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Unsupervised Learning of Non-Rigid Point Set Registration Lingjing Wang, Xiang Li, Jianchun Chen, and Yi Fang

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Unsupervised Learning of Non-Rigid Point Set Registration Lingjing Wang, Xiang Li, Jianchun Chen, and Yi Fang 1 Coherent Point Drift Networks: Unsupervised Learning of Non-Rigid Point Set Registration Lingjing Wang, Xiang Li, Jianchun Chen, and Yi Fang Abstract—Given new pairs of source and target point sets, the registration task turns into to process to determine the standard point set registration methods often repeatedly conduct displacement field that moves the source point set towards the the independent iterative search of desired geometric transfor- target one. Regardless of the model-based or model-free point mation to align the source point set with the target one. This limits their use in applications to handle the real-time point set registration approaches, most existing efforts have to start set registration with large volume dataset. This paper presents over a new iterative optimization process for each pair of input a novel method, named coherent point drift networks (CPD- to determine the geometric transformation. Consequently, the Net), for the unsupervised learning of geometric transformation intensive computation in the iterative routine pose great chal- towards real-time non-rigid point set registration. In contrast to previous efforts (e.g. coherent point drift), CPD-Net can learn displacement field function to estimate geometric transformation from a training dataset, consequently, to predict the desired geometric transformation for the alignment of previously unseen pairs without any additional iterative optimization process. Fur- thermore, CPD-Net leverages the power of deep neural networks to fit an arbitrary function, that adaptively accommodates differ- ent levels of complexity of the desired geometric transformation. Particularly, CPD-Net is proved with a theoretical guarantee to learn a continuous displacement vector function that could fur- ther avoid imposing additional parametric smoothness constraint as in previous works. Our experiments verify the impressive performance of CPD-Net for non-rigid point set registration on various 2D/3D datasets, even in the presence of significant displacement noise, outliers, and missing points. Our code will be available at https://github.com/nyummvc/CPD-Net. Index Terms—3D Shapes, Registration, Matching, Point Cloud, Fig. 1. For a given pair of source and target point sets, CPD-Net directly Deep Learning predicts the coherent drifts to move source point set towards target one. OINT set registration is a fundamental computer vision P task, which has wide applications in many fields such lenges for existing approaches to handle a large scale dataset as image registration, object correspondence, large-scale 3D in real-time application. To address this challenge, we are reconstruction and so on [1, 3–6, 9–14]. Existing approaches motivated to develop a learning based approach for point set often solve the registration problem through an iterative op- registration and to generalize its ability from training process timization process to search the optimal geometric transfor- to predict the desired geometric transformation to register mation to minimize a pre-defined alignment loss between previously unseen pair of point sets. transformed source point set and target point set[2,7,8,28,30]. In this paper, we proposed a novel approach, named co- herent point drift networks (CPD-Net), for the unsupervised arXiv:1906.03039v5 [cs.GR] 28 Jul 2019 The geometric transformation can be modeled by a specific type of transformation (e.g. thin plate spline) [11], or described learning of displacement vector function to estimate a model- as a model-free deformation field [28] where a displacement free geometric transformation for point set registration. Figure vector function is defined. The rotation, translation and scaling 2 illustrates the pipeline of the proposed CPD-Net which are often used to model rigid geometric transformation, while consists of three major components. The first component is the affine and thin plate spline are widely used to model non- “Learning Shape Descriptor’. In this component, the global rigid geometric transformation. For model-based methods, the shape descriptor is learned with a multilayer perceptron registration task turns into a searching process for an optimal (MLP). The second component is “Coherent PointMorph”. set of parameters of the model. For model-free method, In this component, we firstly concatenate the global shape descriptor of the source point set and target point set with the L.Wang is with MMVC Lab, the Department of Mathematics, coordinate of source point to form a geometric transformation New York University, New York, NY, 30332 USA, e-mail: descriptor for the input pair. Three successive MLP are then [email protected]. X.Li is with the MMVC Lab, New York University, New York, NY11201, USA, e-mail: [email protected]. defined and take the geometric transformation descriptor as J.Chen is with the MMVC Lab, New York University, New York, NY11201, input to learn the continuous displacement vector field. The USA, e-mail: [email protected]. Y.Fang is with MMVC Lab, Dept. of third component is “Point Set Alignment”. In this component, ECE, NYU Abu Dhabi, UAE and Dept. of ECE, NYU Tandon School of Engineering, USA, e-mail: [email protected]. a loss function is defined to assess the quality of alignment. Corresponding author: Yi Fang. Email: [email protected] The main contributions of our proposed method as follows: 2 Fig. 2. Our pipeline. The proposed structure includes three parts: learning shape descriptor, coherent PointMorph, and the alignment Loss. For a pair of source point set Si and target point set Gj, we firstly leverage MLPs to learn two global descriptors LSi and LGj . We then concatenate these two descriptors to each coordinate fxkgk=1;2;:::;n of source points as the input ([xk; LSi ; LGj ]) for PointMorph structure. We further use MLPs to learn the drifts for each source point. Finally, we move the source point set by our predicted drifts and define the alignment loss function between target and transformed source point sets for back-propagation. • This paper introduces a novel Coherent Point Drift the ICP algorithm is reported to be vulnerable to the selection Networks (CPD-Net) for the unsupervised learning of of corresponding points for initial transformation estimation, point set registration. The proposed network can be and also incapable of dealing with non-rigid transformation. generalized to predict geometric transformation for To accommodate the deformation (e.g. morphing, articulation) non-rigid point set registration. between a pair of point sets, many efforts were spent on the development of algorithms for a non-rigid transformation. • CPD-Net leverages the power of deep neural networks For non-rigid registration, classical previous methods can to fit an arbitrary function, thus makes it possible to usually be divided into parametric and non-parametric by the accommodate different levels of complexity of the target target transformation. As a robust parametric method, TPS- geometric transformation for best aligning the pair of RSM algorithm was proposed by Chui and Rangarajan [16] point sets. to estimate parameters of non-rigid transformation with a penalization on second-order derivatives. As a classical non- • The CPD-Net theoretically guarantees the continuity of parametric method, coherence point drift (CPD) was proposed predicted displacement field as geometric transformation, by Myronenko et al. [28], which successfully introduced a which naturally eliminates the necessity to impose a process of fitting Gaussian mixture likelihood to align the parametric hand-crafted smoothness constraint. source point set with the target one. With the penalization term on the velocity field, the algorithm enforces the motion of • The CPD-Net is free of specific geometric model se- source point set to be coherent during the registration process. lection for modeling the desired transformation, which Another non-parametric vector field consensus algorithm was avoids the potential mismatch between the transformation proposed by Ma et al. [7]. This algorithm is emphasized described by specific pre-defined geometric models and to be robust to outliers. Ma et al. proposed the importance the actual transformation required for point set registra- to leverage both local and global structures during the non- tion. rigid point set registration in [1]. The existing algorithms achieved great success for the registration task. Even though I. RELATED WORKS all these methods state the registration task as an independent A. Iterative registration methods. optimization process for every single pair of source and target point sets, they greatly inspire us for designing our learning- The development of optimization algorithms to estimate based system. the rigid or non-rigid geometric transformations in an itera- tive routine have been attracting great research attention in decades. The Iterative Closest Point (ICP) algorithm [11] is B. Learning-based registration methods. one successful solution for rigid registration. It initializes an In recent years, learning-based methods achieved great suc- estimation of a rigid function and then iteratively chooses cess in many fields of computer vision [21–27,35]. Especially corresponding points to refine the transformation. However, recent works started a trend of directly learning geometric 3 features from cloud points (especially 3D points), which For a given dataset, we use stochastic gradient descent based motivates us to approach the point set registration problem algorithm to optimize parameters set θ for minimizing
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