Paper:

# Refraction-Based Bundle Adjustment for Scale Reconstructible Structure from Motion

## Akira Shibata^{*,**}, Yukari Okumura^{*}, Hiromitsu Fujii^{*,***}, Atsushi Yamashita^{*}, and Hajime Asama^{*}

^{*}The University of Tokyo

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8654, Japan

^{**}Ricoh Co., Ltd.

2-7-1 Izumi, Ebina-City, Kanagawa 243-0460, Japan

^{***}Chiba Institute of Technology

2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan

Structure from motion is a three-dimensional (3D) reconstruction method that uses one camera. However, the absolute scale of objects cannot be reconstructed by the conventional structure from motion method. In our previous studies, to solve this problem by using refraction, we proposed a scale reconstructible structure from motion method. In our measurement system, a refractive plate is fixed in front of a camera and images are captured through this plate. To overcome the geometrical constraints, we derived an extended essential equation by theoretically considering the effect of refraction. By applying this formula to 3D measurements, the absolute scale of an object could be obtained. However, this method was verified only by a simulation under ideal conditions, for example, by not taking into account real phenomena such as noise or occlusion, which are necessarily caused in actual measurements. In this study, to robustly apply this method to an actual measurement with real images, we introduced a novel bundle adjustment method based on the refraction effect. This optimization technique can reduce the 3D reconstruction errors caused by measurement noise in actual scenes. In particular, we propose a new error function considering the effect of refraction. By minimizing the value of this error function, accurate 3D reconstruction results can be obtained. To evaluate the effectiveness of the proposed method, experiments using both simulations and real images were conducted. The results of the simulation show that the proposed method is theoretically accurate. The results of the experiments using real images show that the proposed method is effective for real 3D measurements.

*J. Robot. Mechatron.*, Vol.30 No.4, pp. 660-670, 2018.

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