Video stabilization aims at removing the undesirable effects of camera motion by estimating its shake and applying a smoothing compensation. This paper proposes a unified mathematical analysis and classification of existing smoothing strategies. We assume that the apparent velocity induced by the camera is estimated as a set of global parametric models, typically those of a homography. We classify the existing smoothing strategies into compositional and additive methods and discuss their technical issues, particularly the definition of the boundary conditions. Our discussion of the various alternatives leads to clear-cut conclusions. It rules out the global compositional methods in favor of local linear methods and finds the adequate boundary conditions. We also show that the best smoothing strategy yields a scale-space analysis of the camera ego-motion parameters. Analyzing this scale-space on examples, we show how it is highly characteristic of the camera path, permitting us to compute ego-motion frequencies and to detect periodic ego-motions like walking or running.

The aim of this paper is to study several strategies for the preservation of flow discontinuities in variational optical flow methods. We analyze the combination of robust functionals and diffusion tensors in the smoothness assumption. Our study includes the use of tensors based on decreasing functions, which has shown to provide good results. However, it presents several limitations and usually does not perform better than other basic approaches. It typically introduces instabilities in the computed motion fields in the form of independent blobs of vectors with large magnitude. We propose two alternatives to overcome these drawbacks: first, a simple approach that combines the decreasing function with a minimum isotropic smoothing, and second, a method that looks for the best parameter configuration that preserves the important motion contours and avoid instabilities. It relies on the input images and the regularization parameter. It is fully automatic, providing a near-optimal value for many sequences, as shown in the experiments. Both proposals allow to detect the contours of the motion field and produce more stable solutions for a large range of parameters. In the experimental results, we present a detailed study and comparison of the different strategies.

The use of decreasing functions, for mitigating the regularization at image contours, is typical in many recent optical flow methods. However, finding the correct parameter for getting the best of this strategy is challenging. Most of the methods use default parameters that are conservative, providing results that are not better than traditional approaches. Configurations that clearly enhance discontinuities may produce instabilities in the computed optical flows. This is due to the fact that the regularization process may get cancelled, yielding an ill-posed problem. In this work, we analyze the problem of instabilities and propose a method for efficiently determining the value of the parameter. We show that this approach allows us to obtain well preserved discontinuities at the same time that it avoids the ill-posed problem. The experiments with synthetic sequences demonstrate that the results are accurate and the selected parameter is close to the optimal value.

The aim of this work is to propose a model for computing the optical flow in a sequence of images with a spatio-temporal regularizer explicitly designed for large displacements. We study the introduction of a temporal regularizer that expands the information beyond two consecutive frames. We use the large optical flow constraint equation in the data term, the Nagel-Enkelmann operator for the spatial smoothness term and a newly designed temporal regularization. Our model is based on an energy functional that yields a partial differential equation (PDE). This PDE is embedded into a multipyramidal strategy to recover large displacements. The numerical experiments show that thanks to this regularizer the results are more stable and accurate.