When the boundary of a familiar object is shown by a series of isolated dots, humans can often recognize the object with ease. This ability can be sustained with addition of distracting dots around the object. However, such capability has not been reproduced algorithmically on computers. We introduce a new algorithm that groups a set of dots into multiple non-disjoint subsets. It connects the dots into a spanning tree using the proximity cue. It then applies the straight polygon transformation to an initial polygon derived from the spanning tree. The straight polygon divides the space into polygons recursively and each polygon can be viewed as grouping of a subset of the dots. The number of polygons generated is O( @math ). We also introduce simple shape selection and recognition algorithms that can be applied to the grouping result. We used both natural and synthetic images to show effectiveness of these algorithms.
For the clustering side, standard clustering algorithms such as k-means, mixture of Gaussian, and ISODATA can be applied. However, these algorithms work on compact clusters that are well separated, and do not work on the dolphin example shown in Figure . Parametric or template based models @cite_16 can be used to isolate specific shapes from background. However, the approaches are not applicable to general shapes.
@cite_16
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The Lloyd’s algorithm, also known as the k-means algorithm, is one of the most popular algorithms for solving the k-means clustering problem in practice. However, it does not give any performance guarantees. This means that there are datasets on which this algorithm can behave very badly. One reason for poor performance on certain datasets is bad initialization. The following simple sampling based seeding algorithm tends to fix this problem: pick the first center randomly from among the given points and then for i ≥ 2, pick a point to be the i th center with probability proportional to the squared distance of this point from the previously chosen centers. This algorithm is more popularly known as the k-means++ seeding algorithm and is known to exhibit some nice properties. These have been studied in a number of previous works [AV07, AJM09, ADK09, BR11]. The algorithm tends to perform well when the optimal clusters are separated in some sense. This is because the algorithm gives preference to further away points when picking centers. [ORSS06] discuss one such separation condition on the data. Jaiswal and Garg [JG12] show that if the dataset satisfies the separation condition of [ORSS06], then the sampling algorithm gives a constant approximation with probability Ω(1 k). Another separation condition that is strictly weaker than [ORSS06] is the approximation stability condition discussed by [BBG09]. In this work, we show that the sampling algorithm gives a constant approximation with probability Ω(1 k) if the dataset satisfies the separation condition of [BBG09] and the optimal clusters are not too small. We give a negative result for datasets that have small optimal clusters. In k-means clustering, we are given a set of n data points in d-dimensional space R sup d and an integer k and the problem is to determine a set of k points in Rd, called centers, so as to minimize the mean squared distance from each data point to its nearest center. A popular heuristic for k-means clustering is Lloyd's (1982) algorithm. We present a simple and efficient implementation of Lloyd's k-means clustering algorithm, which we call the filtering algorithm. This algorithm is easy to implement, requiring a kd-tree as the only major data structure. We establish the practical efficiency of the filtering algorithm in two ways. First, we present a data-sensitive analysis of the algorithm's running time, which shows that the algorithm runs faster as the separation between clusters increases. Second, we present a number of empirical studies both on synthetically generated data and on real data sets from applications in color quantization, data compression, and image segmentation. Bayesian models offer great flexibility for clustering applications---Bayesian nonparametrics can be used for modeling infinite mixtures, and hierarchical Bayesian models can be utilized for sharing clusters across multiple data sets. For the most part, such flexibility is lacking in classical clustering methods such as k-means. In this paper, we revisit the k-means clustering algorithm from a Bayesian nonparametric viewpoint. Inspired by the asymptotic connection between k-means and mixtures of Gaussians, we show that a Gibbs sampling algorithm for the Dirichlet process mixture approaches a hard clustering algorithm in the limit, and further that the resulting algorithm monotonically minimizes an elegant underlying k-means-like clustering objective that includes a penalty for the number of clusters. We generalize this analysis to the case of clustering multiple data sets through a similar asymptotic argument with the hierarchical Dirichlet process. We also discuss further extensions that highlight the benefits of our analysis: i) a spectral relaxation involving thresholded eigenvectors, and ii) a normalized cut graph clustering algorithm that does not fix the number of clusters in the graph. We present a new method for clustering based on compression. The method does not use subject-specific features or background knowledge, and works as follows: First, we determine a parameter-free, universal, similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorov complexity. We propose axioms to capture the real-world setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (ternary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics, we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis.