ABSTRACT
In this chapter, the authors consider the “inverse problem” of rotating vectors, i.e., estimating the rotation from observing multiple vectors and their rotated positions. They show that the solution is also obtained using the quaternion representation introduced in the preceding chapter. Finally, they apply these techniques to optimally correcting those matrices which are supposedly rotation matrices but not exactly so, due to noise. The above argument is for the noiseless case. The standard procedure to deal with noise is to regard noisy data as random variables. A random variable is a variable whose value is not definitive but is specified by a probability distribution. All observations in real situations are definitive values, and regarding them, or “modeling them” as random variables by introducing an artificial probability distribution is only a mathematical convenience.
