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

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Local Regularity and Multifractal Methods for Image and Signal Analysis
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Figure 1113 Paths of SRMPs with g(Z) = Z
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when m is large, X and s will essentially have the same form Second, because of the rst property, it allows us to decide where the process will be irregular and where it will be smooth Figure 1114 displays an example of SRMP with controlled shapes
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Figure 1114 (a) SRMP with g(Z) = Z (black), and controlling shape function (gray); (b) the same SRMP (black) with estimated H lder exponent (gray)
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It is then possible to obtain a ne model for RR traces based on the following ingredients: an s function, that describes the overall shape of the trace, and in particular the nycthemeral cycle; a g function whose role is to ensure that the correct relation between the heart rate and its regularity is maintained at all times The shape s is estimated from the data in the following way; for each RRi time series, histograms of both the signal and its exponent are plotted, and modeled as a sum of two Gaussians, as represented in Figure 1115
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Figure 1115 Histogram of RRi time series, modeled as a sum of two Gaussians
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From these signals the shape functions are inferred They are based on splines and parameterized by: Dn , duration of the night: Dn [6, 10] Dm , duration of the beginning of the measure: Dm [2, 4] Ds , duration of the sleeping phase: Ds [05, 15] Da , duration of the awakening phase: Dr [05, 15] RRid , mean interbeat interval during the day: RRid [06018, 07944] RRin , mean interbeat interval during the night: RRin [07739, 10531] randomly chosen, in each case, in their respective intervals, with uniform probability (see Figure 1116 for a representation)
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Local Regularity and Multifractal Methods for Image and Signal Analysis
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Figure 1116 Shape function of RR intervals
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The g function is estimated in the phase space More precisely, for each trace, the value of H as a function of the RR interval is plotted Representing all these graphs on a single plot, a histogram is obtained, as in Figure 1117
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Figure 1117 Histogram in the phase space
The ridge line of this histogram, seen as a surface in the (RR, ) plane is then extracted (see Figure 1117) It is seen that this ridge line is roughly a straight line, that is, tted using least squares minimization in order to obtain an equation of the form = g(RR) = aRR + b The last step is to synthesize an SRMP with shape function s and regularity function g, as explained in the previous section Paths obtained in this way are shown in Figure 1118 Compare this with the graphs shown in Figure 1112, displaying true RR traces 117 Texture segmentation We will now brie y explain how multifractal analysis may be used for texture segmentation We present an application to 1D signals, namely well logs For an application to images, see [MUL 94, SAU 99]
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Figure 1118 Two forged RR intervals based on SRMP (upper curves) and estimated regularity (lower curves)
The characterization of geological strata with the help of well logs can be used for the interpretation of a sedimentary environment of an area of interest, such as reservoirs Recent progress [SER 87] of electrofacies has enabled us to relate well logs to the sedimentary environment and to extrapolate information coming from the core of any vertical well span Electrofacies predictions are based on multivariate correlations and cluster analysis for which the entry data are conventional well logs (including sonic logs, of density and gamma) as well as the information extracted from the core analysis The microresistivity log (ML) measures the local rock wall resistivity of the wells The measure is obtained by passing an electric current in the rock, to a lateral depth of approximately 1 cm The resistivity varies according to the local porosity function and the connectivity of the pores (normally, the rock is a non-conductor and thus the current passes in the uid contained in the pores) ML contain information not only on the inclination of geological strata but also on the texture of these strata To analyze the irregular variations of ML, we may calculate texture parameters locally and at different depths in the well These can be used to obtain a well segmentation by letting [SAU 97] r(xi ) denote the signal resistivity, where xi are equidistant coordinates which measure the depth in the wells In order to emphasize the vertical variations of r(xi ), a transformed signal s (xi ) is rst de ned s (xi ) = |r(xi+1 ) r(xi )| where > 0 This transformation ampli es the small scales and eliminates any constant component of signal r(xi ) Analysis of well logs from the Oseberg reservoir in the North Sea shows that, typically, a fractal behavior is observed for lengths of about [2 cm, 20 cm] It has been