ABSTRACT
This chapter introduces a number of other transform operations — the multidimensional Fourier transform, the Fourier series, the z transform, the Laplace transform, the Hankel transforms, and the Radon transform. Multi-dimensional signals occur naturally in radiological applications. The chapter details the discrete and continuous Fourier transforms. A method of associating the continuous and discrete variables is the dual relation of the Fourier series. It will lead to a relation between a continuous, periodic frequency domain signal and a discrete, aperiodic time domain signal. The major difference between the Fourier transform and the Laplace transform is that the frequency variable is complex. The Laplace transform of a signal depends both on the signal and the choice of the constant, σ. With s equal to zero, the Laplace transform is equal to the Fourier transform. The Laplace transform is a bridge to considering all of the transform operations in terms of complex independent variables for both the time and frequency domains.
