Apply geometric duality to energy-efficient non-local phenomenon awareness using sensor networks

A powerful concept to cope with resource limitations and information redundancy in wireless sensor networks is the use of collaboration groups to distill information within the network and suppress unnecessary activities. When the phenomena to be monitored have large geographical extents, it is not obvious how to define these collaboration groups. This article presents the application of geometric duality to form such groups for sensor selection and non-local phenomena tracking. Using a dual-space transformation, which maps a non-local phenomenon (e.g., the edge of a half-plane shadow) to a single point in the dual space and maps locations of distributed sensor nodes to a set of lines that partitions the dual space, one can turn off the majority of the sensors to achieve resource preservation without losing detection and tracking accuracy. Since the group so defined may consist of nodes that are far away in physical space, we propose a hierarchical architecture that uses a small number of computationally powerful nodes and a massive number of power constrained motes. By taking advantage of the continuity of physical phenomena and the duality principle, we can greatly reduce the power consumption in non-local phenomena tracking and extend the lifetime of the network.     

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.     

Discovery of Intrinsic Primitives on Triangle Meshes

The discovery of meaningful parts of a shape is required for many geometry processing applications, such as parameterization, shape correspondence, and animation. It is natural to consider primitives such as spheres, cylinders and cones as the building blocks of shapes, and thus to discover parts by fitting such primitives to a given surface. This approach, however, will break down if primitive parts have undergone almost‐isometric deformations, as is the case, for example, for articulated human models. We suggest that parts can be discovered instead by finding intrinsic primitives, which we define as parts that posses an approximate intrinsic symmetry. We employ the recently‐developed method of computing discrete approximate Killing vector fields (AKVFs) to discover intrinsic primitives by investigating the relationship between the AKVFs of a composite object and the AKVFs of its parts. We show how to leverage this relationship with a standard clustering method to extract k intrinsic primitives and remaining asymmetric parts of a shape for a given k. We demonstrate the value of this approach for identifying the prominent symmetry generators of the parts of a given shape. Additionally, we show how our method can be modified slightly to segment an entire surface without marking asymmetric connecting regions and compare this approach to state‐of‐the‐art methods using the Princeton Segmentation Benchmark.     

As‐Killing‐As‐Possible Vector Fields for Planar Deformation

Cartoon animation, image warping, and several other tasks in two‐dimensional computer graphics reduce to the formulation of a reasonable model for planar deformation. A deformation is a map from a given shape to a new one, and its quality is determined by the type of distortion it introduces. In many applications, a desirable map is as isometric as possible. Finding such deformations, however, is a nonlinear problem, and most of the existing solutions approach it by minimizing a nonlinear energy. Such methods are not guaranteed to converge to a global optimum and often suffer from robustness issues. We propose a new approach based on approximate Killing vector fields (AKVFs), first introduced in shape processing. AKVFs generate near‐isometric deformations, which can be motivated as direction fields minimizing an “as‐rigid‐as‐possible” (ARAP) energy to first order. We first solve for an AKVF on the domain given user constraints via a linear optimization problem and then use this AKVF as the initial velocity field of the deformation. In this way, we transfer the inherent nonlinearity of the deformation problem to finding trajectories for each point of the domain having the given initial velocities. We show that a specific class of trajectories — the set of logarithmic spirals — is especially suited for this task both in practice and through its relationship to linear holomorphic vector fields. We demonstrate the effectiveness of our method for planar deformation by comparing it with existing state‐of‐the‐art deformation methods.     

Shape Decomposition using Modal Analysis

We introduce a novel algorithm that decomposes a deformable shape into meaningful parts requiring only a single input pose. Using modal analysis, we are able to identify parts of the shape that tend to move rigidly. We define a deformation energy on the shape, enabling modal analysis to find the typical deformations of the shape. We then find a decomposition of the shape such that the typical deformations can be well approximated with deformation fields that are rigid in each part of the decomposition. We optimize for the best decomposition, which captures how the shape deforms. A hierarchical refinement scheme makes it possible to compute more detailed decompositions for some parts of the shape.
Although our algorithm does not require user intervention, it is possible to control the process by directly changing the deformation energy, or interactively refining the decomposition as necessary. Due to the construction of the energy function and the properties of modal analysis, the computed decompositions are robust to changes in pose as well as meshing, noise, and even imperfections such as small holes in the surface.     

3DPointCaps++: Learning 3D Representations with Capsule Networks

International Journal of Computer Vision - We present 3DPointCaps++ for learning robust, flexible and generalizable 3D object representations without requiring heavy annotation efforts or...     

Learning Fuzzy Set Representations of Partial Shapes on Dual Embedding Spaces

Modeling relations between components of 3D objects is essential for many geometry editing tasks. Existing techniques commonly rely on labeled components, which requires substantial annotation effort and limits components to a dictionary of predefined semantic parts. We propose a novel framework based on neural networks that analyzes an uncurated collection of 3D models from the same category and learns two important types of semantic relations among full and partial shapes: complementarity and interchangeability. The former helps to identify which two partial shapes make a complete plausible object, and the latter indicates that interchanging two partial shapes from different objects preserves the object plausibility. Our key idea is to jointly encode both relations by embedding partial shapes as fuzzy sets in dual embedding spaces. We model these two relations as fuzzy set operations performed across the dual embedding spaces, and within each space, respectively. We demonstrate the utility of our method for various retrieval tasks that are commonly needed in geometric modeling interfaces.     

QuadriFlow: A Scalable and Robust Method for Quadrangulation

QuadriFlow is a scalable algorithm for generating quadrilateral surface meshes based on the Instant Field‐Aligned Meshes of Jakob et al. (ACM Trans. Graph. 34(6):189, 2015). We modify the original algorithm such that it efficiently produces meshes with many fewer singularities. Singularities in quadrilateral meshes cause problems for many applications, including parametrization and rendering with Catmull‐Clark subdivision surfaces. Singularities can rarely be entirely eliminated, but it is possible to keep their number small. Local optimization algorithms usually produce meshes with many singularities, whereas the best algorithms tend to require non‐local optimization, and therefore are slow. We propose an efficient method to minimize singularities by combining the Instant Meshes objective with a system of linear and quadratic constraints. These constraints are enforced by solving a global minimum‐cost network flow problem and local boolean satisfiability problems. We have verified the robustness and efficiency of our method on a subset of ShapeNet comprising 17,791 3D objects in the wild. Our evaluation shows that the quality of the quadrangulations generated by our method is as good as, if not better than, those from other methods, achieving about four times fewer singularities than Instant Meshes. Other algorithms that produce similarly few singularities are much slower; we take less than ten seconds to process each model. Our source code is publicly available.