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

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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.