Alla Sheffer

Cross-parameterization and compatible remeshing of 3D models

Many geometry processing applications, such as morphing, shape blending, transfer of texture or material properties, and fitting template meshes to scan data, require a bijective mapping between two or more models. This mapping, or cross-parameterization, typically needs to preserve the shape and features of the parameterized models, mapping legs to legs, ears to ears, and so on. Most of the applications also require the models to be represented by compatible meshes, i.e. meshes with identical connectivity, based on the cross-parameterization. In this paper we introduce novel methods for shape preserving cross-parameterization and compatible remeshing. Our cross-parameterization method computes a low-distortion bijective mapping between models that satisfies user prescribed constraints. Using this mapping, the remeshing algorithm preserves the user-defined feature vertex correspondence and the shape correlation between the models. The remeshing algorithm generates output meshes with significantly fewer elements compared to previous techniques, while accurately approximating the input geometry. As demonstrated by the examples, the compatible meshes we construct are ideally suitable for morphing and other geometry processing applications.

Periodic global parameterization

We present a new globally smooth parameterization method for the triangulated surfaces of arbitrary topology. Given two orthogonal piecewise linear vector fields defined over the input mesh (typically the estimated principal curvature directions), our method computes two piecewise linear periodic functions, aligned with the input vector fields, by minimizing an objective function. The bivariate function they define is a smooth parameterization almost everywhere on the surface, except in the vicinity of singular vertices, edges, and triangles, where the derivatives of the parameterization vanish. We extract a quadrilateral chart layout from the parameterization function and propose an automatic procedure to detect the singularities, and fix them by splitting and reparameterizing the containing charts. Our method can construct both quasiconformal (angle preserving) and quasi-isometric (angle and area preserving) parameterizations. The more restrictive class of quasi-isometric parameterizations is constructed at the expense of introducing more singularities. The constructed parameterizations can be used for a variety of geometry processing applications. Since we can align the parameterization with the principal curvature directions, our result is particularly suitable for surface fitting and remeshing.

ABF++: fast and robust angle based flattening

Conformal parameterization of mesh models has numerous applicationsin geometry processing. Conformality is desirable for remeshing,surface reconstruction, and many other mesh processingapplications. Subject to the conformality requirement, theseapplications typically benefit from parameterizations with smallerstretch. The Angle Based Flattening (ABF) method, presented a fewyears ago, generates provably valid conformal parameterizationswith low stretch. However, it is quite time-consuming and becomeserror prone for large meshes due to numerical error accumulation.This work presents ABF++, a highly efficientextension of the ABF method, that overcomes these drawbacks whilemaintaining all the advantages of ABF. ABF++ robustly parameterizesmeshes of hundreds of thousands and millions of triangles withinminutes. It is based on three main components: (1) a new numericalsolution technique that dramatically reduces the dimension of thelinear systems solved at each iteration, speeding up the solution;(2) a new robust scheme for reconstructing the 2D coordinates fromthe angle space solution that avoids the numerical instabilitieswhich hindered the ABF reconstruction scheme; and (3) an efficienthierarchical solution technique. The speedup with (1) does not comeat the expense of greater distortion. The hierarchical technique(3) enables parameterization of models with millions of faces inseconds at the expense of a minor increase in parametricdistortion. The parameterization computed by ABF++ are provablyvalid, that is they contain no flipped triangles. As a result ofthese extensions, the ABF++ method is extremely suitable forrobustly and efficiently parameterizing models forgeometry-processing applications.

Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening

Abstract.We propose a new method to compute planar triangulations of faceted surfaces for surface parameterization. In contrast to previous approaches that define the flattening problem as a mapping of the three-dimensional node locations to the plane, our method defines the flattening problem as a constrained optimization problem in terms of angles (only). After applying a scaling that derives from the ‘curvature’ at a node, we minimize the relative deformation of the angles in the plane with respect to their counterparts in the three-dimensional surface. This approach makes the method more stable and robust than previous approaches, which used node locations in their formulations. The new method can handle any manifold surface for which a connected, valid, two-dimensional parameterization exists, including surfaces with large curvature gradients. It does not require the boundary of the flat two-dimensional domain to be prede-fined or convex. We use only the necessary and sufficient constraints for a valid two-dimensional triangulation. As a result, the existence of a theoretical solution to the minimization procedure is guaranteed.

D-Charts: Quasi-Developable Mesh Segmentation

Quasi-developable mesh segmentation is required for many applications in graphics and CAD, including texture atlas generation and the design of patterns for model fabrication from sheets of material. In this work we introduce D-Charts, a simple and robust algorithm for mesh segmentation into (nearly) developable charts. As part of our method we introduce a new metric of developability for mesh surfaces. Thanks to this metric, using our segmentation for texture atlas generation, we can bound the distortion of the atlas directly during the segmentation stage. We demonstrate that by using this bound, we generate more isometric atlases for the same number of charts compared to existing state-of-the-art techniques. Using our segmentation algorithm we also develop a technique for automatic pattern design. To demonstrate the practicality of this technique, we use the patterns produced by our algorithm to make fabric and paper copies of popular computer graphics models.

Deformation-driven shape correspondence

Non‐rigid 3D shape correspondence is a fundamental and difficult problem. Most applications which require a correspondence rely on manually selected markers. Without user assistance, the performances of existing automatic correspondence methods depend strongly on a good initial shape alignment or shape prior, and they generally do not tolerate large shape variations. We present an automatic feature correspondence algorithm capable of handling large, non‐rigid shape variations, as well as partial matching. This is made possible by leveraging the power of state‐of‐the‐art mesh deformation techniques and relying on a combinatorial tree traversal for correspondence search. The search is deformation‐driven, prioritized by a self‐distortion energy measured on meshes deformed according to a given correspondence. We demonstrate the ability of our approach to naturally match shapes which differ in pose, local scale, part decomposition, and geometric detail through numerous examples.

Virtual Garments: A Fully Geometric Approach for Clothing Design

Modeling dressed characters is known as a very tedious process. It usually requires specifying 2D fabric patterns, positioning and assembling the min 3D, and then performing a physically‐based simulation. The latter accounts for gravity and collisions to compute the rest shape of the garment, with the adequate folds and wrinkles.

Markerless garment capture

A lot of research has recently focused on the problem of capturing the geometry and motion of garments. Such work usually relies on special markers printed on the fabric to establish temporally coherent correspondences between points on the garments surface at different times. Unfortunately, this approach is tedious and prevents the capture of off-the-shelf clothing made from interesting fabrics. In this paper, we describe a marker-free approach to capturing garment motion that avoids these downsides. We establish temporally coherent parameterizations between incomplete geometries that we extract at each timestep with a multiview stereo algorithm. We then fill holes in the geometry using a template. This approach, for the first time, allows us to capture the geometry and motion of unpatterned, off-the-shelf garments made from a range of different fabrics.

Upright orientation of man-made objects

Humans usually associate an upright orientation with objects, placing them in a way that they are most commonly seen in our surroundings. While it is an open challenge to recover the functionality of a shape from its geometry alone, this paper shows that it is often possible to infer its upright orientation by analyzing its geometry. Our key idea is to reduce the two-dimensional (spherical) orientation space to a small set of orientation candidates using functionality-related geometric properties of the object, and then determine the best orientation using an assessment function of several functional geometric attributes defined with respect to each candidate. Specifically we focus on obtaining the upright orientation for man-made objects that typically stand on some flat surface (ground, floor, table, etc.), which include the vast majority of objects in our everyday surroundings. For these types of models orientation candidates can be defined according to static equilibrium. For each candidate, we introduce a set of discriminative attributes linking shape to function. We learn an assessment function of these attributes from a training set using a combination of Random Forest classifier and Support Vector Machine classifier. Experiments demonstrate that our method generalizes well and achieves about 90% prediction accuracy for both a 10-fold cross-validation over the training set and a validation with an independent test set.

Pyramid coordinates for morphing and deformation

Many model editing operations, such as morphing, blending, and shape deformation requires the ability to interactively transform the surface of a model in response to some control mechanism. For most computer graphics applications, it is important to preserve the local shape properties of input models during editing operations. We introduce the mesh editing technique that explicitly preserves local shape properties. The method is based on a local shape representation, which we refer to as pyramid coordinates. The pyramid coordinates capture the local shape of the mesh around each vertex and help maintain this shape under various editing operations. They are based on a set of angles and lengths relating a vertex to its immediate neighbors. This representation is invariant under rigid transformations. Using pyramid coordinates, we introduce A technique for mesh deformation and morphing based on a small number of user-specified control vertices. Our algorithm generates natural looking deformations and morphing sequences in seconds with minimal user interaction.