Image and sensor data are not necessarily accurate for a variety of reasons: optic effects, misalignment, electronic noise, poor lighting, etc. To cope with such uncertainty, this book refers to a priori knowledge about the underlying problem, for example, geometric constraints that should be satisfied if no noise is present. The author focuses on two methods: geometric correction (when predetermined geometric constraints should be satisfied) and parametric fitting (when parameters of the geometric constraints are to be estimated). For all the statistical methods described, the author discusses accuracy bounds, reliability in practical implementation, efficiency of computation, and the plausibility of the model. This involves such techniques as regularisation and stochastic relaxation. In contrast to the traditional statistical framework which refers to repeated measurements, here the study of the accuracy in the limit of small noise is very important. The author derives the corresponding bounds for the variance that are conceptually related to perturbation analysis.
The book deals with simple geometric primitives (lines, planes, conics and quadrics in two or three dimensions) that describe the most elementary structure of the scene. Each primitives corresponds to a point in some projective space, which allows one to refer to multivariate statistical methods. The author treats each geometric primitive separately and thoroughly, but a unified approach for a general dimension would be a good addition to the current text.
The material presented in the book provides a non-traditional framework and interpretation for multivariate statistical methods and a good ``visualization'' of statistical and optimisation concepts. The presentation reaches the stage of practical applications when the author applies his algorithms to stereo vision and robot control.
The book starts with a useful introduction to linear algebra, probability and statistical inference (including the Kalman filter, the Cramer-Rao bound, the Akaike information criterion, and probability distributions in manifolds). Then the author discusses numerical representations of geometric objects, where concepts of projective geometry are presented from a computational point of view. The subsequent chapters discuss the general geometric correction problem based on the Mahalanobis distance, three-dimensional reconstruction in stereo vision from the statistical point of view, parametric line fitting and optimal filtering in the Bayesian framework, and the parametric fitting problem for linear equations using the so-called eigenvector fit and renormalisation technique. These results are applied to the fitting of lines, conics and planes. An advantage of the suggested procedure is that in the course of the fitting the reliability of the fit is automatically evaluated in the form of some covariance matrix. Later chapters deal with three-dimensional motion analysis (that in contrast to previous image analysis approaches does not assume that the data are ``exact''), computing the camera motion and the object shape from two views, interpretation of optical flow (or small displacements of points), and information criteria for model selection.
The statistical approach advocated by the author is a very interesting and promising analytical approach for dealing with pattern recognition problems. The book will be of interest both to statisticians who seek new interpretations of theoretical results and to image analysts who are interested in learning about new tools suitable for solving some of their ``everyday'' problems.