Current algorithmic approaches for piecewise affine motion estimation are based on alternating motion segmentation and estimation. We propose a new method to estimate piecewise affine motion fields directly without intermediate segmentation. To this end, we reformulate the problem by imposing piecewise constancy of the parameter field, and derive a specific proximal splitting optimization scheme. A key component of our framework is an efficient 1D piecewise-affine estimator for vector-valued signals. The first advantage of our approach over segmentation-based methods is its absence of initialization. The second advantage is its lower computational cost, which is independent of the complexity of the motion field. In addition to these features, we demonstrate competitive accuracy with other piecewise-parametric methods on standard evaluation benchmarks. Our new regularization scheme also outperforms the more standard use of total variation and total generalized variation.
The importance of initialization when optimizing ) with an alternating scheme is illustrated in @cite_15 @cite_8 , where the optimization is initialized through advanced motion estimation methods @cite_13 @cite_3 . In @cite_34 , an alternating direction method of multipliers (ADMM) approach is used to solve ) without intermediate segmentation steps. However, the underlying model is piecewise-constant and not rich enough in most practical scenarios; it is initialized by a block matching algorithm.
@cite_8 @cite_3 @cite_15 @cite_34 @cite_13
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The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems. We show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules. The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for large-scale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking. In this paper we find the optimal algorithm parameters that minimize the convergence factor of the ADMM iterates in the context of l 2 -regularized minimization and constrained quadratic programming. Numerical examples show that our parameter selection rules significantly outperform existing alternatives in the literature. Alternating direction methods are a common tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image processing, which frequently requires the minimization of nondifferentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are presented to demonstrate the superior performance of the fast methods for a wide variety of problems. The alternating direction method of multipliers (ADMM), also well known as a special split Bregman algorithm in imaging, is being popularly used in many areas including the image processing field. One useful modification is the symmetric version of the original ADMM, which updates the Lagrange multiplier twice at each iteration and thus the variables are treated in a symmetric manner. The symmetric version of ADMM, however, is not necessarily convergent. It was recently found that the convergence of symmetric ADMM can be sufficiently ensured if both the step sizes for updating the Lagrange multiplier are shrunk conservatively. Despite the theoretical significance in ensuring convergence, however, smaller step sizes should be strongly avoided in practice. On the contrary, we actually have the desire of seeking larger step sizes whenever possible in order to accelerate the numerical performance. Another technique leading to numerical acceleration of ADMM is enlarging its step size by a constant originally p...