Section: Scientific Foundations

3D scene modelling based on projective geometry

3d reconstruction is the process of estimating the shape and position of 3d objects from views of these objects. TEMICS deals more specifically with the modelling of large scenes from monocular video sequences. 3d reconstruction using projective geometry is by definition an inverse problem. Some key issues which do not have yet satisfactory solutions are the estimation of camera parameters, especially in the case of a moving camera. Specific problems to be addressed are e.g. the matching of features between images, and the modelling of hidden areas and depth discontinuities. 3d reconstruction uses theory and methods from the areas of computer vision and projective geometry. When the camera is modelled as a perspective projection , the projection equations are : (1)

where is a 3d point with homogeneous coordinates in the scene reference frame , and where are the coordinates of its projection on the image plane Ii . The projection matrix Pi associated to the camera is defined as Pi = K(Ri|ti) . It is function of both the intrinsic parameters K of the camera, and of transformations (rotation Ri and translation ti ) called the extrinsic parameters and characterizing the position of the camera reference frame with respect to the scene reference frame . Intrinsic and extrinsic parameters are obtained through calibration or self-calibration procedures. The calibration is the estimation of camera parameters using a calibration pattern (objects providing known 3d points), and images of this calibration pattern. The self-calibration is the estimation of camera parameters using only image data. These data must have previously been matched by identifying and grouping all the image 2d points resulting from projections of the same 3d point. Solving the 3d reconstruction problem is then equivalent to searching for , given , i.e. to solve Eqn. (1 ) with respect to coordinates . Like any inverse problem, 3d reconstruction is very sensitive to uncertainty. Its resolution requires a good accuracy for the image measurements, and the choice of adapted numerical optimization techniques.

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