GPS has long been the standard for determining position and navigation across a wide range of applications. However, as this technology cannot be relied on in many scenarios, alternative sensing modalities, such as vision, could provide viable alternatives. This work investigated the use of an event-based camera for navigation, within a mobile embedded system. Event-based cameras generate events when pixels asynchronously detect changes in light intensity, and report these changes as binary spikes. Such sensors can provide very high temporal resolution, wide dynamic range and low power consumption. A ground-based rover was developed, hosting a DAVIS346 event-based camera, to test a navigation system based on a visual-inertial odometry (VIO) approach. Processing and communications were implemented on two microcontrollers (Raspberry Pi 5 and ESP32), both communicating with a base station, which tracked the position of the rover and status of onboard sensors. The event-based processing pipeline showed improved accuracy when using frames augmented with event data, as compared to regular frames only. Due to the scarcity of modular, open source, event-based VIO systems, this work focused on how the benefits of an event-based camera could be leveraged in OpenVINS, which is a traditional VIO system designed for conventional frame-based cameras. To the best of our knowledge this is the first demonstration of a framework that can leverage the benefits of an event-based camera in a traditional VIO system of this style.

Polynomial matrices and ideals generated by their minors appear in various domains such as cryptography, polynomial optimization and effective algebraic geometry. When the given matrix is symmetric, this additional structure on top of the determinantal structure, affects computations on the derived ideals. Thus, understanding the complexity of these computations is important. Moreover, this study serves as a stepping stone towards further understanding the effects of structure in determinantal systems, such as those coming from moment matrices. In this paper, we focus on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals. Under a variant of Fröberg's conjecture, we study its complexity for symmetric determinantal ideals and identify the gain of exploiting sparsity in the Sparse-FGLM algorithm compared with the classical FGLM algorithm. For an n×n symmetric matrix with polynomial entries of degree d, we show that the complexity of Sparse-FGLM for zero-dimensional determinantal ideals obtained from this matrix over that of the FGLM algorithm is at least O(1/d). Moreover, for some specific sizes of minors, we prove finer results of at least O(1/nd) and O(1/m3d).