We present PyBDR, a Python toolkit for reachability analysis based on set-boundary techniques, which centralizes on widely-adopted set propagation techniques for formal verification, controller synthesis, and state estimation. By analyzing the boundaries of initial sets, PyBDR mitigates the wrapping effect in computations, thereby enhancing the efficiency of reachability algorithms without incurring substantial computational overhead. Its modular architecture provides a collection of building blocks that streamline the prototyping of new reachability analysis methods. Illustrative examples highlight PyBDR’s key features and demonstrate its advantages in computing reachable sets for large initial conditions and long time horizons.

Abstract Inner-approximate reachability analysis involves calculating subsets of reachable sets, known as inner-approximations. This analysis is crucial in the fields of dynamic systems analysis and control theory as it provides a reliable estimation of the set of states that a system can reach from given initial states at a specific time instant. In this paper, we study the inner-approximate reachability analysis problem based on the set-boundary reachability method for systems modelled by ordinary differential equations, in which the computed inner-approximations are represented with zonotopes. The set-boundary reachability method computes an inner-approximation by excluding states reached from the initial set’s boundary. The effectiveness of this method is highly dependent on the efficient extraction of the exact boundary of the initial set. To address this, we propose methods leveraging boundary and tiling matrices that can efficiently extract and refine the exact boundary of the initial set represented by zonotopes. Additionally, we enhance the exclusion strategy by contracting the outer-approximations in a flexible way, which allows for the computation of less conservative inner-approximations. To evaluate the proposed method, we compare it with state-of-the-art methods against a series of benchmarks. The numerical results demonstrate that our method is not only efficient but also accurate in computing inner-approximations.

Abstract We present PyBDR, a Python reachability analysis toolkit based on set-boundary analysis, which centralizes on widely-adopted set propagation techniques for formal verification, controller synthesis, state estimation, etc. It employs boundary analysis of initial sets to mitigate the wrapping effect during computations, thus improving the performance of reachability analysis algorithms without significantly increasing computational costs. Beyond offering various set representations such as polytopes and zonotopes, our toolkit particularly excels in interval arithmetic by extending operations to the tensor level, enabling efficient parallel interval arithmetic computation and unifying vector and matrix intervals into a single framework. Furthermore, it features symbolic computation of derivatives of arbitrary order and evaluates them as real or interval-valued functions, which is essential for approximating behaviours of nonlinear systems at specific time instants. Its modular architecture design offers a series of building blocks that facilitate the prototype development of reachability analysis algorithms. Comparative studies showcase its strengths in handling verification tasks with large initial sets or long time horizons. The toolkit is available at https://github.com/ASAG-ISCAS/PyBDR .

We present a novel method to produce discernible image mosaics, with relatively large image tiles replaced by images drawn from a database, to resemble a target image. Compared to existing works on image mosaics, the novelty of our method is two-fold. Firstly, believing that the presence of visual edges in the final image mosaic strongly supports image perception, we develop an edge-aware photo retrieval scheme which emphasizes the preservation of visual edges in the target image. Secondly, unlike most previous works which apply a pre-determined partition to an input image, our image mosaics are composed of adaptive tiles, whose sizes are determined based on the available images in the database and the objective of maximizing resemblance to the target image. We show discernible image mosaics obtained by our method, using image collections of only moderate size. To evaluate our method, we conducted a user study to validate that the image mosaics generated present both globally and locally appropriate visual impressions to the human observers. Visual comparisons with existing techniques demonstrate the superiority of our method in terms of mosaic quality and perceptibility.