Unbiased Estimator for Distorted Conics in Camera Calibration

TL;DR

This paper presents a novel formulation for conic-based calibration using moments based on the mathematical finding that the first moment can be estimated without bias even under dis-tortion, which allows us to track moment changes during pro-jection and distortion, ensuring the preservation of the first moment of the distorted conic.

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.

Figures (8)

• Figure 1: Centroid estimation in distorted Images. This figure illustrates the effects of camera projection and lens distortion on circular targets within an image. Due to these distortions, circles lose their conic properties and deform into distorted ellipses, making center point tracking challenging. (Red) failure of conventional control point estimation methods, as indicated by an incorrectly tracked center point. (Green) the proposed unbiased estimator accurately identifies the center point of the transformed ellipse, as derived from closed-form calculations.
• Figure 2: Image projection geometry. Due to projection and lens distortion, there exists some mismatch between the projected center of the circle on the target plane (circled dot), projected center of ellipse on the normalized plane(triangle sign), and the actual centroid of the shape on the image plane (crossed sign). The transformed shape, caused by non-linear distortion in the normalized plane, cannot be analytically described as the original conic. However, despite these distortions, our algorithm successfully locates the true centroid through moment tracking.
• Figure 3: Moment calculation strategy for a arbitrary ellipse $\boldsymbol{Q}_n$. The moments of the rotated ellipse $\boldsymbol{Q}_n$ are obtained from the moments of the un-rotated standard ellipse $\boldsymbol{Q}_s$ using Theorem. \ref{['thm:rot']}. The moments of $\boldsymbol{Q}_s$ are obtained from moments of another standard ellipse $\boldsymbol{Q}_0$, which is located at the origin, using Theorem. \ref{['thm:vr']}.
• Figure 4: Reprojection error comparison in synthetic images. We examined the consistency and magnitude of the reprojection errors of each method as we varied the radius of the circles and the amount of distortion, represented as $d_1$. In the graphs, each curve represents the mean reprojection error and the envelope represents the corresponding standard deviation boundary. The first row uses raw images and the second row applies the Gaussian blur to the images to check robustness. Unlike other circular pattern methods, our method is unbiased, resulting in near-zero errors regardless of distortion and radius changes.
• Figure 5: Real world experiment: Reprojection error. The first row shows sample images from (a) RGB and (b) TIR cameras. The second row shows the distribution of the reprojection error divided by the distance from the camera to the target. The error distribution of the biased methods, such as point and conic-based methods, varies greatly with distance. The unbiased methods, such as checkerboard and ours, show a low and uniform error distribution. In TIR images, our method significantly outperforms the checkerboard method due to the robust control point of circular patterns.

Theorems & Definitions (17)

• Definition 1: Shape
• Definition 2: Moment
• Theorem 1
• proof
• Theorem 2: Rotation Equivariant
• proof
• Lemma 3
• proof
• Lemma 4
• proof