Discrete-Continuous Smoothing and Mapping (2204.11936v3)

Abstract: We describe a general approach for maximum a posteriori (MAP) inference in a class of discrete-continuous factor graphs commonly encountered in robotics applications. While there are openly available tools providing flexible and easy-to-use interfaces for specifying and solving inference problems formulated in terms of either discrete or continuous graphical models, at present, no similarly general tools exist enabling the same functionality for hybrid discrete-continuous problems. We aim to address this problem. In particular, we provide a library, DC-SAM, extending existing tools for inference problems defined in terms of factor graphs to the setting of discrete-continuous models. A key contribution of our work is a novel solver for efficiently recovering approximate solutions to discrete-continuous inference problems. The key insight to our approach is that while joint inference over continuous and discrete state spaces is often hard, many commonly encountered discrete-continuous problems can naturally be split into a "discrete part" and a "continuous part" that can individually be solved easily. Leveraging this structure, we optimize discrete and continuous variables in an alternating fashion. In consequence, our proposed work enables straightforward representation of and approximate inference in discrete-continuous graphical models. We also provide a method to approximate the uncertainty in estimates of both discrete and continuous variables. We demonstrate the versatility of our approach through its application to distinct robot perception applications, including robust pose graph optimization, and object-based mapping and localization.

• The paper presents an alternating minimization strategy for MAP inference in hybrid discrete-continuous factor graphs.
• It leverages conditional independence and incremental algorithms like iSAM2 to reduce complexity and boost computational efficiency.
• Demonstrated on SLAM and pose graph optimization tasks, the approach achieves robust performance with fewer iterations.

An Overview of Discrete-Continuous Smoothing and Mapping

The featured paper introduces a novel approach to hybrid discrete-continuous inference challenges within the domain of robotics, which are pivotal in applications such as semantic SLAM and robust pose estimation. The authors address these challenges by proposing a unified framework, accessible through their developed library, \DCSAM{}, designed to extend the capabilities of existing factor graph tools to encapsulate both discrete and continuous variables effectively. This framework fills a notable gap, as previous methodologies primarily focused on either continuous or discrete models separately.

Core Contributions

The paper's principal contribution is the development of an alternating minimization strategy for tackling maximum a posteriori (MAP) inference in discrete-continuous factor graphs. Recognizing the inherent difficulty in joint optimization over both state spaces, the authors propose an iterative approach to partition the problem into discrete and continuous components, optimizing each separately. This method enables approximate solutions to elaborate hybrid models while ensuring significant efficiency both computationally and in solution quality.

Key insights driving this method include:

1. Conditional Independence Utilization: By leveraging the conditional independence structure often present in these models, discrete variables can be isolated when conditioned on continuous estimates. This reduces problem complexity significantly.
2. Alternating Minimization: This approach involves iterating between discrete and continuous optimization steps, ensuring monotonic descent in objective value. The authors provide theoretical guarantees demonstrating that their method consistently improves upon these objectives.
3. Incremental Algorithm: By using existing incremental techniques available in tools like iSAM2, the approach is able to scale efficiently to large datasets typical in robotics, such as those encountered in SLAM.
4. Uncertainty Approximation: Beyond MAP estimation, they introduce a mechanism to approximate uncertainties in both discrete and continuous variables, crucial for decision-making under uncertainty in autonomous systems.

Numerical Results and Practical Implications

The authors validate their approach through several experiments, notably on problems like point-cloud registration, robust pose graph optimization, and semantic SLAM, utilizing datasets like the Stanford Dragon and KITTI for demonstration. High-quality solutions are achieved with fewer iterations than traditional approaches, substantiating the claim of computational efficiency coupled with the robustness of the solution. This is significant in real-time robotics applications where computational resources and time are often constrained.

The work has profound practical implications:

• Robust Mapping and Localization: In situations where robots operate in dynamic environments, the ability to distinguish outliers from genuine signals becomes paramount. \DCSAM{} enhances robustness by efficiently incorporating multi-hypothesis handling.
• Semantic Understanding: By coupling semantic data with geometric mapping, robots can navigate and interact with their environments more intelligently, enabling “understanding” beyond spatial awareness.

Future Research Directions

The paper paves the way for several interesting lines of inquiry. The efficiency of the approach suggests further exploration into scalability for increasingly complex models in robotics and beyond, such as in medical imaging or environmental monitoring. Extensions to more general forms of hybrid inference problems could be pursued, potentially enhancing applicability across different domains. Additionally, alternative strategies for initializing the optimization process could improve robustness against local minima in the solution space.

In conclusion, the paper presents a robust framework that significantly advances the state of the art for solving hybrid discrete-continuous inference problems within factor graphs. This work represents an important step in the ongoing development of tools that empower complex decision-making processes in robotics and should be of considerable interest to the research community focused on probabilistic graphical models and autonomous system design.