Exploration tasks are embedded in many robotics applications, such as search and rescue and space exploration. Information-based exploration algorithms aim to find the most informative trajectories by maximizing an information-theoretic metric, such as the mutual information between the map and potential future measurements. Unfortunately, most existing information-based exploration algorithms are plagued by the computational difficulty of evaluating the Shannon mutual information metric. In this article, we consider the fundamental problem of evaluating Shannon mutual information between the map and a range measurement. First, we consider 2D environments. We propose a novel algorithm, called the fast Shannon mutual information (FSMI). The key insight behind the algorithm is that a certain integral can be computed analytically, leading to substantial computational savings. Second, we consider 3D environments, represented by efficient data structures, e.g., an OctoMap, such that the measurements are compressed by run-length encoding (RLE). We propose a novel algorithm, called FSMI-RLE, that efficiently evaluates the Shannon mutual information when the measurements are compressed using RLE. For both the FSMI and the FSMI-RLE, we also propose variants that make different assumptions on the sensor noise distribution for the purpose of further computational savings. We evaluate the proposed algorithms in extensive experiments. In particular, we show that the proposed algorithms outperform existing algorithms that compute Shannon mutual information as well as other algorithms that compute the Cauchy–Schwarz quadratic mutual information (CSQMI). In addition, we demonstrate the computation of Shannon mutual information on a 3D map for the first time.

中文翻译：

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FSMI：用于信息论映射的香农互信息的快速计算

探索任务嵌入在许多机器人应用中，例如搜救和太空探索。基于信息的探索算法旨在通过最大化信息理论度量（例如地图和潜在的未来测量之间的互信息）来找到信息量最大的轨迹。不幸的是，大多数现有的基于信息的探索算法都受到评估香农互信息度量的计算困难的困扰。在本文中，我们考虑评估地图和距离测量之间的香农互信息的基本问题。首先，我们考虑 2D 环境。我们提出了一种新算法，称为快速香农互信息（FSMI）。该算法背后的关键见解是可以分析计算某个积分，导致大量的计算节省。其次，我们考虑由高效数据结构（例如 OctoMap）表示的 3D 环境，以便通过运行长度编码 (RLE) 压缩测量值。我们提出了一种称为 FSMI-RLE 的新算法，当使用 RLE 压缩测量值时，该算法可以有效地评估香农互信息。对于 FSMI 和 FSMI-RLE，我们还提出了对传感器噪声分布做出不同假设的变体，以进一步节省计算量。我们在广泛的实验中评估了所提出的算法。特别是，我们表明所提出的算法优于计算香农互信息的现有算法以及计算柯西-施瓦茨二次互信息（CSQMI）的其他算法。此外，