Local Regularity and Multifractal Methods for Image and Signal Analysis in .NET

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Figure 1110 shows two paths of MBM with a linear function H(t) = 02 + 06t and a periodic H(t) = 05 + 03 sin(4 t) We clearly see how regularity evolves over time
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Figure 1110 MBM paths with linear and periodic H functions
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Estimation of the H functions from the traces in Figure 1110, using the so-called generalized quadratic variations are shown in Figure 1111 (the theoretical regularity is in gray and the estimated regularity is in black)
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Figure 1111 Estimation of the local regularity of MBM paths Left: linear H function Right: periodic H function
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24-hour interbeat (RR) interval time series obtained from the PhysioNet database [PHY] along with their estimated local regularity (assuming that the processes may be modeled as MBM) are shown in Figure 1112 These signals were derived from long-term ECG recordings of adults between the ages of 20 and 50 who have no known cardiac abnormalities and typically begin and end in the early morning (within an hour or two of the subject waking) They are composed of around 100, 000 points As we can see from Figure 1112, there is a clear negative correlation between the value of the RR interval and its local regularity: when the black curve moves up, the
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Figure 1112 RR interval time series (upper curves) and estimation of the local regularity (lower curves)
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gray tends to move down In other words, slower heartbeats (higher RR values) are typically more irregular (smaller H lder exponents) than faster ones In order to account for this striking feature, the modeling based on MBM must be re ned Indeed, while MBM allows us to tune regularity at each time, it does so in an exogenous manner This means that the value of H and of WH are independent A better model for RR time series requires us to de ne a modi ed MBM where the regularity would be a function of WH at each time Such a process is called a self-regulating multifractional process (SRMP) It is de ned as follows We give ourselves a deterministic, smooth, one-to-one function g ranging in (0, 1), and we seek a process X such that, at each t, X (t) = g(X(t)) almost surely It is not possible to write an explicit expression for such a process Rather, a xed point approach is used, which we now brie y describe (see [BAR 07] for more details) Start from an MBM WH with an arbitrary function H (for instance a constant) At the second step, set H = g(WH ) Then iterate this process, ie calculate a new WH with this updated H function, and so on We may prove that these iterations will almost surely converge to a well-de ned SRMP X with the desired property, namely the regularity of X at any given time t is equal to g(X(t)) For such a process, there is a functional relation between the amplitude and the regularity However, this does not make precise control of the H lder exponent possible Let us explain this through an example Take de niteness g(t) = t for all t Then, a given realization might result in a low value of X at, say, t = 05 and thus high irregularity at this point, while another realization might give a large X(05), resulting in a path that is smooth at 05 See Figure 1113 for an example of this fact In order to gain more control, the de nition of an SRMP is modi ed as follows First de ne a shape function s, which is a deterministic smooth function Then, at each step, calculate WH , and set H = g(WH + ms), where m is a positive number The function s thus serves two purposes First, it allows us to tune the shape of X:
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