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.

In this paper, we study lens distortion for still images considering two well-known distortion models: the two-parameter polynomial model and the two-parameter division model. We study the invertibility of these models, and we mathematically characterize the conditions for the distortion parameters under which the distortion model defines a one-to-one transformation. This ensures that the inverse transformation is well defined and the distortion-free image can be properly computed, which provides robustness to the distortion models. A new automatic method to correct the radial distortion is proposed, and a comparative analysis for this method is extensively performed using the polynomial and the division models. With the aim of obtaining an accurate estimation of the model, we propose an optimization scheme which iteratively improves the parameters to achieve a better matching between the distorted lines and the edge points. The proposed method estimates two-parameter radial distortion models by detecting the longest distorted lines within the image. This is done by applying the Hough transform extended with a radial distortion parameter. Next, a two-parameter model is estimated using an iterative nonlinear optimization scheme. This scheme aims at minimizing the distance from the edge points to their associated lines by adjusting the two distortion parameters as well as the coordinates of the center of distortion. We present some experiments on real images with significant distortion to show the ability of the proposed approach to correct the radial distortion. A visual and quantitative comparison between both automatic two-parameter model estimations indicates that the division model is more efficient for those images showing strong distortion.

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.