Spectral clustering is one of the most important image processing tools, especially for image segmentation. This specializes at taking local information such as edge weights and globalizing them. Due to its unsupervised nature, it is widely applicable. However, traditional spectral clustering is O(n 3/2). This poses a challenge, especially given the recent trend of large datasets. In this article, we propose an algorithm by using ideas from Γ −convergence, which is an amalgamation of Maximum Spanning Tree (MST) clustering and spectral clustering. This algorithm scales as O(nlog(n)) under certain conditions, while producing solutions which are similar to that of spectral clustering. Several toy examples are used to illustrate the similarities and differences. To validate the proposed algorithm, a recent state-of-the-art technique for segmentation-multiscale combinatorial grouping is used, where the normalized cut is replaced with the proposed algorithm and results are analyzed.

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Watershed technique from mathematical morphology (MM) is one of the most widely used operators for image segmentation. Recently watersheds are adapted to edge weighted graphs, allowing for wider applicability. However, a few questions remain to be answered-(a) How do the boundaries of the watershed operator behave? (b) Which loss function does the watershed operator optimize? (c) How does watershed operator relate with existing ideas from machine learning. In this article, a framework is developed, which allows one to answer these questions. This is achieved by generalizing the maximum margin principle to maximum margin partition and proposing a generic solution, MORPHMEDIAN, resulting in the maximum margin principle. It is then shown that watersheds form a particular class of MORPHMEDIAN classifiers. Using the ensemble technique, watersheds are also extended to ensemble watersheds. These techniques are compared with relevant methods from literature and it is shown that watersheds perform better than SVM on some datasets, and ensemble watersheds usually outperform random forest classifiers.

The problem of clustering is to partition the dataset into groups such that elements belonging to the same group are similar and elements belonging to the different groups are dissimilar. The unsupervised nature of the problem makes it widely applicable and also tough to solve objectively. Clustering in the context of image data is referred to as image segmentation. Distance based methods such as K-means fail to detect the non-globular clusters and hence spectral clustering was proposed to overcome this problem . This method detects the non globular structures by projecting the data set into a subspace, in which the usual clustering methods work well. Gamma convergence is the study of asymptotic behavior of minimizers of a family of minimization problems. Such a limit of minimizers is referred to as the gamma limit. Calculating the gamma limit for various variational problems has been proved useful -giving a different algorithm and insights into why existing methods work. In this article, we calculate the gamma limit of the spectral clustering methods, analyze its properties, and compare them with minimum spanning tree based clustering methods and spectral clustering methods.

In this article, we propose a novel algorithm to obtain a solution to the clustering problem with an additional constraint of connectivity. This is achieved by suitably modifying K-Means algorithm to include connectivity constraints. The modified algorithm involves repeated application of watershed transform, and hence is referred to as iterated watersheds. Detailed analysis of the algorithm is performed using toy examples. Iterated watersheds is compared with several image segmentation algorithms. It has been shown that iterated watersheds performs better than methods such as spectral clustering, isoperimetric partitioning, and K-Means on various measures. To illustrate the applicability of iterated watersheds -a simple problem of placing emergency stations and suitable cost function is considered. Using real world road networks of various cities, iterated watersheds is compared with K-Means and greedy K-center methods. It is observed that iterated watersheds result in 4 -66 percent improvement over K-Means and in 31 -72 percent improvement over Greedy K-Centers in experiments on road networks of various cities.