Non-rigid Structure From Motion Research Articles

A Closed-Form, Pairwise Solution to Local Non-Rigid Structure-from-Motion.

A recent trend in Non-Rigid Structure-from-Motion (NRSfM) is to express local, differential constraints between pairs of images, from which the surface normal at any point can be obtained by solving a system of polynomial equations. While this approach is more successful than its counterparts relying on global constraints, the resulting methods face two main problems: First, most of the equationsystems they formulate are of high degree and must be solved using computationally expensive polynomial solvers. Some methods use polynomial reduction strategies to simplify the system, but this adds some phantom solutions. In any event, an additional mechanism is employed to pick the best solution, which adds to the computation without any guarantees on the reliability of the solution. Second, these methods formulate constraints between a pair of images. Even if there is enough motion between them, they may suffer from local degeneracies that make the resulting estimates unreliable without any warning mechanism. %Unfortunately, these systems are of high degree with up to five real solutions. Hence, a computationally expensive strategy is required to select a unique solution. Furthermore, they suffer from degeneracies that make the resulting estimates unreliable, without any mechanism to identify this situation. In this paper, we solve these problems for isometric/conformal NRSfM. We show that, under widely applicable assumptions, we can derive a new system of equationsin terms of the surface normals, whose two solutions can be obtained in closed-form and can easily be disambiguated locally. Our formalism also allows us to assess how reliable the estimated local normals are and to discard them if they are not. Our experiments show that our reconstructions, obtained from two or more views, are significantly more accurate than those of state-of-the-art methods, while also being faster. %In this paper, we show that, under widely applicable assumptions, we can derive a new system of equationsin terms of the surface normals, whose two solutions can be obtained in closed-form and can easily be disambiguated locally. Our formalism also allows us to assess how reliable the estimated local normals are and to discard them if they are not. Our experiments show that our reconstructions, obtained from two or more views, are significantly more accurate than those of state-of-the-art methods, while also being faster.

Temporal-Aware Neural Network for Dense Non-Rigid Structure from Motion

Modern neural networks addressing dense Non-Rigid Structure from Motion (NRSFM) dilemmas often grapple with intricate a priori constraints, deterring scalability, or overlook the imperative of consistent application of a priori knowledge throughout the entire input sequence. In this paper, an innovative neural network architecture is introduced. Initially, the complete 2D sequence image undergoes embedding into a low-dimensional space. Subsequently, multiple self-attention layers are employed to extract inter-frame features, with the objective of deriving a more continuous and temporally smooth low-dimensional structure closely resembling real data’s intrinsic structure. Moreover, it has been demonstrated by others that gradient descent during the training of multilayer linear networks yields minimum rank solutions, implicitly providing regularization that is equally applicable to this task. Benefiting from the excellence of the proposed network architecture, no additional a priori knowledge is mandated, barring the constraint of temporal smoothness. Extensive experimentation confirms the method’s exceptional performance in addressing dense NRSFM challenges, outperforming recent results across various dense benchmark datasets.

Deep Spatial-Temporal Neural Network for Dense Non-Rigid Structure from Motion

Dense non-rigid structure from motion (NRSfM) has long been a challenge in computer vision because of the vast number of feature points. As neural networks develop rapidly, a novel solution is emerging. However, existing methods ignore the significance of spatial–temporal data and the strong capacity of neural networks for learning. This study proposes a deep spatial–temporal NRSfM framework (DST-NRSfM) and introduces a weighted spatial constraint to further optimize the 3D reconstruction results. Layer normalization layers are applied in dense NRSfM tasks to stop gradient disappearance and hasten neural network convergence. Our DST-NRSfM framework outperforms both classical approaches and recent advancements. It achieves state-of-the-art performance across commonly used synthetic and real benchmark datasets.

Robust Isometric Non-Rigid Structure-From-Motion.

Non-Rigid Structure-from-Motion (NRSfM) reconstructs a deformable 3D object from keypoint correspondences established between monocular 2D images. Current NRSfM methods lack statistical robustness, which is the ability to cope with correspondence errors. This prevents one to use automatically established correspondences, which are prone to errors, thereby strongly limiting the scope of NRSfM. We propose a three-step automatic pipeline to solve NRSfM robustly by exploiting isometry. Step (i) computes the optical flow from correspondences, step (ii) reconstructs each 3D point's normal vector using multiple reference images and integrates them to form surfaces with the best reference and step (iii) rejects the 3D points that break isometry in their local neighborhood. Importantly, each step is designed to discard or flag erroneous correspondences. Our contributions include the robustification of optical flow by warp estimation, new fast analytic solutions to local normal reconstruction and their robustification, and a new scale-independent measure of 3D local isometric coherence. Experimental results show that our robust NRSfM method consistently outperforms existing methods on both synthetic and real datasets.

A Closed-Form Uncertainty Propagation in Non-Rigid Structure From Motion

Semi-Definite Programming (SDP) with low-rank prior has been widely applied in Non-Rigid Structure from Motion (NRSfM). Based on a low-rank constraint, it avoids the inherent ambiguity of basis number selection in conventional base-shape or base-trajectory methods. Despite the efficiency in deformable shape reconstruction, it remains unclear how to assess the uncertainty of the recovered shape from the SDP process. In this paper, we present a statistical inference on the element-wise uncertainty quantification of the estimated deforming 3D shape points in the case of the exact low-rank SDP problem. A closed-form uncertainty quantification method is proposed and tested. Moreover, we extend the exact low-rank uncertainty quantification to the approximate low-rank scenario with a numerical optimal rank selection method, which enables solving practical application in SDP based NRSfM scenario. The proposed method provides an independent module to the SDP method and only requires the statistic information of the input 2D tracked points. Extensive experiments prove that the output 3D points have identical normal distribution to the 2D trackings, the proposed method and quantify the uncertainty accurately, and supports that it has desirable effects on routinely SDP low-rank based NRSfM solver.

Colonoscopic 3D reconstruction by tubular non-rigid structure-from-motion.

The visual examination of colonoscopic images fails to extract precise geometric information of the colonic surface. Reconstructing the 3D surface of the colon from colonoscopic image sequences may thus add valuable clinical information. We address this problem of extracting precise spatio-temporal 3D structure information from colonoscopic images. Using just the intrinsically calibrated monocular image stream, we develop a technique to compute the depth of certain feature points that have been tracked across images. Our method uses the prior knowledge of an approximate geometry of the colon, called the (TTP). It works by fitting a deformable cylindrical model to points reconstructed independently by non-rigid structure-from-motion (NRSfM), compromising between the data term and a novel tubular smoothing prior. Our method represents the first method ever to exploit a very weak topological prior to improve NRSfM. As such, it lies in-between standard NRSfM, which does not use a topological prior beyond the mere plane, and shape-from-template (SfT), which uses a very strong prior as a full deformable 3D object model. We validate our method on both synthetic images of tubular structures and real colonoscopic data. Our method improves the results obtained by existing NRSfM methods by 71.74% on average on synthetic data and succeeds in obtaining 3D reconstruction from a real colonoscopic sequence defeating the existing methods. Colonoscopic 3D reconstruction is a difficult problem, which is yet unresolved by the existing methods from computer vision. Our proposed dedicated NRSfM method and experiments show that the visual motion might be the right visual cue to use in colonoscopy.

Reconstruct as Far as You Can: Consensus of Non-Rigid Reconstruction from Feasible Regions.

Much progress has been made for non-rigid structure from motion (NRSfM) during the last two decades, which made it possible to provide reasonable solutions for synthetically-created benchmark data. In order to utilize these NRSfM techniques in more realistic situations, however, we are now facing two important problems that must be solved: First, general scenes contain complex deformations as well as multiple objects, which violates the usual assumptions of previous NRSfM proposals. Second, there are many unreconstructable regions in the video, either because of the discontinued tracks of 2D trajectories or those regions static towards the camera, which require careful manipulations. In this paper, we show that a consensus-based reconstruction framework can handle these issues effectively. Even though the entire scene is complex, its parts usually have simpler deformations, and even though there are some unreconstructable parts, they can be weeded out to reduce their harmful effect on the entire reconstruction. The main difficulty of this approach lies in identifying appropriate parts, however, it can be effectively avoided by sampling parts stochastically and then aggregate their reconstructions afterwards. Experimental results show that the proposed method renews the state-of-the-art for popular benchmark data under much harsher environments, i.e., narrow camera view ranges, and it can reconstruct video-based real-world data effectively for as many areas as it can without an elaborated user input.

NRSfPP: non-rigid structure-from-perspective projection

A state-of-the-art algorithm for perspective projection reconstruction of non-rigid surfaces from single-view and realistic videos is proposed. It overcomes the limitations arising from the usage of orthographic camera model and also the complexity and non-linearity issues of perspective projection equation. Unlike traditional non-rigid structure-from-motion (NRSfM) methods, which have been studied only on synthetic datasets and controlled lab environments that require some prior constraints (such as manually segmented objects, limited rotations and occlusions, and full-length trajectories); the proposed method can be used in realistic video sequences. In addition, contrary to previous methods that use multiple cameras with different relative viewing angles, only a single-view video is required to reconstruct the 3D structures. By only using the 2D frames of incoming video stream, the proposed method extracts the projective depth coefficients of each point in each input frame, rotation matrix, translation vector, varying camera parameters (such as focal lengths for each input frame), and finally reconstructs the 3D deformable shape. Due to the high number of unknowns, the problem has been divided into two parts of projective depth coefficients extraction and 3D shape reconstruction. Perspective reconstruction of non-rigid surfaces has been extended to be certainly converged which leads to a significant increase in execution frequency of the iterated algorithm that has been presented for projective depth coefficients extraction. As such, it produces promising results for perspective projection reconstruction of non-rigid surfaces from single-view and realistic videos. The accuracy and robustness of the proposed method is demonstrated quantitatively on synthetic data and qualitatively on real image sequences. The experimental results show that NRSfPP provides the state-of-the-art results and resolves the failures of previous approaches by eliminating all of the predefined situations and constraints of applying perspective projection as the camera model.

Local Deformable 3D Reconstruction with Cartan's Connections.

3D reconstruction of deformable objects using inter-image visual motion from monocular images has been studied under Shape-from-Template (SfT) and Non-Rigid Structure-from-Motion (NRSfM). Most methods have been developed for simple deformation models, primarily isometry. They may treat a surface as a discrete set of points and draw constraints from the points only or they may use a non-parametric representation and use both points and differentials to express constraints. We propose a differential framework based on Cartan's theory of connections and moving frames. It is applicable to SfT and NRSfM, and to deformation models other than isometry. It utilises infinitesimal-level assumptions on the surface's geometry and mappings. It has the following properties. 1) It allows one to derive existing solutions in a simpler way. 2) It models SfT and NRSfM in a unified way. 3) It allows us to introduce a new skewless deformation model and solve SfT and NRSfM for it. 4) It facilitates a generic solution to SfT which does not require deformation modeling. Our framework is complete: it solves deformable 3D reconstruction for a whole class of algebraic deformation models including isometry. We compared our solutions with the state-of-the-art methods and show that ours outperform in terms of both accuracy and computation time.

DefSLAM: Tracking and Mapping of Deforming Scenes From Monocular Sequences

Monocular simultaneous localization and mapping (SLAM) algorithms perform robustly when observing rigid scenes; however, they fail when the observed scene deforms, for example, in medical endoscopy applications. In this article, we present DefSLAM, the first monocular SLAM capable of operating in deforming scenes in real time. Our approach intertwines Shape-from-Template (SfT) and Non-Rigid Structure-from-Motion (NRSfM) techniques to deal with the exploratory sequences typical of SLAM. A deformation tracking thread recovers the pose of the camera and the deformation of the observed map, at frame rate, by means of SfT processing a template that models the scene shape-at-rest. A deformation mapping thread runs in parallel with the tracking to update the template, at keyframe rate, by means of an isometric NRSfM processing a batch of full perspective keyframes. In our experiments, DefSLAM processes close-up sequences of deforming scenes, both in a laboratory-controlled experiment and in medical endoscopy sequences, producing accurate 3-D models of the scene with respect to the moving camera.

Solving Rolling Shutter 3D Vision Problems using Analogies with Non-rigidity

We propose an original approach to absolute pose and structure-from-motion (SfM) which handles rolling shutter (RS) effects. Unlike most existing methods which either augment global shutter projection with velocity parameters or impose continuous time and motion through pose interpolation, we use local differential constraints. These are established by drawing analogies with non-rigid 3D vision techniques, namely shape-from-template and non-rigid SfM (NRSfM). The proposed idea is to interpret the images of a rigid surface acquired by a moving RS camera as those of a virtually deformed surface taken by a GS camera. These virtually deformed surfaces are first recovered by relaxing the RS constraint using SfT or NRSfM. Then we upgrade the virtually deformed surface to the actual rigid structure and compute the camera pose and ego-motion by reintroducing the RS constraint. This uses a new 3D-3D registration procedure that minimizes a cost function based on the Euclidean 3D point distance. This is more stable and physically meaningful than the reprojection error or the algebraic distance used in previous work. Experimental results obtained with synthetic and real data show that the proposed methods outperform existing ones in terms of accuracy and stability, even in the known critical configurations.

Deep Non-Rigid Structure From Motion With Missing Data.

Non-rigid structure from motion (NRSfM) refers to the problem of reconstructing cameras and the 3D point cloud of a non-rigid object from an ensemble of images with 2D correspondences. Current NRSfM algorithms are limited from two perspectives: (i) the number of images, and (ii) the type of shape variability they can handle. These difficulties stem from the inherent conflict between the condition of the system and the degrees of freedom needing to be modeled - which has hampered its practical utility for many applications within vision. In this paper we propose a novel hierarchical sparse coding model for NRSFM which can overcome (i) and (ii) to such an extent, that NRSFM can be applied to problems in vision previously thought too ill posed. Our approach is realized in practice as the training of an unsupervised deep neural network (DNN) auto-encoder with a unique architecture that is able to disentangle pose from 3D structure. Using modern deep learning computational platforms allows us to solve NRSfM problems at an unprecedented scale and shape complexity. Our approach has no 3D supervision, relying solely on 2D point correspondences. Further, our approach is also able to handle missing/occluded 2D points without the need for matrix completion. Extensive experiments demonstrate the impressive performance of our approach where we exhibit superior precision and robustness against all available state-of-the-art works in some instances by an order of magnitude. We further propose a new quality measure (based on the network weights) which circumvents the need for 3D ground-truth to ascertain the confidence we have in the reconstructability. We believe our work to be a significant advance over state-of-the-art in NRSFM.

A CSF-Based CNR Approach for Small-Size Image Sequences

For non-rigid structure from motion (NRSFM), the performance of most traditional approaches may decrease significantly when the frame number of the image sequence is relatively small. In this letter, a column space fitting (CSF) based consensus of non-rigid (CNR) reconstruction approach is proposed to deal with the 3D structure estimation problem for small-size image sequences. In the proposed method, a set of trajectory groups are first extracted by utilizing the distance weight of the pairwise points. In order to improve the estimation accuracy, an adaptive rank selection strategy is designed to choose the approximately optimal rank parameter. Corresponding to the trajectory group, the z-coordinates of the observation matrix are estimated by the CSF algorithm due to its good performance. After obtaining the outputs of the CSF-based weak estimators, the final 3D shape is derived by combining the outputs via the alternating directional method of multipliers. Experimental results on several widely used image sequences demonstrate the effectiveness and feasibility of the proposed algorithm.

Structure from Articulated Motion: Accurate and Stable Monocular 3D Reconstruction without Training Data.

Recovery of articulated 3D structure from 2D observations is a challenging computer vision problem with many applications. Current learning-based approaches achieve state-of-the-art accuracy on public benchmarks but are restricted to specific types of objects and motions covered by the training datasets. Model-based approaches do not rely on training data but show lower accuracy on these datasets. In this paper, we introduce a model-based method called Structure from Articulated Motion (SfAM), which can recover multiple object and motion types without training on extensive data collections. At the same time, it performs on par with learning-based state-of-the-art approaches on public benchmarks and outperforms previous non-rigid structure from motion (NRSfM) methods. SfAM is built upon a general-purpose NRSfM technique while integrating a soft spatio-temporal constraint on the bone lengths. We use alternating optimization strategy to recover optimal geometry (i.e., bone proportions) together with 3D joint positions by enforcing the bone lengths consistency over a series of frames. SfAM is highly robust to noisy 2D annotations, generalizes to arbitrary objects and does not rely on training data, which is shown in extensive experiments on public benchmarks and real video sequences. We believe that it brings a new perspective on the domain of monocular 3D recovery of articulated structures, including human motion capture.

Ordered Subspace Clustering for Complex Non-Rigid Motion by 3D Reconstruction

As a fundamental and challenging problem, non-rigid structure-from-motion (NRSfM) has attracted a large amount of research interest. It is worth mentioning that NRSfM has been applied to dynamic scene understanding and motion segmentation. Especially, a motion segmentation approach combining NRSfM with the subspace representation has been proposed. However, the current subspace representation for non-rigid motions clustering do not take into account the inherent sequential property, which has been proved vital for sequential data clustering. Hence this paper proposes a novel framework to segment the complex and non-rigid motion via an ordered subspace representation method for the reconstructed 3D data, where the sequential property is properly formulated in the procedure of learning the affinity matrix for clustering with simultaneously recovering the 3D non-rigid motion by a monocular camera with 2D point tracks. Experiment results on three public sequential action datasets, BU-4DFE, MSR and UMPM, verify the benefits of method presented in this paper for classical complex non-rigid motion analysis and outperform state-of-the-art methods with lowest subspace clustering error (SCE) rates and highest normalized mutual information (NMI) in subspace clustering and motion segmentation fields.

An Accelerated Procrustean Markov Process Model With Coherent Constraint for Non-Rigid Structure From Motion

Non-Rigid Structure from Motion (NRSfM) is the task of reconstructing the 3D point set of a non-rigid object from an ensemble of images with 2D correspondences, which has been a long-lasting challenging research topic. Compared to the state-of-the-art methods for NRSfM, the Procrustean Markov Process (PMP) model has obtained a relatively good performance. However, the estimation error and the convergence time of the PMP model will increase simultaneously when noise is present. To address this problem, in this paper, a coherent constraint is constructed to suppress the noise in the initialization step of the PMP algorithm. Moreover, an Accelerated Expectation Maximization (AEM) algorithm is devised to optimize the PMP estimation model. Experimental results on several widely used sequences demonstrate that our proposed algorithm achieves state-of-the-art performance, as well as its effectiveness and feasibility.

Latent-Smoothness Nonrigid Structure From Motion by Revisiting Multilinear Factorization.

How to implement an effective factorization for nonrigid structure from motion (NRSFM) has attracted much attention in recent years. A straightforward factorization scheme is to multilinearly solve NRSFM in an alternating manner, where each of the unknown variables in NRSFM is updated by fixing the others at each iteration. However, recent works show that most existing multilinear factorization (MLF) methods achieve poorer performances than some state-of-the-art sequential factorization methods. In this paper, we reinvestigate the MLF scheme for improving factorization accuracy, and first propose an MLF method with the only low-rank prior for NRSFM in the presence of missing data. Then, for further improving the performances of such MLF methods, a latent "smoothness" characteristic on unknown 3-D deformable shapes is investigated, which is independent of temporal relations among deformable shapes. Accordingly, a latent-smoothness prior for solving NRSFM is derived from the latent smoothness characteristic, and it is able to effectively recover 3-D deformable shapes from unordered data, which is hard for the traditional temporal-smoothness prior to handle. Finally, a regularized factorization method is proposed by integrating MLF with the explored latent-smoothness prior for further pursuing better performances. Extensive experimental results show the effectiveness of our methods in comparison to eight existing multilinear/sequential methods.

Robust Sparse and Dense Nonrigid Structure From Motion

Modeling deformable three-dimensional (3-D) shape from a video sequence is a fundamental task in computer vision. Nonrigid structure from motion (NRSfM) refers to the problem of recovering the 3-D shape and pose of an object, deforming over time from a monocular video sequence. Presently, dense NRSfM is a research problem of great interest in academia and the industry due to the large demand for 3-D data in various contexts. We provide a robust system for the sparse and dense NRSfM. The strength of our approach comes with the ability to deal with the trajectories corrupted with outliers that serve as an input to NRSfM. To tackle this problem, the input trajectories are processed with a density-based spatial clustering approach that is combined with a RANSAC technique for the outlier' s detection. Processing the trajectories with this process enhances the trajectories by removing the unwanted outliers. Also, extending the work from sparse to dense NRS fM substantially increases the difficulty of the optimization problem. Thus, the proposed system also provides asymptotic improvements to the current optimization approaches by providing an efficient and a novel supervised Gauss-Newton method. Extensive experiments have demonstrated that the proposed method outperforms most of the existing NRSfM methods. The results show that the proposed method reconstructs largely deforming objects accurately and efficiently.

A scalable, efficient, and accurate solution to non-rigid structure from motion

Most Non-Rigid Structure from Motion (NRSfM) solutions are based on factorization approaches that allow reconstructing objects parameterized by a sparse set of 3D points. These solutions, however, are low resolution and generally, they do not scale well to more than a few tens of points. While there have been recent attempts at bringing NRSfM to a dense domain, using for instance variational formulations, these are computationally demanding alternatives which require certain spatial continuity of the data, preventing their use for articulated shapes with large deformations or situations with multiple discontinuous objects. In this paper, we propose incorporating existing point trajectory low-rank models into a probabilistic framework for matrix normal distributions. With this formalism, we can then simultaneously learn shape and pose parameters using expectation maximization, and easily exploit additional priors such as known point correlations. While similar frameworks have been used before to model distributions over shapes, here we show that formulating the problem in terms of distributions over trajectories brings remarkable improvements, especially in generality and efficiency. We evaluate the proposed approach in a variety of scenarios including one or multiple objects, sparse or dense reconstructions, missing observations, mild or sharp deformations, and in all cases, with minimal prior knowledge and low computational cost.

Inextensible Non-Rigid Structure-from-Motion by Second-Order Cone Programming.

We present a global and convex formulation for the template-less 3D reconstruction of a deforming object with the perspective camera. We show for the first time how to construct a Second-Order Cone Programming (SOCP) problem for Non-Rigid Structure-from-Motion (NRSfM) using the Maximum-Depth Heuristic (MDH). In this regard, we deviate strongly from the general trend of using affine cameras and factorization-based methods to solve NRSfM, which do not perform well with complex nonlinear deformations. In MDH, the points' depths are maximized so that the distance between neighbouring points in camera space are upper bounded by the geodesic distance. In NRSfM both geodesic and camera space distances are unknown. We show that, nonetheless, given point correspondences and the camera's intrinsics the whole problem can be solved with SOCP. This is the first convex formulation for NRSfM with physical constraints. We further present how robustness and temporal continuity can be included in the formulation to handle outliers and decrease the problem size, respectively. We show with extensive experiments that our methods accurately reconstruct quasi-isometric objects from partial views under articulated and strong deformations. Compared to the previous methods, our approach gives better or similar accuracy. It naturally handles missing correspondences, non-smooth objects and is very simple to implement compared to previous methods, with only one free parameter (the neighbourhood size).