Shape structure is about the arrangement and relations between shape parts. Structure-aware shape processing goes beyond local geometry and low level processing, and analyzes and processes shapes at a high level. It focuses more on the global inter and intra semantic relations among the parts of shape rather than on their local geometry. With recent developments in easy shape acquisition, access to vast repositories of 3D models, and simple-to-use desktop fabrication possibilities, the study of structure in shapes has become a central research topic in shape analysis, editing, and modeling. A whole new line of structure-aware shape processing algorithms has emerged that base their operation on an attempt to understand such structure in shapes. The algorithms broadly consist of two key phases: an analysis phase, which extracts structural information from input data; and a (smart) processing phase, which utilizes the extracted information for exploration, editing, and synthesis of novel shapes. In this survey paper, we organize, summarize, and present the key concepts and methodological approaches towards efficient structure-aware shape processing. We discuss common models of structure, their implementation in terms of mathematical formalism and algorithms, and explain the key principles in the context of a number of state-of-the-art approaches. Further, we attempt to list the key open problems and challenges, both at the technical and at the conceptual level, to make it easier for new researchers to better explore and contribute to this topic. Our goal is to both give the practitioner an overview of available structure-aware shape processing techniques, as well as identify future research questions in this important, emerging, and fascinating research area.

Figure 1: Our method automatically decomposes any mesh animations like performance captured faces (left) or muscle deformations (right) into sparse and localized deformation modes (shown in blue). Left: a new facial expression is generated by summing deformation components. Our method automatically separates spatially confined effects like separate eyebrow motions from the data. Right: Our algorithm extracts individual muscle and bone deformations. The deformation components can then be used for convenient editing of the captured animation. Here, the deformation component of the clavicle is over-exaggerated to achieve an artistically desired look. We propose a method that extracts sparse and spatially localized de-formation modes from an animated mesh sequence. To this end, we propose a new way to extend the theory of sparse matrix decom-positions to 3D mesh sequence processing, and further contribute with an automatic way to ensure spatial locality of the decompo-sition in a new optimization framework. The extracted dimensions often have an intuitive and clear interpretable meaning. Our method optionally accepts user-constraints to guide the process of discover-ing the underlying latent deformation space. The capabilities of our efficient, versatile, and easy-to-implement method are extensively demonstrated on a variety of data sets and application contexts. We demonstrate its power for user friendly intuitive editing of captured mesh animations, such as faces, full body motion, cloth animations, and muscle deformations. We further show its benefit for statisti-cal geometry processing and biomechanically meaningful anima-tion editing. It is further shown qualitatively and quantitatively that our method outperforms other unsupervised decomposition meth-ods and other animation parameterization approaches in the above use cases.

We present a computational approach for designing wire meshes, i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties of wire meshes to the geometric model of Chebyshev nets. This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design explo-ration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial ap-plications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods.

Figure 1: Subdivision of planar-quad meshes (left) and editing of polyhedral meshes with map prescription (right). We present a novel framework for polyhedral mesh editing with face-based projective maps, that preserves pla-narity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar-preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar-quadrilateral mesh subdivision, a resulting multi-resolution editing algorithm, and novel shape-space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate-preserving projective transformation fields on surfaces. Our shape space directly addresses planar-quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.

This paper builds on recent progress in computing with geometric constraints, which is particularly relevant to architectural geometry. Not only do various kinds of meshes with additional properties (like planar faces, or with equilibrium forces in their edges) become available for interactive geometric modeling, but so do other arrangements of geometric primitives, like honeycomb structures. The latter constitute an important class of geometric objects, with relations to “Lobel” meshes, and to freeform polyhedral patterns. Such patterns are particularly interesting and pose research problems which go beyond what is known for meshes, e.g. with regard to their computing, their flexibility, and the assessment of their fairness.

Shape structure is about the arrangement and relations between shape parts. Structure-aware shape processing goes beyond local geometry and low level processing, and analyzes and processes shapes at a high level. It focuses more on the global inter and intra semantic relations among the parts of shape rather than on their lo-cal geometry. With recent developments in easy shape acquisition, access to vast repositories of 3D models, and simple-to-use desktop fabrication possibilities, the study of structure in shapes has become a central research topic in shape analysis, editing, and modeling. A whole new line of structure-aware shape processing algorithms has emerged that base their operation on an attempt to understand such structure in shapes. The algorithms broadly consist of two key phases: an analysis phase, which extracts structural informa-tion from input data; and a (smart) processing phase, which utilizes the extracted information for exploration, editing, and synthesis of novel shapes. In this survey paper, we organize, summarize, and present the key concepts and methodological approaches towards efficient structure-aware shape processing. We discuss common models of structure, their implementation in terms of mathematical formalism and algorithms, and explain the key principles in the con-text of a number of state-of-the-art approaches. Further, we attempt to list the key open problems and challenges, both at the technical and at the conceptual level, to make it easier for new researchers to better explore and contribute to this topic. Our goal is to both give the practitioner an overview of available structure-aware shape processing techniques, as well as identify future research questions in this important, emerging, and fascinating research area.

Figure 1: Our framework for generating deformation with enforced metrics provides a user-friendly tool for designers to accurately control the metrics in a deformation while well preserving the shape of input models. This function is hard to be realized by constrained deformation. Progressive deformation (by gradually increasing the constrained length by 1 % in each step) cannot generate results as good as ours. Moreover, the formulation of scale-driven deformation investigated in this work can converge in a few iterations. Techniques have been developed to deform a mesh with multiple types of constraints. One limitation of prior methods is that the accuracy of demanded metrics on the resultant model cannot be guaranteed. Adding metrics directly as hard constraints to an optimization functional often leads to unexpected distortion when target metrics differ significant from what are on the input model. In this paper, we present an effective framework to deform mesh models by enforcing demanded metrics on length, area and volume. To approach target metrics stably and minimize distortion, an iterative scale-driven deformation is investigated, and a global optimization functional is exploited to balance the scaling effect at different parts of a model. Examples demonstrate that our approach provides a user-friendly tool for designers who are used to semantic input.

Figure 1: (Left) Random models generated from a probabilistic building grammar. Although these models are visually plausible, they do not comply with a design scenario which also requires architectural plausibility, i.e. matching styles of ground floors, upper floors, and roofs (B1, see Table 2). (Right) Our framework takes user specified preference scores as input and learns a new model probability density function (pdf) which samples models (with consistent style) proportionally to their predicted preference scores. In this design scenario, office buildings received a higher preference score. A shape grammar defines a procedural shape space containing a variety of models of the same class, e.g. buildings, trees, furniture, airplanes, bikes, etc. We present a framework that enables a user to interactively design a probability density function (pdf) over such a shape space and to sample models according to the designed pdf. First, we propose a user interface that enables a user to quickly provide preference scores for selected shapes and suggest sampling strategies to decide which models to present to the user to evaluate. Second, we propose a novel kernel function to encode the similarity

(a) Where should new faces be inserted? (b) How should adjacent faces be updated, keeping them planar? (c) How should edge collapses be handled? (d) Example showing all features Figure 1: There are multiple challenges when a PushPull operation is performed on a face or edge. Case (a): New faces can either be inserted for all edges (left) or not at all by adjusting adjacent faces (middle). In addition, our solution can adaptively add new faces where needed (right). New faces are blue and modified adjacent faces are green. In (b-d), the left figure is the input, the middle is the degenerate result by previous approaches, and the right is our result. Non-planar or self-intersecting faces are red and edge collapses are blue dots. PushPull tools are implemented in most commercial 3D modeling suites. Their purpose is to intuitively transform a face, edge, or ver-tex, and then to adapt the polygonal mesh locally. However, pre-vious approaches have limitations: Some allow adjustments only when adjacent faces are orthogonal; others support slanted surfaces but never create new details. Moreover, self-intersections and edge-collapses during editing are either ignored or work only partially for solid geometry. To overcome these limitations, we introduce the PushPull++ tool for rapid polygonal modeling. In our solution, we contribute novel methods for adaptive face insertion, adjacent face updates, edge collapse handling, and an intuitive user interface that automatically proposes useful drag directions. We show that PushPull++ reduces the complexity of common modeling tasks by up to an order of magnitude when compared with existing tools.

max min Figure 1: Comparison of different types of eigenfunctions of the Laplace-Beltrami operator of the input mesh (leftmost). Top Row: the manifold harmonics have global support. Bottom Row: the proposed compressed manifold modes (CMMs) have local support and are confined to specific local features like protrusions and ridges. Here, 8 of the CMMs were found for the 8 protrusions at the corners (1 to 8, only 2 shown here), 6 concentrate at each of the dents (2 shown here), and 12 CMMs automatically form at the valleys between the protrusions. This paper introduces compressed eigenfunctions of the Laplace-Beltrami operator on 3D manifold surfaces. They constitute a novel functional basis, called the compressed manifold basis, where each function has local support. We derive an algorithm, based on the alternating direction method of multipliers (ADMM), to compute this basis on a given triangulated mesh. We show that compressed manifold modes identify key shape features, yielding an intuitive understanding of the basis for a human observer, where a shape can be processed as a collection of parts. We evaluate compressed manifold modes for potential applications in in shape matching and mesh abstraction. Our results show that this basis has distinct advantages over existing alternatives, indicating high potential for a wide range of use-cases in mesh processing.