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This chapter discusses image registration techniques based on a finite set of parameters and/or a finite set of so-called image features. The basic idea is to determine the desired transformation for a finite number of features, any feature of the template image is mapped onto the corresponding feature on the reference image. This general notion of image features is made concrete using image landmarks. A naive approach to landmark based registration is formalized and theoretical issues such as existence and uniqueness are discussed. The naive approach is replaced by a proper regularized approach, and analytical solutions are derived using the theory of representers, radial basis functions, and thin-plate-splines. Using this advanced theoretical framework, the strict interpolation approach is further extended to a more flexible approximation approach. The various approaches (linear, quadratic, interpolating, and approximating thin-plate splines) are compared and illustrated. It is shown that landmark-based registration may not result in a meaningful one-to-one overall transformation.

As noted In Chapter 1, the general class of NARMAX models is extremely broad and includes almost all of the other discrete-time model classes discussed in this book, linear and nonlinear. Because of its enormity, little can be said about the qualitative behavior of the NARMAX family in general, but it is surprising how much can be said about the behavior of some important subclasses. In particular, there are sharp qualitative distinctions between nonlinear moving average models (NMAX models) and nonlinear autoregressive models (NARX models). Since these representations are equivalent for linear models (c.f. Chapter 2), this observation highlights one important difference between linear and nonlinear discrete-time dynamic models. Consequently, this chapter focuses primarily on the structure and qualitative behavior of various subclasses of the NARMAX family, particularly the NMAX and NARX classes. More specifically, Sec. 4.1 presents a brief discussion of the general NARMAX class, defining five important subclasses that are discussed in subsequent sections: the NMAX class (Sec. 4.2), the NARX class (Sec. 4.3), the class of (structurally) additive NARMAX models (Sec. 4.4), the class of polynomial NARMAX models (Sec. 4.5), and the class of rational NARMAX models (Sec. 4.6). More complex NARMAX models are then discussed briefly in Sec. 4.7 and Sec. 4.8 concludes the chapter with a brief summary of the NARMAX class. The general class of NARMAX models was defined and discussed in a series of papers by Billings and various co-authors (Billings and Voon, 1983, 1986a; Leontartis and Billings, 1985; Zhu and Billings, 1993) and is defined by the equation . . . y(k) = F(y(k – 1), . . . , y(k – p), u(k), . . . , u(k – q), e(k – 1), . . . , e(k – r)) + e(k). (4.1) . . . Because this book is primarily concerned with the qualitative input/output behavior of these models, it will be assumed that e(k) = 0 identically as in the class of linear ARMAX models discussed in Chapter 2.

This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists. Boundary value problems for the Poisson equation and initial value problems for the diffusion equation are solved using FDM. Numerical instability of explicit schemes are explained physically and implicit schemes introduced. A discrete version of theClebsch formulation of incompressible Euler equations is proposed. The chapter concludes with the radial basis function method and its application to a discrete version of the Lagrangian formulation of Navier–Stokes.