In classical mathematics, models of dynamic systems are derived by relating the cause-effect or energy-balance relationships between the inputs (excitation) and the outputs (response). Obtaining a complete solution to the resulting integrodifferential equations requires the derivations of the complementary function (free response) and the particular integral (forced response). For systems modeled by linear differential equations, the Laplace transformation offers an algebraic framework for the derivation of the output response for a given excitation. The digital counterpart for modeling consists of the difference equations and pulse transfer function. Since many low-level image processing algorithms use well-established analog and digital signal-processing mathematics, some of the essential concepts associated with these are outlined in this appendix. For conformation with the standard literature, the derivations in this appendix are given with time as the independent variable.
D.1 Laplace Transform
Since any real-life analog signal f(t) may not always be absolutely integrable, the formal derivations of Fourier transforms in Appendix A, with f(t) as the parent function, may not apply to a broad range of engineering analysis and design tasks. To exploit the conceptual framework of transforming the independent variable into the frequency provided by the Fourier transform, an exponentially decaying function e-σt is factored such that the augmented function fσ(t) = e-σt f(t) is absolutely integrable. The Fourier transform of this augmented function is then derived as