Categories of partitions are combinatorial structures arising from the representation theory of certain compact quantum groups and are linked to classical diagram algebras such as the Temperley-Lieb algebra. In this paper, we present efficient algorithms and data structures for partitions of sets and their corresponding category operations, including a concrete implementation in the computer algebra system OSCAR. Moreover, we show that there exists a category of partitions for which the natural computational problems of deciding membership of a given partition as well as counting partitions of a given size are algorithmically undecidable.

Since two years there is a recent trend in optical flow estimation to improve the results of state-of-the-art variational methods by applying additional filtering steps such as median filters, bilateral filters, and non-local techniques. So far, however, the application of such filters has been restricted to two-frame optical flow methods. In this paper, we go beyond this two-frame case and investigate the usefulness of such filtering steps for multi-frame optical flow estimation. Thereby we consider both the application to single flow fields as well as the filtering of the entire spatio-temporal flow volume. In this context, we propose the use of a joint trilateral filter that processes all flow fields simultaneously while imposing consistency of joint flow structures at the same time. Evaluations on the Middlebury benchmark clearly demonstrate the success of our filtering strategy. Achieving rank 3, our method yields state-of-the art results and significantly outperforms the baseline method providing considerably sharper results.

Despite the fact that temporal coherence is undeniably one of the key aspects when processing video data, this concept has hardly been exploited in recent optical flow methods. In this paper, we will present a novel parametrization for multi-frame optical flow computation that naturally enables us to embed the assumption of a temporally coherent spatial flow structure, as well as the assumption that the optical flow is smooth along motion trajectories. While the first assumption is realized by expanding spatial regularization over multiple frames, the second assumption is imposed by two novel first- and second-order trajectorial smoothness terms. With respect to the latter, we investigate an adaptive decision scheme that makes a local (per pixel) or global (per sequence) selection of the most appropriate model possible. Experiments show the clear superiority of our approach when compared to existing strategies for imposing temporal coherence. Moreover, we demonstrate the state-of-the-art performance of our method by achieving Top 3 results at the widely used Middlebury benchmark.