FSMI: Fast computation of Shannon Mutual Information for information-theoretic mapping (1905.02238v1)

Abstract: 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 paper, 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.

• The paper presents FSMI algorithms that significantly accelerate the computation of Shannon Mutual Information for occupancy mapping in 2D and 3D environments.
• The approaches, including FSMI, Approx-FSMI, and FSMI-RLE variants, achieve speed improvements up to an order of magnitude compared to traditional numerical methods.
• These algorithms enable real-time robotic exploration by balancing computational efficiency with the accuracy required for dynamic, information-driven mapping.

A Comprehensive Analysis of FSMI for Efficient Mutual Information Computation

The paper "FSMI: Fast computation of Shannon Mutual Information for information-theoretic mapping" presents novel algorithms to address computational challenges in robot exploration tasks, particularly those reliant on information-based exploration strategies. The focus of the paper is the computation of Shannon Mutual Information (MI) between potential measurements and an occupancy map, which is central to selecting optimal exploratory paths in robotic applications such as search and rescue or space exploration.

Traditionally, evaluating the Shannon MI involved computationally intensive numerical approximations, especially in complex environments. The authors introduce a series of algorithms called Fast Shannon Mutual Information (FSMI) to alleviate these computational barriers.

Overview of Proposed Algorithms

1. FSMI for 2D Mapping: The foundational FSMI algorithm leverages the analytical computation of certain integrals inherent to the Shannon MI formula, thus reducing reliance on discrete numerical integration. Considering Gaussian sensor noise models, FSMI computes MI exactly in $O(n^2)$ time complexity, where $n$ is the number of intersecting occupancy cells. The algorithm circumvents the need for high-resolution integrals and estimates MI more quickly than traditional numerical methods.
2. Approx-FSMI: An approximation of FSMI through Gaussian truncation reduces computational complexity to $O(n\Delta)$, where $\Delta$ is a constant truncation length. This approximation results in a balance between computational efficiency and accuracy, making it suitable for real-time applications.
3. Uniform-FSMI: By assuming uniform sensor noise, the authors derive an even more computationally efficient algorithm with $O(n)$ complexity, leveraging the piecewise constant nature of occupancy models.
4. FSMI-RLE for 3D Mapping: Extending FSMI to three-dimensional environments, the FSMI-RLE algorithm employs run-length encoding (RLE) of occupancy sequences to compress data and minimize data processing time, specifically in OctoMap-based mappings.
5. Approx-FSMI-RLE: Tailored for large 3D environments, this approximation method further accelerates MI computation using RLE while maintaining computational tractability, particularly when managing extensive datasets in memory-limited platforms.

Key Contributions and Comparative Analysis

The paper provides extensive theoretical analyses and a structured evaluation of FSMI against existing algorithms like Approx-CSQMI. Experimental results showcase significant improvements in computational speed without sacrificing necessary accuracy for real-world applicable scenarios. The FSMI and its variants deliver notable speed advantages over the typical Shannon MI computation approaches, extending up to an order of magnitude faster in some scenarios, while effectively managing approximation errors.

Theoretical and Practical Implications

The introduction of FSMI variants reflects a significant advancement in robot mapping and exploration. FSMI allows for higher frequency map updates and faster decision-making, critical factors for efficient autonomous navigation. The practical implications of this are considerable, enabling more dynamic and responsive robotic exploration systems that can operate within rapidly changing environments, a crucial improvement for applications such as disaster relief or unmanned planetary exploration.

Speculations and Future Directions

Looking forward, the developments made in FSMI could spark further investigation into adaptive exploration strategies. As computation paradigms shift, integrating FSMI algorithms with machine learning models for adaptive sensing strategies may yield even further efficiencies. Additionally, extending these algorithms to other sensor types or integrating them with multi-modal sensory data could expand their utility in a wider range of robotics and AI applications.

In conclusion, the proposed FSMI algorithms offer a substantial leap in computing mutual information metrics vital for information-theoretic mapping in robotics, marrying theoretical innovation with practical applicability. These algorithms provide a foundation upon which future advancements in autonomous exploration can build, pushing the boundaries of what is possible in computational efficiency and autonomy.