Discrete-Time continuous-dilation construction of linear scale-invariant systems and multi-dimensional self-similar signals

This dissertation presents novel models for purely discrete-time self-similar processes and scale-invariant systems. The results developed are based on the definition of a discrete-time scaling (dilation) operation through amapping between discrete and continuous frequencies. It is shown that it is possible to have continuous scaling factors through this operation even though the signal itself is discrete-time. Both deterministic and stochastic discrete-time self-similar signals are studied. Conditions of existence for self-similar signals are provided. Construction of discrete-time linear scale-invariant (LSI) systems and white noise driven models of self-similar stochastic processes are discussed. It is shown that unlike continuoustime self-similar signals, a wide class of non-trivial discrete-time self-similar signals can be constructed through these models. The results obtained in the one-dimensional case are extended to multi-dimensional case. Constructions of discrete-space self-similar ran dom fields are shown to be potentially useful for the generation, modeling and analysis of multi-dimensional self-similar signals such as textures. Constructions of discrete-time and discrete-space self-similar signals presented in the dissertation provide potential tools

[1]  R.F. Voss,et al.  1/f (Flicker) Noise: A Brief Review , 1979, 33rd Annual Symposium on Frequency Control.

[2]  Shinozuka,et al.  Numerical method for colored-noise generation and its application to a bistable system. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[3]  Thor O. Gulsrud,et al.  Optimal filter for detection of stellate lesions and circumscribed masses in mammograms , 1996, Other Conferences.

[4]  M. Selby,et al.  Invariant Spectroscopic Pattern Recognition Using Mellin Transforms , 1994 .

[5]  Peter H. Richter,et al.  The Beauty of Fractals , 1988, 1988.

[6]  N. Magnenat-Thalmann,et al.  Image Synthesis Theory and Practice , 1988 .

[7]  Patrick Flandrin,et al.  Wavelet analysis and synthesis of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[8]  Devesh Patel,et al.  Page segmentation for document image analysis using a neural network , 1996 .

[9]  B. Onaral,et al.  Time domain characteristics of rational systems with scale-invariant frequency response , 1996, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications.

[10]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

[11]  Veljko Radeka,et al.  1/|f| Noise in Physical Measurements , 1969 .

[12]  Abhijit Mahalanobis,et al.  Correlation filters for texture recognition and applications to terrain-delimitation in wide-area surveillance , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[13]  A. van der Ziel,et al.  Unified presentation of 1/f noise in electron devices: fundamental 1/f noise sources , 1988, Proc. IEEE.

[14]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[15]  A. Papoulis.  Linear systems, Fourier transforms, and optics , 1981, Proceedings of the IEEE.

[16]  Mita D. Desai,et al.  Localized fractal dimension measurement in digital mammographic images , 1993, Other Conferences.

[17]  C.-C. Jay Kuo,et al.  Texture Roughness Analysis and Synthesis via Extended Self-Similar (ESS) Model , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Ronald N. Bracewell,et al.  The Fourier Transform and Its Applications , 1966 .

[19]  L. M. Linnett,et al.  Remote sensing of the sea-bed using fractal techniques , 1991 .

[20]  G. Carlson.  Signal and Linear System Analysis , 1992 .

[21]  Benoit B. Mandelbrot,et al.  Some noises with I/f spectrum, a bridge between direct current and white noise , 1967, IEEE Trans. Inf. Theory.

[22]  Przemyslaw Prusinkiewicz,et al.  Graphical applications of L-systems , 1986 .

[23]  Donald S. Fussell,et al.  Computer rendering of stochastic models , 1982, Commun. ACM.

[24]  Malvin Carl Teich,et al.  Power-law shot noise , 1990, IEEE Trans. Inf. Theory.

[25]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[26]  Alex Pentland,et al.  Fractal-Based Description of Natural Scenes , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  Azriel Rosenfeld,et al.  A Comparative Study of Texture Measures for Terrain Classification , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[28]  Hartmut Jürgens,et al.  Introduction to fractals and chaos , 1992 .

[29]  H. Takayasu.  f -β Power Spectrum and Stable Distribution , 1987 .

[30]  Elias Masry,et al.  The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion , 1993, IEEE Trans. Inf. Theory.

[31]  Ioannis Pitas,et al.  A texture-based approach to the segmentation of seismic images , 1992, Pattern Recognit..

[32]  Ingrid Daubechies,et al.  Where do wavelets come from? A personal point of view , 1996, Proc. IEEE.

[33]  B. Mandelbrot.  A Fast Fractional Gaussian Noise Generator , 1971 .

[34]  Y. Fisher.  Fractal image compression: theory and application , 1995 .

[35]  J. R. Wallis,et al.  Some long‐run properties of geophysical records , 1969 .

[36]  D. W. Allan,et al.  A statistical model of flicker noise , 1966 .

[37]  Neal,et al.  Using Peano Curves for Bilevel Display of Continuous-Tone Images , 1982, IEEE Computer Graphics and Applications.

[38]  J Duvernoy,et al.  Circular-Fourier-radial-Mellin transform descriptors for pattern recognition. , 1986, Journal of the Optical Society of America. A, Optics and image science.

[39]  Dietmar Saupe,et al.  Cayley’s problem and Julia sets , 1984 .

[40]  Y. Meyer,et al.  Wavelets and Filter Banks , 1991 .

[41]  C. Chui.  Wavelets: A Tutorial in Theory and Applications , 1992 .

[42]  H. Piaggio.  Mathematical Analysis , 1955, Nature.

[43]  M. Fox,et al.  Fractal feature analysis and classification in medical imaging. , 1989, IEEE transactions on medical imaging.

[44]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[45]  Wei Zhao,et al.  Continuous-dilation discrete-time self-similar signals and linear scale-invariant systems , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[46]  Azriel Rosenfeld,et al.  Picture Processing and Psychopictorics , 1970 .

[47]  Benoit B. Mandelbrot,et al.  Self-Similar Error Clusters in Communication Systems and the Concept of Conditional Stationarity , 1965 .

[48]  Gregory W. Wornell,et al.  Wavelet-based representations for a class of self-similar signals with application to fractal modulation , 1992, IEEE Trans. Inf. Theory.

[49]  B. H. Kaye.  A random walk through fractal dimensions , 1989 .

[50]  N. Jeremy Usdin,et al.  Discrete Simulation of Colored Noise and Stochastic Processes and llf" Power Law Noise Generation , 1995 .

[51]  M. Hassner,et al.  The use of Markov Random Fields as models of texture , 1980 .

[52]  Rangasami L. Kashyap,et al.  A class of second-order stationary self-similar processes for 1/f phenomena , 1997, IEEE Trans. Signal Process..

[53]  Masoud Salehi,et al.  Communication Systems Engineering , 1994 .

[54]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[55]  James M. Keller,et al.  Characteristics of Natural Scenes Related to the Fractal Dimension , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[56]  Luiz Velho,et al.  Digital halftoning with space filling curves , 1991, SIGGRAPH.

[57]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[58]  Brett Ninness,et al.  Estimation of 1/f Noise , 1998, IEEE Trans. Inf. Theory.

[59]  Anthony B. Davis,et al.  Multifractal characterizations of nonstationarity and intermittency in geophysical fields: Observed, retrieved, or simulated , 1994 .

[60]  Chan S. Park.  The Mellin Transform in Probabilistic Cash Flow Modeling , 1986 .

[61]  Robert M. Haralick,et al.  Textural Features for Image Classification , 1973, IEEE Trans. Syst. Man Cybern..

[62]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[63]  Walter Willinger,et al.  On the Self-Similar Nature of Ethernet Traffic ( extended version ) , 1995 .

[64]  A. W. M. van den Enden,et al.  Discrete Time Signal Processing , 1989 .

[65]  W. Kilmer.  A Friendly Guide To Wavelets , 1998, Proceedings of the IEEE.

[66]  I. Daubechies.  Orthonormal bases of compactly supported wavelets II: variations on a theme , 1993 .

[67]  H. Vincent Poor,et al.  Signal detection in fractional Gaussian noise , 1988, IEEE Trans. Inf. Theory.

[68]  N. Lu,et al.  Fractal imaging , 1997 .

[69]  C.-C. Jay Kuo,et al.  An improved method for 2-D self-similar image synthesis , 1996, IEEE Trans. Image Process..

[70]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[71]  M. A. Caloyannides,et al.  Microcycle spectral estimates of 1/f noise in semiconductors , 1974 .

[72]  Przemyslaw Prusinkiewicz,et al.  Applications of L-systems to computer imagery , 1986, Graph-Grammars and Their Application to Computer Science.

[73]  E. Montroll,et al.  On 1/f noise and other distributions with long tails. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[74]  Wei Zhao,et al.  On modeling of self-similar random processes in discrete-time , 1998, Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380).

[75]  S. Kay,et al.  Fractional Brownian Motion: A Maximum Likelihood Estimator and Its Application to Image Texture , 1986, IEEE Transactions on Medical Imaging.

[76]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[77]  K. Shanmugan,et al.  Random Signals: Detection, Estimation and Data Analysis , 1988 .

[78]  Gregory W. Wornell,et al.  Estimation of fractal signals from noisy measurements using wavelets , 1992, IEEE Trans. Signal Process..

[79]  George Sugihara,et al.  Fractals: A User's Guide for the Natural Sciences , 1993 .

[80]  Melba M. Crawford,et al.  Cloud type discrimination via multispectral textural analysis , 1993, Defense, Security, and Sensing.

[81]  Nirupam Sarkar,et al.  Improved fractal geometry based texture segmentation technique , 1993 .

[82]  Mohamed A. Deriche,et al.  Maximum likelihood estimation of the parameters of discrete fractionally differenced Gaussian noise process , 1993, IEEE Trans. Signal Process..

[83]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[84]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[85]  J. Rogers.  Chaos , 1876 .

[86]  T. Vicsek,et al.  Fractals in natural sciences , 1994 .

[87]  A.K. Krishnamurthy,et al.  Multidimensional digital signal processing , 1985, Proceedings of the IEEE.

[88]  A.H. Tewfik,et al.  Correlation structure of the discrete wavelet coefficients of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[89]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[90]  Joseph Naor,et al.  Multiple Resolution Texture Analysis and Classification , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[91]  J. R. Wallis,et al.  Computer Experiments With Fractional Gaussian Noises: Part 1, Averages and Variances , 1969 .

[92]  Todd K. Moon.  Similarity methods in signal processing , 1996, IEEE Trans. Signal Process..

[93]  H. E. Derksen,et al.  Fluctuation phenomena in nerve membrane , 1968 .

[94]  R. Kumaresan,et al.  Isotropic two-dimensional Fractional Brownian Motion and its application in Ultrasonic analysis , 1992, 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[95]  Gregory W. Wornell,et al.  A Karhunen-Loève-like expansion for 1/f processes via wavelets , 1990, IEEE Trans. Inf. Theory.

[96]  I. Good,et al.  Fractals: Form, Chance and Dimension , 1978 .

[97]  Anil K. Jain,et al.  Markov Random Field Texture Models , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[98]  R.N. Bracewell,et al.  Signal analysis , 1978, Proceedings of the IEEE.

[99]  Lyman P. Hurd,et al.  Fractal image compression , 1993 .

[100]  Ofer Zeitouni,et al.  On the wavelet transform of fractional Brownian motion , 1991, IEEE Trans. Inf. Theory.

[101]  M. S. Keshner.  1/f noise , 1982, Proceedings of the IEEE.

[102]  R. Voss,et al.  ’’1/f noise’’ in music: Music from 1/f noise , 1978 .

[103]  Mohamed A. Deriche,et al.  Signal modeling with filtered discrete fractional noise processes , 1993, IEEE Trans. Signal Process..

[104]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[105]  R. Rao,et al.  Discrete-time continuous-dilation wavelet transforms , 1998, Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380).