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.
Zahn used a minimum spanning tree from a dot pattern, as we do in our algorithm, and break the tree into a forrest by removing edges that are significantly longer than the others @cite_26 . The method of Bajcsy and Ahuja @cite_3 is similar to Zahn's method, but exploits maximum intra-cluster similarity and inter-cluster dissimilarity. Ahuja and Tuceryan @cite_27 used the Voronoi diagram to derive the neighbor relation and various local geometric structures from the diagram to classify dots into interior, border, curve, and isolated. Globally consistent classification is encouraged by relaxation labeling @cite_6 @cite_31 . They use 7 different geometric structures and the classification method is complex and requires many free parameters.
@cite_26 @cite_6 @cite_3 @cite_27 @cite_31
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We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an n-node graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n^ o(1) ) worst-case update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen Wulff-Nilsen16a with update time O(n^ 0.5-&#x2265;ilon ) for some constant &#x2265;ilon 0 and, independently, by Nanongkai and Saranurak NanongkaiS16 with update time O(n^ 0.494 ) (the latter works only for maintaining a spanning forest).Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n^ 0.5-&#x2265;ilon ) in Wulff-Nilsen16a to O(n^ o(1) ) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the contraction technique by Henzinger and King HenzingerK97b and HolmLT01, we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1+o(1))n) edges. This significantly improves the previous approach in Wulff-Nilsen16a, NanongkaiS16 which is based on Fredericksons 2-dimensional topology tree Frederickson85 and illustrates a new application to this old technique. Previous experiments with low dimensional data sets have shown that Gabriel graph methods for instance-based learning are among the best machine learning algorithms for pattern classification applications. However, as the dimensionality of the data grows large, all data points in the training set tend to become Gabriel neighbors of each other, bringing the efficacy of this method into question. Indeed, it has been conjectured that for high-dimensional data, proximity graph methods that use sparser graphs, such as relative neighbor graphs (RNG) and minimum spanning trees (MST) would have to be employed in order to maintain their privileged status. Here the performance of proximity graph methods, in instance-based learning, that employ Gabriel graphs, relative neighborhood graphs, and minimum spanning trees, are compared experimentally on high-dimensional data sets. These methods are also compared empirically against the traditional k-NN rule and support vector machines (SVMs), the leading competitors of proximity graph methods. A family of graph-theoretical algorithms based on the minimal spanning tree are capable of detecting several kinds of cluster structure in arbitrary point sets; description of the detected clusters is possible in some cases by extensions of the method. Development of these clustering algorithms was based on examples from two-dimensional space because we wanted to copy the human perception of gestalts or point groupings. On the other hand, all the methods considered apply to higher dimensional spaces and even to general metric spaces. Advantages of these methods include determinacy, easy interpretation of the resulting clusters, conformity to gestalt principles of perceptual organization, and invariance of results under monotone transformations of interpoint distance. Brief discussion is made of the application of cluster detection to taxonomy and the selection of good feature spaces for pattern recognition. Detailed analyses of several planar cluster detection problems are illustrated by text and figures. The well-known Fisher iris data, in four-dimensional space, have been analyzed by these methods also. PL 1 programs to implement the minimal spanning tree methods have been fully debugged.