Unbiased Estimator for Distorted Conics in Camera Calibration

Abstract: In the literature, points and conics have been major features for camera geometric calibration. Although conics are more informative features than points, the loss of the conic property under distortion has critically limited the utility of conic features in camera calibration. Many existing approaches addressed conic-based calibration by ignoring distortion or introducing 3D spherical targets to circumvent this limitation. In this paper, we present a novel formulation for conic-based calibration using moments. Our derivation is based on the mathematical finding that the first moment can be estimated without bias even under distortion. This allows us to track moment changes during projection and distortion, ensuring the preservation of the first moment of the distorted conic. With an unbiased estimator, the circular patterns can be accurately detected at the sub-pixel level and can now be fully exploited for an entire calibration pipeline, resulting in significantly improved calibration. The entire code is readily available from https://github.com/ChaehyeonSong/discocal.

• The paper introduces a moment-based approach that preserves the first moment of distorted conics, enabling unbiased calibration even under severe lens distortion.
• The paper derives analytic solutions using recursive transformation formulas to compute the moments of conics, ensuring computational efficiency.
• The paper demonstrates through extensive experiments that the proposed method significantly reduces reprojection errors and enhances calibration accuracy in both synthetic and real-world scenarios.

Overview of "Unbiased Estimator for Distorted Conics in Camera Calibration"

The paper "Unbiased Estimator for Distorted Conics in Camera Calibration" introduces a novel approach to camera calibration, which is a crucial step in 3D computer vision. Traditionally, camera calibration methods rely heavily on planar patterns such as checkerboards, which guarantee unbiased estimation under projective transformations but are limited to pixel-level detection accuracy. In contrast, circular patterns have been underutilized due to a biased projection model, leading to poor calibration results when considering lens distortions.

This research proposes an innovative calibration method by leveraging the moments of conics to develop an unbiased estimator, particularly focusing on the first moment, over traditional reliance on the geometric properties that distort under non-linear transformations. The central thesis of the paper is that circular patterns, which excel in sub-pixel level accuracy, can be fully exploited by ensuring an unbiased estimation under lens distortion, a commonly encountered issue in practical camera systems.

Methodological Contributions

1. Moment-Based Approach: The paper presents a key theoretical insight that the first moment of a projected conic under distortion can be preserved and accurately estimated regardless of the image distortion levels. By using the first moment, the centroid of distorted conics can be tracked precisely, thus enabling the exploitation of circular patterns throughout the calibration pipeline.
2. Analytic Solution for Moments: The paper provides a mathematic derivation for calculating the first and higher-order moments of distorted conics. This derivation allows the computation of the actual center of the distorted ellipse using polynomial mapping through recursion formulas and transformation invariants, ensuring computational efficiency.
3. Calibration Performance: Through extensive synthetic and real-world experiments, the authors validate their approach against traditional methods. Their findings demonstrate that the proposed method can outperform conventional approaches, especially under conditions of high lens distortion and image blur, such as those encountered when using TIR cameras.

Experimental Insights

The experimental setup compares the proposed moment-based estimator with existing point-based and conic-based estimators across various distortion levels and target sizes. It is shown that the proposed estimator consistently results in lower reprojection errors and more accurate intrinsic and extrinsic parameter estimations than existing methods. The authors emphasize that their method not only addresses distortion bias successfully but also operates efficiently, making it practical for real-time applications.

Additionally, the research showcases the superior robustness of the proposed method when applied to TIR camera images. The inherent challenges of low resolution and blurred boundaries typical of TIR images pose difficulties for checkerboard methods, which are effectively mitigated by the presented moment-based approach.

Theoretical and Practical Implications

The paper extends the theoretical framework of camera calibration by introducing a method that remains unbiased under typical radial distortions. Practically, this research has significant implications for enhancing the precision of camera calibration, crucial for applications in robotics, autonomous vehicles, and augmented reality systems, where reliable camera geometry is indispensable.

Speculations for Future Developments

The success of the moment-based unbiased estimator suggests potential extensions into calibrating non-standard distortion models or even across varying camera systems with non-central projection models. Additionally, as computational and algorithmic capacities improve, real-time calibration of moving cameras, continuous recalibration in changing environmental conditions, and integration into SLAM systems could all benefit from this refined estimator approach.

In conclusion, the technique introduced in this paper provides a robust solution to the longstanding challenge of exploiting circular patterns in camera calibration. By meticulously addressing the distortion biases, this research paves the way for high-accuracy calibration tasks, crucial for advancing camera-dependent technology in rapidly evolving fields.