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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 Im1 $\#119966 _i$ is modelled as a perspective projection , the projection equations are :

Im2 ${\mover p_i\#732 =P_i\mover x\#732 ,}$(1)

where Im3 $\mover x\#732 $ is a 3d point with homogeneous coordinates Im4 ${\mover x\#732 ={(x~y~z~1)}^t}$ in the scene reference frame Im5 $\#8475 _0$ , and where Im6 ${\mover p_i\#732 ={(X_i~Y_i~1)}^t}$ are the coordinates of its projection on the image plane Ii . The projection matrix Pi associated to the camera Im1 $\#119966 _i$ 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 Im7 $\#8475 _i$ with respect to the scene reference frame Im5 $\#8475 _0$ . 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 Im3 $\mover x\#732 $ , given Im8 $\mover p_i\#732 $ , i.e. to solve Eqn. (1 ) with respect to coordinates Im3 $\mover x\#732 $ . 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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