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Robust Estimation of Nonrigid Transformation for Point Set Registration Robust Estimation of Nonrigid Transformation for Point Set Registration Jiayi Ma1;2, Ji Zhao3, Jinwen Tian1, Zhuowen Tu4, and Alan L. Yuille2 1Huazhong University of Science and Technology, 2Department of Statistics, UCLA 3The Robotics Institute, CMU, 4Lab of Neuro Imaging, UCLA jyma2010, zhaoji84, zhuowen.tu @gmail.com, [email protected], [email protected] f g Abstract widely studied [5,4,7, 19, 14]. By contrast, nonrigid reg- istration is more difficult because the underlying nonrigid We present a new point matching algorithm for robust transformations are often unknown, complex, and hard to nonrigid registration. The method iteratively recovers the model [6]. But nonrigid registration is very important be- point correspondence and estimates the transformation be- cause it is required for many real world tasks including tween two point sets. In the first step of the iteration, fea- hand-written character recognition, shape recognition, de- ture descriptors such as shape context are used to establish formable motion tracking and medical image registration. rough correspondence. In the second step, we estimate the In this paper, we focus on the nonrigid case and present transformation using a robust estimator called L2E. This is a robust algorithm for nonrigid point set registration. There the main novelty of our approach and it enables us to deal are two unknown variables we have to solve for in this prob- with the noise and outliers which arise in the correspon- lem: the correspondence and the transformation. Although dence step. The transformation is specified in a functional solving for either variable without information regarding space, more specifically a reproducing kernel Hilbert space. the other is difficult, an iterated estimation framework can We apply our method to nonrigid sparse image feature cor- be used [4,3,6]. In this iterative process, the estimate of the respondence on 2D images and 3D surfaces. Our results correspondence is used to refine the estimate of the trans- quantitatively show that our approach outperforms state-of- formation, and vice versa. But a problem arises if there the-art methods, particularly when there are a large num- are errors in the correspondence which occurs in many ap- ber of outliers. Moreover, our method of robustly estimating plications particularly if the transformation is large and/or transformations from correspondences is general and has there are outliers in the data (e.g., data points that are not many other applications. undergoing the non-rigid transformation). In this situation, the estimate of the transformation will degrade badly un- less it is performed robustly. The main contribution of our 1. Introduction approach is to robustly estimate the transformations from the correspondences using a robust estimator named the L - Point set registration is a fundamental problem which 2 Minimizing Estimate (L E)[20,2]. frequently arises in computer vision, medical image analy- 2 More precisely, our approach iteratively recovers the sis, and pattern recognition [5,4,6]. Many tasks in these point correspondences and estimates the transformation be- fields – such as stereo matching, shape matching, image tween two point sets. In the first step of the iteration, fea- registration and content-based image retrieval – can be for- ture descriptors such as shape context are used to estab- mulated as a point matching problems because point repre- lish correspondence. In the second step, we estimate the sentations are general and easy to extract [5]. The points transformation using the robust estimator L E. This esti- in these tasks are typically the locations of interest points 2 mator enable us to deal with the noise and outliers in the extracted from an image, or the edge points sampled from correspondences. The nonrigid transformation is modeled a shape contour. The registration problem then reduces to in a functional space, called the reproducing kernel Hilbert determining the correct correspondence and to find the un- space (RKHS) [1], in which the transformation function has derlying spatial transformation between two point sets ex- an explicit kernel representation. tracted from the input data. The registration problem can be categorized into rigid or 1.1. Related Work nonrigid registration depending on the application and the form of the data. Rigid registration, which only involves a The iterated closest point (ICP) algorithm [4] is one small number of parameters, is relatively easy and has been of the best known point registration approaches. It uses 1 nearest-neighbor relationships to assign a binary correspon- 2.1. Problem Formulation Using L2E: Robust Esti- dence, and then uses estimated correspondence to refine the mation transformation. Belongie et al.[3] introduced a method for Parametric estimation is typically done using maximum registration based on the shape context descriptor, which likelihood estimation (MLE). It can be shown that MLE incorporates the neighborhood structure of the point set is the optimal estimator if it is applied to the correct proba- and thus helps establish correspondence between the point bility model for the data (or to a good approximation). But sets. But these methods ignore robustness when they re- MLE can be badly biased if the model is not a sufficiently cover the transformation from the correspondence. In re- good approximation or, in particular, there are a significant lated work, Chui and Rangarajan [6] established a general fraction of outliers. In many point matching problems, it framework for estimating correspondence and transforma- is desirable to have a robust estimator of the transformation tions for nonrigid point matching. They modeled the trans- f because the point correspondence set S usually contains formation as a thin-plate spline and did robust point match- outliers. There are two choices: (i) to build a more complex ing by an algorithm (TRS-RPM) which involved determin- model that includes the outliers – which is complex since it istic annealing and soft-assignment. Alternatively, the co- involves modeling the outlier process using extra (hidden) herence point drift (CPD) algorithm [17] uses Gaussian ra- variables which enable us to identify and reject outliers, dial basis functions instead of thin-plate splines. Another or (ii) to use an estimator which is different from MLE interesting point matching approach is the kernel correla- but less sensitive to outliers, as described in Huber’s robust tion (KC) based method [22], which was later improved statistics [8]. In this paper, we use the second method and in [9]. Zheng and Doermann [27] introduced the notion adopt the L E estimator [20,2], a robust estimator which of a neighborhood structure for the general point matching 2 minimizes the L distance between densities, and is par- problem, and proposed a matching method, the robust point 2 ticularly appropriate for analyzing massive data sets where matching-preserving local neighborhood structures (RPM- data cleaning (to remove outliers) is impractical. More for- LNS) algorithm. Other related work includes the relaxation mally, L E estimator for model f(x θ) recommends esti- labeling method, generalized in [11], and the graph match- 2 j mating the parameter θ by minimizing the criterion: ing approach for establishing feature correspondences [21]. n The main contributions of our work include: (i) we pro- Z 2 X L E(θ) = f(x θ)2dx f(x θ): (1) pose a new robust algorithm to estimate a spatial transfor- 2 j − n ij mation/mapping from correspondences with noise and out- i=1 liers; (ii) we apply the robust algorithm to nonrigid point To get some intuition for why L2E is robust, observe that its set registration and also to sparse image feature correspon- penalty for a low probability point x is f(x θ) which is i − ij dence. much less than the penalty of log f(x θ) given by MLE − ij (which becomes infinite as f(x θ) tends to 0). Hence ij 2. Estimating Transformation from Corre- MLE is reluctant to assign low probability to any points, spondences by L2E including outliers, and hence tends to be biased by out- liers. By contrast, L E can assign low probabilities to Given a set of point correspondences S = (x ; y ) n , 2 f i i gi=1 many points, hopefully to the outliers, without paying too which are typically perturbed by noise and by outlier points high a penalty. To demonstrate the robustness of L E, we which undergo different transformations, the goal is to esti- 2 present a line-fitting example which contrasts the behavior mate a transformation f : y = f(x ) and fit the inliers. i i of MLE and L E, see Fig.1. The goal is to fit a linear re- In this paper we make the assumption that the noise on 2 gression model, y = αx + , with residual N(0; 1), the inliers is Gaussian on each component with zero mean ∼ by estimating α using MLE and L2E. This gives, re- and uniform standard deviation σ (our approach can be di- Pn spectively, α^MLE = arg maxα log φ(yi αxi 0; 1) rectly applied to other noise models). More precisely, an h i=1 − j i p1 2 Pn inlier point correspondence (x ; y ) satisfies y f(x ) and α^L2E = arg minα 2 π n i=1 φ(yi αxi 0; 1) , i i i − i ∼ − − j N(0; σ2I), where I is an identity matrix of size d d, with d where φ(x µ, Σ) denotes the normal density. × j being the dimension of the point. The data y f(x ) n As shown in Fig.1, L E is very resistant when we con- f i − i gi=1 2 can then be thought of as a sample set from a multivariate taminate the data by outliers, but MLE does not show this 2 normal density N(0; σ I) which is contaminated by out- desirable property.
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