Linear Reconstruction with Error Analysis for Image Sequences

This paper deals with the problem of reconstruction from a sequence of images, by linear methods. The reconstruction is made directly, without calculating the fundamental matrices or the trilinear tensors. In a general setting it is shown that it is possible to do such a reconstruction of six points from four images up to a projective transformation. Furthermore, when more point matches are available, it is possible to reconstruct subsets of six points linearly and then make a linear reconstruction of the whole scene. This can be done in the same way using more than four images. The possibility to make direct shape recovery was pointed out in 1], where it was shown that there is a duality between reconstruction of the object and reconstruction of the motion of the camera. A similar approach has been outlined in 6], with the diierence that they are using seven points each time. Furthermore we are using an aanely reduced framework, introduced in 3], which simpliies our approach. This new reconstruction is then analysed when measurement errors are introduced. Their aaect can be derived analytically in a rst order approximation if the errors are assumed to be normally distributed. 1 Summary The aim of this work is to reconstruct an object consisting of n points from m images. The reconstruction is made by linear methods, wich means that the impact of noise can be analysed. Furthermore the reconstruction is obtained directly, without calculating fundamental matrices or tensors. This is possible because of the duality between reconstruction of the object and reconstruction of the camera movement. Instead six corresponding points in each image are used to estimate coeecients of a homogeneous polynomial in three variable of degree three. Pairs of such polynomials are used to eliminate one of the three variables, using resultants, giving a polynomial in one variable of degree three, after some spurious roots have been factored out. The coeecients of this polynomial can be caluclated directly from image measurement and then this simgle variabel can be solved for, by linear methods. Then it is easy to calculate the other eliminated variables, also linearly, and make reconstruction. We also provide an error analysis of our reconstruction. The importance of this reconstruction technique is that the reconstruction is made directly, without calculating fundamental matrices or tensors, and by linear methods. An error analysis of the reconstruction is also of great importance. Closely related …