Flag manifolds encode nested sequences of subspaces and serve as powerful structures for various computer vision and machine learning applications. Despite their utility in tasks such as dimensionality reduction, motion averaging, and subspace clustering, current applications are often restricted to extracting flags using common matrix decomposition methods like the singular value decomposition. Here, we address the need for a general algorithm to factorize and work with hierarchical datasets. In particular, we propose a novel, flag-based method that decomposes arbitrary hierarchical real-valued data into a hierarchy-preserving flag representation in Stiefel coordinates. Our work harnesses the potential of flag manifolds in applications including denoising, clustering, and few-shot learning.

We propose Dyn-HaMR, to the best of our knowledge, the first approach to reconstruct 4D global hand motion from monocular videos recorded by dynamic cameras in the wild. Reconstructing accurate 3D hand meshes from monocular videos is a crucial task for understanding human behaviour, with significant applications in augmented and virtual reality (AR/VR). However, existing methods for monocular hand reconstruction typically rely on a weak perspective camera model, which simulates hand motion within a limited camera frustum. As a result, these approaches struggle to recover the full 3D global trajectory and often produce noisy or incorrect depth estimations, particularly when the video is captured by dynamic or moving cameras, which is common in egocentric scenarios. Our DynHaMR consists of a multi-stage, multi-objective optimization pipeline, that factors in (i) simultaneous localization and mapping (SLAM) to robustly estimate relative camera motion, (ii) an interacting-hand prior for generative infilling and to refine the interaction dynamics, ensuring plausible recovery under (self-)occlusions, and (iii) hierarchical initialization through a combination of state-of-the-art hand tracking methods. Through extensive evaluations on both in-the-wild and indoor datasets, we show that our approach significantly outperforms state-of-the-art methods in terms of 4D global mesh recovery. This establishes a new benchmark for hand motion reconstruction from monocular video with moving cameras. Our project page is at https://dyn-hamr.github.io/.

We present a hierarchical, control theory inspired method for variational inference (VI) for neural stochastic differential equations (SDEs).While VI for neural SDEs is a promising avenue for uncertaintyaware reasoning in time-series, it is computationally challenging due to the iterative nature of maximizing the ELBO.In this work, we propose to decompose the control term into linear and residual non-linear components and derive an optimal control term for linear SDEs, using stochastic optimal control.Modeling the non-linear component by a neural network, we show how to efficiently train neural SDEs without sacrificing their expressive power.Since the linear part of the control term is optimal and does not need to be learned, the training is initialized at a lower cost and we observe faster convergence.* MO acknowledges funding

Modeling the rotation of moving objects is a fundamental task in computer vision, yet $SO(3)$ extrapolation still presents numerous challenges: (1) unknown quantities such as the moment of inertia complicate dynamics, (2) the presence of external forces and torques can lead to non-conservative kinematics, and (3) estimating evolving state trajectories under sparse, noisy observations requires robustness. We propose modeling trajectories of noisy pose estimates on the manifold of 3D rotations in a physically and geometrically meaningful way by leveraging Neural Controlled Differential Equations guided with $SO(3)$ Savitzky-Golay paths. Existing extrapolation methods often rely on energy conservation or constant velocity assumptions, limiting their applicability in real-world scenarios involving non-conservative forces. In contrast, our approach is agnostic to energy and momentum conservation while being robust to input noise, making it applicable to complex, non-inertial systems. Our approach is easily integrated as a module in existing pipelines and generalizes well to trajectories with unknown physical parameters. By learning to approximate object dynamics from noisy states during training, our model attains robust extrapolation capabilities in simulation and various real-world settings. Code is available at https://github.com/bastianlb/forecasting-rotational-dynamics

We present CuMPerLay, a novel differentiable vectorization layer that enables the integration of Cubical Multiparameter Persistence (CMP) into deep learning pipelines. While CMP presents a natural and powerful way to topologically work with images, its use is hindered by the complexity of multifiltration structures as well as the vectorization of CMP. In face of these challenges, we introduce a new algorithm for vectorizing MP homologies of cubical complexes. Our CuMPerLay decomposes the CMP into a combination of individual, learnable single-parameter persistence, where the bifiltration functions are jointly learned. Thanks to the differentiability, its robust topological feature vectors can be seamlessly used within state-of-the-art architectures such as Swin Transformers. We establish theoretical guarantees for the stability of our vectorization under generalized Wasserstein metrics. Our experiments on benchmark medical imaging and computer vision datasets show the benefit CuMPerLay on classification and segmentation performance, particularly in limited-data scenarios. Overall, CuMPerLay offers a promising direction for integrating global structural information into deep networks for structured image analysis.

Solving non-convex, NP-hard optimization problems is crucial for training machine learning models, including neural networks. However, non-convexity often leads to black-box machine learning models with unclear inner workings. While convex formulations have been used for verifying neural network robustness, their application to training neural networks remains less explored. In response to this challenge, we reformulate the problem of training infinite-width two-layer ReLU networks as a convex completely positive program in a finite-dimensional (lifted) space. Despite the convexity, solving this problem remains NP-hard due to the complete positivity constraint. To overcome this challenge, we introduce a semidefinite relaxation that can be solved in polynomial time. We then experimentally evaluate the tightness of this relaxation, demonstrating its competitive performance in test accuracy across a range of classification tasks.

Grokking, or sudden generalization that occurs after prolonged overfitting, is a surprising phenomenon that has challenged our understanding of deep learning. While a lot of progress has been made in understanding grokking, it is still not clear why generalization is delayed and why grokking often does not happen without regularization. In this work we argue that without regularization, grokking tasks push models to the edge of numerical stability, introducing floating point errors in the Softmax that we refer to as _Softmax Collapse_ (SC). We show that SC prevents grokking and that mitigating SC leads to grokking _without_ regularization. Investigating the root cause of SC, we find that beyond the point of overfitting, the gradients strongly align with what we call the _naïve loss minimization_ (NLM) direction. This component of the gradient does not change the predictions of the model but decreases the loss by scaling the logits, usually through the scaling of the weights along their current direction. We show that this scaling of the logits explains the delay in generalization characteristic of grokking, and eventually leads to SC, stopping learning altogether. To validate these hypotheses, we introduce two key contributions that mitigate the issues faced in grokking tasks: (i) $\mathrm{StableMax}$, a new activation function that prevents SC and enables grokking without regularization, and (ii) $\perp\mathrm{Grad}$, a training algorithm that leads to quick generalization in grokking tasks by preventing NLM altogether. These contributions provide new insights into grokking, shedding light on its delayed generalization, reliance on regularization, and the effectiveness of known grokking-inducing methods.

We introduce copresheaf topological neural networks (CTNNs), a powerful unifying framework that encapsulates a wide spectrum of deep learning architectures, designed to operate on structured data, including images, point clouds, graphs, meshes, and topological manifolds. While deep learning has profoundly impacted domains ranging from digital assistants to autonomous systems, the principled design of neural architectures tailored to specific tasks and data types remains one of the field's most persistent open challenges. CTNNs address this gap by formulating model design in the language of copresheaves, a concept from algebraic topology that generalizes most practical deep learning models in use today. This abstract yet constructive formulation yields a rich design space from which theoretically sound and practically effective solutions can be derived to tackle core challenges in representation learning, such as long-range dependencies, oversmoothing, heterophily, and non-Euclidean domains. Our empirical results on structured data benchmarks demonstrate that CTNNs consistently outperform conventional baselines, particularly in tasks requiring hierarchical or localized sensitivity. These results establish CTNNs as a principled multi-scale foundation for the next generation of deep learning architectures.

Accurate 3D reconstruction of hands and instruments is critical for vision-based analysis of ophthalmic microsurgery, yet progress has been hampered by the lack of realistic, large-scale datasets and reliable annotation tools. In this work, we introduce OphNet-3D, the first extensive RGB-D dynamic 3D reconstruction dataset for ophthalmic surgery, comprising 41 sequences from 40 surgeons and totaling 7.1 million frames, with fine-grained annotations of 12 surgical phases, 10 instrument categories, dense MANO hand meshes, and full 6-DoF instrument poses. To scalably produce high-fidelity labels, we design a multi-stage automatic annotation pipeline that integrates multi-view data observation, data-driven motion prior with cross-view geometric consistency and biomechanical constraints, along with a combination of collision-aware interaction constraints for instrument interactions. Building upon OphNet-3D, we establish two challenging benchmarks—bimanual hand pose estimation and hand–instrument interaction reconstruction—and propose two dedicated architectures: H-Net for dual-hand mesh recovery and OH-Net for joint reconstruction of two-hand–two-instrument interactions. These models leverage a novel spatial reasoning module with weak-perspective camera modeling and collision-aware center-based representation. Both architectures outperform existing methods by substantial margins, achieving improvements of over 2mm in Mean Per Joint Position Error (MPJPE) and up to 23\% in ADD-S metrics for hand and instrument reconstruction, respectively.

We present *Fractional Diffusion Bridge Models* (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_\alpha$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.

Object Pose Estimation is a crucial component in robotic grasping and augmented reality. Learning based approaches typically require training data from a highly accurate CAD model or labeled training data acquired using a complex setup. We address this by learning to estimate pose from weakly labeled data without a known CAD model. We propose to use a NeRF to learn object shape implicitly which is later used to learn view-invariant features in conjunction with CNN using a contrastive loss. While NeRF helps in learning features that are view-consistent, CNN ensures that the learned features respect symmetry. During inference, CNN is used to predict view-invariant features which can be used to establish correspondences with the implicit 3d model in NeRF. The correspondences are then used to estimate the pose in the reference frame of NeRF. Our approach can also handle symmetric objects unlike other approaches using a similar training setup. Specifically, we learn viewpoint invariant, discriminative features using NeRF which are later used for pose estimation. We evaluated our approach on LM, LM-Occlusion, and T-Less dataset and achieved benchmark accuracy despite using weakly labeled data.

Faithfully modeling the space of articulations is a cru-cial task that allows recovery and generation of realistic poses, and remains a notorious challenge. To this end, we introduce Neural Riemannian Distance Fields (NRDFs), data-driven priors modeling the space of plausible articu-lations, represented as the zero-level-set of a neural field in a high-dimensional product-quaternion space. To train NRDFs only on positive examples, we introduce a new sam-pling algorithm, ensuring that the geodesic distances fol-low a desired distribution, yielding a principled distance field learning paradigm. We then devise a projection al-gorithm to map any random pose onto the level-set by an adaptive-step Riemannian optimizer, adhering to the product manifold of joint rotations at all times. NRDFs can compute the Riemannian gradient via backpropagation and by mathematical analogy, are related to Rieman-nian flow matching, a recent generative model. We conduct a comprehensive evaluation of NRDF against other pose priors in various downstream tasks, i.e., pose generation, image-based pose estimation, and solving inverse kinematics, highlighting NRDF's superior performance. Besides humans, NRDF's versatility extends to hand and animal poses, as it can effectively represent any articulation.

3D shape generation from text is a fundamental task in 3D representation learning. The text-shape pairs exhibit a hierarchical structure, where a general text like “chair” covers all 3D shapes of the chair, while more detailed prompts refer to more specific shapes. Furthermore, both text and 3D shapes are inherently hierarchical structures. However, existing Text2Shape methods, such as SDFusion, do not exploit that. In this work, we propose HyperSD-Fusion, a dual-branch diffusion model that generates 3D shapes from a given text. Since hyperbolic space is suitable for handling hierarchical data, we propose to learn the hierarchical representations of text and 3D shapes in hyperbolic space. First, we introduce a hyperbolic text-image encoder to learn the sequential and multi-modal hierarchical features of text in hyperbolic space. In addition, we design a hyperbolic text-graph convolution module to learn the hierarchical features of text in hyperbolic space. In order to fully utilize these text features, we introduce a dual-branch structure to embed text features in 3D feature space. At last, to endow the generated 3D shapes with a hierarchical structure, we devise a hyperbolic hierarchical loss. Our method is the first to explore the hyperbolic hierarchical representation for text-to-shape generation. Experimental results on the existing text-to-shape paired dataset, Text2Shape, achieved state-of-the-art results. We release our implementation under HyperSDFusion.github.io.

Principal component analysis (PCA), along with its ex-tensions to manifolds and outlier contaminated data, have been indispensable in computer vision and machine learning. In this work, we present a unifying formalism for PCA and its variants, and introduce a framework based on the flags of linear subspaces, i.e. a hierarchy of nested linear subspaces of increasing dimension, which not only allows for a common implementation but also yields novel variants, not explored previously. We begin by generalizing traditional PCA methods that either maximize variance or minimize reconstruction error. We expand these interpre-tations to develop a wide array of new dimensionality re-duction algorithms by accounting for outliers and the data manifold. To devise a common computational approach, we recast robust and dual forms of peA as optimization problems on flag manifolds. We then integrate tangent space approximations of principal geodesic analysis (tangent-PCA) into this flag-based framework, creating novel robust and dual geodesic PCA variations. The remarkable flexibility offered by the ‘flagification’ introduced here enables even more algorithmic variants identified by specific flag types. Last but not least, we propose an effective convergent solver for these flag-formulations employing the Stiefel manifold. Our empirical results on both real-world and synthetic sce-narios, demonstrate the superiority of our novel algorithms, especially in terms of robustness to outliers on manifolds.

Adiabatic quantum computing (AQC) is a promising approach for discrete and often NP-hard optimization prob-lems. Current AQCs allow to implement problems of re-search interest, which has sparked the development of quan-tum representations for many computer vision tasks. De-spite requiring multiple measurements from the noisy AQC, current approaches only utilize the best measurement, dis-carding information contained in the remaining ones. In this work, we explore the potential of using this information for probabilistic balanced k-means clustering. Instead of discarding non-optimal solutions, we propose to use them to compute calibrated posterior probabilities with little ad-ditional compute cost. This allows us to identify ambiguous solutions and data points, which we demonstrate on a D-Wave AQC on synthetic tasks and real visual data.

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression for optimal approximation coefficients and propose to use neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,—an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

We present a novel set of rigorous and computationally efficient topology-based complexity notions that exhibit a strong correlation with the generalization gap in modern deep neural networks (DNNs). DNNs show remarkable generalization properties, yet the source of these capabilities remains elusive, defying the established statistical learning theory. Recent studies have revealed that properties of training trajectories can be indicative of generalization. Building on this insight, state-of-the-art methods have leveraged the topology of these trajectories, particularly their fractal dimension, to quantify generalization. Most existing works compute this quantity by assuming continuous- or infinite-time training dynamics, complicating the development of practical estimators capable of accurately predicting generalization without access to test data. In this paper, we respect the discrete-time nature of training trajectories and investigate the underlying topological quantities that can be amenable to topological data analysis tools. This leads to a new family of reliable topological complexity measures that provably bound the generalization error, eliminating the need for restrictive geometric assumptions. These measures are computationally friendly, enabling us to propose simple yet effective algorithms for computing generalization indices. Moreover, our flexible framework can be extended to different domains, tasks, and architectures. Our experimental results demonstrate that our new complexity measures exhibit a strong correlation with generalization error in industry-standard architectures such as transformers and deep graph networks. Our approach consistently outperforms existing topological bounds across a wide range of datasets, models, and optimizers, highlighting the practical relevance and effectiveness of our complexity measures.

Seams, distortions, wasted UV space, vertex-duplication, and varying resolution over the surface are the most prominent issues of the standard UV-based texturing of meshes. These issues are particularly acute when automatic UV-unwrapping techniques are used. For this reason, instead of generating textures in automatically generated UV-planes like most state-of-the-art methods, we propose to represent textures as coloured point-clouds whose colours are generated by a denoising diffusion probabilistic model constrained to operate on the surface of 3D objects. Our sampling and resolution agnostic generative model heavily relies on heat diffusion over the surface of the meshes for spatial communication between points. To enable processing of arbitrarily sampled point-cloud textures and ensure long-distance texture consistency we introduce a fast re-sampling of the mesh spectral properties used during the heat diffusion and introduce a novel heat-diffusion-based self-attention mechanism. Our code and pre-trained models are available at github.com/simofoti/UV3-TeD.

We introduce the first continuous-time score-based generative model that leverages fractional diffusion processes for its underlying dynamics. Although diffusion models have excelled at capturing data distributions, they still suffer from various limitations such as slow convergence, mode-collapse on imbalanced data, and lack of diversity. These issues are partially linked to the use of light-tailed Brownian motion (BM) with independent increments. In this paper, we replace BM with an approximation of its non-Markovian counterpart, fractional Brownian motion (fBM), characterized by correlated increments and Hurst index $H \in (0,1)$, where $H=0.5$ recovers the classical BM. To ensure tractable inference and learning, we employ a recently popularized Markov approximation of fBM (MA-fBM) and derive its reverse-time model, resulting in *generative fractional diffusion models* (GFDM). We characterize the forward dynamics using a continuous reparameterization trick and propose *augmented score matching* to efficiently learn the score function, which is partly known in closed form, at minimal added cost. The ability to drive our diffusion model via MA-fBM offers flexibility and control. $H \leq 0.5$ enters the regime of *rough paths* whereas $H>0.5$ regularizes diffusion paths and invokes long-term memory. The Markov approximation allows added control by varying the number of Markov processes linearly combined to approximate fBM. Our evaluations on real image datasets demonstrate that GFDM achieves greater pixel-wise diversity and enhanced image quality, as indicated by a lower FID, offering a promising alternative to traditional diffusion models

We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io.

We propose VidStyleODE, a spatiotemporally continuous disentangled video representation based upon StyleGAN and Neural-ODEs. Effective traversal of the latent space learned by Generative Adversarial Networks (GANs) has been the basis for recent breakthroughs in image editing. However, the applicability of such advancements to the video domain has been hindered by the difficulty of representing and controlling videos in the latent space of GANs. In particular, videos are composed of content (i.e., appearance) and complex motion components that require a special mechanism to disentangle and control. To achieve this, VidStyleODE encodes the video content in a pre-trained StyleGAN ${\mathcal{W}_ + }$ space and benefits from a latent ODE component to summarize the spatiotemporal dynamics of the input video. Our novel continuous video generation process then combines the two to generate high-quality and temporally consistent videos with varying frame rates. We show that our proposed method enables a variety of applications on real videos: text-guided appearance manipulation, motion manipulation, image animation, and video interpolation and extrapolation. Project website: https://cyberiada.github.io/VidStyleODE

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems.To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis.

We present SyNoRiM, a novel way to jointly register multiple non-rigid shapes by synchronizing the maps that relate learned functions defined on the point clouds. Even though the ability to process non-rigid shapes is critical in various applications ranging from computer animation to 3D digitization, the literature still lacks a robust and flexible framework to match and align a collection of real, noisy scans observed under occlusions. Given a set of such point clouds, our method first computes the pairwise correspondences parameterized via functional maps. We simultaneously learn potentially non-orthogonal basis functions to effectively regularize the deformations, while handling the occlusions in an elegant way. To maximally benefit from the multi-way information provided by the inferred pairwise deformation fields, we synchronize the pairwise functional maps into a cycle-consistent whole thanks to our novel and principled optimization formulation. We demonstrate via extensive experiments that our method achieves a state-of-the-art performance in registration accuracy, while being flexible and efficient as we handle both non-rigid and multi-body cases in a unified framework and avoid the costly optimization over point-wise permutations by the use of basis function maps.

Regressing rotations on SO(3) manifold using deep neural networks is an important yet unsolved problem. The gap between the Euclidean network output space and the non-Euclidean SO(3) manifold imposes a severe challenge for neural network learning in both forward and backward passes. While several works have proposed different regression-friendly rotation representations, very few works have been devoted to improving the gradient back-propagating in the backward pass. In this paper, we propose a manifold-aware gradient that directly backpropagates into deep network weights. Leveraging Riemannian optimization to construct a novel projective gradient, our proposed regularized projective manifold gradient (RPMG) method helps networks achieve new state-of-the-art performance in a variety of rotation estimation tasks. Our proposed gradient layer can also be applied to other smooth manifolds such as the unit sphere. Our project page is at https://jychen18.github.io/RPMG.

We propose Point2Cyl, a supervised network transforming a raw 3D point cloud to a set of extrusion cylinders. Reverse engineering from a raw geometry to a CAD model is an essential task to enable manipulation of the 3D data in shape editing software and thus expand their usages in many downstream applications. Particularly, the form of CAD models having a sequence of extrusion cylinders - a 2D sketch plus an extrusion axis and range - and their boolean combinations is not only widely used in the CAD community/software but also has great expressivity of shapes, compared to having limited types of primitives (e.g., planes, spheres, and cylinders). In this work, we introduce a neural network that solves the extrusion cylinder decomposition problem in a geometry-grounded way by first learning underlying geometric proxies. Precisely, our approach first predicts per-point segmentation, base/barrel labels and normals, then estimates for the underlying extrusion parameters in differentiable and closed-form formulations. Our experiments show that our approach demonstrates the best performance on two recent CAD datasets, Fusion Gallery and DeepCAD, and we further showcase our approach on reverse engineering and editing.

for learning robust, flexible and generalizable 3D object representations without requiring heavy annotation efforts or supervision. Unlike conventional 3D generative models, our algorithm aims for building a structured latent space where certain factors of shape variations, such as object parts, can be disentangled into independent sub-spaces. Our novel decoder then acts on these individual latent sub-spaces (i.e. capsules) using deconvolution operators to reconstruct 3D points in a self-supervised manner. We further introduce a cluster loss ensuring that the points reconstructed by a single capsule remain local and do not spread across the object uncontrollably. These contributions allow our network to tackle the challenging tasks of part segmentation, part interpolation/replacement as well as correspondence estimation across rigid / non-rigid shape, and across / within category. Our extensive evaluations on ShapeNet objects and human scans demonstrate that our network can learn generic representations that are robust and useful in many applications.

We present QuantumSync, the first quantum algorithm for solving a synchronization problem in the context of computer vision. In particular, we focus on permutation synchronization which involves solving a non-convex optimization problem in discrete variables. We start by formulating synchronization into a quadratic unconstrained binary optimization problem (QUBO). While such formulation respects the binary nature of the problem, ensuring that the result is a set of permutations requires extra care. Hence, we: (i) show how to insert permutation constraints into a QUBO problem and (ii) solve the constrained QUBO problem on the current generation of the adiabatic quantum computers D-Wave. Thanks to the quantum annealing, we guarantee global optimality with high probability while sampling the energy landscape to yield confidence estimates. Our proof-of-concepts realization on the adiabatic D-Wave computer demonstrates that quantum machines offer a promising way to solve the prevalent yet difficult synchronization problems.

We propose a data-driven scene flow estimation algorithm exploiting the observation that many 3D scenes can be explained by a collection of agents moving as rigid bodies. At the core of our method lies a deep architecture able to reason at the object-level by considering 3D scene flow in conjunction with other 3D tasks. This object level abstraction enables us to relax the requirement for dense scene flow supervision with simpler binary background segmentation mask and ego-motion annotations. Our mild supervision requirements make our method well suited for recently released massive data collections for autonomous driving, which do not contain dense scene flow annotations. As output, our model provides low-level cues like pointwise flow and higher-level cues such as holistic scene understanding at the level of rigid objects. We further propose a test-time optimization refining the predicted rigid scene flow. We showcase the effectiveness and generalization capacity of our method on four different autonomous driving datasets. We release our source code and pre-trained models under github.com/zgojcic/Rigid3DSceneFlow.

We present MultiBodySync, a novel, end-to-end trainable multi-body motion segmentation and rigid registration framework for multiple input 3D point clouds. The two non-trivial challenges posed by this multi-scan multibody setting that we investigate are: (i) guaranteeing correspondence and segmentation consistency across multiple input point clouds capturing different spatial arrangements of bodies or body parts; and (ii) obtaining robust motion-based rigid body segmentation applicable to novel object categories. We propose an approach to address these issues that incorporates spectral synchronization into an iterative deep declarative network, so as to simultaneously recover consistent correspondences as well as motion segmentation. At the same time, by explicitly disentangling the correspondence and motion segmentation estimation modules, we achieve strong generalizability across different object categories. Our extensive evaluations demonstrate that our method is effective on various datasets ranging from rigid parts in articulated objects to individually moving objects in a 3D scene, be it single-view or full point clouds. Code at https://github.com/huangjh-pub/multibody-sync.

We introduce HuMoR: a 3D Human Motion Model for Robust Estimation of temporal pose and shape. Though substantial progress has been made in estimating 3D human motion and shape from dynamic observations, recovering plausible pose sequences in the presence of noise and occlusions remains a challenge. For this purpose, we propose an expressive generative model in the form of a conditional variational autoencoder, which learns a distribution of the change in pose at each step of a motion sequence. Furthermore, we introduce a flexible optimization-based approach that leverages HuMoR as a motion prior to robustly estimate plausible pose and shape from ambiguous observations. Through extensive evaluations, we demonstrate that our model generalizes to diverse motions and body shapes after training on a large motion capture dataset, and enables motion reconstruction from multiple input modalities including 3D keypoints and RGB(-D) videos. See the project page at geometry.stanford.edu/projects/humor.

Disobeying the classical wisdom of statistical learning theory, modern deep neural networks generalize well even though they typically contain millions of parameters. Recently, it has been shown that the trajectories of iterative optimization algorithms can possess \emph{fractal structures}, and their generalization error can be formally linked to the complexity of such fractals. This complexity is measured by the fractal's \emph{intrinsic dimension}, a quantity usually much smaller than the number of parameters in the network. Even though this perspective provides an explanation for why overparametrized networks would not overfit, computing the intrinsic dimension (\eg, for monitoring generalization during training) is a notoriously difficult task, where existing methods typically fail even in moderate ambient dimensions. In this study, we consider this problem from the lens of topological data analysis (TDA) and develop a generic computational tool that is built on rigorous mathematical foundations. By making a novel connection between learning theory and TDA, we first illustrate that the generalization error can be equivalently bounded in terms of a notion called the 'persistent homology dimension' (PHD), where, compared with prior work, our approach does not require any additional geometrical or statistical assumptions on the training dynamics. Then, by utilizing recently established theoretical results and TDA tools, we develop an efficient algorithm to estimate PHD in the scale of modern deep neural networks and further provide visualization tools to help understand generalization in deep learning. Our experiments show that the proposed approach can efficiently compute a network's intrinsic dimension in a variety of settings, which is predictive of the generalization error.

The majority of descriptor-based methods for geometric processing of non-rigid shape rely on hand-crafted descriptors. Recently, learning-based techniques have been shown effective, achieving state-of-the-art results in a variety of tasks. Yet, even though these methods can in principle work directly on raw data, most methods still rely on handcrafted descriptors at the input layer. In this work, we wish to challenge this practice and use a neural network to learn descriptors directly from the raw mesh. To this end, we introduce two modules into our neural architecture. The first is a local reference frame (LRF) used to explicitly make the features invariant to rigid transformations. The second is continuous convolution kernels that provide robustness to sampling. We show the efficacy of our proposed network in learning on raw meshes using two cornerstone tasks: shape matching, and human body parts segmentation. Our results show superior results over baseline methods that use handcrafted descriptors.

We introduce a new paradigm, `measure synchronization', for synchronizing graphs with measure-valued edges. We formulate this problem as maximization of the cycle-consistency in the space of probability measures over relative rotations. In particular, we aim at estimating marginal distributions of absolute orientations by synchronizing the `conditional' ones, which are defined on the Riemannian manifold of quaternions. Such graph optimization on distributions-on-manifolds enables a natural treatment of multimodal hypotheses, ambiguities and uncertainties arising in many computer vision applications such as SLAM, SfM, and object pose estimation. We first formally define the problem as a generalization of the classical rotation graph synchronization, where in our case the vertices denote probability measures over rotations. We then measure the quality of the synchronization by using Sinkhorn divergences, which reduces to other popular metrics such as Wasserstein distance or the maximum mean discrepancy as limit cases. We propose a nonparametric Riemannian particle optimization approach to solve the problem. Even though the problem is non-convex, by drawing a connection to the recently proposed sparse optimization methods, we show that the proposed algorithm converges to the global optimum in a special case of the problem under certain conditions. Our qualitative and quantitative experiments show the validity of our approach and we bring in new perspectives to the study of synchronization.

We present a novel, end-to-end learnable, multiview 3D point cloud registration algorithm. Registration of multiple scans typically follows a two-stage pipeline: the initial pairwise alignment and the globally consistent refinement. The former is often ambiguous due to the low overlap of neighboring point clouds, symmetries and repetitive scene parts. Therefore, the latter global refinement aims at establishing the cyclic consistency across multiple scans and helps in resolving the ambiguous cases. In this paper we propose, to the best of our knowledge, the first end-to-end algorithm for joint learning of both parts of this two-stage problem. Experimental evaluation on well accepted benchmark datasets shows that our approach outperforms the state-of-the-art by a significant margin, while being end-to-end trainable and computationally less costly. Moreover, we present detailed analysis and an ablation study that validate the novel components of our approach. The source code and pretrained models are publicly available under https://github.com/zgojcic/3D_multiview_reg.

We propose CaSPR, a method to learn object-centric Canonical Spatiotemporal Point Cloud Representations of dynamically moving or evolving objects. Our goal is to enable information aggregation over time and the interrogation of object state at any spatiotemporal neighborhood in the past, observed or not. Different from previous work, CaSPR learns representations that support spacetime continuity, are robust to variable and irregularly spacetime-sampled point clouds, and generalize to unseen object instances. Our approach divides the problem into two subtasks. First, we explicitly encode time by mapping an input point cloud sequence to a spatiotemporally-canonicalized object space. We then leverage this canonicalization to learn a spatiotemporal latent representation using neural ordinary differential equations and a generative model of dynamically evolving shapes using continuous normalizing flows. We demonstrate the effectiveness of our method on several applications including shape reconstruction, camera pose estimation, continuous spatiotemporal sequence reconstruction, and correspondence estimation from irregularly or intermittently sampled observations.

We present a novel and effective method for detecting 3D primitives in cluttered, unorganized point clouds, without axillary segmentation or type specification. We consider the quadric surfaces for encapsulating the basic building blocks of our environments - planes, spheres, ellipsoids, cones or cylinders, in a unified fashion. Moreover, quadrics allow us to model higher degree of freedom shapes, such as hyperboloids or paraboloids that could be used in non-rigid settings. We begin by contributing two novel quadric fits targeting 3D point sets that are endowed with tangent space information. Based upon the idea of aligning the quadric gradients with the surface normals, our first formulation is exact and requires as low as four oriented points. The second fit approximates the first, and reduces the computational effort. We theoretically analyze these fits with rigor, and give algebraic and geometric arguments. Next, by re-parameterizing the solution, we devise a new local Hough voting scheme on the null-space coefficients that is combined with RANSAC, reducing the complexity from O(N4) to O(N3) (three points). To the best of our knowledge, this is the first method capable of performing a generic cross-type multi-object primitive detection in difficult scenes without segmentation. Our extensive qualitative and quantitative results show that our method is efficient and flexible, as well as being accurate.

We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the relaxed version of the combinatorially intractable cycle consistency loss in a principled manner, (2) Birkhoff-Riemannian Langevin Monte Carlo for generating samples on the Birkhoff Polytope and estimating the confidence of the found solutions. To this end, we first introduce the very recently developed Riemannian geometry of the Birkhoff Polytope. Next, we introduce a new probabilistic synchronization model in the form of a Markov Random Field (MRF). Finally, based on the first order retraction operators, we formulate our problem as simulating a stochastic differential equation and devise new integrators. We show on both synthetic and real datasets that we achieve high quality multi-graph matching results with faster convergence and reliable confidence/uncertainty estimates.

We present a novel, data driven approach for solving the problem of registration of two point cloud scans. Our approach is direct in the sense that a single pair of corresponding local patches already provides the necessary transformation cue for the global registration. To achieve that, we first endow the state of the art PPF-FoldNet auto-encoder (AE) with a pose-variant sibling, where the discrepancy between the two leads to pose-specific descriptors. Based upon this, we introduce RelativeNet, a relative pose estimation network to assign correspondence-specific orientations to the keypoints, eliminating any local reference frame computations. Finally, we devise a simple yet effective hypothesize-and-verify algorithm to quickly use the predictions and align two point sets. Our extensive quantitative and qualitative experiments suggests that our approach outperforms the state of the art in challenging real datasets of pairwise registration and that augmenting the keypoints with local pose information leads to better generalization and a dramatic speed-up.

In this paper, we propose 3D point-capsule networks, an auto-encoder designed to process sparse 3D point clouds while preserving spatial arrangements of the input data. 3D capsule networks arise as a direct consequence of our unified formulation of the common 3D auto-encoders. The dynamic routing scheme and the peculiar 2D latent space deployed by our capsule networks bring in improvements for several common point cloud-related tasks, such as object classification, object reconstruction and part segmentation as substantiated by our extensive evaluations. Moreover, it enables new applications such as part interpolation and replacement.

3D object detection and pose estimation from a single image are two inherently ambiguous problems. Oftentimes, objects appear similar from different viewpoints due to shape symmetries, occlusion and repetitive textures. This ambiguity in both detection and pose estimation means that an object instance can be perfectly described by several different poses and even classes. In this work we propose to explicitly deal with these ambiguities. For each object instance we predict multiple 6D pose outcomes to estimate the specific pose distribution generated by symmetries and repetitive textures. The distribution collapses to a single outcome when the visual appearance uniquely identifies just one valid pose. We show the benefits of our approach which provides not only a better explanation for pose ambiguity, but also a higher accuracy in terms of pose estimation.

In this survey we carefully analyze the characteristics of higher order rigid body motion interpolation methods to obtain a continuous trajectory from a discrete set of poses. We first discuss the tradeoff between continuity, local control and approximation of classical Euclidean interpolation schemes such as Bezier and B-splines. The benefits of the manifold of unit quaternions SU(2), a double-cover of rotation matrices SO(3), as rotation parameterization are presented, which allow for an elegant formulation of higher order orientation interpolation with easy analytic derivatives, made possible through the Lie Algebra su(2) of pure quaternions and the cumulative form of cubic B-splines. The same construction scheme is then applied for joint interpolation in the full rigid body pose space, which had previously been done for the matrix representation SE(3) and its twists, but not for the more efficient unit dual quaternion DH1 and its screw motions. Both suffer from the effects of coupling translation and rotation that have mostly been ignored by previous work. We thus conclude that split interpolation in ℝ3× SU(2) is preferable for most applications. Our final runtime experiments show that joint interpolation in SE(3) is 2 times and in DH11.3 times slower - which furthermore justifies our suggestion from a practical point of view.

This paper proposes a segmentation-free, automatic and efficient procedure to detect general geometric quadric forms in point clouds, where clutter and occlusions are inevitable. Our everyday world is dominated by man-made objects which are designed using 3D primitives (such as planes, cones, spheres, cylinders, etc.). These objects are also omnipresent in industrial environments. This gives rise to the possibility of abstracting 3D scenes through primitives, thereby positions these geometric forms as an integral part of perception and high level 3D scene understanding. As opposed to state-of-the-art, where a tailored algorithm treats each primitive type separately, we propose to encapsulate all types in a single robust detection procedure. At the center of our approach lies a closed form 3D quadric fit, operating in both primal & dual spaces and requiring as low as 4 oriented-points. Around this fit, we design a novel, local null-space voting strategy to reduce the 4-point case to 3. Voting is coupled with the famous RANSAC and makes our algorithm orders of magnitude faster than its conventional counterparts. This is the first method capable of performing a generic cross-type multi-object primitive detection in difficult scenes. Results on synthetic and real datasets support the validity of our method.

We present PPFNet - Point Pair Feature NETwork for deeply learning a globally informed 3D local feature descriptor to find correspondences in unorganized point clouds. PPFNet learns local descriptors on pure geometry and is highly aware of the global context, an important cue in deep learning. Our 3D representation is computed as a collection of point-pair-features combined with the points and normals within a local vicinity. Our permutation invariant network design is inspired by PointNet and sets PPFNet to be ordering-free. As opposed to voxelization, our method is able to consume raw point clouds to exploit the full sparsity. PPFNet uses a novel N-tuple loss and architecture injecting the global information naturally into the local descriptor. It shows that context awareness also boosts the local feature representation. Qualitative and quantitative evaluations of our network suggest increased recall, improved robustness and invariance as well as a vital step in the 3D descriptor extraction performance.

We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of solutions. We devise theoretical convergence guarantees and extensively evaluate our method on synthetic and real benchmarks. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data.

We present an efficient and automatic approach for accurate instance reconstruction of big 3D objects from multiple, unorganized and unstructured point clouds, in presence of dynamic clutter and occlusions. In contrast to conventional scanning, where the background is assumed to be rather static, we aim at handling dynamic clutter where the background drastically changes during object scanning. Currently, it is tedious to solve this problem with available methods unless the object of interest is first segmented out from the rest of the scene. We address the problem by assuming the availability of a prior CAD model, roughly resembling the object to be reconstructed. This assumption almost always holds in applications such as industrial inspection or reverse engineering. With aid of this prior acting as a proxy, we propose a fully enhanced pipeline, capable of automatically detecting and segmenting the object of interest from scenes and creating a pose graph, online, with linear complexity. This allows initial scan alignment to the CAD model space, which is then refined without the CAD constraint to fully recover a high fidelity 3D reconstruction, accurate up to the sensor noise level. We also contribute a novel object detection method, local implicit shape models (LISM) and give a fast verification scheme. We evaluate our method on multiple datasets, demonstrating the ability to accurately reconstruct objects from small sizes up to 125m3.

We present a 3D mesh re-sampling algorithm, carefully tailored for 3D object detection using point pair features (PPF). Computing a sparse representation of objects is critical for the success of state-of-the-art object detection, recognition and pose estimation methods. Yet, sparsity needs to preserve fidelity. To this end, we develop a simple, yet very effective point sampling strategy for detection of any CAD model through geometric hashing. Our approach relies on rendering the object coordinates from a set of views evenly distributed on a sphere. Actual sampling takes place on 2D domain over these renderings; the resulting samples are efficiently merged in 3D with the aid of a special voxel structure and relaxed with Lloyd iterations. The generated vertices are not concentrated only on critical points, as in many keypoint extraction algorithms, and there is even spacing between selected vertices. This is valuable for quantization based detection methods, such as geometric hashing of point pair features. The algorithm is fast and can easily handle the elongated/acute triangles and sharp edges typically existent in industrial CAD models, while automatically pruning the invisible structures. We do not introduce structural changes such as smoothing or interpolation and sample the normals on the original CAD model, achieving the maximum fidelity. We demonstrate the strength of this approach on 3D object detection in comparison to similar sampling algorithms.

In this paper we design a novel planar 2D fiducial marker and develop fast detection algorithm aiming easy camera calibration and precise 3D reconstruction at the marker locations via the bundle adjustment. Even though an abundance of planar fiducial markers have been made and used in various tasks, none of them has properties necessary to solve the aforementioned tasks. Our marker, X-tag, enjoys a novel design, coupled with very efficient and robust detection scheme, resulting in a reduced number of false positives. This is achieved by constructing markers with random circular features in the image domain and encoding them using two true perspective invariants: cross-ratios and intersection preservation constraints. To detect the markers, we developed an effective search scheme, similar to Geometric Hashing and Hough Voting, in which the marker decoding is cast as a retrieval problem. We apply our system to the task of camera calibration and bundle adjustment. With qualitative and quantitative experiments, we demonstrate the robustness and accuracy of X-tag in spite of blur, noise, perspective and radial distortions, and showcase camera calibration, bundle adjustment and 3d fusion of depth data from precise extrinsic camera poses.

The raising standards in manufacturing demands reliable and fast industrial quality control mechanisms. This paper proposes an accurate, yet easy to install multi-view, close range optical metrology system, which is suited to online operation. The system is composed of multiple static, locally overlapping cameras forming a network. Initially, these cameras are calibrated to obtain a global coordinate frame. During run-time, the measurements are performed via a novel geometry extraction techniques coupled with an elegant projective registration framework, where 3D to 2D fitting energies are minimized. Finally, a non-linear regression is carried out to compensate for the uncontrollable errors. We apply our pipeline to inspect various geometrical structures found on automobile parts. While presenting the implementation of an involved 3D metrology system, we also demonstrate that the resulting inspection is as accurate as 0.2 mm, repeatable and much faster, compared to the existing methods such as coordinate measurement machines (CMM) or ATOS.

We present a revised pipe-line of the existing 3D object detection and pose estimation framework based on point pair feature matching. This framework proposed to represent 3D target object using self-similar point pairs, and then matching such model to 3D scene using efficient Hough-like voting scheme operating on the reduced pose parameter space. Even though this work produces great results and motivated a large number of extensions, it had some general shortcoming like relatively high dimensionality of the search space, sensitivity in establishing 3D correspondences, having performance drops in presence of many outliers and low density surfaces. In this paper, we explain and address these drawbacks and propose new solutions within the existing framework. In particular, we propose to couple the object detection with a coarse-to-fine segmentation, where each segment is subject to disjoint pose estimation. During matching, we apply a weighted Hough voting and an interpolated recovery of pose parameters. Finally, all the generated hypothesis are tested via an occlusion-aware ranking and sorted. We argue that such a combined pipeline simultaneously boosts the detection rate and reduces the complexity, while improving the accuracy of the resulting pose. Thanks to such enhanced pose retrieval, our verification doesn't necessitate ICP and thus achieves better compromise of speed vs. accuracy. We demonstrate our method on existing datasets as well as on our scenes. We conclude that via the new pipe-line, point pair features can now be used in more challenging scenarios.

Both consumer market and manufacturing industry makes heavy use of 1D (linear) barcodes. From helping the visually impaired to identifying the products to industrial automated industry management, barcodes are the prevalent source of item tracing technology. Because of this ubiquitous use, in recent years, many algorithms have been proposed targeting barcode decoding from high-accessibility devices such as cameras. However, the current methods have at least one of the two major problems: 1) they are sensitive to blur, perspective/ lens distortions, and non-linear deformations, which often occur in practice, 2) they are specifically designed for a specific barcode symbology (such as UPC-A) and cannot be applied to other symbologies. In this paper, we aim to address these problems and present a dynamic Bayesian network in order to robustly model all kinds of linear progressive barcodes. We apply our method on various barcode datasets and compare the performance with the state-of-the-art. Our experiments show that, as well as being applicable to all progressive barcode types, our method provides competitive results in clean UPC-A datasets and outperforms the state-of-the-art in difficult scenarios.