Shape Matching via Quotient Spaces

We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state‐of‐the‐art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated.     

Approximate Symmetry Detection in Partial 3D Meshes

Symmetry is a common characteristic in natural and man‐made objects. Its ubiquitous nature can be exploited to facilitate the analysis and processing of computational representations of real objects. In particular, in computer graphics, the detection of symmetries in 3D geometry has enabled a number of applications in modeling and reconstruction. However, the problem of symmetry detection in incomplete geometry remains a challenging task. In this paper, we propose a vote‐based approach to detect symmetry in 3D shapes, with special interest in models with large missing parts. Our algorithm generates a set of candidate symmetries by matching local maxima of a surface function based on the heat diffusion in local domains, which guarantee robustness to missing data. In order to deal with local perturbations, we propose a multi‐scale surface function that is useful to select a set of distinctive points over which the approximate symmetries are defined. In addition, we introduce a vote‐based scheme that is aware of the partiality, and therefore reduces the number of false positive votes for the candidate symmetries. We show the effectiveness of our method in a varied set of 3D shapes and different levels of partiality. Furthermore, we show the applicability of our algorithm in the repair and completion of challenging reassembled objects in the context of cultural heritage.     

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.     

Symmetry Measure Computation for Convex Polyhedra

The paper discusses measures and indices of different kind of symmetry (central, reflection, rotation, mirror rotation) for 3D convex shapes. The measures are based on Minkowski addition and inequalities for volume and mixed volume and are valid for convex shapes only. Symmetry index computation in 3D case is a complicated optimization problem. Taking advantage of convex polyhedra we investigate the situations when these indices can be computed efficiently.     

Convex Set Symmetry Measurement via Minkowski Addition

We introduce and investigate measures of rotation and reflectionsymmetries for compact convex sets. The appropriate symmetrization transformations are used to transform original sets into symmetrical ones. Symmetry measures are defined as the ratio of volumes (Lebesgue measure) of the original set and the corresponding symmetrical set.For the case of rotation symmetry we use as a symmetrizationtransformation a generalization of the Minkowski symmetric set (a difference body) for a cyclic group of rotations.For the case of reflection symmetry we investigate Blaschke symmetrization. Given a convex set and a hyperplane E in we get a set symmetrical with respect to this hyperplane. Analyzing all hyperplanes containing the coordinate center one gets the measure of reflection symmetry. We discuss the lower bounds of this symmetry measure and also a derived symmetry measure.In the two dimensional case a perimetric measure representation of convexsets is applied for convex sets symmetrization as well as for the symmetry measure calculation. The perimetric measure allows also to perform a decomposition of a compact convex set into Minkowski sum of two sets. The first one is rotationally symmetrical and the second one is completely asymmetrical in the sense that it does not allow such a decomposition.We discuss a problem of the fast computation of symmetrization transformations. Minkowski addition of two sets is reduced to the convolution of their characteristic functions. Therefore, in the case of the discrete plane , Fast Fourier Transformation can be applied for the fast computation of symmetrization transformations.     

Symmetry in 3D Geometry: Extraction and Applications

The concept of symmetry has received significant attention in computer graphics and computer vision research in recent years. Numerous methods have been proposed to find, extract, encode and exploit geometric symmetries and high‐level structural information for a wide variety of geometry processing tasks. This report surveys and classifies recent developments in symmetry detection. We focus on elucidating the key similarities and differences between existing methods to gain a better understanding of a fundamental problem in digital geometry processing and shape understanding in general. We discuss a variety of applications in computer graphics and geometry processing that benefit from symmetry information for more effective processing. An analysis of the strengths and limitations of existing algorithms highlights the plenitude of opportunities for future research both in terms of theory and applications.     

Object Completion using k‐Sparse Optimization

We present a new method for the completion of partial globally‐symmetric 3D objects, based on the detection of partial and approximate symmetries in the incomplete input dataset. In our approach, symmetry detection is formulated as a constrained sparsity maximization problem, which is solved efficiently using a robust RANSAC‐based optimizer. The detected partial symmetries are then reused iteratively, in order to complete the missing parts of the object. A global error relaxation method minimizes the accumulated alignment errors and a non‐rigid registration approach applies local deformations in order to properly handle approximate symmetry. Unlike previous approaches, our method does not rely on the computation of features, it uniformly handles translational, rotational and reflectional symmetries and can provide plausible object completion results, even on challenging cases, where more than half of the target object is missing. We demonstrate our algorithm in the completion of 3D scans with varying levels of partiality and we show the applicability of our approach in the repair and completion of heavily eroded or incomplete cultural heritage objects.     

Skeleton-based intrinsic symmetry detection on point clouds

We present a skeleton-based algorithm for intrinsic symmetry detection on imperfect 3D point cloud data. The data imperfections such as noise and incompleteness make it difficult to reliably compute geodesic distances, which play essential roles in existing intrinsic symmetry detection algorithms. In this paper, we leverage recent advances in curve skeleton extraction from point clouds for symmetry detection. Our method exploits the properties of curve skeletons, such as homotopy to the input shape, approximate isometry-invariance, and skeleton-to-surface mapping, for the detection task. Starting from a curve skeleton extracted from an input point cloud, we first compute symmetry electors, each of which is composed of a set of skeleton node pairs pruned with a cascade of symmetry filters. The electors are used to vote for symmetric node pairs indicating the symmetry map on the skeleton. A symmetry correspondence matrix (SCM) is constructed for the input point cloud through transferring the symmetry map from skeleton to point cloud. The final symmetry regions on the point cloud are detected via spectral analysis over the SCM. Experiments on raw point clouds, captured by a 3D scanner or the Microsoft Kinect, demonstrate the robustness of our algorithm. We also apply our method to repair incomplete scans based on the detected intrinsic symmetries.     

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.     

Nonrigid Matching of Undersampled Shapes via Medial Diffusion

We introduce medial diffusion for the matching of undersampled shapes undergoing a nonrigid deformation. We construct a diffusion process with respect to the medial axis of a shape, and use the quantity of heat diffusion as a measure which is both tolerant of missing data and approximately invariant to nonrigid deformations. A notable aspect of our approach is that we do not define the diffusion on the shape's medial axis, or similar medial representation. Instead, we construct the diffusion process directly on the shape. This permits the diffusion process to better capture surface features, such as varying spherical and cylindrical parts, as well as combine with other surface‐based diffusion processes. We show how to use medial diffusion to detect intrinsic symmetries, and for computing correspondences between pairs of shapes, wherein shapes contain substantial missing data.     

Feature Curve Co‐Completion in Noisy Data

Feature curves on 3D shapes provide important hints about significant parts of the geometry and reveal their underlying structure. However, when we process real world data, automatically detected feature curves are affected by measurement uncertainty, missing data, and sampling resolution, leading to noisy, fragmented, and incomplete feature curve networks. These artifacts make further processing unreliable. In this paper we analyze the global co‐occurrence information in noisy feature curve networks to fill in missing data and suppress weakly supported feature curves. For this we propose an unsupervised approach to find meaningful structure within the incomplete data by detecting multiple occurrences of feature curve configurations (co‐occurrence analysis). We cluster and merge these into feature curve templates, which we leverage to identify strongly supported feature curve segments as well as to complete missing data in the feature curve network. In the presence of significant noise, previous approaches had to resort to user input, while our method performs fully automatic feature curve co‐completion. Finding feature reoccurrences however, is challenging since naïve feature curve comparison fails in this setting due to fragmentation and partial overlaps of curve segments. To tackle this problem we propose a robust method for partial curve matching. This provides us with the means to apply symmetry detection methods to identify co‐occurring configurations. Finally, Bayesian model selection enables us to detect and group re‐occurrences that describe the data well and with low redundancy.     

Fast and Exact (Poisson) Solvers on Symmetric Geometries

In computer graphics, numerous geometry processing applications reduce to the solution of a Poisson equation. When considering geometries with symmetry, a natural question to consider is whether and how the symmetry can be leveraged to derive an efficient solver for the underlying system of linear equations. In this work we provide a simple representation‐theoretic analysis that demonstrates how symmetries of the geometry translate into block diagonalization of the linear operators and we show how this results in efficient linear solvers for surfaces of revolution with and without angular boundaries.     

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

We construct a family of barycentric coordinates for 2D shapes including non‐convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater‐Hormann‐Kós family of barycentric coordinates for 2D convex shapes. We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co‐dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well‐defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.     

Parallel Globally Consistent Normal Orientation of Raw Unorganized Point Clouds

A mandatory component for many point set algorithms is the availability of consistently oriented vertex‐normals (e.g. for surface reconstruction, feature detection, visualization). Previous orientation methods on meshes or raw point clouds do not consider a global context, are often based on unrealistic assumptions, or have extremely long computation times, making them unusable on real‐world data. We present a novel massively parallelized method to compute globally consistent oriented point normals for raw and unsorted point clouds. Built on the idea of graph‐based energy optimization, we create a complete kNN‐graph over the entire point cloud. A new weighted similarity criterion encodes the graph‐energy. To orient normals in a globally consistent way we perform a highly parallel greedy edge collapse, which merges similar parts of the graph and orients them consistently. We compare our method to current state‐of‐the‐art approaches and achieve speedups of up to two orders of magnitude. The achieved quality of normal orientation is on par or better than existing solutions, especially for real‐world noisy 3D scanned data.     

Equi-affine Invariant Geometry for Shape Analysis

Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it. These geometries are typically defined using either the shortest path length (geodesic distance), or properties of heat diffusion (diffusion distance) on the surface. Both measures are implicitly derived from the metric induced by the ambient Euclidean space. In this paper, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We apply equi-affine geometry for analyzing arbitrary shapes with positive Gaussian curvature. The potential of the proposed framework is explored in a range of applications such as shape matching and retrieval, symmetry detection, and computation of Voroni tessellation. We show that in some shape analysis tasks, equi-affine-invariant intrinsic geometries often outperform their Euclidean-based counterparts. We further explore the potential of this metric in facial anthropometry of newborns. We show that intrinsic properties of this homogeneous group are better captured using the equi-affine metric.     

Skeleton‐Intrinsic Symmetrization of Shapes

Enhancing the self‐symmetry of a shape is of fundamental aesthetic virtue. In this paper, we are interested in recovering the aesthetics of intrinsic reflection symmetries, where an asymmetric shape is symmetrized while keeping its general pose and perceived dynamics. The key challenge to intrinsic symmetrization is that the input shape has only approximate reflection symmetries, possibly far from perfect. The main premise of our work is that curve skeletons provide a concise and effective shape abstraction for analyzing approximate intrinsic symmetries as well as symmetrization. By measuring intrinsic distances over a curve skeleton for symmetry analysis, symmetrizing the skeleton, and then propagating the symmetrization from skeleton to shape, our approach to shape symmetrization is skeleton‐intrinsic. Specifically, given an input shape and an extracted curve skeleton, we introduce the notion of a backbone as the path in the skeleton graph about which a self‐matching of the input shape is optimal. We define an objective function for the reflective self‐matching and develop an algorithm based on genetic programming to solve the global search problem for the backbone. The extracted backbone then guides the symmetrization of the skeleton, which in turn, guides the symmetrization of the whole shape. We show numerous intrinsic symmetrization results of hand drawn sketches and artist‐modeled or reconstructed 3D shapes, as well as several applications of skeleton‐intrinsic symmetrization of shapes.     

Data‐Driven Sparse Priors of 3D Shapes

We present a sparse optimization framework for extracting sparse shape priors from a collection of 3D models. Shape priors are defined as point‐set neighborhoods sampled from shape surfaces which convey important information encompassing normals and local shape characterization. A 3D shape model can be considered to be formed with a set of 3D local shape priors, while most of them are likely to have similar geometry. Our key observation is that the local priors extracted from a family of 3D shapes lie in a very low‐dimensional manifold. Consequently, a compact and informative subset of priors can be learned to efficiently encode all shapes of the same family. A comprehensive library of local shape priors is first built with the given collection of 3D models of the same family. We then formulate a global, sparse optimization problem which enforces selecting representative priors while minimizing the reconstruction error. To solve the optimization problem, we design an efficient solver based on the Augmented Lagrangian Multipliers method (ALM). Extensive experiments exhibit the power of our data‐driven sparse priors in elegantly solving several high‐level shape analysis applications and geometry processing tasks, such as shape retrieval, style analysis and symmetry detection.