Estimating the region of attraction (RoA) for a robot controller is essential for safe application and controller composition. Many existing methods require a closed-form expression that limit applicability to data-driven controllers. Methods that operate only over trajectory rollouts tend to be data-hungry. In prior work, we have demonstrated that topological tools based on Morse Graphs (directed acyclic graphs that combinatorially represent the underlying nonlinear dynamics) offer data-efficient RoA estimation without needing an analytical model. They struggle, however, with high-dimensional systems as they operate over a state-space discretization. This paper presents Morse Graph-aided discovery of Regions of Attraction in a learned Latent Space (MORALS)**. The approach combines auto-encoding neural networks with Morse Graphs. MORALS shows promising predictive capabilities in estimating attractors and their RoAs for data-driven controllers operating over high-dimensional systems, including a 67-dim humanoid robot and a 96-dim 3-fingered manipulator. It first projects the dynamics of the controlled system into a learned latent space. Then, it constructs a reduced form of Morse Graphs representing the bistability of the underlying dynamics, i.e., detecting when the controller results in a desired versus an undesired behavior. The evaluation on high-dimensional robotic datasets indicates data efficiency in RoA estimation.

This paper proposes an integration of surrogate modeling and topology to significantly reduce the amount of data required to describe the underlying global dynamics of robot controllers, including closed-box ones. A Gaussian Process (GP), trained with randomized short trajectories over the state-space, acts as a surrogate model for the underlying dynamical system. Then, a combinatorial representation is built and used to describe the dynamics in the form of a directed acyclic graph, known as Morse graph. The Morse graph is able to describe the system's attractors and their corresponding regions of attraction (RoA). Furthermore, a pointwise confidence level of the global dynamics estimation over the entire state space is provided. In contrast to alternatives, the framework does not require estimation of Lyapunov functions, alleviating the need for high prediction accuracy of the GP. The framework is suit-able for data-driven controllers that do not expose an analytical model as long as Lipschitz-continuity is satisfied. The method is compared against established analytical and recent machine learning alternatives for estimating Roas, outperforming them in data efficiency without sacrificing accuracy. Link to code: https://go.rutgers.edu/49hy35en

Large programs of dynamic gene expression, like cell cyles and circadian rhythms, are controlled by a relatively small "core" network of transcription factors and post-translational modifiers, working in concerted mutual regulation. Recent work suggests that system-independent, quantitative features of the dynamics of gene expression can be used to identify core regulators. We introduce an approach of iterative network hypothesis reduction from time-series data in which increasingly complex features of the dynamic expression of individual, pairs, and entire collections of genes are used to infer functional network models that can produce the observed transcriptional program. The culmination of our work is a computational pipeline, Iterative Network Hypothesis Reduction from Temporal Dynamics (Inherent dynamics pipeline), that provides a priority listing of targets for genetic perturbation to experimentally infer network structure. We demonstrate the capability of this integrated computational pipeline on synthetic and yeast cell-cycle data.

Since the seminal 1961 paper of Monod and Jacob, mathematical models of biomolecular circuits have guided our understanding of cell regulation. Model-based exploration of the functional capabilities of any given circuit requires systematic mapping of multidimensional spaces of model parameters. Despite significant advances in computational dynamical systems approaches, this analysis remains a nontrivial task. Here, we use a nonlinear system of ordinary differential equations to model oocyte selection in Drosophila, a robust symmetry-breaking event that relies on autoregulatory localization of oocyte-specification factors. By applying an algorithmic approach that implements symbolic computation and topological methods, we enumerate all phase portraits of stable steady states in the limit when nonlinear regulatory interactions become discrete switches. Leveraging this initial exact partitioning and further using numerical exploration, we locate parameter regions that are dense in purely asymmetric steady states when the nonlinearities are not infinitely sharp, enabling systematic identification of parameter regions that correspond to robust oocyte selection. This framework can be generalized to map the full parameter spaces in a broad class of models involving biological switches.

We demonstrate a modeling and computational framework that allows for rapid screening of thousands of potential network designs for particular dynamic behavior. To illustrate this capability we consider the problem of hysteresis, a prerequisite for construction of robust bistable switches and hence a cornerstone for construction of more complex synthetic circuits. We evaluate and rank most three node networks according to their ability to robustly exhibit hysteresis where robustness is measured with respect to parameters over multiple dynamic phenotypes. Focusing on the highest ranked networks, we demonstrate how additional robustness and design constraints can be applied. We compare our results to more traditional methods based on specific parameterization of ordinary differential equation models and demonstrate a strong qualitative match at a small fraction of the computational cost.

We present a new modeling and computational tool that computes rigorous summaries of network dynamics over large sets of parameter values. These summaries, organized in a database, can be searched for observed dynamics, e.g., bistability and hysteresis, to discover parameter regimes over which they are supported. We illustrate our approach on several networks underlying the restriction point of the cell cycle in humans and yeast. We rank networks by how robustly they support hysteresis, which is the observed phenotype. We find that the best 6-node human network and the yeast network share similar topology and robustness of hysteresis, in spite of having no homology between the corresponding nodes of the network. Our approach provides a new tool linking network structure and dynamics.

Existing topology-based vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits, and separatrices that are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. We advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of a vector field in the form of a directed graph, called a Morse connection graph (MCG). While an MCG exists for every vector field, it need not be unique. Previous techniques for computing MCG's, while fast, are overly conservative and usually results in MCG's that are too coarse to be useful for the applications. To address this issue, we present a new technique for performing Morse decomposition based on the concept of tau-maps, which typically provides finer MCG's than existing techniques. Furthermore, the choice of tau provides a natural tradeoff between the fineness of the MCG's and the computational costs. We provide efficient implementations of Morse decomposition based on tau-maps, which include the use of forward and backward mapping techniques and an adaptive approach in constructing better approximations of the images of the triangles in the meshes used for simulation.. Furthermore, we propose the use of spatial tau-maps in addition to the original temporal tau-maps. These techniques provide additional trade-offs between the quality of the MCGs and the speed of computation. We demonstrate the utility of our technique with various examples in the plane and on surfaces including engine simulation data sets.

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.

Simple point detection is an important task for several problems in discrete geometry, such as topology preserving thinning in image processing to compute discrete skeletons. In this paper, the approach to simple point detection is based on techniques from cubical homology, a framework ideally suited for problems in image processing. A (d-dimensional) unitary cube (for a d-dimensional digital image) is associated with every discrete picture element, instead of a point in epsilon(d) (the d-dimensional Euclidean space) as has been done previously. A simple point in this setting then refers to the removal of a unitary cube without changing the topology of the cubical complex induced by the digital image. The main result is a characterization of a simple point p (i.e., simple unitary cube) in terms of the homology groups of the (3d - 1) neighborhood of p for arbitrary, finite dimensions

Vector field design on surfaces is necessary for many graphics applications: example-based texture synthesis, nonphotorealistic rendering, and fluid simulation. For these applications, singularities contained in the input vector field often cause visual artifacts. In this article, we present a vector field design system that allows the user to create a wide variety of vector fields with control over vector field topology, such as the number and location of singularities. Our system combines basis vector fields to make an initial vector field that meets user specifications.The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated due to the Poincaré-Hopf index theorem. To reduce the visual artifacts caused by these singularities, our system allows the user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations offer topological guarantees for the vector field in that they only affect user-specified singularities. We develop efficient implementations of these operations based on Conley index theory . Our system also provides other editing operations so that the user may change the topological and geometric characteristics of the vector field.To create continuous vector fields on curved surfaces represented as meshes, we make use of the ideas of geodesic polar maps and parallel transport to interpolate vector values defined at the vertices of the mesh. We also use geodesic polar maps and parallel transport to create basis vector fields on surfaces that meet the user specifications. These techniques enable our vector field design system to work for both planar domains and curved surfaces.We demonstrate our vector field design system for several applications: example-based texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces.

Surface parameterization is necessary for many graphics tasks: texture-preserving simplification, remeshing, surface painting, and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a three-stage feature-based patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distance-based surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature's surface points. To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the Green-Lagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles. We demonstrate our feature-based patch creation and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we describe an image-based error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.

We segment and topologically classify brain vessel data obtained from magnetic resonance angiography (MRA). The segmentation is done adaptively and the classification by means of cubical homology, i.e. the computation of homology groups. In this way the number of connected components; (measured by H/sub 0/), the tunnels (given by H/sub 1/) and the voids (given by H/sub 2/) are determined, resulting in a topological characterization of the blood vessels.

There are a number of tasks in low level vision and image processing that involve computing certain topological characteristics of objects in a given image including connectivity and the number of holes. We combine a new combinatorial topology method to compute the number of connected components and holes of objects in a given image with fast segmentation methods to extract the objects.