Freeform surfaces are popular in architectural design for their expressive aesthetics, yet they remain challenging to fabricate. A common strategy is to approximate them with assemblies of planar polygonal panels in a process we term panelization, enabling cost-effective fabrication from sheet materials such as wood, metal, or glass. Our goal is to compute panelizations that are both geometrically faithful and structurally robust. A two-stage pipeline, i.e. geometric panelization followed by structural optimization, can be effective, but joint optimization may yield better outcomes by letting mechanics influence the panelization itself. The challenges are that the panelization task is inherently combinatorial, and structural objectives depend on stresses that have a complex, global dependence on the geometry and are costly to evaluate. Moreover, improving structural performance must be balanced against preserving the input surface's design intent. To address these challenges, we formulate panelization as a continuous surface flow coupled to stress analysis, by modeling candidates as creased elastic shells. We define a global objective by integrating an arbitrary function of stress over the shell and derive efficient gradients with respect to the surface shape to enable optimization. We further introduce flow terms that better preserve the input shape and show that running BFGS in a Sobolev space substantially outperforms standard BFGS and Sobolev gradient descent. We evaluate our method on architectural models, showing that incorporating stress during panelization yields substantial stress reductions and more robust designs compared to optimizing stress after fixing the panelization.

We introduce a high-performance framework for solving nonconvex optimization problems arising in simulation and geometry processing applications. Our framework especially benefits high-dimensional problems whose unknowns are spatial coordinates in ℝ d and whose objectives are expressed as a sum of element energies over small stencils. The Hessians of these problems have a block-sparse structure, where the nonzero entries are grouped into dense d × d blocks. Many works have exploited this structure to speed up the Hessian-vector products and nodal smoothers of iterative solvers. In contrast, accelerating direct sparse linear solvers, preferred for applications requiring high accuracy on unstructured grids, has been less explored. Our framework leverages block structure to accelerate all phases of a Newton-type minimization algorithm, from sparsity pattern construction to system assembly, matrix factorization, and solves. A fundamental piece of the framework is BlockCatamari, our highly tuned, block-accelerated multifrontal sparse Cholesky code that achieves significantly faster factorizations than existing direct solvers on shared-memory systems. We combine this solver with our fast parallel Hessian assembly routine, heuristics for conditionally enabling per-element Hessian projections, and strategies for minimizing symbolic factorization re-computations. We show that this leads to dramatic speedups on a large suite of benchmark problems involving injective surface parametrization and elasticity simulation with contact, and we further demonstrate the ease with which new energies can be defined at the different levels of abstraction offered by our framework.

A stretch sensor is a device that attaches to objects and measures the amount by which they deform. These sensors have shown great promise as an alternative to vision-based motion-capture systems, and for robotic sensing. Currently, they are generally limited to linear designs, and require a somewhat challenging calibration process. Our goal is to enable inverse design of such sensors, and to largely eliminate the calibration process. To this end, we introduce an accurate, differentiable simulator for capacitive stretch sensors, that treats both the elasto- and electro -static parts of the system. Differentiability allows optimizing the geometry of the sensor in order to improve its design for specific applications. We demonstrate the accuracy of our simulator and the effectiveness of our sensor optimization process for various use cases, such as human interfaces and robotics.

Collision-avoiding motion planning for articulated robotic arms is one of the major challenges in robotics. The difficulty of the problem arises from its high dimensionality and the intricate geometry of the feasible space. Our goal is to seek large convex domains in configuration space, which contain no obstacles. In these domains, simple linear trajectories are guaranteed to be collision free, and can be leveraged for further optimization. To find such domains, practitioners have harnessed a methodology known as Sum-Of-Squares (SOS) Programming. SOS programs, however, are notorious for their poor scaling properties, which makes it challenging to employ them for complex problems. In this paper, we explore a simple formulation for a two-dimensional arm, which results in smaller SOS programs than previous suggested ones. We show that this formulation can express a variety of scenarios in a unified manner.

Mobile robots are becoming safer and more affordable, and their presence in the workspace is increasing. However, many tasks that involve reasoning, long-term planning or human preferences are still hard to automate. While some solutions in specialised areas slowly emerge, an alternative to full autonomy can be to actively leverage intuition and experience of human operators. To do this, suitable interfaces and modes of interaction have to be explored. Inspired by Real-Time Strategy games, we implement a Mixed Reality interface that can be used with either a Microsoft HoloLens 2 headset or a tablet. The interface allows users to interact with multiple mobile robots simultaneously. We conduct a user study to compare the headset and tablet versions of the interface in different scenarios inspired by a real-world construction setting. We show that, while performance and preference of interface are dependent on the task and the complexity of the required interaction, users are able to solve non-trivial tasks on both platforms using our system.

Task and motion planning is one of the key problems in robotics today. It is often formulated as a discrete task allocation problem combined with continuous motion planning. Many existing approaches to TAMP involve explicit descriptions of task primitives that cause discrete changes in the kinematic relationship between the actor and the objects. In this work we propose an alternative, fully differentiable approach which supports a large number of TAMP problem instances. Rather than explicitly enumerating task primitives, actions are instead represented implicitly as part of the solution to a nonlinear optimization problem. We focus on decision making for robotic manipulators, specifically for pick and place tasks, and explore the efficacy of the model through a number of simulated experiments including multiple robots, objects and interactions with the environment. We also show several possible extensions.

Many flexible structures are characterized by a small number of compliant modes , i.e., large-deformation paths that can be traversed with little mechanical effort, whereas resistance to other deformations is much stiffer. Predicting the compliant modes for a given flexible structure, however, is challenging. While linear eigenmodes capture the small-deformation behavior, they quickly divert into states of unrealistically high energy for larger displacements. Moreover, they are inherently unable to predict nonlinear phenomena such as buckling, stiffening, multistability, and contact. To address this limitation, we propose Nonlinear Compliant Modes —a physically principled extension of linear eigenmodes for large-deformation analysis. Instead of constraining the entire structure to deform along a given eigenmode, our method only prescribes the projection of the system’s state onto the linear mode while all other degrees of freedom follow through energy minimization. We evaluate the potential of our method on a diverse set of flexible structures, ranging from compliant mechanisms to topology-optimized joints and structured materials. As validated through experiments on physical prototypes, our method correctly predicts a broad range of nonlinear effects that linear eigenanalysis fails to capture.

Multi-Robot Systems (MRS) present many advantages over single robots, e.g. improved stability and payload capacity. Being able to operate or teleoperate these systems is therefore of high interest in industries such as construction or logistics. However, controlling the collective motion of a MRS can place a significant cognitive burden on the operator. We present a Mixed Reality (MR) control interface, which allows an operator to specify payload target poses for a MRS in real-time, while effectively keeping the system away from unfavorable configurations. To this end, we solve the inverse kinematics problem for each arm individually and leverage redundant degrees of freedom to optimize for a secondary objective. Using the manipulability index as a secondary objective in particular, allows us to significantly improve the tracking and singularity avoidance capabilities of our MRS in comparison to the unoptimized scenario. This enables more secure and intuitive teleoperation. We simulate and test our approach on different setups and over different input trajectories, and analyse the convergence properties of our method. Finally, we show that the method also works well when deployed on to a dual-arm ABB YuMi robot.

Many path planning algorithms are based on sampling the state space. While this approach is very simple, it can become costly when the obstacles are unknown, since samples hitting these obstacles are wasted. The goal of this paper is to efficiently identify obstacles in a map and remove them from the sampling space. To this end, we propose a pre-processing algorithm for space exploration that enables more efficient sampling. We show that it can boost the performance of other space sampling methods and path planners. Our approach is based on the fact that a convex obstacle can be approximated provably well by its minimum volume enclosing ellipsoid (MVEE), and a non-convex obstacle may be partitioned into convex shapes. Our main contribution is an al-gorithm that strategically finds a small sample, called the active-coreset, that adaptively samples the space via membership-oracle such that the MVEE of the coreset approximates the MVEE of the obstacle. Experimental results confirm the ef-fectiveness of our approach across multiple planners based on rapidly-exploring random trees, showing significant improve-ment in terms of time and path length.

A central aspect of robotic motion planning is collision avoidance, where a multitude of different approaches are currently in use. Optimization-based motion planning is one method, that often heavily relies on distance computations between robots and obstacles. These computations can easily become a bottleneck, as they do not scale well with the complexity of the robots or the environment. To improve performance, many different methods suggested to use collision primitives, i.e. simple shapes that approximate the more complex rigid bodies, and that are simpler to compute distances to and from. However, each pair of primitives requires its own specialized code, and certain pairs are known to suffer from numerical issues. In this paper, we propose an easy-to-use, unified treatment of a wide variety of primitives. We formulate distance computation as a minimization problem, which we solve iteratively. We show how to take derivatives of this minimization problem, allowing it to be seamlessly integrated into a trajectory optimization method. We demonstrate that the resulting method can be used to plan smooth and collision-free paths based on a variety of single- and multi-robot scenarios with different obstacles.

Design rationalization is the process of simplifying a 3D shape to enable cost-efficient manufacturing. A common approach is to approximate the input shape by a collection of simple units, such as flat or spherical panels, that are easy to manufacture and simple to assemble. This panelization process typically involves a segmentation step, with each surface patch intended to be replaced by a single unit, followed by an approximation stage, where the final shapes and locations of the units are determined. While optimal panel parameters for given segments are readily determined, the discrete nature of segmentation—assigning surface elements to segments—prevents a continuous design optimization workflow. In this work, we propose a differentiable reformulation of panelization that enables its use in gradient-based design optimization. Our approach is to treat panelization as a smooth optimization problem, whose objective function encourages the surface to locally deform towards best-matching units. This formulation enables a fully-differentiable rationalization process with implicit segmentation in which panels emerge automatically. We integrate our rationalization process in a simple user interface allowing the designer to guide the optimization towards desired panelizations. We demonstrate the potential of our approach on a diverse set of complex shapes and different panel types.

Estimating 3D hand meshes from RGB images robustly is a highly desirable task, made challenging due to the numerous degrees of freedom, and issues such as self-similarity and occlusions. Previous methods generally either use parametric 3D hand models or follow a model-free approach. While the former can be considered more robust, e.g. to occlusions, they are less expressive. We propose a hybrid approach, utilizing a deep neural network and differential rendering based optimization to demonstrably achieve the best of both worlds. In addition, we explore Virtual Reality (VR) as an application. Most VR headsets are nowadays equipped with multiple cameras, which we can leverage by extending our method to the egocentric stereo domain. This extension proves to be more resilient to the above mentioned issues. Finally, as a use-case, we show that the improved image-model alignment can be used to acquire the user’s hand texture, which leads to a more realistic virtual hand representation.

Teleoperation provides a way for human operators to guide robots in situations where full autonomy is challenging or where direct human intervention is required. It can also be an important tool to teach robots in order to achieve autonomous behaviour later on. The increased availability of collaborative robot arms and Virtual Reality (VR) devices, provides ample opportunity for development of novel teleoperation methods. Since robot arms are often kinematically different from human arms, mapping human motions to a robot in real-time is not trivial. Additionally, a human operator might steer the robot arm toward singularities or its workspace limits, which can lead to undesirable behaviour. This is further accentuated for the orchestration of multiple robots. In this paper, we present a VR interface targeted to multi-arm payload manipulation, which can closely match real-time input motion. Allowing the user to manipulate the payload rather than mapping their motions to individual arms we are able to simultaneously guide multiple collaborative arms. By releasing a single rotational degree of freedom, and by using a local optimization method, we can improve each arm’s manipulability index, which in turn lets us avoid kinematic singularities and workspace limitations. We apply our approach to predefined trajectories as well as real-time teleoperation on different robot arms and compare performance in terms of end-effector position error and relevant joint motion metrics.

Teleoperating robotic arms is a challenging task that requires years of training to master. It is mentally demanding, as the operator must internally compute transformations, or rely on muscle memory, to perform even the simplest tasks. Alternative methods that rely on embodiment –the immersive, first person experience of controlling the robot from its point of view are recently becoming more popular, thanks to the emergence of mixed reality devices. These methods create an intuitive experience by tracking the users motions, and retargetting them to the robot. However, even recent hardware fails at achieving total immersion, due to inherent discrepancies such as latency, imperfect tracking, and the differences between human and robot motor systems. Thus, performing even simple pick-and-place tasks with these systems, while more intuitive, is still cumbersome, and far from the level of human performance.In this paper we propose an immersive system that aims to bridge this gap. The system tracks the user’s motion and retargets them to the robot as usual, but it also detects the user’s intent, that is, the task they wish to perform. Based on this knowledge, the system can autocorrect the motion when it is about to fail, in a seamless manner, such that the task is successfully performed. We evaluate the efficacy of our autocorrection system in a user study. The results show a statistically significant performance improvement in terms of operation accuracy and time.

We combine Boston Dynamics Spot®with a light-weight, external robot arm to perform dynamic grasping maneuvers. While Spot is a reliable, robust and easy-to-control mobile robot, these highly desirable qualities come with the price that the control access granted to the user is restricted. Consequently Spot’s behavior must largely be treated as a black box, which causes difficulties when combined with a moving payload such as a robotic arm. We overcome the arising challenges by building a model of the combined platform, fitting the corresponding model parameters using experimental data and a straight-forward optimization framework. We use this model to generate control commands for the physical platform using trajectory optimization. We demonstrate that even with a simple model, and control trajectories deployed in a feed-forward manner, the combined platform is capable of executing grasping tasks in a dynamic fashion. Furthermore, we show how the platform can use the additional degrees of freedom of the legs to extend the reachability of the arm.

We present an algorithmic approach to designing animatronic figures - expressive robotic characters whose movements are driven by a large number of actuators. The input to our design system provides a high-level specification of the space of motions the character should be able to perform. The output consists of a fully functional mechatronic blueprint. We cast the design task as a search problem in a vast combinatorial space of possible solutions. To find an optimal design in this space, we propose an efficient best-first search algorithm that is guided by an admissible heuristic. The objectives guiding the search process demand that the design remains free of singularities and self-collisions at any point in the high-dimensional space of motions the character is expected to be able to execute. To identify worst-case self-collision scenarios for multi degree-of-freedom closed-loop mechanisms, we additionally develop an elegant technique inspired by the concept of adversarial attacks. We demonstrate the efficacy of our approach by creating designs for several animatronic figures of varying complexity.

Hot-wire cutting is a subtractive fabrication technique used to carve foam and similar materials. Conventional machines rely on straight wires and are thus limited to creating piecewise ruled surfaces. In this work, we propose a method that exploits a dual-arm robot setup to actively control the shape of a flexible, heated rod as it cuts through the material. While this setting offers great freedom of shape, using it effectively requires concurrent reasoning about three tightly coupled sub-problems: 1) modeling the way in which the shape of the rod and the surface it sweeps are governed by the robot's motions; 2) approximating a target shape through a sequence of surfaces swept by the equilibrium shape of an elastic rod; and 3) generating collision-free motion trajectories that lead the robot to create desired sweeps with the deformable tool. We present a computational framework for robotic hot wire cutting that addresses all three sub-problems in a unified manner. We evaluate our approach on a set of simulated results and physical artefacts generated with our robotic fabrication system.

We present a computational framework for robotic animation of real-world string puppets. Also known as marionettes, these articulated figures are typically brought to life by human puppeteers. The puppeteer manipulates rigid handles that are attached to the puppet from above via strings. The motions of the marionette are therefore governed largely by gravity, the pull forces exerted by the strings, and the internal forces arising from mechanical articulation constraints. This seemingly simple setup conceals a very challenging and nuanced control problem, as marionettes are, in fact, complex coupled pendulum systems. Despite this, in the hands of a master puppeteer, marionette animation can be nothing short of mesmerizing. Our goal is to enable autonomous robots to animate marionettes with a level of skill that approaches that of human puppeteers. To this end, we devise a predictive control model that accounts for the dynamics of the marionette and kinematics of the robot puppeteer. The input to our system consists of a string puppet design and a target motion, and our trajectory planning algorithm computes robot control actions that lead to the marionette moving as desired. We validate our methodology through a series of experiments conducted on an array of marionette designs and target motions. These experiments are performed both in simulation and using a physical robot, the human-sized, dual arm ABB YuMi ® IRB 14000.

In this paper, we address the challenge of robotic manipulation of elastically deforming objects. To this end, we model elastic objects using the Finite Element Method. Through a quasi-static assumption, we leverage sensitivity analysis to mathematically model how changes in the robot's configuration affect the deformed shape of the object being manipulated. This enables an interactive, simulation-based control methodology, wherein user-specified deformations for the elastic objects are automatically mapped to joint angle commands. The optimization formulation we introduce is general, operates directly within a robot's workspace and can readily incorporate joint limits as well as collision avoidance between the links. We validate our control methodology on a YuMi® IRB 14000, which we use to manipulate a variety of elastic objects.

Abstract Laser cutting is an appealing fabrication process due to the low cost of materials and extremely fast fabrication. However, the design space afforded by laser cutting is limited, since only flat panels can be cut. Previous methods for manufacturing from flat sheets usually roughly approximate 3D objects by polyhedrons or cross sections. Computational design methods for connecting, interlocking, or folding several laser cut panels have been introduced; to obtain a good approximation, these methods require numerous parts and long assembly times. In this paper, we propose a radically different approach: Our approximation is based on cutting thin, planar spirals out of flat panels. When such spirals are pulled apart, they take on the shape of a 3D spring whose contours are similar to the input object. We devise an optimization problem that aims to minimize the number of required parts, thus reducing costs and fabrication time, while at the same time ensuring that the resulting spring mimics the shape of the original object. In addition to rapid fabrication and assembly, our method enables compact packaging and storage as flat parts. We also demonstrate its use for creating armatures for sculptures and moulds for filling, with potential applications in architecture or construction.

We present a computation-driven approach to design optimization and motion synthesis for robotic creatures that locomote using arbitrary arrangements of legs and wheels. Through an intuitive interface, designers first create unique robots by combining different types of servomotors, 3D printable connectors, wheels and feet in a mix-and-match manner. With the resulting robot as input, a novel trajectory optimization formulation generates walking, rolling, gliding and skating motions. These motions emerge naturally based on the components used to design each individual robot. We exploit the particular structure of our formulation and make targeted simplifications to significantly accelerate the underlying numerical solver without compromising quality. This allows designers to interactively choreograph stable, physically-valid motions that are agile and compelling. We furthermore develop a suite of user-guided, semi-automatic, and fully-automatic optimization tools that enable motion-aware edits of the robot's physical structure. We demonstrate the efficacy of our design methodology by creating a diverse array of hybrid legged/wheeled mobile robots which we validate using physics simulation and through fabricated prototypes.

Fabrication from developable parts is the basis for arts such as papercraft and needlework, as well as modern architecture and CAD in general, and it has inspired much research. We observe that the assembly of complex 3D shapes created by existing methods often requires first fabricating many small parts and then carefully following instructions to assemble them together. Despite its significance, this error prone and tedious process is generally neglected in the discussion. We present the concept of zippables - single , two dimensional, branching, ribbon-like pieces of fabric that can be quickly zipped up without any instructions to form 3D objects. Our inspiration comes from the so-called zipit bags [zipit 2017], which are made of a single, long ribbon with a zipper around its boundary. In order to "assemble" the bag, one simply needs to zip up the ribbon. Our method operates in the same fashion, but it can be used to approximate a wide variety of shapes. Given a 3D model, our algorithm produces plans for a single 2D shape that can be laser cut in few parts from fabric or paper. A zipper can then be attached along the boundary by sewing, or by gluing using a custom-built fastening rig. We show physical and virtual results that demonstrate the capabilities of our method and the ease with which shapes can be assembled.

We propose a UV mapping algorithm that jointly optimizes for cuts and distortion, sidestepping heuristics for placing the cuts. The energy we minimize is a state-of-the-art geometric distortion measure, generalized to take seams into account. Our algorithm is designed to support an interactive workflow: it optimizes UV maps on the fly, while the user can interactively move vertices, cut mesh parts, join seams, separate overlapping regions, and control the placement of the parameterization patches in the UV space. Our UV maps are of high quality in terms of both geometric distortion and cut placement, and compare favorably to those designed with traditional modeling tools. The UV maps can be created in a fraction of the time as existing methods, since our algorithm drastically alleviates the trial-and-error, iterative procedures that plague traditional UV mapping approaches.

We present a scalable approach for the optimization of flip-preventing energies in the general context of simplicial mappings and specifically for mesh parameterization. Our iterative minimization is based on the observation that many distortion energies can be optimized indirectly by minimizing a family of simpler proxy energies. Minimization of these proxies is a natural extension of the local/global minimization of the ARAP energy. Our algorithm is simple to implement and scales to datasets with millions of faces. We demonstrate our approach for the computation of maps that minimize a conformal or isometric distortion energy, both in two and three dimensions. In addition to mesh parameterization, we show that our algorithm can be applied to mesh deformation and mesh quality improvement.

Many algorithms on meshes require the minimization of composite objectives, i.e. , energies that are compositions of simpler parts. Canonical examples include mesh parameterization and deformation. We propose a second order optimization approach that exploits this composite structure to efficiently converge to a local minimum. Our main observation is that a convex-concave decomposition of the energy constituents is simple and readily available in many cases of practical relevance in graphics. We utilize such convex-concave decompositions to define a tight convex majorizer of the energy, which we employ as a convex second order approximation of the objective function. In contrast to existing approaches that largely use only local convexification, our method is able to take advantage of a more global view on the energy landscape. Our experiments on triangular meshes demonstrate that our approach outperforms the state of the art on standard problems in geometry processing, and potentially provide a unified framework for developing efficient geometric optimization algorithms.

We introduce a method for computing seamless bijective mappings between two surface-meshes that interpolates a given set of correspondences. A common approach for computing a map between surfaces is to cut the surfaces to disks, flatten them to the plane, and extract the mapping from the flattenings by composing one flattening with the inverse of the other. So far, a significant drawback in this class of techniques is that the choice of cuts introduces a bias in the computation of the map that often causes visible artifacts and wrong correspondences. In this paper we develop a surface mapping technique that is indifferent to the particular cut choice. This is achieved by a novel type of surface flattenings that encodes this cut-invariance, and when optimized with a suitable energy functional results in a seamless surface-to-surface map. We show the algorithm enables producing high-quality seamless bijective maps for pairs of surfaces with a wide range of shape variability and from a small number of prescribed correspondences. We also used this framework to produce three-way, consistent and seamless mappings for triplets of surfaces.

Polyhedral meshes (PM)-meshes having planar faces-have enjoyed a rise in popularity in recent years due to their importance in architectural and industrial design. However, they are also notoriously difficult to generate and manipulate. Previous methods start with a smooth surface and then apply elaborate meshing schemes to create polyhedral meshes approximating the surface. In this paper, we describe a reverse approach: given the topology of a mesh, we explore the space of possible planar meshes having that topology. Our approach is based on a complete characterization of the maximal linear spaces of polyhedral meshes contained in the curved manifold of polyhedral meshes with a given topology. We show that these linear spaces can be described as nullspaces of differential operators, much like harmonic functions are nullspaces of the Laplacian operator. An analysis of this operator provides tools for global and local design of a polyhedral mesh, which fully expose the geometric possibilities and limitations of the given topology.

This paper introduces an algorithm for computing low-distortion, bijective mappings between surface meshes. The algorithm recieves as input a coarse set of corresponding pairs of points on the two surfaces, and follows three steps: (i) cutting the two meshes to disks in a consistent manner; (ii) jointly flattening the two disks via a novel formulation for minimizing isometric distortion while guaranteeing local injectivity (the flattenings can overlap, however); and (iii) computing a unique continuous bijection that is consistent with the flattenings. The construction of the algorithm stems from two novel observations: first, bijections between disk-type surfaces can be uniquely and efficiently represented via consistent locally injective flattenings that are allowed to be globally overlapping. This observation reduces the problem of computing bijective surface mappings to the task of computing locally injective flattenings, which is shown to be easier. Second, locally injective flattenings that minimize isometric distortion can be efficiently characterized and optimized in a convex framework. Experiments that map a wide baseline of pairs of surface meshes using the algorithm are provided. They demonstrate the ability of the algorithm to produce high-quality continuous bijective mappings between pairs of surfaces of varying isometric distortion levels.

We consider the problem of finding a geometrically consistent set of point matches between two images. We assume that local descriptors have provided a set of candidate matches, which may include many outliers. We then seek the largest subset of these correspondences that can be aligned perfectly using a nonrigid deformation that exerts a bounded distortion. We formulate this as a constrained optimization problem and solve it using a constrained, iterative reweighted least-squares algorithm. In each iteration of this algorithm we solve a convex quadratic program obtaining a globally optimal match over a subset of the bounded distortion transformations. We further prove that a sequence of such iterations converges monotonically to a critical point of our objective function. We show experimentally that this algorithm produces excellent results on a number of test sets, in comparison to several state-of-the-art approaches.

The problem of planar mapping and deformation is central in computer graphics. This paper presents a framework for adapting general, smooth, function bases for building provably good planar mappings. The term "good" in this context means the map has no fold-overs (injective), is smooth, and has low isometric or conformal distortion. Existing methods that use mesh-based schemes are able to achieve injectivity and/or control distortion, but fail to create smooth mappings, unless they use a prohibitively large number of elements, which slows them down. Meshless methods are usually smooth by construction, yet they are not able to avoid fold-overs and/or control distortion. Our approach constrains the linear deformation spaces induced by popular smooth basis functions, such as B-Splines, Gaussian and Thin-Plate Splines, at a set of collocation points, using specially tailored convex constraints that prevent fold-overs and high distortion at these points. Our analysis then provides the required density of collocation points and/or constraint type, which guarantees that the map is injective and meets the distortion constraints over the entire domain of interest. We demonstrate that our method is interactive at reasonably complicated settings and compares favorably to other state-of-the-art mesh and meshless planar deformation methods.

Abstract Constraining 3D meshes to restricted classes is necessary in architectural and industrial design, but it can be very challenging to manipulate meshes while staying within these classes. Specifically, polyhedral meshes—those having planar faces—are very important, but also notoriously difficult to generate and manipulate efficiently. We describe an interactive method for computing, optimizing and editing polyhedral meshes. Efficiency is achieved thanks to a numerical procedure combining an alternating least‐squares approach with the penalty method. This approach is generalized to manipulate other subsets of polyhedral meshes, as defined by a variety of other constraints.

Abstract Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed‐form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed‐form expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.

Abstract We define a generalized distance function on an unoriented 3D point set and describe how it may be used to reconstruct a surface approximating these points. This distance function is shown to be a Mahalanobis distance in a higher‐dimensional embedding space of the points, and the resulting reconstruction algorithm a natural extension of the classical Radial Basis Function (RBF) approach. Experimental results show the superiority of our reconstruction algorithm to RBF and other methods in a variety of practical scenarios.