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Correct-by-Construction Vision-based Pose Estimation using Geometric Generative Models

Ulices Santa Cruz, Mahmoud Elfar, Yasser Shoukry

TL;DR

This work tackles the absence of certifiable guarantees in vision-based pose estimation by integrating physics-based image formation with learning-based perception through a Geometric Generative Model (GGM). It designs target-specific GGMs that reproduce ideal images from camera poses and uses a decoder-encoder architecture to provide deterministic pose-estimation bounds, even for unseen data. The framework is extended to cluttered environments via forward-reachability-based detectors and a multi-stage perception pipeline that employs spatial filtering and region-wise detectors with bounded clutter intrusion. Empirically, GGMs can generate images that closely resemble real camera views, and the approach yields certified pose estimates and robust target detection in both synthetic and event-based camera settings, signaling strong potential for safety-critical autonomous systems.

Abstract

We consider the problem of vision-based pose estimation for autonomous systems. While deep neural networks have been successfully used for vision-based tasks, they inherently lack provable guarantees on the correctness of their output, which is crucial for safety-critical applications. We present a framework for designing certifiable neural networks (NNs) for perception-based pose estimation that integrates physics-driven modeling with learning-based estimation. The proposed framework begins by leveraging the known geometry of planar objects commonly found in the environment, such as traffic signs and runway markings, referred to as target objects. At its core, it introduces a geometric generative model (GGM), a neural-network-like model whose parameters are derived from the image formation process of a target object observed by a camera. Once designed, the GGM can be used to train NN-based pose estimators with certified guarantees in terms of their estimation errors. We first demonstrate this framework in uncluttered environments, where the target object is the only object present in the camera's field of view. We extend this using ideas from NN reachability analysis to design certified object NN that can detect the presence of the target object in cluttered environments. Subsequently, the framework consolidates the certified object detector with the certified pose estimator to design a multi-stage perception pipeline that generalizes the proposed approach to cluttered environments, while maintaining its certified guarantees. We evaluate the proposed framework using both synthetic and real images of various planar objects commonly encountered by autonomous vehicles. Using images captured by an event-based camera, we show that the trained encoder can effectively estimate the pose of a traffic sign in accordance with the certified bound provided by the framework.

Correct-by-Construction Vision-based Pose Estimation using Geometric Generative Models

TL;DR

This work tackles the absence of certifiable guarantees in vision-based pose estimation by integrating physics-based image formation with learning-based perception through a Geometric Generative Model (GGM). It designs target-specific GGMs that reproduce ideal images from camera poses and uses a decoder-encoder architecture to provide deterministic pose-estimation bounds, even for unseen data. The framework is extended to cluttered environments via forward-reachability-based detectors and a multi-stage perception pipeline that employs spatial filtering and region-wise detectors with bounded clutter intrusion. Empirically, GGMs can generate images that closely resemble real camera views, and the approach yields certified pose estimates and robust target detection in both synthetic and event-based camera settings, signaling strong potential for safety-critical autonomous systems.

Abstract

We consider the problem of vision-based pose estimation for autonomous systems. While deep neural networks have been successfully used for vision-based tasks, they inherently lack provable guarantees on the correctness of their output, which is crucial for safety-critical applications. We present a framework for designing certifiable neural networks (NNs) for perception-based pose estimation that integrates physics-driven modeling with learning-based estimation. The proposed framework begins by leveraging the known geometry of planar objects commonly found in the environment, such as traffic signs and runway markings, referred to as target objects. At its core, it introduces a geometric generative model (GGM), a neural-network-like model whose parameters are derived from the image formation process of a target object observed by a camera. Once designed, the GGM can be used to train NN-based pose estimators with certified guarantees in terms of their estimation errors. We first demonstrate this framework in uncluttered environments, where the target object is the only object present in the camera's field of view. We extend this using ideas from NN reachability analysis to design certified object NN that can detect the presence of the target object in cluttered environments. Subsequently, the framework consolidates the certified object detector with the certified pose estimator to design a multi-stage perception pipeline that generalizes the proposed approach to cluttered environments, while maintaining its certified guarantees. We evaluate the proposed framework using both synthetic and real images of various planar objects commonly encountered by autonomous vehicles. Using images captured by an event-based camera, we show that the trained encoder can effectively estimate the pose of a traffic sign in accordance with the certified bound provided by the framework.
Paper Structure (33 sections, 6 theorems, 71 equations, 24 figures, 4 tables, 5 algorithms)

This paper contains 33 sections, 6 theorems, 71 equations, 24 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notation
  4. Target Model
  5. Camera Model
  6. Problem Formulation
  7. NN-based Generative Geometric Model
  8. Pinhole Camera Model for Single Point
  9. Generalized Camera Model for Convex Polygons
  10. Compositional Model for $\mathit{N}$-Polygon Targets
  11. Geometric Generative Model (GGM) for N-Polygon Target
  12. Certified Vision-Based Pose Estimation for Uncluttered Environments
  13. Problem Overview
  14. GGM-based Decoder-Encoder Framework
  15. Formal Guarantees on Encoder Performance
  16. ...and 18 more sections

Key Result

Theorem 1

Consider a system $(\mathcal{T}, \overline{\mathcal{T}}, \mathcal{C}_{\mathcal{T}}, \Xi)$ where $\overline{\mathcal{T}} = \varnothing$, i.e., an uncluttered environment. Given a neural network encoder $\mathcal{{N\!\!N}}_{\!\triangleright}$ trained using Algorithm alg:dec_enc_framework. Let Assumpti where $\mathit{L}_{\triangleleft}$ and $\mathit{L}_{\triangleright}$ are the Lipschitz constants of

Figures (24)

  • Figure 1: Example of (left) a taxiway planar marking, and (right) its known geometry (source: faa2020advisory) that can be used for vision-based pose estimation.
  • Figure 2: Sequence of camera images of an aircraft approaching a runway for autonomously landing. The known geometry of the planar runway markings can be used for vision-based pose estimation.
  • Figure 3: Example of (left) a stop sign as a target; (center) its image generated by the camera; and (right) its target model composed of 8 polygons representing a non-convex shape.
  • Figure 4: Ideal pinhole camera illustration, showing the target (TCF), camera (CCF), and pixel (PCF) coordinate frames. $\lambda_i$ denotes the projection of point $\mathrm{v}_i$ onto the camera sensor.
  • Figure 5: Orientation test for determining whether each pixel $\mathit{q}$ lies within the convex hull of the projected vertices $\lambda_1, \lambda_2, \lambda_3$ of the convex polygon from Fig. \ref{['fig:pinhole_camera']}. The sign of the cross products are shown for $C_{\mathit{q}}(1,2)$, $C_{\mathit{q}}(2,3)$, and $C_{\mathit{q}}(3,1)$.
  • ...and 19 more figures

Theorems & Definitions (24)

  • Definition 1: Target Model
  • Example 1
  • Definition 2: Ideal Camera Model
  • Definition 3: Image Clutter
  • Definition 4: System Model
  • Definition 5: Analog Behavior
  • Definition 6: Digital Behavior
  • Definition 7: Digital Behavior for Convex Polygons
  • Example 2
  • Definition 8: Injectivity Condition
  • ...and 14 more