Some Applications of Mathematics in Multidimensional Signal Processing
- Date: 12/05/2006
Len Bruton (University of Calgary)
Calgary Place Tower (Shell)
Signals are embedded in physical media and they codify information within the spatio-temporal structure of that media. By definition, multidimensional (MD) signals are mathematically represented as a function of more than one independent dimension (or variable). A simple example is the 2D signal embedded in the surface material of a rectangular photograph, from which the intensity of the emitted light may be represented as a 2D scalar continuous-domain mathematical function of two orthogonal spatial dimensions: the so-called x and y dimensions. Similarly, a displayed movie is a spatio-temporal 3D signal that may be modelled as a continuous-domain scalar function of two spatial variables and one time variable. Such MD spatio-temporal signals are ubiquitous. They include the signals acquired and processed by television and biomedical systems, as well as by 1D, 2D or 3D spatial arrays of sensors in seismic, radio, ultrasound, sonar, speech, holographic and photographic systems, to name a few. Such signals are often sampled in both space and time, resulting in MD discrete-domain signals, most of which are quantized and digitized, yielding purely numerical signals that are typically embedded on the surface of silicon-based computational devices.
In many cases, the mathematics that is used for understanding the modelling, processing and analysis of MD signals is a generalization of the 1D case, typically involving MD extensions of the well known contributions of Laplace, Heaviside, Fourier, Gauss, Cauchy and many others. Real or complex functions of M independent complex variables are widely encountered. For practical purposes, MD signal analysis and MD system design require a thorough understanding of the spectral properties of the signal and also of the input-to-output transfer properties of the system. In such cases, the MD version of Parseval’s Theorem plays a key role, as do the MD Fourier transform and the MD Z transform. Further, Nyquist’s Sampling Theorem, extended to MD, allows us to understand the important practical problem of aliasing in space-time. Finally, the use of feedback is of practical importance in MD systems but is often fraught with mathematical difficulties and practical challenges.
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