Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Signals in one dimension
- 3 Signals in two dimensions
- 4 Optical imaging systems
- 5 Antenna systems
- 6 The ambiguity function
- 7 Radar imaging systems
- 8 Diffraction imaging systems
- 9 Construction and reconstruction of images
- 10 Tomography
- 11 Likelihood and information methods
- 12 Radar search systems
- 13 Passive and baseband surveillance systems
- 14 Data combination and tracking
- 15 Phase noise and phase distortion
- References
- Index
3 - Signals in two dimensions
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Signals in one dimension
- 3 Signals in two dimensions
- 4 Optical imaging systems
- 5 Antenna systems
- 6 The ambiguity function
- 7 Radar imaging systems
- 8 Diffraction imaging systems
- 9 Construction and reconstruction of images
- 10 Tomography
- 11 Likelihood and information methods
- 12 Radar search systems
- 13 Passive and baseband surveillance systems
- 14 Data combination and tracking
- 15 Phase noise and phase distortion
- References
- Index
Summary
The Fourier transform of a two-dimensional function — or of an n-dimensional function — can be defined by analogy with the Fourier transform of a one-dimensional function. A multidimensional Fourier transform is a mathematical concept. Because many engineering applications of the two-dimensional Fourier transform deal with two-dimensional images, it is common practice, and ours, to refer to the variables of a two-dimensional function as “spatial coordinates” and to the variables of its Fourier transform as “spatial frequencies.”
The study of the two-dimensional Fourier transform closely follows the study of the one-dimensional Fourier transform. As the study develops, however, the two-dimensional Fourier transform displays a richness beyond that of the one-dimensional Fourier transform.
The two-dimensional Fourier transform
A function, s(x, y), possibly complex, of two variables x and y is called a two-dimensional signal or a two-dimensional function (or, more correctly, a function of a two-dimensional variable). A common example is an image, such as a photographic image, wherein the variables x and y are the coordinates of the image, and s(x, y) is the amplitude. In a photographic image, the amplitude is a nonnegative real number. In other examples, the function s(x, y) may also take on negative or even complex values. Figure 3.1 shows a graphical representation of a two-dimensional complex signal in terms of the real and imaginary parts. Figure 3.2 shows the magnitude of the function depicted in two different ways: one as a three-dimensional graph, and one as a plan view.
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- Theory of Remote Image Formation , pp. 67 - 110Publisher: Cambridge University PressPrint publication year: 2004