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Chapter 3:
Mathematics
Abstract

3.1 Fundamental Mathematics for Modeling the Imaging Chain

Before delving into the imaging chain models, it is critical to first understand some of the mathematical principles and methodologies that are fundamental to the development of the imaging chain models. We will first look at some very useful functions that help describe the behavior of light through the elements of the imaging chain. Next we will discuss the properties of linear shift-invariant systems that will simplify much of the modeling, including convolution operations and Fourier transforms.

3.2 Useful Functions

Many objects, such as waves, points, and circles, have simple mathematical representations that will prove very useful for mathematically modeling the imaging chain.1-3 The following functions are generalized for any shifted location (x0, y0) and scale factors wx and wy, and in general follow the form used by Gaskill.1

Figure 3.1 illustrates a simple one-dimensional wave stationary in time that can be described by a cosine function with amplitude A, wavelength λ, and phase Φ, given by

(3.1)

where ζ0 is the spatial frequency of the wave, i.e., the number of cycles that occur per unit distance.

A point is mathematically represented by the Dirac delta function, which is zero everywhere except at the location x0 and has the properties

(3.2)

and

(3.3)

Defining the delta function as infinite at x = x0 is not mathematically rigorous, so the delta function is typically defined as the limit of some function, e.g., a Gaussian, that has a constant area as the width goes to zero. The delta function is drawn as an arrow, and the height of the arrow represents the weighting of the function.