Matching 3D rigid point clouds in complex environments robustly and accurately is still a core technique used in many applications. This paper proposes a new architecture combining error estimation from sample covariances and dual global probability alignment based on the convolution of adaptive Gaussian Mixture Models (GMM) from point clouds. Firstly, a novel adaptive GMM is defined using probability distributions from the corresponding points. Then rigid point cloud alignment is performed by maximizing the global probability from the convolution of dual adaptive GMMs in the whole 2D or 3D space, which can be efficiently optimized and has a large zone of accurate convergence. Thousands of trials have been conducted on 200 models from public 2D and 3D datasets to demonstrate superior robustness and accuracy in complex environments with unpredictable noise, outliers, occlusion, initial rotation, shape and missing points.
We can use a method called registration to integrate some point clouds that represent the shape of the real world. In this paper, we propose highly accurate and stable registration method. Our method detects keypoints from point clouds and generates triplets using multiple descriptors. Furthermore, our method evaluates the consistency of rigid transformation parameters of each triplet with histograms and obtains the rigid transformation between the point clouds. In the experiment of this paper, our method had minimul errors and no major failures. As a result, we obtained sufficiently accurate …
National University of Defense Technology（国防科技大学）；Technical University of Munich（慕尼黑工业大学）
We study the problem of outlier correspondence pruning for non-rigid point cloud registration. In rigid registration, spatial consistency has been a commonly used criterion to discriminate outliers from inliers. It measures the compatibility of two correspondences by the discrepancy between the respective distances in two point clouds. However, spatial consistency no longer holds in non-rigid cases and outlier rejection for non-rigid registration has not been well studied. In this work, we propose Graph-based Spatial Consistency Network (GraphSCNet) to filter outliers for non-rigid registratio…
Non-rigid point cloud registration is a key component in many computer vision and computer graphics applications. The high complexity of the unknown non-rigid motion make this task a challenging problem. In this paper, we break down this problem via hierarchical motion decomposition. Our method called Neural Deformation Pyramid (NDP) represents non-rigid motion using a pyramid architecture. Each pyramid level, denoted by a Multi-Layer Perception (MLP), takes as input a sinusoidally encoded 3D point and outputs its motion increments from the previous level. The sinusoidal function starts with a…
We present SyNoRiM, a novel way to jointly register multiple non-rigid shapes by synchronizing the maps relating 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 n…
In this paper, we unify popular non-rigid registration methods for point sets and surfaces under our general framework, GiNGR. GiNGR builds upon Gaussian Process Morphable Models (GPMM) and hence separates modeling the deformation prior from model adaptation for registration. In addition, it provides explainable hyperparameters, multi-resolution registration, trivial inclusion of expert annotation, and the ability to use and combine analytical and statistical deformation priors. But more importantly, the reformulation allows for a direct comparison of registration methods. Instead of using a g…
Point Cloud Registration is the problem of aligning the corresponding points of two 3D point clouds referring to the same object. The challenges include dealing with noise and partial match of real-world 3D scans. For non-rigid objects, there is an additional challenge of accounting for deformations in the object shape that happen to the object in between the two 3D scans. In this project, we study the problem of non-rigid point cloud registration for use cases in the Augmented/Mixed Reality domain. We focus our attention on a special class of non-rigid deformations that happen in rigid object…
The Discriminative Optimization (DO) algorithm has been proved much successful in 3D point cloud registration. In the original DO, the feature (descriptor) of two point cloud was defined as a histogram, and the element of histogram indicates the weights of scene points in "front" or "back" side of a model point. In this paper, we extended the histogram which indicate the sides from "front-back" to "front-back", "up-down", and "clockwise-anticlockwise". In addition, we reweighted the extended histogram according to the model points' distribution. We evaluated the proposed Improved DO on the Sta…
This paper considers online object-level mapping using partial point-cloud observations obtained online in an unknown environment. We develop and approach for fully Convolutional Object Retrieval and Symmetry-AIded Registration (CORSAIR). Our model extends the Fully Convolutional Geometric Features model to learn a global object-shape embedding in addition to local point-wise features from the point-cloud observations. The global feature is used to retrieve a similar object from a category database, and the local features are used for robust pose registration between the observed and the retri…
SynBench: A Synthetic Benchmark for Non-rigid 3D Point Cloud Registration
Mannheim Institute for Intelligent Systems in Medicine (MIISM), Medical Faculty Mannheim, Heidelberg University, Mannheim 68167, Germany（海德堡大学曼海姆医学院智能医学研究所（MIISM））；Department of Radiation Oncology, Brigham and Women’s Hospital, Dana-Farber Cancer Institute, Harvard Medical School, Boston, MA, USA（放射肿瘤科，布里格斯妇女医院，达纳-法伯癌症研究所，哈佛医学院，马萨诸塞州波士顿，美国）；Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany（科学计算跨学科中心（IWR），海德堡大学，海德堡，德国）；Central Institute for Computer Engineering (ZITI), Heidelberg University, Heidelberg, Germany（计算机工程中央研究所（ZITI），海德堡大学，海德堡，德国）；CZS Heidelberg Center for Model-Based AI, Heidelberg University, Mannheim, Germany（海德堡模型驱动人工智能中心（CZS），海德堡大学，曼海姆，德国）；Department of Otorhinolaryngology, Head and Neck Surgery, University Hospital Mannheim, Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany（耳鼻喉科及头颈外科，曼海姆大学医院，曼海姆医学院，海德堡大学，曼海姆，德国）；AI Health Innovation Cluster, Heidelberg-Mannheim Health and Life Science Alliance, Heidelberg University, Heidelberg, Germany（人工智能健康创新集群，海德堡-曼海姆健康与生命科学联盟，海德堡大学，海德堡，德国）
Occlusion-aware Non-Rigid Point Cloud Registration via Unsupervised Neural Deformation Correntropy
Hong Kong Institute of Science & Innovation, Chinese Academy of Sciences（中国科学院香港创新研究院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）
SemICP: Semantic Non-Rigid Point Cloud Registration with Elastic Energy Regularization
Department of Electrical and Computer Engineering, University of British Columbia（电气与计算机工程系，不列颠哥伦比亚大学）；Faculty of Medicine, University of British Columbia（医学院，不列颠哥伦比亚大学）
Laboratório Nacional de Computação Científica（国家计算科学实验室）；Universidade Federal de Juiz de Fora（茹伊斯迪福拉联邦大学）
In the last decades, some literature appeared using the Lie groups theory to solve problems in computer vision. On the other hand, Lie algebraic representations of the transformations therein were introduced to overcome the difficulties behind group structure by mapping the transformation groups to linear spaces. In this paper we focus on application of Lie groups and Lie algebras to find the rigid transformation that best register two surfaces represented by point clouds. The so called pairwise rigid registration can be formulated by comparing intrinsic second-order orientation tensors that e…
Inria（法国国家信息与自动化研究所）；Mines ParisTech（巴黎矿业学院）；PSL Research University（巴黎文理研究大学）；LIGM（巴黎马恩河谷信息、数学与系统实验室）；Ecole des Ponts（巴黎路桥学院）；Univ Gustave Eiffel（古斯塔夫·埃菲尔大学）；CNRS（法国国家科学研究中心）
Rigid registration of point clouds with partial overlaps is a longstanding problem usually solved in two steps: (a) finding correspondences between the point clouds; (b) filtering these correspondences to keep only the most reliable ones to estimate the transformation. Recently, several deep nets have been proposed to solve these steps jointly. We built upon these works and propose PCAM: a neural network whose key element is a pointwise product of cross-attention matrices that permits to mix both low-level geometric and high-level contextual information to find point correspondences. These cro…
DefTransNet: A Transformer-based Method for Non-Rigid Point Cloud Registration in the Simulation of Soft Tissue Deformation
Mannheim Institute for Intelligent Systems in Medicine (MIISM), Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany（曼海姆智能医学研究所（MIISM）、曼海姆医学院、海德堡大学）；Department of Radiation Oncology, Brigham and Women's Hospital, Dana-Farber Cancer Institute, Harvard Medical School, Boston, MA, USA（放射肿瘤科、布里格姆和女性医院、达纳-法伯癌症研究所、哈佛医学院）；International Institute of Information Technology, Bangalore, India（国际信息技术研究所，班加罗尔，印度）；Department of Otorhinolaryngology, Head and Neck Surgery, Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany（耳鼻喉科与头颈外科、曼海姆医学院、海德堡大学）；Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany（跨学科科学计算中心（IWR）、海德堡大学）；Central Institute for Computer Engineering (ZITI), Heidelberg University, Heidelberg, Germany（计算机工程研究中心（ZITI）、海德堡大学）；CZS Heidelberg Center for Model-Based AI, Heidelberg University, Mannheim, Germany（海德堡模型驱动人工智能中心（CZS）、海德堡大学）
TU Kaiserslautern（凯泽斯劳滕工业大学）；German Research Center for Artificial Intelligence (DFKI GmbH)（德国人工智能研究中心（DFKI GmbH））
We propose RPSRNet - a novel end-to-end trainable deep neural network for rigid point set registration. For this task, we use a novel $2^D$-tree representation for the input point sets and a hierarchical deep feature embedding in the neural network. An iterative transformation refinement module in our network boosts the feature matching accuracy in the intermediate stages. We achieve an inference speed of 12-15ms to register a pair of input point clouds as large as 250K. Extensive evaluation on (i) KITTI LiDAR odometry and (ii) ModelNet-40 datasets shows that our method outperforms prior state…
School of Engineering, The University of Tokyo（东京大学工学部）
Traditional algorithms of point set registration minimizing point-to-plane distances often achieve a better estimation of rigid transformation than those minimizing point-to-point distances. Nevertheless, recent deep-learning-based methods minimize the point-to-point distances. In contrast to these methods, this paper proposes the first deep-learning-based approach to point-to-plane registration. A challenging part of this problem is that a typical solution for point-to-plane registration requires an iterative process of accumulating small transformations obtained by minimizing a linearized en…
Depth Augmented and FE Free 3D/2D Liver Registration for Laparoscopic Liver AR
UCL Hawkes Institute, University College London, London WC1E 6BT, UK（伦敦大学学院UCL哈维斯研究所）；Division of Surgery and Interventional Science, University College London, London WC1E 6BT, UK（伦敦大学学院UCL外科与介入科学系）；Unit for Lifelong Health and Ageing at UCL, University College London, London WC1E 7HB, UK（伦敦大学学院UCL终身健康与老龄化单位）；Medtronic plc., London, UK（伦敦梅脱利克公司）
University of Maryland, College Park（马里兰大学帕克分校）；University of Maryland Institute for Advanced Computer Studies (UMIACS), University of Maryland, College Park（马里兰大学帕克分校马里兰高级计算机研究所）
In this paper, we introduce a non-rigid registration pipeline for pairs of unorganized point clouds that may be topologically different. Standard warp field estimation algorithms, even under robust, discontinuity-preserving regularization, tend to produce erratic motion estimates on boundaries associated with `close-to-open' topology changes. We overcome this limitation by exploiting backward motion: in the opposite motion direction, a `close-to-open' event becomes `open-to-close', which is by default handled correctly. At the core of our approach lies a general, topology-agnostic warp field e…
University of Michigan（密歇根大学）；Toyota Research Institute（丰田研究院）；Cornell University（康奈尔大学）
This paper reports on a novel nonparametric rigid point cloud registration framework that jointly integrates geometric and semantic measurements such as color or semantic labels into the alignment process and does not require explicit data association. The point clouds are represented as nonparametric functions in a reproducible kernel Hilbert space. The alignment problem is formulated as maximizing the inner product between two functions, essentially a sum of weighted kernels, each of which exploits the local geometric and semantic features. As a result of the continuous models, analytical gr…
Point cloud registration is a key problem for computer vision applied to robotics, medical imaging, and other applications. This problem involves finding a rigid transformation from one point cloud into another so that they align. Iterative Closest Point (ICP) and its variants provide simple and easily-implemented iterative methods for this task, but these algorithms can converge to spurious local optima. To address local optima and other difficulties in the ICP pipeline, we propose a learning-based method, titled Deep Closest Point (DCP), inspired by recent techniques in computer vision and n…
Washington University in St. Louis（圣路易斯华盛顿大学）；Megvii Inc.（旷视科技）
In this paper, we propose a novel learning-based pipeline for partially overlapping 3D point cloud registration. The proposed model includes an iterative distance-aware similarity matrix convolution module to incorporate information from both the feature and Euclidean space into the pairwise point matching process. These convolution layers learn to match points based on joint information of the entire geometric features and Euclidean offset for each point pair, overcoming the disadvantage of matching by simply taking the inner product of feature vectors. Furthermore, a two-stage learnable poin…
Deep learning based Non-Rigid Volume-to-Surface Registration for Brain Shift compensation Using Point Cloud
Austrian Center for Medical Innovation and Technology（奥地利医学创新与技术中心）；Medical University of Vienna（维也纳医学大学）；University of Cambridge（剑桥大学）；Danube Private University (DPU)（多瑙私人大学（DPU））
Northwestern Polytechnical University（西北工业大学）；Baidu Research（百度研究院）；National Engineering Laboratory of Deep Learning Technology and Application（深度学习技术及应用国家工程实验室）
3D point cloud registration in remote sensing field has been greatly advanced by deep learning based methods, where the rigid transformation is either directly regressed from the two point clouds (correspondences-free approaches) or computed from the learned correspondences (correspondences-based approaches). Existing correspondences-free methods generally learn the holistic representation of the entire point cloud, which is fragile for partial and noisy point clouds. In this paper, we propose a correspondences-free unsupervised point cloud registration (UPCR) method from the representation se…
German Research Center for Artificial Intelligence（德国人工智能研究中心）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
This article introduces a new physics-based method for rigid point set alignment called Fast Gravitational Approach (FGA). In FGA, the source and target point sets are interpreted as rigid particle swarms with masses interacting in a globally multiply-linked manner while moving in a simulated gravitational force field. The optimal alignment is obtained by explicit modeling of forces acting on the particles as well as their velocities and displacements with second-order ordinary differential equations of motion. Additional alignment cues (point-based or geometric features, and other boundary co…
We provide a dynamical perspective on the classical problem of 3D point cloud registration with correspondences. A point cloud is considered as a rigid body consisting of particles. The problem of registering two point clouds is formulated as a dynamical system, where the dynamic model point cloud translates and rotates in a viscous environment towards the static scene point cloud, under forces and torques induced by virtual springs placed between each pair of corresponding points. We first show that the potential energy of the system recovers the objective function of the maximum likelihood e…
Image registration algorithms can be generally categorized into two groups: non-rigid and rigid. Recently, many deep learning-based algorithms employ a neural net to characterize non-rigid image registration function. However, do they always perform better? In this study, we compare the state-of-art deep learning-based non-rigid registration approach with rigid registration approach. The data is generated from Kaggle Dog vs Cat Competition \url{https://www.kaggle.com/c/dogs-vs-cats/} and we test the algorithms' performance on rigid transformation including translation, rotation, scaling, shear…
University of Science and Technology of China（中国科学技术大学）；Cardiff University（卡迪夫大学）；Zhejiang University（浙江大学）
Imperfect data (noise, outliers and partial overlap) and high degrees of freedom make non-rigid registration a classical challenging problem in computer vision. Existing methods typically adopt the $\ell_{p}$ type robust estimator to regularize the fitting and smoothness, and the proximal operator is used to solve the resulting non-smooth problem. However, the slow convergence of these algorithms limits its wide applications. In this paper, we propose a formulation for robust non-rigid registration based on a globally smooth robust estimator for data fitting and regularization, which can handl…
Northwestern Polytechnical University（西北工业大学）；Chang’an University（长安大学）；Hubei University（湖北大学）；Huazhong University of Science and Technology（华中科技大学）
This paper focuses on developing efficient and robust evaluation metrics for RANSAC hypotheses to achieve accurate 3D rigid registration. Estimating six-degree-of-freedom (6-DoF) pose from feature correspondences remains a popular approach to 3D rigid registration, where random sample consensus (RANSAC) is a de-facto choice to this problem. However, existing metrics for RANSAC hypotheses are either time-consuming or sensitive to common nuisances, parameter variations, and different application scenarios, resulting in performance deterioration in overall registration accuracy and speed. We alle…
Robust Rigid and Non-Rigid Medical Image Registration Using Learnable Edge Kernels
Department of Mathematics, University of Innsbruck（因斯布鲁克大学数学系）；Department of Radiology, Medical University of Innsbruck（因斯布鲁克医学大学放射学系）；Neuroimaging Research Core Facility, Medical University of Innsbruck（因斯布鲁克医学大学神经影像研究核心设施）
This paper addresses the issue of matching rigid 3D object points with 2D image points through point registration based on maximum likelihood principle in computer simulated images. Perspective projection is necessary when transforming 3D coordinate into 2D. The problem then recasts into a missing data framework where unknown correspondences are handled via mixture models. Adopting the Expectation Conditional Maximization for Point Registration (ECMPR), two different rotation and translation optimization algorithms are compared in this paper. We analyze in detail the associated consequences in…
Images taken at different times or positions undergo transformations such as rotation, scaling, skewing, and more. The process of aligning different images which have undergone transformations can be done via registration. Registration is desirable when analyzing time-series data for tracking, averaging, or differential diagnoses of diseases. Efficient registration methods exist for rigid (including linear or affine) transformations; however, for non-rigid (also known as non-affine) transformations, current methods are computationally expensive and time-consuming. In this report, I will explor…
In this paper, we propose a novel approach to solve the 3D non-rigid registration problem from RGB images using Convolutional Neural Networks (CNNs). Our objective is to find a deformation field (typically used for transferring knowledge between instances, e.g., grasping skills) that warps a given 3D canonical model into a novel instance observed by a single-view RGB image. This is done by training a CNN that infers a deformation field for the visible parts of the canonical model and by employing a learned shape (latent) space for inferring the deformations of the occluded parts. As result of …
Cardiff University（卡迪夫大学）；University of Science and Technology of China（中国科学技术大学）；Università della Svizzera italiana（瑞士意大利语区大学）
Non-rigid registration computes an alignment between a source surface with a target surface in a non-rigid manner. In the past decade, with the advances in 3D sensing technologies that can measure time-varying surfaces, non-rigid registration has been applied for the acquisition of deformable shapes and has a wide range of applications. This survey presents a comprehensive review of non-rigid registration methods for 3D shapes, focusing on techniques related to dynamic shape acquisition and reconstruction. In particular, we review different approaches for representing the deformation field, an…
We present a new fully-automatic non-rigid 3D shape registration (morphing) framework comprising (1) a new 3D landmarking and pose normalisation method; (2) an adaptive shape template method to accelerate the convergence of registration algorithms and achieve a better final shape correspondence and (3) a new iterative registration method that combines Iterative Closest Points with Coherent Point Drift (CPD) to achieve a more stable and accurate correspondence establishment than standard CPD. We call this new morphing approach Iterative Coherent Point Drift (ICPD). Our proposed framework is eva…
The non-rigid registration between CT data and ultrasonic images of liver can facilitate the diagnosis and treatment, which has been widely studied in recent years. To improve the registration accuracy of the Demons model on the non-rigid registration between 3D CT liver data and 2D ultrasonic images, a novel boundary extraction and enhancement method based on radial directional local intuitionistic fuzzy entropy in the polar coordinates has been put forward, and a new registration workflow has been provided. Experiments show that our method can acquire high-accuracy registration results. Expe…
INRIA Grenoble Rhône-Alpes（法国国立信息学与自动化研究院格勒诺布尔罗讷-阿尔卑斯分院）；University of Hong Kong（香港大学）
This paper addresses the issue of matching rigid and articulated shapes through probabilistic point registration. The problem is recast into a missing data framework where unknown correspondences are handled via mixture models. Adopting a maximum likelihood principle, we introduce an innovative EM-like algorithm, namely the Expectation Conditional Maximization for Point Registration (ECMPR) algorithm. The algorithm allows the use of general covariance matrices for the mixture model components and improves over the isotropic covariance case. We analyse in detail the associated consequences in t…
Perelman School of Medicine, University of Pennsylvania（宾夕法尼亚大学佩雷尔曼医学院）
A novel non-rigid image registration algorithm is built upon fully convolutional networks (FCNs) to optimize and learn spatial transformations between pairs of images to be registered in a self-supervised learning framework. Different from most existing deep learning based image registration methods that learn spatial transformations from training data with known corresponding spatial transformations, our method directly estimates spatial transformations between pairs of images by maximizing an image-wise similarity metric between fixed and deformed moving images, similar to conventional image…
Image registration for stack-based HDR photography is challenging. If not properly accounted for, camera motion and scene changes result in artifacts in the composite image. Unfortunately, existing methods to address this problem are either accurate, but too slow for mobile devices, or fast, but prone to failing. We propose a method that fills this void: our approach is extremely fast---under 700ms on a commercial tablet for a pair of 5MP images---and prevents the artifacts that arise from insufficient registration quality.
Non-rigid registration is challenging because it is ill-posed with high degrees of freedom and is thus sensitive to noise and outliers. We propose a robust non-rigid registration method using reweighted sparsities on position and transformation to estimate the deformations between 3-D shapes. We formulate the energy function with position and transformation sparsity on both the data term and the smoothness term, and define the smoothness constraint using local rigidity. The double sparsity based non-rigid registration model is enhanced with a reweighting scheme, and solved by transferring the …
Chalmers University of Technology（查尔姆斯理工大学）；MedTech West（西部医疗技术中心）
We study the relation between the target functions of low-resolution and high-resolution intensity-based registration for the class of rigid transformations. Our results show that low resolution target values can tightly bound the high-resolution target function in natural images. This can help with analyzing and better understanding the process of multiresolution image registration. It also gives a guideline for designing multiresolution algorithms in which the search space in higher resolution registration is restricted given the fitness values for lower resolution image pairs. To demonstrat…
Learning accurate rigid registration for longitudinal brain MRI from synthetic data
1 Division of Biomedical Imaging, KTH Royal Institute of Technology, Huddinge, Sweden 2 Athinoula A.\ Martinos Center for Biomedical Imaging, Charlestown, USA 3 Department of Radiology, Massachusetts General Hospital, Boston, USA 4 Department of Radiology, Harvard Medical School, Boston, USA 5 Computer Science \& Artificial Intelligence Laboratory, MIT, Cambridge, USA
This work addresses the problem of non-rigid registration of 3D scans, which is at the core of shape modeling techniques. Firstly, we propose a new kernel based on geodesic distances for the Gaussian Process Morphable Models (GPMMs) framework. The use of geodesic distances into the kernel makes it more adapted to the topological and geometric characteristics of the surface and leads to more realistic deformations around holes and curved areas. Since the kernel possesses hyperparameters we have optimized them for the task of face registration on the FaceWarehouse dataset. We show that the Geode…
MSR:Hybrid Field Modeling for CT-MRI Rigid-Deformable Registration of the Cervical Spine with an Annotated Dataset
School of Biomedical Engineering, Southern Medical University, Guangzhou, 510515, China.（南方医科大学生物医学工程学院）；Guangdong Provincial Key Laboratory of Medical Image Processing, Guangzhou, 510515, China.（广东省医学图像处理重点实验室）；Guangdong Province Engineering Laboratory for Medical Imaging（广东省医学影像与诊断技术工程实验室）；Information Center, Nanfang Hospital, Southern Medical University, Guangzhou, 510515, China.（南方医科大学南华医院信息中心）；Division of Spine Surgery, Department of Orthopaedics, Nanfang hospital, Southern Medical University, Guangzhou, Guangdong, 510515, China.（南方医科大学南华医院骨科脊柱外科）
University of Stuttgart（斯图加特大学）；ShanghaiTech University（上海科技大学）；RMIT University（皇家墨尔本理工大学）
Physiological motion, such as cardiac and respiratory motion, during Magnetic Resonance (MR) image acquisition can cause image artifacts. Motion correction techniques have been proposed to compensate for these types of motion during thoracic scans, relying on accurate motion estimation from undersampled motion-resolved reconstruction. A particular interest and challenge lie in the derivation of reliable non-rigid motion fields from the undersampled motion-resolved data. Motion estimation is usually formulated in image space via diffusion, parametric-spline, or optical flow methods. However, im…
