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Subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure. Such methods with sparsity prior, such as sparse subspace clustering (SSC) (Elhamifar and Vidal in IEEE Trans Pattern Anal Mach Intell 35(11):2765-2781, 2013) with the sparsity induced by the ℓ1-norm, are demonstrated to be effective in subspace clustering. Most of those methods require certain assumptions, e.g. independence or disjointness, on the subspaces. However, these assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods. In this paper, we propose ℓ0-induced sparse subspace clustering (ℓ0-SSC). In contrast to the required assumptions, such as independence or disjointness, on subspaces for most existing sparse subspace clustering methods, we prove that ℓ0-SSC guarantees the subspace-sparse representation, a key element in subspace clustering, for arbitrary distinct underlying subspaces almost surely under the mild i.i.d. assumption on the data generation. We also present the “no free lunch” theorem which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding ℓ0 sparse representation problem of ℓ0-SSC. A novel approximate algorithm named Approximate ℓ0-SSC (Aℓ0-SSC) is developed which employs proximal gradient descent to obtain a sub-optimal solution to the optimization problem of ℓ0-SSC with theoretical guarantee. The sub-optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results. Extensive experimental results on various data sets demonstrate the superiority of Aℓ0-SSC compared to other competing clustering methods. Furthermore, we extend ℓ0-SSC to semi-supervised learning by performing label propagation on the sparse similarity matrix learnt by Aℓ0-SSC and demonstrate the effectiveness of the resultant semi-supervised learning method termed ℓ0-sparse subspace label propagation (ℓ0-SSLP). [ABSTRACT FROM AUTHOR]

Optimization with graph cuts became very popular in recent years. While exact optimization is possible in a few cases, many useful energy functions are NP hard to optimize. One approach to approximate optimization is the so-called move making algorithms. At each iteration, a move-making algorithm makes a proposal (move) for a pixel p to either keep its old label or switch to a new label. Two move-making algorithms based on graph cuts are in wide use, namely the swap and expansion. Both of these moves are binary in nature, that is they give each pixel a choice of only two labels. An evaluation of optimization techniques shows that the expansion and swap algorithms perform very well for energies where the underlying MRF has the Potts prior. However for more general priors, the swap and expansion algorithms do not perform as well. The main contribution of this paper is to develop multi-label moves. A multi-label move, unlike expansion and swap, gives each pixel has a choice of more than two labels to switch to. In particular, we develop several multi-label moves for truncated convex priors. We evaluate our moves on image restoration, inpainting, and stereo correspondence. We get better results than expansion and swap algorithms, both in terms of the energy value and accuracy. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding duality-based approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality. [ABSTRACT FROM AUTHOR]

In this paper, a class of deformable contour methods using a constrained optimization approach of minimizing a contour energy function satisfying an interior homogeneity constraint is proposed. The class is defined by any positive potential function describing the contour interior characterization. An evolutionary strategy is used to derive the algorithm. A similarity threshold Tv can be used to determine the interior size and shape of the contour. Sensitivity and significance of Tv and σ (a spreadness measure) are also discussed and shown. Experiments on noisy images and the convergence to a minimum energy gap contour are included. The developed method has been applied to a variety of medical images from CT abdominal section, MRI image slices of brain, brain tumor, a pig heart ultrasound image sequence to visual blood cell images. As the results show, the algorithm can be adapted to a broad range of medical images containing objects with vague, complex and/or irregular shape boundary, inhomogeneous and noisy interior, and contour with small gaps. [ABSTRACT FROM AUTHOR]