an algorithm for the same problem that uses clever heuristics to overcome the numerical inaccuracies under finite precision arithmetic.

Article Free Access Share on Triangulation and CSG representation of polyhedra with arbitrary genus Author: Tamal K. Dey Department of Computer Science, Purdue University, West Lafayette, IN Department of Computer Science, Purdue University, West Lafayette, INView Profile Authors Info & Claims SCG '91: Proceedings of the seventh annual symposium on Computational geometryJune 1991 Pages 364–371https://doi.org/10.1145/109648.109689Online:01 June 1991Publication History 9citation378DownloadsMetricsTotal Citations9Total Downloads378Last 12 Months16Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF