T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....biorthogonal basis As we have already pointed out, in order to use the adapted basis, one must know a priori the value of H . Thus the estimator based on this basis is theoretical and cannot be used in practical situations. As we will see, it asymptotically saturates the Cramer Rao limit [25] [58], 59] for the estimation of ln M q (a) Thus it will be used mainly as a reference estimator. B.1 An estimation of ln(M q (a) q. In this Section, we consider that we have computed N wavelet coecients f d(j; k)g 1 k N at the scale a = 2 j . A natural estimator of ln(M q (a) q is simply ....

T. Lundhal, W.J. Ohley, S.M. Kay, and R. Siert, \Fractional brownian motion : maximum likelihood estimator and its application to image texture," IEEE Trans. Medical Imaging, vol. MI-5, pp. 152-161, 1986.

....(variance time plots) and queuing behavior. 1. INTRODUCTION Fractals models arise frequently in a variety of scientific disciplines, such as physics, chemistry, astronomy, and biology. In DSP, fractals have long proven useful for applications such as computer graphics and texture modeling [1]. More recently, fractal models have had a major impact on the analysis of data communication networks such as the Internet. In their landmark paper [2] Leland et al. demonstrated that network traffic exhibits fractal properties such as self similarity, burstiness, and long range dependence ....

....point of view, for practical computations and simulations, we often work with sampled continuous time fBm. The increments process of sampled fBm X[n] B(n) Gamma B(n Gamma 1) 3) defines a stationary Gaussian sequence known as discrete fractional Gaussian noise (fGn) with covariance behavior [1] rX [k] jkj 2H Gamma2 ; for jkj large: 4) For 1=2 H 1, the covariance of fGn is strictly positive and decays so slowly that it is non summable (i.e. P k rX [k] 1) This non summability, corresponding to positive, slowly decaying covariances over large time lags, defines LRD. The LRD ....

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

....several commonly used alternatives for synthesizing the Gaussian 1=f processes corresponding to sampled fBm and dfGn. It is simple to convert between these processes using (10) so a synthesis algorithm for one can synthesize the other as well. 2.4. 1 Cholesky factorization This method is exact [5, 27], but slow, requiring O(N 2 ) computations to form a length N dfGN vector X. From (9) we form the Toeplitz covariance matrix RXH for dfGn and then factor it into RXH = QQ T via the Levinson Durbin algorithm (direct Cholesky factorization is even slower O(N 3 ) We then form X = ....

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

....of the Cram er Rao lower bound we gauge the convergence rate of the maximum likelihood estimator and show that it slows as the hyperbolicity of the spectra increases from 1=f to 1=f 2 . This is in accordance with the simulation results presented by Wornell and Oppenheim [14] and others [8]. 2 Fractional Brownian Motions Noise referred to as 1=f noise is a stochastic process fx k g whose sample spectral density, or periodogram 1 , jb xN ( j 2 is of the form jb xN ( j 2 oe 2 x j j Gammafl for some finite non zero oe x and fl. Such noise has been found to occur in ....

....n b o = O 1 N 1 Gammafl log 2 N as M 1 Var n b oe 2 o = O 1 N as M 1 Therefore, we should expect estimates of ff and to be more accurate for smaller fl which corresponds to near 1. This is in agreement with the simulation results presented by Lundahl et.al. [8] and Wornell and Oppenheim [13] and is also in accordance with intuition. For fl small near 1=2 the spectrum of the process is nearly flat so its size and slope at high frequencies is highly indicative of it s overall behaviour. For fl large near 1, the spectrum of the process is much more ....

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, Fractional Brownian motion:A maximum likelihood estimator and its application to image texture, IEEE Transactions on Medical Imaging, MI-5 (1986), pp. 152-- 161.

.... Brownian motion (fBm) and fractional Gaussian noise (fGn) 15, 14, 33, 38, 41, 5, 11] A large part of the impetus for such work has been the problem of dealing with so called 1=f stochastic processes which have become of growing importance to physicists and the signal processing community [24, 2, 28, 37, 42], and more recently to the control theory community [27] To be more specific, 1=f noise is the colloquial term given to a stochastic process fx k g whose sample spectral density, or periodogram 1 , jbx N ( j 2 is of the form jbx N ( j 2 This work was supported by the Australian ....

.... (which are sometimes also called flicker noise ) have been empirically observed in a wide variety of physical processes such as [24, 31] currents in semiconductors, oscillation of quartz crystals, geophysical records, rate of insulin uptake, economic data, traffic flow rates, image texture [28] and heart rate variability [19] In these areas, for the purposes of prediction, control or diagnosis, it is of great interest to be able to estimate the spectral exponent fl from an observed sample path. Many methods to achieve this have been proposed. They range from least squares estimation ....

[Article contains additional citation context not shown here]

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, Fractional Brownian motion: A maximum likelihood estimator and its application to image texture, IEEE Transactions on Medical Imaging, MI-5 (1986), pp. 152--161.

.... (48) is difficult to infer analytically, the result implies that the variance of the estimates converge as O(N Gamma1 ) The asymptotic limit (Cramer Rao bound) on d can be obtained easily in case of an fARIMA(0,d,0) process by substituting (13) in (48) In this case, var( d) 6 2 N [15]. Table 4 shows the performance results of Whittle s estimator when applied to fARIMA(1,d,1) database. The main advantage of Whittle s method over any other method is that the estimates are reasonably accurate even when smaller data records are used [15] However, it needs greater computational ....

....(48) In this case, var( d) 6 2 N [15] Table 4 shows the performance results of Whittle s estimator when applied to fARIMA(1,d,1) database. The main advantage of Whittle s method over any other method is that the estimates are reasonably accurate even when smaller data records are used [15]. However, it needs greater computational requirements. Also, presence of linear or polynomial trends in the data affects the performance [50] d = 0:1 a(1) Gamma0:5 b(1) 0:5 mean( 0:1210 Gamma0:4664 0:5101 var( 0:0039 0:0083 0:0005 d = 0:2 a(1) Gamma0:5 b(1) 0:5 mean( ....

T. Lundahl, W.J. Ohley, S. M. Kay and R. Siffert. Fractional Brownian Motion: A Maximum Likelihood Estimator and Its Application to Image Texture. IEEE Transactions on Medical Imaging, MI-5(3):152-161, 1986.

.... the structure of moments of non linear functions of Gaussian random variables and linear processes [34,94,158,182,280,385,398] Some of the results have been extended to random fields, that is, to processes where the time parameter is viewed as a space parameter and is multidimensional [11, 148,179,185,208,278,279,341,375]. Besides the statistical and practical aspects of self similar or fractal models, there is the ever present desire for a physical or phenomenological explanation for the fractal nature of empirically observed data. For recent work on this topic in the context of high speed network traffic ....

T. Lundahl, W. J. Ohley, S. M. Kay, and R. Siffert. Fractional Brownian motion: A maximum likelihood estimator and its application to image texture. IEEE Transactions on Pattern Analysis and Machine Intelligence, A Bibliographical Guide 25 MI-5(3):152--161, 1986.

.... [4] rate of traffic flow [1] voltage or current fluctuations in metal films and semiconductor devices [4] loudness fluctuations in speech and music [5] heart rate and blood pressure [6, 7] DNA structure [8, 9] natural phenomena such as the shape of terrain [10, 11] clouds [12] textures [13, 14], and medical images [15] The long term correlations observed in the above cases correspond to a power spectrum of the form 1=f ff , where f denotes the frequency and ff 0. The fact that 1=f noise is encountered in such a wide variety of systems has led to speculation that there might exist ....

....of the power spectrum integrated over logarithmically spaced bands. 3. The historically precedent Hurst exponent [17] 4. A recently proposed statistic by Peng et al. 18] motivated by the Hurst exponent [17] 5. A maximum likelihood estimator proposed by Deriche and Tewfik [19] See also [14]. It should be noted that these 5 estimators do not by any means exhaust the set of proposed estimators. See Taqqu et al. 16] and the monograph by Beran [20] Bassingthwaighte et al. have conducted extensive investigations of the Hurst exponents and refinements thereto [21, 22, 23] 2.1 Method ....

T. Lundahl, W.J. Ohley, S.M. Kay, and R. Siffert (1986) Fractional brownian motion: a maximum likelihood estimator and its application to image texture, IEEE Trans. Med. Imaging MI-5(3):152-161

.... as outlined in (5) in which the likelihood maximization is performed using standard nonlinear techniques (e.g. the section search method of Matlab) 4 Experimental Results Sixty four fBm sample paths, each having a length of 2048 samples, were generated using the Cholesky decomposition method of [7], precisely the same approach as in Kaplan and Kuo[4] whose experimental results form the basis of comparison with ours. The performance of three fBm estimators is compared in Table 3. The bias in the estimator of [11] for low H, as was argued earlier based on Table 1, is evident. Also recall ....

T. Lundahl, W. Ohley, S. Kay, R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture", IEEE Trans. Med. Im. (5), pp.152--161, 1986

.... in a variety of scientific disciplines, including physics, chemistry, astronomy, biology, meteorology, hydrology, and soil science [1, 2] In signal and image processing, fractals have been applied in fields such as computer graphics, texture modeling, image compression, and pattern recognition [3, 4]. Fractal models have made a major impact in the area of communications recently, particularly in the area of computer data networks. As the work of Leland et al. 5] and subsequent studies have demonstrated, network traffic loads exhibit fractal properties such as self similarity, burstiness, ....

....to a tractable analysis. The fBm is not stationary, but its increments form the stationary fractional Gaussian noise (fGn) process. When the Hurst parameter H 1=2, fGn exhibits LRD. N samples of fGn can be simulated exactly via direct Cholesky factorization (O(N 3 ) computational complexity) [4] or Levinson s recursion (O(N 2 ) complexity) 8] These costs can become overbearing, especially in networking applications where often N AE 10 6 . For such large problems, approximate synthesis techniques (O(N) complexity) based on wavelets have been developed. The discrete wavelet transform ....

[Article contains additional citation context not shown here]

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

.... a variety of scientific disciplines, including physics, chemistry, astronomy, biology, meteorology, hydrology, and soil science [1, 2] In signal and image processing, fractals have been applied in applications such as computer graphics, texture modeling, image compression, and pattern recognition [3, 4]. Fractal models have made a major impact in the area of communications recently, particularly in the area of computer data networks. As the work of Leland et al. 5] and subsequent studies have demonstrated, network traffic loads exhibit fractal properties such as self similarity, burstiness, ....

....to a tractable analysis. The fBm is not stationary, but its increments form the stationary fractional Gaussian noise (fGn) process. When the Hurst parameter H 1=2, fGn exhibits LRD. N samples of fGn can be simulated exactly via direct Cholesky factorization (O(N 3 ) computational complexity) [4] or Levinson s recursion (O(N 2 ) complexity) 8] These costs can become overbearing, however, especially in networking applications where often N AE 10 6 . For such large problems, O(N) approximate synthesis techniques based on wavelets have been developed. The discrete wavelet transform ....

[Article contains additional citation context not shown here]

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

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T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

No context found.

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

No context found.

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

No context found.

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

No context found.

T. Lundahl, W. Ohley, S. Kay, and R. Siffert, "Fractional Brownian motion: A maximum likelihood estimator and its application to image texture," IEEE Trans. on Medical Imaging, vol. 5, pp. 152--161, Sep. 1986.

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T. Lundahl, W. J. Ohley, S. M. Kay, and R. Siffert, Fractional Brownian motion: A maximum likelihood estimator and its applica- tion to image texture, IEEE Trans. Med. 1986,152-161.

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