In many image processing applications, identification of nonlinear structures in image data is of particular interest. Examples include fitting multiple ellipse patterns to image data, estimation and segmentation of multiple motions in subsequent images in video, and fitting nonlinear patterns to cell images for cancer detection in biomedical applications. This chapter introduces a novel approach to calculate a first order approximation for point distances from general nonlinear structures. We also propose an accelerated sampling method for robust segmentation of multiple structures. Our sampling method is substantially faster than random sampling used in the well-known RANSAC method as it effectively makes use of the spatial proximity of the points belonging to each structure. A fast highbreakdown robust estimator called Accelerated-LKS (A-LKS) is devised using the accelerated search to minimize the kth order statistics of squared distances. A number of experiments on homography estimation problems are presented. Those experiments include cases with up to eight different motions and we benchmark the performance of the proposed estimator in comparison with a number of stateof-the-art robust estimators. We also show the result of applying A-LKS to solve ellipse fitting and motion segmentation in practical applications.
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