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## Description

We propose a new objective function for superpixel segmentation. This objective function consists of two components: entropy rate of a random walk on a graph and a balancing term. The entropy rate favors formation of compact and homogeneous clusters, while the balancing function encourages clusters with similar sizes. We present a novel graph construction for images and show that this construction induces a matroid--- a combinatorial structure that generalizes the concept of linear independence in vector spaces. The segmentation is then given by the graph topology that maximizes the objective function under the matroid constraint. By exploiting submodular and monotonic properties of the objective function, we develop an efficient greedy algorithm. Furthermore, we prove an approximation bound of $\frac{1}{2}$ for the optimality of the solution. Extensive experiments on the Berkeley segmentation benchmark show that the proposed algorithm outperforms the state of the art in all the standard evaluation metrics.