In this paper, we propose an efficient algorithm, called vector field consensus, for establishing robust point correspondences between two sets of points. Our algorithm starts by creating a set of putative correspondences which can contain a very large number of false correspondences, or outliers, in addition to a limited number of true correspondences (inliers). Next, we solve for correspondence by interpolating a vector field between the two point sets, which involves estimating a consensus of inlier points whose matching follows a nonparametric geometrical constraint. We formulate this a maximum a posteriori (MAP) estimation of a Bayesian model with hidden/latent variables indicating whether matches in the putative set are outliers or inliers. We impose nonparametric geometrical constraints on the correspondence, as a prior distribution, using Tikhonov regularizers in a reproducing kernel Hilbert space. MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We illustrate this method on data sets in 2D and 3D and demonstrate that it is robust to a very large number of outliers (even up to 90%). We also show that in the special case where there is an underlying parametric geometrical model (e.g., the epipolar line constraint) that we obtain better results than standard alternatives like RANSAC if a large number of outliers are present. This suggests a two-stage strategy, where we use our nonparametric model to reduce the size of the putative set and then apply a parametric variant of our approach to estimate the geometric parameters. Our algorithm is computationally efficient and we provide code for others to use it. In addition, our approach is general and can be applied to other problems, such as learning with a badly corrupted training data set.

Feature matching, which refers to establishing reliable correspondence between two sets of features (particularly point features), is a critical prerequisite in feature-based registration. In this paper, we propose a flexible and general algorithm, which is called locally linear transforming (LLT), for both rigid and nonrigid feature matching of remote sensing images. We start by creating a set of putative correspondences based on the feature similarity and then focus on removing outliers from the putative set and estimating the transformation as well. We formulate this as a maximum-likelihood estimation of a Bayesian model with hidden/latent variables indicating whether matches in the putative set are outliers or inliers. To ensure the well-posedness of the problem, we develop a local geometrical constraint that can preserve local structures among neighboring feature points, and it is also robust to a large number of outliers. The problem is solved by using the expectation-maximization algorithm (EM), and the closed-form solutions of both rigid and nonrigid transformations are derived in the maximization step. In the nonrigid case, we model the transformation between images in a reproducing kernel Hilbert space (RKHS), and a sparse approximation is applied to the transformation that reduces the method computation complexity to linearithmic. Extensive experiments on real remote sensing images demonstrate accurate results of LLT, which outperforms current state-of-the-art methods, particularly in the case of severe outliers (even up to 80%).

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