Paper:

# Relative Magnitude of Gaussian Curvature via Self-Calibration

## Yi Ding^{*1}, Yuji Iwahori^{*2}, Takashi Nakagawa^{*2}, Tsuyoshi Nakamura^{*1}, Lifeng He^{*3}, Robert J. Woodham^{*4}, and Hidenori Itoh^{*1}

^{*1}Nagoya Institute of Technology, Gokiso-cho Showa-ku Nagoya 466-8555, Japan

^{*2}Chubu University, Matsumoto-cho 1200 Kasugai 487-8501, Japan

^{*3}Faculty of Information Science and Technology, Aichi Prefectural University, Aichi 480-1198, Japan

^{*4}University of British Columbia, Vancouver, B.C. Canada, V6T 1Z4

Gaussian curvature encodes important information about object shape. This paper presents a technique to recover the relative magnitude of Gaussian curvature from multiple images acquired under different conditions of illumination. Previous approaches make use of a separate calibration sphere. Here, we require no distinct calibration object. The novel idea is to use controlled motion of the target object itself for self-calibration. The target object is rotated in fixed steps in both the vertical and the horizontal directions. A distinguished point on the object serves as a marker. Neural network training data are obtained from the predicted geometric positions of the marker under known rotations. Four light sources with different directions are used. An RBF neural network learns the mapping of image intensities to marker position coordinates along a virtual sphere. Neural network maps four image irradiances on the target object onto a point on a virtual sphere. The area value surrounded by four mapped points onto a sphere gives an approximate value of Gaussian curvature. The modification neural network is learned for the basis function to obtain more accurate Gaussian curvature. Spatially varying albedo is allowed since the effect of albedo can be removed. It is shown that self-calibration makes it possible to recover the relative magnitude of Gaussian curvature at each point without a separate calibration object. No particular functional model of surface reflectance is assumed. Experiments with real data are demonstrated. Quantitative error analysis is provided for a synthetic example.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.14, No.1, pp. 99-109, 2010.

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