Fourier-based interpretation of the algebraic spectrum properties

Abstract

In the previous chapter we analyzed the properties of the spectrum convolution (i.e., the spectrum of the layer superposition) from a pure algebraic point of view, concentrating only on the spectrum support, and ignoring the impulse amplitudes. In the present chapter we will “augment” these algebraic foundations by reintroducing the impulse amplitudes on top of their geometric locations in the spectrum. We will investigate in Secs. 6.2–6.3 the properties of the impulse amplitudes that are associated with the algebraic structures, and through the Fourier theory, we will see how both the structural and the amplitude properties of the spectrum are related to properties of the layer superposition and its moiré effects back in the image domain. Finally, in Secs. 6.4–6.8 we will analyze the layer superpositions when their moirés become singular, and we will see what happens in the Fourier expressions when each of the impulse clusters collapses down into a single compound impulse. This chapter is, in fact, a generalization of the basic ideas developed in Chapter 4, based on the new algebraic notions of Chapter 5.