Global Intrinsic Symmetries of Shapes

Although considerable attention in recent years has been given to the problem of symmetry detection in general shapes, few methods have been developed that aim to detect and quantify the intrinsic symmetry of a shape rather than its extrinsic, or pose‐dependent symmetry. In this paper, we present a novel approach for efficiently computing symmetries of a shape which are invariant up to isometry preserving transformations. We show that the intrinsic symmetries of a shape are transformed into the Euclidean symmetries in the signature space defined by the eigenfunctions of the Laplace‐Beltrami operator. Based on this observation, we devise an algorithm which detects and computes the isometric mappings from the shape onto itself. We show that our approach is both computationally efficient and robust with respect to small non‐isometric deformations, even if they include topological changes.     

A Condition Number for Non‐Rigid Shape Matching

Despite the large amount of work devoted in recent years to the problem of non‐rigid shape matching, practical methods that can successfully be used for arbitrary pairs of shapes remain elusive. In this paper, we study the hardness of the problem of shape matching, and introduce the notion of the shape condition number, which captures the intuition that some shapes are inherently more difficult to match against than others. In particular, we make a connection between the symmetry of a given shape and the stability of any method used to match it while optimizing a given distortion measure. We analyze two commonly used classes of methods in deformable shape matching, and show that the stability of both types of techniques can be captured by the appropriate notion of a condition number. We also provide a practical way to estimate the shape condition number and show how it can be used to guide the selection of landmark correspondences between shapes. Thus we shed some light on the reasons why general shape matching remains difficult and provide a way to detect and mitigate such difficulties in practice.     

An Operator Approach to Tangent Vector Field Processing

In this paper, we introduce a novel coordinate‐free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace‐Beltrami and symmetry operators, as well as to a straight‐forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.     

Robust Structure‐Based Shape Correspondence

We present a robust method to find region‐level correspondences between shapes, which are invariant to changes in geometry and applicable across multiple shape representations. We generate simplified shape graphs by jointly decomposing the shapes, and devise an adapted graph‐matching technique, from which we infer correspondences between shape regions. The simplified shape graphs are designed to primarily capture the overall structure of the shapes, without reflecting precise information about the geometry of each region, which enables us to find correspondences between shapes that might have significant geometric differences. Moreover, due to the special care we take to ensure the robustness of each part of our pipeline, our method can find correspondences between shapes with different representations, such as triangular meshes and point clouds. We demonstrate that the region‐wise matching that we obtain can be used to find correspondences between feature points, reveal the intrinsic self‐similarities of each shape and even construct point‐to‐point maps across shapes. Our method is both time and space efficient, leading to a pipeline that is significantly faster than comparable approaches. We demonstrate the performance of our approach through an extensive quantitative and qualitative evaluation on several benchmarks where we achieve comparable or superior performance to existing methods.     

Control of Listeria monocytogenes with combined antimicrobials on beef franks stored at 4 degrees C

Contamination of ready-to-eat meat products such as beef franks with Listeria monocytogenes has become a major concern for the meat processing industry and an important food safety issue. The objective of this study was to determine the effectiveness of combinations of antimicrobials as aqueous dipping solutions to control L. monocytogenes on vacuum-packaged beef franks stored at 4 degrees C for 3 weeks. Commercial beef franks were dipped for 5 min in three antimicrobial solutions: pediocin (6,000 AU), 3% sodium diacetate and 6% sodium lactate combined, and a combination of the three antimicrobials. Samples were then inoculated with 10(7) CFU/g of either four L. monocytogenes strains individually or a cocktail of the four strains, vacuum packaged, and stored at 4 degrees C for 3 weeks. Sampling was carried out at day 0 and after 2 and 3 weeks of storage. Individual strains, as well as the cocktail, exhibited different responses to the antimicrobial treatments. After 2 and 3 weeks of storage at 4 degrees C, pediocin-treated beef franks showed a less than 1-log reduction for all bacterial strains. Samples treated with the sodium diacetate-sodium lactate combination showed about a 1-log reduction after 2 weeks of storage for all strains and between a 1- and 2-log reduction after 3 weeks of storage, depending on the bacterial strain. When the three antimicrobials were combined, reductions ranged between 1 and 1.5 log units and 1.5 to 2.5 log units after 2 and 3 weeks of storage, respectively, at 4 degrees C. These results indicate that the use of combined antimicrobial solutions for dipping treatments is more effective at inhibiting L. monocytogenes than treatments using antimicrobials such as pediocin separately.     

Limit Shapes – A Tool for Understanding Shape Differences and Variability in 3D Model Collections

We propose a novel construction for extracting a central or limit shape in a shape collection, connected via a functional map network. Our approach is based on enriching the latent space induced by a functional map network with an additional natural metric structure. We call this shape‐like dual object the limit shape and show that its construction avoids many of the biases introduced by selecting a fixed base shape or template. We also show that shape differences between real shapes and the limit shape can be computed and characterize the unique properties of each shape in a collection – leading to a compact and rich shape representation. We demonstrate the utility of this representation in a range of shape analysis tasks, including improving functional maps in difficult situations through the mediation of limit shapes, understanding and visualizing the variability within and across different shape classes, and several others. In this way, our analysis sheds light on the missing geometric structure in previously used latent functional spaces, demonstrates how these can be addressed and finally enables a compact and meaningful shape representation useful in a variety of practical applications.     

Structured Regularization of Functional Map Computations

We consider the problem of non‐rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace‐Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well‐justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one‐parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly‐used refinement techniques.     

Adjoint Map Representation for Shape Analysis and Matching

In this paper, we propose to consider the adjoint operators of functional maps, and demonstrate their utility in several tasks in geometry processing. Unlike a functional map, which represents a correspondence simply using the pull‐back of function values, the adjoint operator reflects both the map and its distortion with respect to given inner products. We argue that this property of adjoint operators and especially their relation to the map inverse under the choice of different inner products, can be useful in applications including bi‐directional shape matching, shape exploration, and pointwise map recovery among others. In particular, in this paper, we show that the adjoint operators can be used within the cycle‐consistency framework to encode and reveal the presence or lack of consistency between distortions in a collection, in a way that is complementary to the previously used purely map‐based consistency measures. We also show how the adjoint can be used for matching pairs of shapes, by accounting for maps in both directions, can help in recovering point‐to‐point maps from their functional counterparts, and describe how it can shed light on the role of functional basis selection.     

A Survey on Data‐driven Dictionary‐based Methods for 3D Modeling

Dictionaries are very useful objects for data analysis, as they enable a compact representation of large sets of objects through the combination of atoms. Dictionary‐based techniques have also particularly benefited from the recent advances in machine learning, which has allowed for data‐driven algorithms to take advantage of the redundancy in the input dataset and discover relations between objects without human supervision or hard‐coded rules. Despite the success of dictionary‐based techniques on a wide range of tasks in geometric modeling and geometry processing, the literature is missing a principled state‐of‐the‐art of the current knowledge in this field. To fill this gap, we provide in this survey an overview of data‐driven dictionary‐based methods in geometric modeling. We structure our discussion by application domain: surface reconstruction, compression, and synthesis. Contrary to previous surveys, we place special emphasis on dictionary‐based methods suitable for 3D data synthesis, with applications in geometric modeling and design. Our ultimate goal is to enlight the fact that these techniques can be used to combine the data‐driven paradigm with design intent to synthesize new plausible objects with minimal human intervention. This is the main motivation to restrict the scope of the present survey to techniques handling point clouds and meshes, making use of dictionaries whose definition depends on the input data, and enabling shape reconstruction or synthesis through the combination of atoms.     

Improved Functional Mappings via Product Preservation

In this paper, we consider the problem of information transfer across shapes and propose an extension to the widely used functional map representation. Our main observation is that in addition to the vector space structure of the functional spaces, which has been heavily exploited in the functional map framework, the functional algebra (i.e., the ability to take pointwise products of functions) can significantly extend the power of this framework. Equipped with this observation, we show how to improve one of the key applications of functional maps, namely transferring real‐valued functions without conversion to point‐to‐point correspondences. We demonstrate through extensive experiments that by decomposing a given function into a linear combination consisting not only of basis functions but also of their pointwise products, both the representation power and the quality of the function transfer can be improved significantly. Our modification, while computationally simple, allows us to achieve higher transfer accuracy while keeping the size of the basis and the functional map fixed. We also analyze the computational complexity of optimally representing functions through linear combinations of products in a given basis and prove NP‐completeness in some general cases. Finally, we argue that the use of function products can have a wide‐reaching effect in extending the power of functional maps in a variety of applications, in particular by enabling the transfer of high‐frequency functions without changing the representation size or complexity.