# FSMI: Fast computation of Shannon Mutual Information for information-theoretic mapping

arXiv: https://arxiv.org/abs/1905.02238 · HTML: https://mlsanity.com/paper/1905.02238/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2019.

## Built on by (most cited first, 10 of 10)

- [GauSS-MI: Gaussian Splatting Shannon Mutual Information for Active 3D Reconstruction](https://mlsanity.com/paper/2504.21067/) (2025)
- [Deep Reinforcement Learning-based Large-scale Robot Exploration](https://mlsanity.com/paper/2403.10833/) (2024)
- [Incremental Multimodal Surface Mapping via Self-Organizing Gaussian Mixture Models](https://mlsanity.com/paper/2309.10900/) (2023)
- [Learning Continuous Control Policies for Information-Theoretic Active Perception](https://mlsanity.com/paper/2209.12427/) (2022)
- [SEAL: Simultaneous Exploration and Localization in Multi-Robot Systems](https://mlsanity.com/paper/2306.12623/) (2023)
- [HEADER: Hierarchical Robot Exploration via Attention-Based Deep Reinforcement Learning with Expert-Guided Reward](https://mlsanity.com/paper/2510.15679/) (2025)
- [GO-PRE: Goal-Oriented Next-Best-View Selection via Predictive Rendering Entropy for Active 3D Reconstruction](https://mlsanity.com/paper/2607.29037/) (2026)
- [involve-MI: Informative Planning with High-Dimensional Non-Parametric Beliefs](https://mlsanity.com/paper/2209.11591/) (2022)
- [CARE: Confidence-rich Autonomous Robot Exploration using Bayesian Kernel Inference and Optimization](https://mlsanity.com/paper/2309.05200/) (2023)
- [2D GauSS-MI: Efficient Active Scene Reconstruction with Balanced Visual and Geometric Quality](https://mlsanity.com/paper/2609.21516/) (2026)

## Abstract (arXiv)

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.