《Principles Of Magnetic Resonance Imaging》教学Chapter 2.pdfVIP

Chapter 2 Mathematical Fundamentals If the only tool you have is a hammer, you tend to treat everything as ifit were a nail. Abraham Maslow This chapter reviews some mathematical concepts essential for understanding MRI principles. We will begin with a brief description of vector quantities and then review the definitions of some commonly used mathematical functions and two integral transforms: the Fourier transform and the Radon transform. Before we proceed, a word on the notation is in order. Throughout this text, we will use f(x),g(x), h(x), and so on, to represent continuous-variable functions, and f[n], g[n],h[n], or In, gn, hn , and the like to represent discrete- variable functions (or sequences) with the understanding that n is an integer vari- able. In many situations when there is no confusion, we will also use f[n] or fn to represent data samples from f(x) such that i« = f[n] = f(ndx) for some dx. Multivariable functions are sometimes expressed in vector notation such that f(r) may refer to f(x, y) or f(x, y, z), depending on the context. 2.1 Vectors Many physical quantities, such as velocity, magnetic field, and angular momen- tum, have both magnitude and direction. These quantities, called vectors, are represented in this book by two types of symbols: symbols with an overhead ar- row such as A (called explicit vector notations) and boldface symbols such as A (called implicit vector notations). To understand the subtle difference between 13 14 Chapter 2 Mathematical Fundamentals these two notations, consider a three-dimensional vector X