Image Registration and Unknown Coordinate Systems

Abstract

This chapter deals with statistical problems involving unknown coordinate systems, either in Euclidean 3-space or on the unit sphere Ω3 in. We also consider the simpler cases of Euclidean 2-space and the unit circle Ω2. The chapter has five major sections. Although other problems of unknown coordinate systems have arisen, a very important problem of this class is the problem of image registration from landmark data. In this problem we have two images of the same object (such as satellite images taken at different times) or an image of a prototypical object and an actual object. It is desired to find the rotation, translation, and possibly scale change, which will best align the two images. Whereas many problems of this type are two-dimensional, it should be noted that medical imaging is often three-dimensional. After introducing some mathematical preliminaries we introduce the concept of M-estimators, a generalization of least squares estimation. In least squares estimation, the registration that minimizes the sum of squares of the lengths of the deviations is chosen; in M estimation, the sum of squares of the lengths of the deviations is replaced by some other objective function. An important case is L1 estimation, which minimizes the sum of the lengths of the deviations; L1 estimation is often used when the possibility of outliers in the data is suspected. The second section of this chapter deals with the calculation of least squares estimates. Then, in the third section, we introduce an iterative modification of the least squares algorithm to calculate other M-estimates. Note that minimization usually involves some form of differentiation and hence this section starts with a short introduction to the geometry of the group of rotations and differentiation in the rotation group. Many statistical techniques are based upon approximation by derivatives and hence a little understanding of geometry is necessary to understand the later statistical sections. The fourth section discusses the statistical properties of M-estimates. A great deal of emphasis is placed upon the relationship between the geometric configuration of the landmarks and the statistical errors in the image registration. It is shown that these statistical errors are determined, up to a constant, by the geometry of the landmarks. The constant of proportionality depends upon the objective function and the distribution of the errors in the data. General statistical theory indicates that, if the data error distribution is (anisotropic) multivariate normal, least squares estimation is optimal. An important result of this section is that, even in this case when least squares estimation is theoretically the most efficient, the use of L1 estimation can guard against outliers with a very modest cost in efficiency. Here optimality and efficiency refer to the expected size of the statistical errors. In practice, data is often long-tailed and L1 estimation yields smaller statistical errors than least squares estimation. This will be the case with the three-dimensional image registration example given here. Finally, in the fifth section, we discuss diagnostics that can be used to determine which data points are most influential upon the registration. Thus, if the registration is unsatisfactory, these diagnostics can be used to determine which data points are most responsible and should be reexamined.