Prentice Hall, 1995. - 192 pages.
Observations of signals, time series in telecommunication, or pictures in medical imaging reveal selfsimilarity, i.e. , the signal/picture looks the same as the scale varies. Hence, the name "fractal" ! Pictures in the Intro make this scale-similarity visually apparent. But it is made precise in mathematical statistics, and the book further makes the connection to the engineering of signal/image processing. That's a main point of the book!
Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and/or space. Fractional Brownian motion or Brownian sheets are the best known of these. They were found by Kolmogorov long ago, but made popular by Mandelbrot and Ness in 1968. We now speak of 1/f processes. More recent use of wavelet bases in telecommunication and in stochastic integration has revived interest. Other even more recent applications include finance.
While the underlying idea behind all of this is quite simple, and can be traced back to Kolmogorov in the 1930ties, it is only recently, with the advent of wavelet methods, that the computational power has been better appreciated. The idea is analogous to that of random Fourier series: Instead of treating the Fourier coefficients as random variables, it is now wavelet coefficients that are analyzed statistically. Since wavelets have computational advantages, it is not surprising that the engineering applications abound. This little book is well written, and should be attractive both to members of the math community and to engineers.
Observations of signals, time series in telecommunication, or pictures in medical imaging reveal selfsimilarity, i.e. , the signal/picture looks the same as the scale varies. Hence, the name "fractal" ! Pictures in the Intro make this scale-similarity visually apparent. But it is made precise in mathematical statistics, and the book further makes the connection to the engineering of signal/image processing. That's a main point of the book!
Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and/or space. Fractional Brownian motion or Brownian sheets are the best known of these. They were found by Kolmogorov long ago, but made popular by Mandelbrot and Ness in 1968. We now speak of 1/f processes. More recent use of wavelet bases in telecommunication and in stochastic integration has revived interest. Other even more recent applications include finance.
While the underlying idea behind all of this is quite simple, and can be traced back to Kolmogorov in the 1930ties, it is only recently, with the advent of wavelet methods, that the computational power has been better appreciated. The idea is analogous to that of random Fourier series: Instead of treating the Fourier coefficients as random variables, it is now wavelet coefficients that are analyzed statistically. Since wavelets have computational advantages, it is not surprising that the engineering applications abound. This little book is well written, and should be attractive both to members of the math community and to engineers.