## Abstract

The most typical applications of multidimensional signals concern *images*, of both the still (i.e., photography, fax) and dynamic (i.e., movie, television) types. This chapter introduces the fundamentals of images, using multidimensional signals as their mathematical model. In the most general case, 3D images with motion, the source signal may be expressed as *i*(*x*,*y*,*z*,*t*), (*x*,*y*,*z*,*t*)∈ℝ^{4}, where *x*,*y*,*z* are space coordinates, *t* is time, and *i* represents information about (*x*,*y*,*z*) at time *t*. This information may refer to luminous intensity (luminance) or to color (chrominance), in which case the information signal *i* is vector valued. Both space and time coordinates are continuous parameters, and therefore the domain is ℝ^{4}. Often the *z* coordinate is disregarded, and, for the purpose of the chapter, chrominance is neglected. In this way, instead of a 4D signal, we consider a 3D signal *ℓ*(*x*,*y*,*t*),(*x*,*y*,*t*)∈ℝ^{3}, where *ℓ* is the luminance. At first, still images, constant in time, are considered, where the corresponding source signal is *ℓ*(*x*,*y*), (*x*,*y*)∈ℝ^{2}.

In the second part of the chapter images will be considered in a less standard framework, investigating the possibility of an *image reconstruction from its projections*. This problem, born in astronomy, finds its most important application in medicine (computer-aided tomography) and nowadays is investigated and applied in several other disciplines.

## Keywords

Polar Function Video Signal Hankel Operator Separable Lattice Polar Representation## References

- 1.E.R. Andrew, Nuclear magnetic resonance imaging: the multiple sensitive point method. IEEE Trans. Nucl. Sci.
**NS-27**, 1232–1238 (1980) CrossRefGoogle Scholar - 2.R.N. Bracewell,
*Two-Dimensional Imaging*(Prentice Hall, Englewood Cliffs, 1995) MATHGoogle Scholar - 3.R.N. Bracewell, S.J. Wernecke, Image reconstruction over a finite field of view. J. Opt. Soc. Am.
**65**, 1342–1346 (1975) CrossRefGoogle Scholar - 4.G. Cariolaro, Exact evaluation of projections and of related representations of the reference example, in
*Solutions of Problems of Unified Signal Theory*. Available at http://springer.com/CariolaroUST - 5.G. Cariolaro, Scanning theory with application to HDTV. Available at http://dei.unipd.it/~cariolar/USTdocuments
- 6.S.R. Deans,
*The Radon Transform and Some of Its Applications*(Wiley, New York, 1983) MATHGoogle Scholar - 7.J.O. Drewery, The filtering of luminance and chrominance signals to avoid cross-colour in a PAL colour system. BBC Eng., 8–39 (1976) Google Scholar
- 8.E. Dubois, The sampling and reconstruction of time-varying imagery with application in video systems. Proc. IEEE
**73**, 502–523 (1985) CrossRefGoogle Scholar - 9.E.P. Hansen, Theory of circular harmonic image reconstruction. J. Opt. Soc. Am.
**71**, 304–308 (1981) CrossRefGoogle Scholar - 10.A.K. Jain,
*Fundamentals of Digital Image Progressing*(Prentice Hall, Englewood Cliffs, 1999) Google Scholar - 11.B. Javidi, F. Okano (eds.),
*Three-Dimensional Television, Video, and Display Technologies*(Springer, New York, 2003) Google Scholar - 12.P.C. Lauterbur, Image formation by induced local interaction: examples employing nuclear magnetic resonance. Nature
**242**, 190–191 (1973) CrossRefGoogle Scholar - 13.P.C. Lauterbur, C.-M. Lai, Zeugmatography by reconstruction from projections. IEEE Trans. Nucl. Sci.
**NS-27**, 1227–1231 (1980) CrossRefGoogle Scholar - 14.P. Mertz, F. Gray, A theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television. Bell Syst. Tech. J.
**13**, 464–515 (1934) Google Scholar - 15.H.M. Ozaktas, L. Onural (eds.),
*Three-Dimensional Television: Capture, Transmission, Display (Signals and Communication Technology)*(Springer, New York, 2008) Google Scholar - 16.A. Papoulis,
*Systems and Transforms with Applications in Optics*(McGraw–Hill, New York, 1968) Google Scholar - 17.J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, in
*Berichte über die Verhandlungen der Königlichen Sächsichen Gesellschaft der Wissenschaften zu Leipzig (Reports on the proceedings of the Saxony Academy of Science)*, vol. 69 (Mathematisch-Physikalische Klasse, 1917), pp. 262–277 Google Scholar - 18.A.H. Robinson, Multidimensional Fourier transform and image processing with finite scanning apertures. Appl. Opt.
**12**, 2344–2352 (1973) CrossRefGoogle Scholar - 19.O. Schreer, P. Kauff, T. Sikora (eds.),
*3D Videocommunication: Algorithms, Concepts and Real-Time Systems in Human Centred Communication*(Wiley, New York, 2005) Google Scholar - 20.D. Taubman, M.W. Marcellin,
*JPEG2000, Image Compression Fundamentals and Practice*(Kluwer Academic, Boston, 2002) CrossRefGoogle Scholar - 21.G.J. Tonge, The sampling of television images, Independent Broadcasting Authority, Experimental and Development Rep. 112/81, May 1981 Google Scholar
- 22.G.J. Tonge, Three-dimensional filters for television sampling, Independent Broadcasting Authority, Experimental and Development Rep. 117/82, June 1982 Google Scholar
- 23.G.J. Tonge, The television scanning process. SMPTE J.
**93**, 657–666 (1984) CrossRefGoogle Scholar