By Dr. Zhengyou Zhang, Dr. Olivier Faugeras (auth.)
he challenge of studying sequences of pictures to extract third-dimensional T movement and constitution has been on the middle of the examine in machine vi sion for a few years. it is important to due to the fact that its good fortune or failure will ascertain even if imaginative and prescient can be utilized as a sensory approach in reactive structures. The massive learn curiosity during this box has been stimulated no less than via the subsequent issues: 1. The redundancy of data contained in time-varying pictures can over come a number of problems encountered in analyzing a unmarried picture. 2. there are various vital purposes together with computerized motor vehicle driv ing, site visitors regulate, aerial surveillance, clinical inspection and international version development. even if, there are various new difficulties which can be solved: tips to effi ciently procedure the ample details contained in time-varying pictures, how one can version the swap among photos, find out how to version the uncertainty inherently linked to the imaging procedure and the way to resolve inverse difficulties that are in most cases ill-posed. there are naturally many percentages for attacking those difficulties and lots of extra stay to be explored. We talk about some of them during this publication in response to paintings performed over the past 5 years within the machine imaginative and prescient and Robotics crew at INRIA (Institut nationwide de Recherche en Informatique et en Automatique).
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V , 4. 1 Rigid Motion 41 of a matrix, -1 denotes the inverse of a matrix, IIII denotes the norm of a vector, . denotes the inner (dot) product of two vectors, and 1\ denotes the cross product of two vectors. 2 Representations The rotation matrix gives a simple way of representing a 3D rotation, but leads to a high dimensional space of constraints. 2]. Here, we present two of them: using the rotation axis and using a quaternion. Representing a rotation by its rotation axis A rotation can be defined as a 3-dimensional vector r = [a, b, cf whose direction is that of the rotation axis and whose norm is equal to the rotation angle.
More rigorously, there exists a correlation between 1jJ and m, but this correlation is negligible. 01. 17] to represent 2D segments obtained by projecting 3D segments on the ground plane. 15) is used. Considering the unreliability of the midpoint or the endpoints of a segment, a lengthened covariance matrix along the direction of the segment is added to the original covariance matrix before using it and is removed later. 3 51 Summary In this chapter, we have investigated the representations of rigid motions and 3D line segments, which will be used in the following chapters.
One of them is the Singular Value Decomposition (SVD) method. In fact, the m x n matrix H can be put through SVD in the form H = USV T , where U is an m X m orthogonal matrix containing the left singular vectors of H, V is an n X n orthogonal matrix containing the right singular vectors of H, and S is an m x n quasi-diagonal matrix containing the singular values of H. Without 26 2. Uncertainty Manipulation and Parameter Estimation loss of generality, the diagonal entries of S are arranged to be nonincreasing.