Topological Guarantees Research Articles

TopoMap++: A faster and more space efficient technique to compute projections with topological guarantees.

High-dimensional data, characterized by many features, can be difficult to visualize effectively. Dimensionality reduction techniques, such as PCA, UMAP, and t-SNE, address this challenge by projecting the data into a lower-dimensional space while preserving important relationships. TopoMap is another technique that excels at preserving the underlying structure of the data, leading to interpretable visualizations. In particular, TopoMap maps the high-dimensional data into a visual space, guaranteeing that the 0-dimensional persistence diagram of the Rips filtration of the visual space matches the one from the high-dimensional data. However, the original TopoMap algorithm can be slow and its layout can be too sparse for large and complex datasets. In this paper, we propose three improvements to TopoMap: 1) a more space-efficient layout, 2) a significantly faster implementation, and 3) a novel TreeMap-based representation that makes use of the topological hierarchy to aid the exploration of the projections. These advancements make TopoMap, now referred to as TopoMap++, a more powerful tool for visualizing high-dimensional data which we demonstrate through different use case scenarios.

Robust Low-Poly Meshing for General 3D Models

We propose a robust re-meshing approach that can automatically generate visual-preserving low-poly meshes for any high-poly models found in the wild. Our method can be seamlessly integrated into current mesh-based 3D asset production pipelines. Given an input high-poly, our method proceeds in two stages: 1) Robustly extracting an offset surface mesh that is feature-preserving, and guaranteed to be watertight, manifold, and self-intersection free; 2) Progressively simplifying and flowing the offset mesh to bring it close to the input. The simplicity and the visual-preservation of the generated low-poly is controlled by a user-required target screen size of the input: decreasing the screen size reduces the element count of the low-poly but enlarges its visual difference from the input. We have evaluated our method on a subset of the Thingi10K dataset that contains models created by practitioners in different domains, with varying topological and geometric complexities. Compared to state-of-the-art approaches and widely used software, our method demonstrates its superiority in terms of the element count, visual preservation, geometry, and topology guarantees of the generated low-polys.

A Novel Efficient Hybrid Compensation Topology for Wireless Power Transfer

Wireless power transfer systems require compensators to improve efficiency by increasing transfer power between transmitter and receiver. The most commonly used compensators are series-series (SS), series–parallel, and double-sided LCC. This article analyzes the comparison between typical compensators focusing on transfer efficiency. Due to these characteristics of typical compensators, a hybrid compensation topology is designed based on two compensators. Hybrid compensation topology is capable of switching between SS and double-sided LCC topologies. Two ac switches are used on the transmitter and receiver for switching mode in hybrid compensation topology with wireless communication links, and its variation algorithm is clarified. A novel compensation topology guarantees system’s efficiency for a wide range of loads based on the maximum efficiency of two compensation topologies. The possibility of a proposed hybrid compensator is verified with an experimental prototype used for massive vehicles.

Efficient tetrahedralization of multi-material images with quality, fidelity, and topological guarantees

An algorithm is proposed for generating three-dimensional unstructured tetrahedral meshes of multi-material images. The algorithm produces a mesh whose boundary is proved to be homeomorphic to the object surface. In addition, it provides a guaranteed dihedral angle bound of up to 19 . 4 7 ∘ for the output tetrahedra. Moreover, it allows for user-specified guaranteed bounds on the distance between the boundaries of the mesh and the boundaries of the materials. It produces a small number of mesh elements that comply with these guarantees, and is compatible in performance with other software. The algorithm offers all of the following properties: bounded dihedral angles, customized two-sided Hausdorff distance, topological faithfulness, a small number of elements, and run-time performance. The experimental evaluation on synthetic and real medical data illustrates the efficiency and effectiveness of the method. • The mesh offers a faithful topology to the materials. • Elements with arbitrarily small angles do not appear in the mesh. • The mesh offers a close representation (fidelity) of the underlying materials. • The mesh contains a small number of elements that comply with the guarantees above. • The mesh can be constructed within tight time constraints.

Stripped halfedge data structure for parallel computation of arrangements of segments

Computing an arrangement of segments with some geometrical and topological guarantees is a critical step in many geometry processing applications. In this paper, we propose a method to efficiently compute arrangements of segments using a strip-based data structure. Thanks to this new data structure, the arrangement computation algorithm can easily be parallelized as the per strip computations are independent. Another interest of our approach is that we can propose an out-of-core and streamed construction for large datasets, while keeping a low memory footprint. We prove the correctness of our structure and provide a complete comparative evaluation with respect to state-of-the-art demonstrating the interest of our construction for the computation of an exact arrangement.

TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data.

Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to favor different aspects of the data, most multidimensional projection methods strive to preserve dissimilarity measures that encapsulate geometric properties such as distances or the proximity relation between data objects. However, geometric relations are not the only interesting property to be preserved in a projection. For instance, the analysis of particular structures such as clusters and outliers could be more reliably performed if the mapping process gives some guarantee as to topological invariants such as connected components and loops. This paper introduces TopoMap, a novel projection technique which provides topological guarantees during the mapping process. In particular, the proposed method performs the mapping from a high-dimensional space to a visual space, while preserving the 0-dimensional persistence diagram of the Rips filtration of the high-dimensional data, ensuring that the filtrations generate the same connected components when applied to the original as well as projected data. The presented case studies show that the topological guarantee provided by TopoMap not only brings confidence to the visual analytic process but also can be used to assist in the assessment of other projection methods.

Adaptive Metrics for Adaptive Samples

We generalize the local-feature size definition of adaptive sampling used in surface reconstruction to relate it to an alternative metric on Euclidean space. In the new metric, adaptive samples become uniform samples, making it simpler both to give adaptive sampling versions of homological inference results and to prove topological guarantees using the critical points theory of distance functions. This ultimately leads to an algorithm for homology inference from samples whose spacing depends on their distance to a discrete representation of the complement space.

Conjugate shape simplification via precise algebraic planar sweeps toward gear design

Gears play a pivotal role in machine design. This paper proposes an algorithm to simplify the shapes of planar gears. This is achieved via iterative conjugation, using precise algebraic sweeps. The notion of shape simplification is introduced in a mathematically rigorous manner and it is shown that the conjugation process converges, yielding a pair of meshing gears that follow the desired motion. Simplified gear shapes may lead to improved mechanical characteristics and reduction in manufacturing costs. The generality of algebraic sweeps allows precise design of gears with freeform shapes and non-uniform motion transmission. Moreover, the computational framework proposed in this paper is versatile, with applications beyond gear design. A variety of examples from an implementation of our algorithm, that offers topological guarantees, are presented, which demonstrate the robustness and efficacy of our approach.

Stochastic Online Learning with Probabilistic Graph Feedback

We consider a problem of stochastic online learning with general probabilistic graph feedback, where each directed edge in the feedback graph has probability pij. Two cases are covered. (a) The one-step case, where after playing arm i the learner observes a sample reward feedback of arm j with independent probability pij. (b) The cascade case where after playing arm i the learner observes feedback of all arms j in a probabilistic cascade starting from i – for each (i,j) with probability pij, if arm i is played or observed, then a reward sample of arm j would be observed with independent probability pij. Previous works mainly focus on deterministic graphs which corresponds to one-step case with pij ∈ {0,1}, an adversarial sequence of graphs with certain topology guarantees, or a specific type of random graphs. We analyze the asymptotic lower bounds and design algorithms in both cases. The regret upper bounds of the algorithms match the lower bounds with high probability.

Numerical Integration Over Implicitly Defined Domains with Topological Guarantee

Numerical integration over the implicitly defined domains is challenging due to topological variances of implicit functions. In this paper, we use interval arithmetic to identify the boundary of the integration domain exactly, thus getting the correct topology of the domain. Furthermore, a geometry-based local error estimate is explored to guide the hierarchical subdivision and save the computation cost. Numerical experiments are presented to demonstrate the accuracy and the potential of the proposed method.

Two-stage transformerless dual-buck PV grid-connected inverters with high efficiency

A semi-two-stage common-ground-type transformerless dual-buck-based grid-connected inverter is proposed in this paper. The common-ground-type topology can eliminate common mode (CM) leakage current by connecting the negative terminal of the photovoltaic(PV) directly to the neutral point of the grid, which bypasses the PV array's stray capacitance. The dual-buck-based topology guarantees increased robustness since the dc-link cannot be short-circuited by a shoot-through event. The semi-two-stage topology features multi-level characteristic, which has a lower forward-voltage drop and smaller dv/dt. Compared with the conventional two-stage inverter, the proposed topology achieves higher efficiency and higher reliability. Experimental results of a 1.5kW prototype show that the proposed inverter is able to achieve high efficiency and low leakage currents.

Stabilizing embedology: Geometry-preserving delay-coordinate maps.

Delay-coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time-series outputs. The efficacy of delay-coordinate mapping has long been supported by Takens' embedding theorem, which guarantees that delay-coordinate maps use the time-series output to provide a reconstruction of the hidden state space that is a one-to-one embedding of the system's attractor. While this topological guarantee ensures that distinct points in the reconstruction correspond to distinct points in the original state space, it does not characterize the quality of this embedding or illuminate how the specific parameters affect the reconstruction. In this paper, we extend Takens' result by establishing conditions under which delay-coordinate mapping is guaranteed to provide a stable embedding of a system's attractor. Beyond only preserving the attractor topology, a stable embedding preserves the attractor geometry by ensuring that distances between points in the state space are approximately preserved. In particular, we find that delay-coordinate mapping stably embeds an attractor of a dynamical system if the stable rank of the system is large enough to be proportional to the dimension of the attractor. The stable rank reflects the relation between the sampling interval and the number of delays in delay-coordinate mapping. Our theoretical findings give guidance to choosing system parameters, echoing the tradeoff between irrelevancy and redundancy that has been heuristically investigated in the literature. Our initial result is stated for attractors that are smooth submanifolds of Euclidean space, with extensions provided for the case of strange attractors.

Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees

Implicitly described domains are a well-established tool in the simulation of time-dependent problems, for example, using level-set methods. To solve partial differential equations on such domains, a range of numerical methods was developed, for example, the Immersed Boundary method, the Unfitted Finite Element or Unfitted Discontinuous Galerkin methods, and the eXtended or Generalised Finite Element methods, just to name a few. Many of these methods involve integration over cut-cells or their boundaries, as they are described by sub-domains of the original level-set mesh. We present a new algorithm to geometrically evaluate the integrals over domains described by a first-order, conforming level-set function. The integration is based on a polyhedral reconstruction of the implicit geometry, following the concepts of the marching cubes algorithm. The algorithm preserves various topological properties of the implicit geometry in its polyhedral reconstruction, making it suitable for Finite Element computations. Numerical experiments show second-order accuracy of the integration. An implementation of the algorithm is available as free software, which allows for an easy incorporation into other projects. The software is in productive use within the DUNE framework (Bastian et al. 2008a).

Minkowski sum computation of B-spline surfaces

We propose a method for computing the Minkowski sum of two free-form surfaces, given in a tensor product B-spline representation in R3. The Minkowski sum (typically a three dimensional volume), is represented by its boundary surface(s), also referred to as the envelope. The envelope is obtained via an algebraic equation solving approach (in the parameter space of the input geometries), followed by mapping the parametric solution to Euclidean space and filtering of redundant solution patches. The suggested method is applicable to a fairly general class of input surfaces, allowing non-convex regions, boundary curves and C1 discontinuities, while providing a solution with topological guarantee. Test results are provided, demonstrating the suggested method using a triangular mesh approximation of the Minkowski sum envelope surface.

Topologically guaranteed bivariate solutions of under-constrained multivariate piecewise polynomial systems

We present a subdivision based algorithm to compute the solution of an under-constrained piecewise polynomial system of n−2 equations with n unknowns, exploiting properties of B-spline basis functions. The solution of such systems is, typically, a two-manifold in Rn. To guarantee the topology of the approximated solution in each sub-domain, we provide subdivision termination criteria, based on the (known) topology of the univariate solution on the domain’s boundary, and the existence of a one-to-one projection of the unknown solution on a two dimensional plane, in Rn. We assume the equation solving problem is regular, while sub-domains containing points that violate the regularity assumption are detected, bounded, and returned as singular locations of small (subdivision tolerance) size. This work extends (and makes extensive use of) topological guarantee results for systems with zero and one dimensional solution sets. Test results in R3 and R4 are also demonstrated, using error-bounded piecewise linear approximations of the two-manifolds.

Reconstruction of water-tight surfaces through Delaunay sculpting

Given a finite set of points S⊆R2, we define a proximity graph called as shape-hull graph (SHG(S)) that contains all Gabriel edges and a few non-Gabriel edges of Delaunay triangulation of S. For any S, SHG(S) is topologically regular with its boundary (referred to as shape-hull (SH)) homeomorphic to a simple closed curve. We introduce the concept of divergent concavity for simple, closed, planar curves based on the alignment of curves in concave portions and discuss various measures to characterize curves having divergent concavity. Under sufficiently dense sampling, we prove that SH(S), where S is sampled from a divergent concave curve ΣD, represents a piece-wise linear approximation of ΣD. We extend this result to provide a sculpting algorithm for closed surface reconstruction from a set of raw samples. The surface is constructed through a repeated elimination of Delaunay tetrahedra subjected to circumcenter and topological constraints. Theoretically, we justify our algorithm by establishing a topological guarantee on the 3D shape-hull with the help of topological rules. We demonstrate the effectiveness of our approach with experimental results on models with sharp features and sparsely distributed point clouds. Compared to existing sculpting approaches for surface reconstruction that require either a parameter tuning or several stages, our approach is simple, non-parametric, single stage and reconstructs topologically correct piece-wise linear approximation for divergent concave surfaces.

Geometric Tomography with Topological Guarantees

We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in $$\mathbb R ^3$$ from its cross-sections $$\mathcal{S }$$ with a given set of cutting planes $$\mathcal P $$ having arbitrary orientations. In this paper, we analyse a very natural reconstruction strategy: a point $$x \in \mathbb R ^3$$ belongs to the reconstructed object if (at least one of) its nearest point(s) in $$\mathcal P $$ belongs to $$\mathcal{S }$$ . We prove that under appropriate sampling conditions, the output of such an algorithm preserves the homotopy type of the original object. Using the homotopy equivalence, we also show that the reconstructed object is homeomorphic (and isotopic) to the original object. This is the first time that 3-dimensional shape reconstruction from cross-sections comes with theoretical guarantees.

Modelling and Simulation of 3-phase Transformerless Split Inductor Multilevel Inverter for Grid Connected Photovoltaic System

A Split inductor is used with Exiting Inverter to interconnect inverter with grid connected system which avoids the usage of transformer. While using split inductor NPC-MLI shoot-through problems are producing in the bridge legs of an NPCTLI, its operation stability is ruined. Hysteresis Current Control (HCC) offers an excellent current control performance to NPCTLI. It acts based on the error current value i) and hysteresis band value (h). The proposed topology guarantees for no common-mode voltage, shoot-through possibility and capacitor balancing problem. Finally, the simulation results of a proposed SI-NPCTLI system verified using MATLAB SIMULINK.

Generalized Topological Simplification of Scalar Fields on Surfaces.

We present a combinatorial algorithm for the general topological simplification of scalar fields on surfaces. Given a scalar field f, our algorithm generates a simplified field g that provably admits only critical points from a constrained subset of the singularities of f, while guaranteeing a small distance ||f - g||∞ for data-fitting purpose. In contrast to previous algorithms, our approach is oblivious to the strategy used for selecting features of interest and allows critical points to be removed arbitrarily. When topological persistence is used to select the features of interest, our algorithm produces a standard ϵ-simplification. Our approach is based on a new iterative algorithm for the constrained reconstruction of sub- and sur-level sets. Extensive experiments show that the number of iterations required for our algorithm to converge is rarely greater than 2 and never greater than 5, yielding O(n log(n)) practical time performances. The algorithm handles triangulated surfaces with or without boundary and is robust to the presence of multi-saddles in the input. It is simple to implement, fast in practice and more general than previous techniques. Practically, our approach allows a user to arbitrarily simplify the topology of an input function and robustly generate the corresponding simplified function. An appealing application area of our algorithm is in scalar field design since it enables, without any threshold parameter, the robust pruning of topological noise as selected by the user. This is needed for example to get rid of inaccuracies introduced by numerical solvers, thereby providing topological guarantees needed for certified geometry processing. Experiments show this ability to eliminate numerical noise as well as validate the time efficiency and accuracy of our algorithm. We provide a lightweight C++ implementation as supplemental material that can be used for topological cleaning on surface meshes.

Topologically Enhanced Slicing of MLS Surfaces

Growing use of massive scan data in various engineering applications has necessitated research on point-set surfaces. A point-set surface is a continuous surface, defined directly with a set of discrete points. This paper presents a new approach that extends our earlier work on slicing point-set surfaces into planar contours for rapid prototyping usage. This extended approach can decompose a point-set surface into slices with guaranteed topology. Such topological guarantee stems from the use of Morse theory based topological analysis of the slicing operation. The Morse function for slicing is a height function restricted to the point-set surface, an implicitly defined moving least-squares (MLS) surface. We introduce a Lagrangian multiplier formulation for critical point identification from the restricted surface. Integral lines are constructed to form Morse-Smale complex and the enhanced Reeb graph. This graph is then used to provide seed points for forming slicing contours, with the guarantee that the sliced model has the same topology as the input point-set surface. The extension of this approach to degenerate functions on point-set surface is also discussed.