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UBC Theses and Dissertations

Solving for relative orientation and depth McReynolds, Daniel Peter


A translating and rotating camera acquires images of a static scene from disparate viewpoints. Given a minimum number of corresponding image tokens from two or more images, the rotation and translation (referred to as the relative orientation) of the camera, and the relative depths of the tokens, can be computed between the views. Image tokens, which can be points or lines, are deemed corresponding if they arise from the same physical entity. The process of determining corresponding tokens is assumed to be known. This thesis poses the problem of solving for relative orientation and depth. The solution to this problem has applications in building object or scene models from multiple views and in passive navigation. A minimum of five corresponding pairs of points, measured from two perspective projection images, are required. In the case of lines, the measurements of six corresponding sets of lines, from a minimum of three images, are necessary for a unique solution. The algorithm simultaneously determines the rotation and translation of the camera between the images and the three dimensional position of the matched scene tokens. The translation and depth can only be determined up to a global scale factor. It is assumed that the motion of the camera or object is rigid. The equations that describe the motion and scene structure are nonlinear in the camera model parameters. Newton's method is used to perform the nonlinear parameter estimation. The camera model is based on the collinearity condition, well known in the area of photogrammetry. This condition states that the focal point, image point and object point must all lie on the same line under perspective projection. This approach differs from other similar approaches in the choice of model parameters and in the formulation of the collinearity condition. The formulation for line correspondences turns out to be an easy extension of the point formulation. To improve the range of convergence and the robustness of the algorithm, the Levenberg-Marquardt extension for discrete least squares is applied to the over-determined system. Results from a Monte Carlo simulation with noise-free images indicate a range of convergence for rotation of approximately plus or minus sixty degrees. Simulations with noisy images (image measurements perturbed by plus or minus three pixels) yield a range of convergence for rotation of approximately plus or minus forty degrees. The range of convergence in translation is in the order of twice the length of the translation vector in any direction parallel to the image plane. For translation orthogonal to the image plane, the range of convergence is approximately eighty percent of the length of the translation vector. Simulations with the same noisy images generated for the rotation tests, indicate that the range of convergence for translation is not appreciably affected by these noise levels. Tests with real images, in which a 3D model of an object is derived from multiple views with human assistance in determining line correspondences, yield results that agree reasonably well with the noisy image simulations.

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