Abstract-Motion is a strong cue for unsupervised object-level grouping. In this paper, we demonstrate that motion will be exploited most effectively, if it is regarded over larger time windows. Opposed to classical two-frame optical flow, point trajectories that span hundreds of frames are less susceptible to short term variations that hinder separating different objects. As a positive side effect, the resulting groupings are temporally consistent over a whole video shot, a property that requires tedious post-processing in the vast majority of existing approaches. We suggest working with a paradigm that starts with semi-dense motion cues first and that fills up textureless areas afterwards based on color. This paper also contributes the Freiburg-Berkeley motion segmentation (FBMS) dataset, a large, heterogeneous benchmark with 59 sequences and pixel-accurate ground truth annotation of moving objects.

Abstract. In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly non-convex) and a convex (possibly non-differentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a non-smooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on non-convex problems. The convergence result is obtained based on the Kurdyka-Lojasiewicz inequality. This is a very weak restriction, which was used to prove convergence for several other gradient methods. First, an abstract convergence theorem for a generic algorithm is proved, and, then iPiano is shown to satisfy the requirements of this theorem. Furthermore, a convergence rate is established for the general problem class. We demonstrate iPiano on computer vision problems: image denoising with learned priors and diffusion based image compression.Key words. non-convex optimization, Heavy-ball method, inertial forward-backward splitting, Kurdyka-Lojasiewicz inequality, proof of convergence 1. Introduction. The gradient method is certainly one of the most fundamental but also one of the most simple algorithms to solve smooth convex optimization problems. In the last decades, the gradient method has been modified in many ways. One of those improvements is to consider so-called multi-step schemes [38,35]. It has been shown that such schemes significantly boost the performance of the plain gradient method. Triggered by practical problems in signal processing, image processing and machine learning, there has been an increased interest in so-called composite objective functions, where the objective function is given by the sum of a smooth function and a non-smooth function with an easy to compute proximal map. This initiated the development of the so-called proximal gradient or forward-backward method [28], that combines explicit (forward) gradient steps w.r.t. the smooth part with proximal (backward) steps w.r.t. the non-smooth part.In this paper, we combine the concepts of multi-step schemes and the proximal gradient method to efficiently solve a certain class of non-convex, non-smooth optimization problems. Although, the transfer of knowledge from convex optimization to non-convex problems is very challenging, it aspires to find efficient algorithms for certain non-convex problems. Therefore, we consider the subclass of non-convex problems

No abstract

Point trajectories have emerged as a powerful means to obtain high quality and fully unsupervised segmentation of objects in video shots. They can exploit the long term motion difference between objects, but they tend to be sparse due to computational reasons and the difficulty in estimating motion in homogeneous areas. In this paper we introduce a variational method to obtain dense segmentations from such sparse trajectory clusters. Information is propagated with a hierarchical, nonlinear diffusion process that runs in the continuous domain but takes superpixels into account. We show that this process raises the density from 3% to 100% and even increases the average precision of labels.

Natural image statistics indicate that we should use nonconvex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge of optimization. Recently, iteratively reweighed 1 minimization (IRL1) has been proposed as a way to tackle a class of nonconvex functions by solving a sequence of convex 2-1 problems. We extend the problem class to the sum of a convex function and a (nonconvex) nondecreasing function applied to another convex function. The proposed algorithm sequentially optimizes suitably constructed convex majorizers. Convergence to a critical point is proved when the Kurdyka-Lojasiewicz property and additional mild restrictions hold for the objective function. The efficiency and practical importance of the algorithm are demonstrated in computer vision tasks such as image denoising and optical flow. Most applications seek smooth results with sharp discontinuities. These are achieved by combining nonconvexity with higher order regularization.

We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate minimizer of the model function yields a descent direction, along which the next iterate is found. Complemented with an Armijo-like line search strategy, we obtain a flexible algorithm for which we prove (subsequential) convergence to a stationary point under weak assumptions on the growth of the model function error. Special instances of the algorithm with a Euclidean distance function are, for example, Gradient Descent, Forward-Backward Splitting, ProxDescent, without the common requirement of a "Lipschitz continuous gradient". In addition, we consider a broad class of Bregman distance functions (generated by Legendre functions), replacing the Euclidean distance. The algorithm has a wide range of applications including many linear and non-linear inverse problems in signal/image processing and machine learning.arXiv:1707.02278v4 [math.OC] 25 Jun 2018 Contributions and Related WorkSince sequential minimization of model functions does not require smoothness of the objective or the model function, non-smoothness is handled naturally. The crucial aspect is the "approximation quality" of the model function, which is controlled by a growth function, that describes the approximation error around the current iterate. Drusvyatskiy et al. [19] refer to such model functions as Taylor-like models. The difference among algorithms lies in the properties of such a growth function, rather than the specific choice of a model function.For the example of the Gradient Descent model function (linearization around the current iterate) for a continuously differentiable function, the value and the derivative of the growth function (approximation error) vanish at the current iterate. In this case, a line search strategy is required to determine a suitable step size that reduces the objective value. If the gradient of the objective function is additionally L-Lipschitz continuous, then the growth function satisfies a quadratic growth globally, and step sizes can be controlled analytically.A large class of algorithms, which are widely popular in machine learning, statistics, computer vision, signal and image processing can be cast in the same framework. This includes algorithms such as Forward-Backward Splitting [26] (Proximal Gradient Descent), ProxDescent [25,20] (or proximal Gauss-Newton method), and many others. They all obey the same growth function as Gradient Descent. This allows for a unified analysis of all these algorithms, which is a key contribution of this paper. Moreover, we allow for a broad class of (iteration dependent) Bregman proximity functions (e.g., generated by common entropies such as Boltzmann-Shannon, Fermi-Dirac, and Burg's entropy), which leads to new algorithms. To be generic in the choice of the objective, the model, and the Bregman functions, the algorithm is complemented with...

In this paper, we present a forward-backward splitting algorithm with additional inertial term for solving a strongly convex optimization problem of a certain type. The strongly convex objective function is assumed to be a sum of a non-smooth convex and a smooth convex function. This additional knowledge is used for deriving a worst-case convergence rate for the proposed algorithm. It is proved to be an optimal algorithm with linear rate of convergence. For certain problems this linear rate of convergence is better than the provably optimal worst-case rate of convergence for smooth strongly convex functions. We demonstrate the efficiency of the proposed algorithm in numerical experiments and examples from image processing.

Abstract. We consider a bilevel optimization approach for parameter learning in nonsmooth variational models. Existing approaches solve this problem by applying implicit differentiation to a sufficiently smooth approximation of the nondifferentiable lower level problem. We propose an alternative method based on differentiating the iterations of a nonlinear primal-dual algorithm. Our method computes exact (sub)gradients and can be applied also in the nonsmooth setting. We show preliminary results for the case of multi-label image segmentation.