Pierre Kestener, André Khalil, Alain Arneodo and the Astrophysics Research Group of TCD, Dublin. CEA-Saclay, IRFU, SEDI, France. Service d'Electronique ...
wtmm P. Kestener
Characterizing Complexity in Solar Magnetogram Data using a Wavelet-based Segmentation Method
Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
Pierre Kestener, André Khalil, Alain Arneodo and the Astrophysics Research Group of TCD, Dublin CEA-Saclay, IRFU, SEDI, France Service d’Electronique, Informatique et Détecteurs
ADA6, Monastir, Tunisia, May 2010
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Contents...
wtmm P. Kestener Solar magnetogram complexity: brief overview
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Brief overview of previous related works on characterizing solar magnetogram complexity • Cancelation exponent and sign-singular measure • flatness
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a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
A multifractal image segmentation tool based on local self-similar properties • WTMM method to quantity local self-similarity • example of multifractal segmentation
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Application to solar magnetogram (quiet-Sun and active regions)
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Other Applications
• characterizing an evolving active region
• Turbulence in interstellar medium simulations.
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Contents
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Solar magnetogram complexity: brief overview
Application to solar magnetogram data
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
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Solar magnetogram complexity
wtmm P. Kestener Solar magnetogram complexity: brief overview
Solar magnetogram
a multifractal image segmentation tool The WTMM method
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SOHO/MDI magnetogram of active region NOAA-9077 (Abramenko et al., ApJ 619, 1160, (2005))
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Use continuous wavelet transform to perform quiet-Sun versus active region segmentation (the most objective as possible)
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Disentangle the underlying scale invariant components of a compound system displaying a phase transition in its singularity spectra
A segmentation example
Application to solar magnetogram data
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Statistical tools to quantify complexity
wtmm P. Kestener Solar magnetogram complexity: brief overview
Numerous works loosely or strongly connected to multifractal ideas
a multifractal image segmentation tool The WTMM method A segmentation example
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Cancelation exponent of current helicity [1] : χ(r) = ΣLi (r) |µi (r)| ∼ r −κ
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scaling behavior of the structure function [2] : Sq (r) =< |B∥ (x + r) − B∥ (x)| >∼ r ζ(q)
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box-counting fractal study [3]
Application to solar magnetogram data
V.B. Yurchyshyn et al. ApJ, 538:968-979, (2000). V.I. Abramenko et al. ApJ, 577:487, (2002). V.I. Abramenko. Sol. Phys., 228:29, (2005).
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Contents
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Solar magnetogram complexity: brief overview
Application to solar magnetogram data
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
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wtmm
WTMM method I
P. Kestener Solar magnetogram complexity: brief overview
Basics of the 2D WTMM method 1. 2D continuous wavelet transform of the input image function f (x, y) Tψ [f ](b, a)
=
¡ ¢ R ¶ µ Tψ1 [f ] = a−2 R d2 x ψ1 ¡a−1 (x − b)¢f (x) , −2 2 −1 d x ψ2 a (x − b) f (x) Tψ2 [f ] = a
=
∇{Tφ [f ](b, a)}
=
∇{φb,a ∗ f },
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
(1)
where ψ1 = ∂φ/∂x, ψ2 = ∂φ/∂y and φ(x) = exp(−|x|2 /2). 2. For each scale a, extract the WTMM edges defined as the locations b where the WT modulus Mψ [f ](b, a) = |Tψ [f ](b, a)| is locally maximum in the direction of the WT vector Tψ [f ](b, a). These WTMM points lie on connected maxima chains . Along each of these maxima chains, locate the local maxima previously called WTMMM for WTMM maxima. 7 / 21
wtmm
WTMM method II
P. Kestener Solar magnetogram complexity: brief overview
Basics of the 2D WTMM method 3. Extract the WT skeleton which is the set of maxima lines L x0 obtained by connecting these WTMMM from scale to scale. Mψ [f ][L x0 (a)] ∼ ah(x0 ) ,
(2)
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
where h(x0 ) is the Hölder exponent, i.e. the strength of the singularity of the function f at the point x0 . 4. From the WT skeleton compute the partition function: Z (q, a) =
X L ∈L (a)
£ ¤q Mψ [f ](x ∈ L , a) ∼ aτ(q) , a → 0+ .
(3)
5. Compute the τ(q) spectrum by performing linear regression fits of ln Z (q, a) vs ln a and finally compute the D(h) singularity spectrum by Legendre transforming τ(q): £ ¤ D(h) = min qh − τ(q) . q
(4) 8 / 21
WTMM segmentation of a compound system
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
Segmentation Ï
Sum of two scale invariant components: f (x) = f I (x) + f II (x)
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study a mixture of two unknown scale invariant components with different multifractal properties
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WTMM method give access to the dashed τ(q) and D(h) curves
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Z (q, a) = Z I (q, a) + Z II (q, a) = CI (q)aτ
I (q)
+ CII (q)aτ
II (q)
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WTMM segmentation of a compound system
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
Segmentation Ï
segmentation strategy : discriminate the maxima lines L I (a) associated with singularities of f I (x) the maxima lines L II (a) associated with singularities of f II (x)
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WTMM segmentation of a synthetic data
wtmm P. Kestener Solar magnetogram complexity: brief overview
Academic example of a compound system
a multifractal image segmentation tool The WTMM method
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f I (x) is a fractional Brownian noise, Hölder exponent h = −0.7, D = 2
A segmentation example
Application to solar magnetogram data
f II (x) is an IFS fractal Dragon, the border of this set is characterized by h = 0, DF ' 1.52 f (x) = (1 − λ)f I (x) + λf II (x), λ is a mixture parameter
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WTMM segmentation of a synthetic data
wtmm P. Kestener Solar magnetogram complexity: brief overview
Academic example of a compound system Ï
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WT skeleton maxima lines pointing to noise features at small scale are characterized by a WTMMM power law behavior Mψ [f ](a) ∼ a−0.7
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
WT skeleton maxima lines pointing to fractal Dragon boundary at small scale are characterized by a WTMMM power law behavior Mψ [f ](a) ∼ a0.0
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WTMM segmentation of a synthetic data Log-log plot of WT Mψ [f ](b, a) modulus along the skeleton maxima lines versus scale a λ = 0.5
λ = 0.2
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
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WTMM segmentation of a synthetic data Segmentation in the WT skeleton Each maxima line is characterized by: Ï a length, i.e. its maximun scale amax Ï Ï Ï
the WT modulus Mψ [f ](amax ) at scale amax .
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
log2 Mψ [f ](amax ) ≥ hs log2 amax + log2 Ms segmentation parameters: slope hs and intercept Ms
Parameter adjustment Ï
By eye: wrong parameters destroy the quality of the partition functions scaling properties
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automatic: implement a clustering algorithm (TODO) 14 / 21
Contents
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Solar magnetogram complexity: brief overview
Application to solar magnetogram data
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
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Multifractal properties of MDI/SOHO quiet-Sun magnetogram Quiet Sun magnetograms analysis
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method
Ï Ï
Multifractal analysis of a set of 30 quiet Sun images (505×505) quiet Sun images are everywhere singular (D(q = 0) = 2) with a corresponding Hölder exponent h(q = 0) ' −0.75.
A segmentation example
Application to solar magnetogram data
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Multifractal properties of MDI/SOHO quiet-Sun magnetogram Quiet Sun magnetograms analysis
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method
Ï Ï
Multifractal analysis of a set of 30 quiet Sun images (505×505) quiet Sun images are everywhere singular (D(q = 0) = 2) with a corresponding Hölder exponent h(q = 0) ' −0.75.
A segmentation example
Application to solar magnetogram data
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Multifractal segmentation of active region
wtmm P. Kestener Solar magnetogram complexity: brief overview
Active region segmentation Ï
Disentangle maxima chains associated with the ‚ and those corresponding to the quiet Sun. WT skeleton maxima lines are sorted according to scaling behavior described by Mψ [f ](a) ∼ ah .
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quiet Sun: log2 Mψ [f ](amax ) ≤ hQ log2 amax + log2 MQ ,
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active region: log2 Mψ [f ](amax ) ≥ hA log2 amax + log2 MA ,
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
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Multifractal segmentation of active region
wtmm P. Kestener Solar magnetogram complexity: brief overview
Comparative multifractal analysis of quiet and active region Ï
a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
active region multifractal D(h) approximated by the log-normal model : c0 = 2, c1 = 0.38 and c2 = 0.12 .
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Multifractal segmentation of active region Comparative multifractal analysis of quiet and active region
wtmm P. Kestener Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
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active region multifractal D(h) approximated by the
Application to solar magnetogram data
log-normal model : c0 = 2, c1 = 0.38 and c2 = 0.12 .
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wtmm
TODO
P. Kestener
Further work : revisiting existing study of intermittency in the photosphere using the active region segmentation first.
Measure of intermittency: flatness 2
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FSF (r) = S4 (r)/|S2 (r)|
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FWTMM (r) = Z4 (r)/Z2 (r)2
Solar magnetogram complexity: brief overview a multifractal image segmentation tool The WTMM method A segmentation example
Application to solar magnetogram data
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