⚡ electrical engineering

Uncertainty Representation in a SOTIF-Related Use Case with Dempster-Shafer Theory for LiDAR Sensor-Based Object Detection

This paper proposes a systematic framework for SOTIF-compliant LiDAR object detection that utilizes Dempster-Shafer Theory with Yager's Rule to model and combine uncertainty sources, while employing variance-based sensitivity analysis to quantify their specific impacts on detection accuracy.

Original authors: Milin Patel, Rolf Jung

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Milin Patel, Rolf Jung

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a self-driving car on a rainy, foggy country road. You are trying to overtake a slow cyclist while a truck is coming the other way. Your car's "eyes" are LiDAR sensors, which shoot out invisible laser beams to map the world. But just like human eyes struggle in a heavy downpour, these lasers get confused by rain, fog, and wet roads. They might see a ghost (a false alarm) or miss a real danger (a missed cyclist).

This paper is about teaching the car's computer how to say, "I'm not 100% sure what I'm seeing, and here is exactly why I'm unsure."

Here is a simple breakdown of what the authors did:

1. The Problem: The "Foggy Window" of Uncertainty

The authors focus on a specific safety goal called SOTIF (Safety of the Intended Functionality). Think of SOTIF as the car's promise to stay safe even when the rules of the road get weird or the sensors get confused.

The problem is that LiDAR sensors make mistakes because of two types of "fog":

The Weather Fog (Aleatoric): This is the rain, the wet road, and the actual fog outside. The car can't control this; it just happens.
The Knowledge Fog (Epistemic): This is the car's own limitations. Maybe its "brain" (the AI model) hasn't seen enough rainy days in its training, or the sensor just isn't perfect.

2. The Solution: A New Way to Count "Maybe"

Usually, computers use standard math (like flipping a coin) to guess probabilities. But if you don't have enough data, a coin flip is a bad guess.

Instead, the authors used a special math tool called Dempster-Shafer Theory (DST).

The Analogy: Imagine you are a detective trying to solve a crime.
- Standard Math says: "There is a 60% chance the butler did it."
- DST says: "I have some evidence pointing to the butler, some evidence pointing to the gardener, and a big pile of evidence that just says 'I don't know yet.' I can keep all these possibilities open at the same time without forcing a single answer."

This is perfect for self-driving cars because it allows the system to admit, "I am unsure," rather than making a risky guess.

3. The Map: Connecting the Dots

The authors built a map (called a DAG) to show how different problems connect.

Rain makes the road wet.
A wet road makes the laser beams bounce weirdly.
Weird bounces confuse the sensor.
A confused sensor makes the AI brain less sure about where the cyclist is.

They used this map to trace how a little bit of rain can turn into a big safety risk.

4. The Experiment: The "Country Road" Test

They tested this on a specific scenario: A car driving at 60 km/h trying to pass a cyclist on a narrow, rainy road with a truck coming the other way.

They asked the system to identify four possible outcomes:

I see the cyclist correctly.
I see the truck correctly.
I miss the cyclist (False Negative).
I see a cyclist where there isn't one (False Positive).

Using their special math (DST), they combined all the confusing signals (rain, fog, wet roads) to see how much the car's confidence dropped.

5. The Results: What Matters Most?

By running a "sensitivity analysis" (basically, a stress test), they found out which factors were the biggest troublemakers:

The Big Bad: Rain intensity and the quality of the sensor were the biggest causes of uncertainty. If it rains hard, the car's confidence plummets.
The Medium Trouble: Wet roads and noise in the data were also significant.
The Minor Nuisance: Fog density and signal scattering were less critical than rain, but still mattered.

6. The Fix: How to Make it Safer

Based on their findings, the authors suggest three main ways to fix the problem:

Smarter Software: Create algorithms that can automatically adjust when it starts raining, kind of like how your car's wipers speed up when the rain gets heavier.
Cleaner Data: Use better tools to scrub the "noise" (static) out of the sensor data so the car sees a clearer picture.
Better Training: Teach the AI by showing it thousands of simulated rainy and foggy days so it learns what to expect before it ever hits a real storm.

Summary

This paper doesn't invent a new sensor or a new car. Instead, it invents a new way of thinking about uncertainty. It gives the self-driving car a vocabulary to say, "I am seeing a cyclist, but because it is raining hard, I am only 70% sure, and here is the math to prove it." This helps engineers know exactly where to focus their efforts to make autonomous driving safer in bad weather.

Technical Summary: Uncertainty Representation in a SOTIF-Related Use Case with Dempster-Shafer Theory for LiDAR Sensor-Based Object Detection

Problem Statement
The safety of intended functionality (SOTIF) in automated driving systems (ADS) requires addressing hazards arising from functional insufficiencies, particularly in environments with unpredictable conditions where operational parameters are ill-defined. LiDAR-based object detection, while critical for 3D environmental perception, is susceptible to significant uncertainty caused by environmental variability (e.g., rain, fog) and sensor performance limitations. These uncertainties manifest as both aleatoric (inherent environmental variability) and epistemic (system knowledge/model limitations) types. Existing methods often struggle to represent these uncertainties without relying on biased prior probabilities or failing to account for conflicting evidence. This paper addresses the need for a systematic approach to identify, classify, and represent these uncertainties to ensure the reliability of ADS perception functions.

Methodology
The authors propose a structured workflow to model uncertainty within a specific SOTIF-related use case: a LiDAR-equipped vehicle navigating a two-lane country road in adverse weather (rain and fog) while interacting with a cyclist and an oncoming truck.

Uncertainty Identification and Mapping:
- Sources of uncertainty are categorized into aleatoric (e.g., rain intensity, fog density, wet road conditions, signal scattering) and epistemic (e.g., deep learning model training quality, sensor limitations).
- Dependencies between these sources are mapped using a Directed Acyclic Graph (DAG), illustrating how environmental factors (like rain) influence intermediate states (wet roads, signal scattering) which ultimately affect detection accuracy.
Dempster-Shafer Theory (DST) Framework:
- Frame of Discernment (FoD): Defined as $\Theta = \{\theta_1, \theta_2, \theta_3, \theta_4\}$ , representing four detection outcomes: detection of a cyclist, detection of a truck, false negative (object present but undetected), and false positive (incorrect detection).
- Basic Probability Assignments (BPAs): Conditional BPAs are assigned to subsets of the FoD based on the states of identified uncertainty sources (e.g., low/medium/high rain intensity). These are derived from simulation data to capture both aleatoric and epistemic uncertainties.
- Evidence Combination: To handle conflicting evidence from multiple sources (e.g., when one source suggests a high probability of detection while another suggests a high probability of noise), the paper employs Yager's Rule of Combination. Unlike Dempster's rule, Yager's rule redistributes conflicting mass to the universal set $\Theta$ without normalization, preserving information about ignorance and conflict in complex systems.
- Quantification: The uncertainty for each source is quantified using the interval between the Belief ($Bel$) and Plausibility ($Pl$) functions. The difference $U(A) = Pl(A) - Bel(A)$ serves as a metric for the level of uncertainty.
Sensitivity Analysis and Mitigation:
- Variance-Based Sensitivity Analysis (VBSA): Applied to quantify the contribution of each uncertainty source to the total variance in detection performance (specifically in $Bel$ and $Pl$ values).
- Prioritization: Sources are classified into Low, Moderate, and High impact categories based on the magnitude of their uncertainty intervals and variance contributions.
- Mitigation Strategies: Specific measures are proposed based on the high-impact sources identified.

Key Contributions
The paper makes two primary contributions:

Systematic Approach: It presents a structured methodology for identifying, classifying, and representing uncertainties in LiDAR-based object detection specifically within a SOTIF context.
DST Application: It demonstrates the application of Dempster-Shafer Theory to manage these uncertainties. Unlike previous works that focused on performance evaluation, this study shifts focus to uncertainty representation to develop targeted mitigation strategies. The use of Yager's rule is highlighted as a mechanism to resolve conflicts without forcing a single probabilistic outcome, which is crucial when evidence is contradictory or insufficient.

Results

Uncertainty Quantification: The application of DST revealed that certain environmental factors significantly degrade detection confidence. For instance, rain intensity was found to increase the probability of False Positives, while fog density and signal scattering correlated with increased False Negatives.
Impact Classification: Using the $Pl - Bel$ metric, uncertainty sources were categorized:
- High Impact: Rain intensity, LiDAR sensor performance, and surface type.
- Moderate Impact: Wet road conditions, noise in LiDAR data, DL model training quality, and object proximity.
- Low Impact: Fog density, scattering of LiDAR signals, and reflection variability.
Sensitivity Analysis: VBSA results indicated that rain intensity and scattering of LiDAR signals generate the highest variance in detection performance, followed by wet road conditions and sensor performance.
Mitigation Proposals: Based on these findings, the authors propose:
- Developing adaptive algorithms to adjust to varying weather conditions.
- Applying noise reduction techniques to minimize environmental interference.
- Expanding training datasets with simulated extreme weather data to improve model generalization.

Significance and Limitations
The paper claims significance in providing a structured framework to evaluate how uncertainties affect detection accuracy, moving beyond simple performance metrics to a deeper analysis of why performance varies. By utilizing DST, the authors argue for a more nuanced representation of uncertainty that accommodates both lack of knowledge (epistemic) and inherent variability (aleatoric) without requiring difficult-to-justify prior probabilities.

The authors modestly acknowledge several limitations:

Subjectivity: The assignment of BPAs involves a degree of subjectivity which may introduce bias.
Static Nature: The current analysis is static and does not capture temporal dynamics, which are critical for real-time decision-making in dynamic environments.
Sensitivity to Assumptions: Conclusions are sensitive to initial assumptions and the specific choice of combination rules in DST.

Future work is suggested to extend this static framework into a dynamic one, potentially integrating temporal models like Hidden Markov Models (HMMs) or Bayesian Networks to track uncertainty propagation over time, and validating these models against real-world data.