Julien Tierny

Partial 3D Shape Retrieval by Reeb Pattern Unfolding

This paper presents a novel approach for fast and efficient partial shape retrieval on a collection of 3D shapes. Each shape is represented by a Reeb graph associated with geometrical signatures. Partial similarity between two shapes is evaluated by computing a variant of their maximum common sub‐graph.

Interactive Exploration and Analysis of Large-Scale Simulations Using Topology-Based Data Segmentation

Large-scale simulations are increasingly being used to study complex scientific and engineering phenomena. As a result, advanced visualization and data analysis are also becoming an integral part of the scientific process. Often, a key step in extracting insight from these large simulations involves the definition, extraction, and evaluation of features in the space and time coordinates of the solution. However, in many applications, these features involve a range of parameters and decisions that will affect the quality and direction of the analysis. Examples include particular level sets of a specific scalar field, or local inequalities between derived quantities. A critical step in the analysis is to understand how these arbitrary parameters/decisions impact the statistical properties of the features, since such a characterization will help to evaluate the conclusions of the analysis as a whole. We present a new topological framework that in a single-pass extracts and encodes entire families of possible features definitions as well as their statistical properties. For each time step we construct a hierarchical merge tree a highly compact, yet flexible feature representation. While this data structure is more than two orders of magnitude smaller than the raw simulation data it allows us to extract a set of features for any given parameter selection in a postprocessing step. Furthermore, we augment the trees with additional attributes making it possible to gather a large number of useful global, local, as well as conditional statistic that would otherwise be extremely difficult to compile. We also use this representation to create tracking graphs that describe the temporal evolution of the features over time. Our system provides a linked-view interface to explore the time-evolution of the graph interactively alongside the segmentation, thus making it possible to perform extensive data analysis in a very efficient manner. We demonstrate our framework by extracting and analyzing burning cells from a large-scale turbulent combustion simulation. In particular, we show how the statistical analysis enabled by our techniques provides new insight into the combustion process.

Panorama weaving: fast and flexible seam processing

A fundamental step in stitching several pictures to form a larger mosaic is the computation of boundary seams that minimize the visual artifacts in the transition between images. Current seam computation algorithms use optimization methods that may be slow, sequential, memory intensive, and prone to finding suboptimal solutions related to local minima of the chosen energy function. Moreover, even when these techniques perform well, their solution may not be perceptually ideal (or even good). Such an inflexible approach does not allow the possibility of user-based improvement. This paper introduces the Panorama Weaving technique for seam creation and editing in an image mosaic. First, Panorama Weaving provides a procedure to create boundaries for panoramas that is fast, has low memory requirements and is easy to parallelize. This technique often produces seams with lower energy than the competing global technique. Second, it provides the first interactive technique for the exploration of the seam solution space. This powerful editing capability allows the user to automatically extract energy minimizing seams given a sparse set of constraints. With a variety of empirical results, we show how Panorama Weaving allows the computation and editing of a wide range of digital panoramas including unstructured configurations.

Invariant High Level Reeb Graphs of 3D Polygonal Meshes

Many applications in computer graphics need high level shape descriptions, in order to benefit from a global understanding of shapes. Topological approaches enable pertinent surface decompositions, providing structural descriptions of 3D polygonal meshes; but in practice, their use raises several difficulties. In this paper, we present a novel method for the construction of invariant high level Reeb graphs, topological entities that give a good overview of the shape structure. With this aim, we propose an accurate and straightforward feature point extraction algorithm for the computation of an invariant and meaningful quotient function. Moreover, we propose a new graph construction algorithm, based on an analysis of the connectivity evolutions of discrete level lines. This algorithm brings a practical solution for the suppression of non-significant critical points over piecewise continuous functions, providing meaningful Reeb graphs. Presented method gives accurate results, with satisfactory execution times and without input parameter. The geometrical invariance of resulting graphs and their robustness to variation in model pose and mesh sampling make them good candidates for several applications, like shape deformation (experimented in this paper), recognition, compression, indexing, etc.

Topological Methods in Data Analysis and Visualization III

We propose a method for visualizing two-dimensional symmetric tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. The time parameter of the heat kernel allows a multiscale analysis in a natural way. By considering a positive definite tensor field as a Riemannian metric the definition of the HKS can be applied directly. To investigate how this measure can be used to visualize more general tensor fields we apply mappings to obtain positive definite tensor fields. The resulting scalar quantity is used for the visualization of tensor fields. For short times it is closely related to Gaussian curvature, i. e. it is quite different to usual tensor invariants like the trace or the determinant.When scientists analyze datasets in a search for underlying phenomena, patterns or causal factors, their first step is often an automatic or semi-automatic search for structures in the data. Of these feature-extraction methods, topological ones stand out due to their solid mathematical foundation. Topologically defined structuresas found in scalar, vector and tensor fieldshave proven their merit in a wide range of scientific domains, and scientists have found them to be revealing in subjects such as physics, engineering, and medicine. Full of state-of-the-art research and contemporary hot topics in the subject, this volume is a selection of peer-reviewed papers originally presented at the fourth Workshop on Topology-Based Methods in Data Analysis and Visualization, TopoInVis 2011, held in Zurich, Switzerland. The workshop brought together many of the leading lights in the field for a mixture of formal presentations and discussion. One topic currently generating a great deal of interest, and explored in several chapters here, is the search for topological structures in time-dependent flows, and their relationship with Lagrangian coherent structures. Contributors also focus on discrete topologies of scalar and vector fields, and on persistence-based simplification, among other issues of note. The new research results included in this volume relate to all three key areas in data analysistheory, algorithms and applications.

Fast and precise kinematic skeleton extraction of 3D dynamic meshes

Shape skeleton extraction is a fundamental pre-processing task in shape-based pattern recognition. This paper presents a new algorithm for fast and precise extraction of kinematic skeletons of 3D dynamic surface meshes. Unlike previous approaches, surface motions are characterized by the mesh edge-length deviation induced by its transformation through time. Then a static skeleton extraction algorithm based on Reeb graphs exploits this latter information to extract the kinematic skeleton. This hybrid static and dynamic shape analysis enables the precise detection of objects¿ articulations as well as shape topological transitions corresponding to possibly-articulated immobile objects¿ features. Experiments show that the proposed algorithm is faster than previous techniques and still achieves better accuracy.

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.

Interactive Quadrangulation with Reeb Atlases and Connectivity Textures

Creating high-quality quad meshes from triangulated surfaces is a highly nontrivial task that necessitates consideration of various application specific metrics of quality. In our work, we follow the premise that automatic reconstruction techniques may not generate outputs meeting all the subjective quality expectations of the user. Instead, we put the user at the center of the process by providing a flexible, interactive approach to quadrangulation design. By combining scalar field topology and combinatorial connectivity techniques, we present a new framework, following a coarse to fine design philosophy, which allows for explicit control of the subjective quality criteria on the output quad mesh, at interactive rates. Our quadrangulation framework uses the new notion of Reeb atlas editing, to define with a small amount of interactions a coarse quadrangulation of the model, capturing the main features of the shape, with user prescribed extraordinary vertices and alignment. Fine grain tuning is easily achieved with the notion of connectivity texturing, which allows for additional extraordinary vertices specification and explicit feature alignment, to capture the high-frequency geometries. Experiments demonstrate the interactivity and flexibility of our approach, as well as its ability to generate quad meshes of arbitrary resolution with high-quality statistics, while meeting the users own subjective requirements.

CageR: Cage-Based Reverse Engineering of Animated 3D Shapes

We present CageR: A novel framework for converting animated 3D shape sequences into compact and stable cage‐based representations. Given a raw animated sequence with one‐to‐one point correspondences together with an initial cage embedding, our algorithm automatically generates smoothly varying cage embeddings which faithfully reconstruct the enclosed object deformation. Our technique is fast, automatic, oblivious to the cage coordinate system, provides controllable error and exploits a GPU implementation. At the core of our method, we introduce a new algebraic algorithm based on maximum volume sub‐matrices (maxvol) to speed up and stabilize the deformation inversion. We also present a new spectral regularization algorithm that can apply arbitrary regularization terms on selected subparts of the inversion spectrum. This step allows to enforce a highly localized cage regularization, guaranteeing its smooth variation along the sequence. We demonstrate the speed, accuracy and robustness of our framework on various synthetic and acquired data sets. The benefits of our approach are illustrated in applications such as animation compression and post‐editing.

Conforming Morse-Smale Complexes

Morse-Smale (MS) complexes have been gaining popularity as a tool for feature-driven data analysis and visualization. However, the quality of their geometric embedding and the sole dependence on the input scalar field data can limit their applicability when expressing application-dependent features. In this paper we introduce a new combinatorial technique to compute an MS complex that conforms to both an input scalar field and an additional, prior segmentation of the domain. The segmentation constrains the MS complex computation guaranteeing that boundaries in the segmentation are captured as separatrices of the MS complex. We demonstrate the utility and versatility of our approach with two applications. First, we use streamline integration to determine numerically computed basins/mountains and use the resulting segmentation as an input to our algorithm. This strategy enables the incorporation of prior flow path knowledge, effectively resulting in an MS complex that is as geometrically accurate as the employed numerical integration. Our second use case is motivated by the observation that often the data itself does not explicitly contain features known to be present by a domain expert. We introduce edit operations for MS complexes so that a user can directly modify their features while maintaining all the advantages of a robust topology-based representation.