We present an analytical framework for exploring the design space of hinged kirigami structures that deploy rigidly and uniformly. A hinged kirigami structure consists of rigid planar faces connected by hinges, and is deployable if the faces can rotate about the hinges without deformation. Such deployability depends on both the topology and the geometry, i.e., the combinatorial connectivity of the faces and their spatial embedding. Prior work studies geometric constraints for uniform deployability in restricted settings, most notably quadrilateral kirigami patterns and structures derived from 2-colorable planar graphs. However, existing analysis does not readily generalize to arbitrary kirigami structures, nor does it provide a systematic approach for constructing deployable kirigami from non-2-colorable planar graphs (e.g., graphs with non-manifold embeddings). Moreover, while geometric conditions for deployability have been partially investigated, the structure of the full deployable design space and its associated degrees of freedom remain largely unexplored. In this work, we propose a new framework that enables the derivation of hinged kirigami structures from arbitrary planar graphs, with the key feature that multiple distinct kirigami structures can be generated from the same graph. We derive geometric constraints that ensure uniform rigid deployability and analytically characterize the full design space of deployable embeddings, including its associated degrees of freedom. This characterization allows continuous navigation of the design space and provides a systematic foundation for the design of kirigami-based mechanical metamaterials.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

We present a computational framework for designing geometric metamaterials capable of approximating freeform 3D surfaces via rotationally deployable kirigami patterns. While prior inverse design methods typically rely on standard, well-studied patterns, such as equilateral triangles or quadrilaterals, we step back to examine the broader design space of the patterns themselves. Specifically, we derive principled rules to determine whether a given planar tiling can be cut into a rotationally deployable hinged kirigami structure with possible curvature adaptation. These insights allow us to generate and validate a broad family of novel tiling patterns beyond traditional examples. We further analyze two key deployment states of a general pattern: the commonly used maximal area expansion, and the maximal rotation angle reached just before face collisions occur, which we adopt as the default for inverse design as it allows for simple deployment in practice, i.e., rotating the faces to their natural limit. Finally, we solve the inverse problem: given a target 3D surface, we compute a planar tiling that, when cut and deployed to its maximal rotation angle, approximates the input geometry. We show that for a subset of patterns, the deployed configurations are hole-free, demonstrating that curvature can be achieved from planar sheets through local combinatorial changes. Our experiments, including physical fabrications, demonstrate the effectiveness of our approach and validate a wide range of previously unexplored patterns that are both physically realizable and geometrically expressive.

Abstract We formalize Italian smocking, an intricate embroidery technique that gathers flat fabric into pleats along meandering lines of stitches, resulting in pleats that fold and gather where the stitching veers. In contrast to English smocking, characterized by colorful stitches decorating uniformly shaped pleats, and Canadian smocking, which uses localized knots to form voluminous pleats, Italian smocking permits the fabric to move freely along the stitched threads following curved paths, resulting in complex and unpredictable pleats with highly diverse, irregular structures, achieved simply by pulling on the threads. We introduce a novel method for digital previewing of Italian smocking results, given the thread stitching path as input. Our method uses a coarse‐grained mass‐spring system to simulate the interaction between the threads and the fabric. This configuration guides the fine‐level fabric deformation through an adaptation of the state‐of‐the‐art simulator, C‐IPC [LKJ21]. Our method models the general problem of fabric‐thread interaction and can be readily adapted to preview Canadian smocking as well. We compare our results to baseline approaches and physical fabrications to demonstrate the accuracy of our method.

Distortion-minimizing surface parameterization is an essential step for computing 2D pieces necessary to fabricate a target 3D shape from flat material. Garment design and textile fabrication are a prominent application example. Common distortion measures quantify length, angle or area preservation in an isotropic manner, so that when applied to woven textile fabrication, they implicitly assume fabric behaves like paper, which is inextensible in all directions and does not permit shearing. However, woven fabric differs significantly from paper: it exhibits anisotropy along the yarn directions and allows for some degree of shearing. We propose a novel distortion energy based on Chebyshev nets that anisotropically penalizes shearing and stretching. Our energy formulation can be used as an optimization objective for surface parameterization and is simple to minimize via a local-global algorithm. We demonstrate its advantages in modeling nets or woven fabric behavior over the commonly used isotropic distortion energies.

We develop an optimization-based method to model smocking , a surface embroidery technique that provides decorative geometric texturing while maintaining stretch properties of the fabric. During smocking, multiple pairs of points on the fabric are stitched together, creating non-manifold geometric features and visually pleasing textures. Designing smocking patterns is challenging, because the outcome of stitching is unpredictable: The final texture is often revealed only when the whole smocking process is completed, necessitating painstaking physical fabrication and time consuming trial-and-error experimentation. This motivates us to seek a digital smocking design method. Straightforward attempts to compute smocked fabric geometry using surface deformation or cloth simulation methods fail to produce realistic results, likely due to the intricate structure of the designs, the large number of contacts and high-curvature folds. We instead formulate smocking as a graph embedding and shape deformation problem. We extract a coarse graph representing the fabric and the stitching constraints and then derive the graph structure of the smocked result. We solve for the three-dimensional embedding of this graph, which in turn reliably guides the deformation of the high-resolution fabric mesh. Our optimization based method is simple, efficient, and flexible, which allows us to build an interactive system for smocking pattern exploration. To demonstrate the accuracy of our method, we compare our results to real fabrications on a large set of smocking patterns.

We present a novel method for realizing freeform surfaces with pieces of flat fabric, where curvature is created by stitching together points on the fabric using a technique known as smocking. Smocking is renowned for producing intricate geometric textures with voluminous pleats. However, it has been mostly used to realize flat shapes or manually designed, limited classes of curved surfaces. Our method combines the computation of directional fields with continuous optimization of a Tangram graph in the plane, which together allow us to realize surfaces of arbitrary topology and curvature with smocking patterns of diverse symmetries. Given a target surface and the desired smocking pattern, our method outputs a corresponding 2D smocking pattern that can be fabricated by sewing specified points together. The resulting textile fabrication approximates the target shape and exhibits visually pleasing pleats. We validate our method through physical fabrication of various smocked examples.

We introduce pointwise map smoothness via the Dirich-let energy into the functional map pipeline, and propose an algorithm for optimizing it efficiently, which leads to highquality results in challenging settings. Specifically, we first formulate the Dirichlet energy of the pulled-back shape coordinates, as a way to evaluate smoothness of a pointwise map across discrete surfaces. We then extend the recently proposed discrete solver and show how a strategy based on auxiliary variable reformulation allows us to optimize pointwise map smoothness alongside desirable functional map properties such as bijectivity. This leads to an efficient map refinement strategy that simultaneously improves functional and point-to-point correspondences, obtaining smooth maps even on non-isometric shape pairs. Moreover, we demonstrate that several previously proposed methods for computing smooth maps can be reformulated as variants of our approach, which allows us to compare different formulations in a consistent framework. Finally, we compare these methods both on existing benchmarks and on a new rich dataset that we introduce, which contains non-rigid, non-isometric shape pairs with inter-category and cross-category correspondences. Our work leads to a general framework for optimizing and analyzing map smoothness both conceptually and in challenging practical settings.

Among the various approaches for producing point distributions with blue noise spectrum, we argue for an optimization framework using Gaussian kernels. We show that with a wise selection of optimization parameters, this approach attains unprecedented quality, provably surpassing the current state of the art attained by the optimal transport (BNOT) approach. Further, we show that our algorithm scales smoothly and feasibly to high dimensions while maintaining the same quality, realizing unprecedented high-quality high-dimensional blue noise sets. Finally, we show an extension to adaptive sampling.

In this paper, we provide a theoretical foundation for pointwise map recovery from functional maps and highlight its relation to a range of shape correspondence methods based on spectral alignment. With this analysis in hand, we develop a novel spectral registration technique: Fast Sinkhorn Filters, which allows for the recovery of accurate and bijective pointwise correspondences with a superior time and memory complexity in comparison to existing approaches. Our method combines the simple and concise representation of correspondence using functional maps with the matrix scaling schemes from computational optimal transport. By exploiting the sparse structure of the kernel matrices involved in the transport map computation, we provide an efficient trade-off between acceptable accuracy and complexity for the problem of dense shape correspondence, while promoting bijectivity.1

Abstract We propose a novel discrete solver for optimizing functional map‐based energies, including descriptor preservation and promoting structural properties such as area‐preservation, bijectivity and Laplacian commutativity among others. Unlike the commonly‐used continuous optimization methods, our approach enforces the functional map to be associated with a pointwise correspondence as a hard constraint, which provides a stronger link between optimized properties of functional and point‐to‐point maps. Under this hard constraint, our solver obtains functional maps with lower energy values compared to the standard continuous strategies. Perhaps more importantly, the recovered pointwise maps from our discrete solver preserve the optimized for functional properties and are thus of higher overall quality. We demonstrate the advantages of our discrete solver on a range of energies and shape categories, compared to existing techniques for promoting pointwise maps within the functional map framework. Finally, with this solver in hand, we introduce a novel Effective Functional Map Refinement (EFMR) method which achieves the state‐of‐the‐art accuracy on the SHREC'19 benchmark.

We propose a novel and flexible roof modeling approach that can be used for constructing planar 3D polygon roof meshes. Our method uses a graph structure to encode roof topology and enforces the roof validity by optimizing a simple but effective planarity metric we propose. This approach is significantly more efficient than using general purpose 3D modeling tools such as 3ds Max or SketchUp, and more powerful and expressive than specialized tools such as the straight skeleton. Our optimization-based formulation is also flexible and can accommodate different styles and user preferences for roof modeling. We showcase two applications. The first application is an interactive roof editing framework that can be used for roof design or roof reconstruction from aerial images. We highlight the efficiency and generality of our approach by constructing a mesh-image paired dataset consisting of 2539 roofs. Our second application is a generative model to synthesize new roof meshes from scratch. We use our novel dataset to combine machine learning and our roof optimization techniques, by using transformers and graph convolutional networks to model roof topology, and our roof optimization methods to enforce the planarity constraint.

Abstract In this paper, we propose a novel method, which we call C onsistent Z oom O ut , for efficiently refining correspondences among deformable 3D shape collections, while promoting the resulting map consistency. Our formulation is closely related to a recent unidirectional spectral refinement framework, but naturally integrates map consistency constraints into the refinement. Beyond that, we show further that our formulation can be adapted to recover the underlying isometry among near‐isometric shape collections with a theoretical guarantee, which is absent in the other spectral map synchronization frameworks. We demonstrate that our method improves the accuracy compared to the competing methods when synchronizing correspondences in both near‐isometric and heterogeneous shape collections, but also significantly outperforms the baselines in terms of map consistency.

In this paper we propose an approach for computing multiple high-quality near-isometric dense correspondences between a pair of 3D shapes. Our method is fully automatic and does not rely on user-provided landmarks or descriptors. This allows us to analyze the full space of maps and extract multiple diverse and accurate solutions, rather than optimizing for a single optimal correspondence as done in most previous approaches. To achieve this, we propose a compact tree structure based on the spectral map representation for encoding and enumerating possible rough initializations, and a novel efficient approach for refining them to dense pointwise maps. This leads to a new method capable of both producing multiple high-quality correspondences across shapes and revealing the symmetry structure of a shape without a priori information. In addition, we demonstrate through extensive experiments that our method is robust and results in more accurate correspondences than state-of-the-art for shape matching and symmetry detection.

We propose a novel framework for computing descriptors for characterizing points on three-dimensional surfaces. First, we present a new non-learned feature that uses graph wavelets to decompose the Dirichlet energy on a surface. We call this new feature Wavelet Energy Decomposition Signature (WEDS). Second, we propose a new Multiscale Graph Convolutional Network (MGCN) to transform a non-learned feature to a more discriminative descriptor. Our results show that the new descriptor WEDS is more discriminative than the current state-of-the-art non-learned descriptors and that the combination of WEDS and MGCN is better than the state-of-the-art learned descriptors. An important design criterion for our descriptor is the robustness to different surface discretizations including triangulations with varying numbers of vertices. Our results demonstrate that previous graph convolutional networks significantly overfit to a particular resolution or even a particular triangulation, but MGCN generalizes well to different surface discretizations. In addition, MGCN is compatible with previous descriptors and it can also be used to improve the performance of other descriptors, such as the heat kernel signature, the wave kernel signature, or the local point signature.

Abstract We consider the problem of non‐rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace‐Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well‐justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one‐parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly‐used refinement techniques.

We present a simple and efficient method for refining maps or correspondences by iterative upsampling in the spectral domain that can be implemented in a few lines of code. Our main observation is that high quality maps can be obtained even if the input correspondences are noisy or are encoded by a small number of coefficients in a spectral basis. We show how this approach can be used in conjunction with existing initialization techniques across a range of application scenarios, including symmetry detection, map refinement across complete shapes, non-rigid partial shape matching and function transfer. In each application we demonstrate an improvement with respect to both the quality of the results and the computational speed compared to the best competing methods, with up to two orders of magnitude speed-up in some applications. We also demonstrate that our method is both robust to noisy input and is scalable with respect to shape complexity. Finally, we present a theoretical justification for our approach, shedding light on structural properties of functional maps.

We propose a method for efficiently computing orientation-preserving and approximately continuous correspondences between non-rigid shapes, using the functional maps framework. We first show how orientation preservation can be formulated directly in the functional (spectral) domain without using landmark or region correspondences and without relying on external symmetry information. This allows us to obtain functional maps that promote orientation preservation, even when using descriptors, that are invariant to orientation changes. We then show how higher quality, approximately continuous and bijective pointwise correspondences can be obtained from initial functional maps by introducing a novel refinement technique that aims to simultaneously improve the maps both in the spectral and spatial domains. This leads to a general pipeline for computing correspondences between shapes that results in high-quality maps, while admitting an efficient optimization scheme. We show through extensive evaluation that our approach improves upon state-of-the-art results on challenging isometric and non-isometric correspondence benchmarks according to both measures of continuity and coverage as well as producing semantically meaningful correspondences as measured by the distance to ground truth maps.

We present a novel and efficient approach for computing joint graph layouts and then use it to visualize collections of segmented meshes. Our joint graph layout algorithm takes as input the adjacency matrices for a set of graphs along with partial, possibly soft, correspondences between nodes of different graphs. We then use a two stage procedure, where in the first step, we extend spectral graph drawing to include a consistency term so that a collection of graphs can be handled jointly. Our second step extends metric multi-dimensional scaling with stress majorization to the joint layout setting, while using the output of the spectral approach as initialization. Further, we discuss a user interface for exploring a collection of graphs. Finally, we show multiple example visualizations of graphs stemming from collections of segmented meshes and we present qualitative and quantitative comparisons with previous work.