The Wavelet Packets Equalization Technique: Applications on LASCO Images M.Mierla, R. Schwenn, G. Stenborg

Contents 1. Motivation 2. Objectives 3. The Data 4. The Wavelet Packets Equalization Technique 5. Applications on LASCO Images 6. Conclusions and Perspectives

What are we looking for?  Observational signatures that would allow us to quantify the coronal outflow from regions close to the limb up to larger distances

Motivation  Studying the near-sun solar wind; sources and topology  Focusing on the slow solar wind since the fast solar wind is much better known

 Unambiguous tracking of the motion of the coronal material  Quantification of such motions Objectives Approach Selective contrast-enhancement of internal structures of both close-to limb features and streamers Temporal correlation of isolated features (HT diagrams)

LASCO/SOHO LASCO = Large Angle and Spectrometric COronagraph SOHO = The Solar and Heliospheric Observatory

The Data C1 ( LASCO/SOHO) T~2*10 6 K T~2*10 6 K C2 (LASCO/SOHO) white light C3 (LASCO/SOHO) white light Fe XIV

The Wavelet Packets Equalization Technique  The technique consists in decomposing a given image in the so called wavelet scales or wavelet planes, the first scales containing the higher (spatial) frequency components and the last ones containing the lower (spatial) frequency signatures.  Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent scales.  By assigning different weight to these levels and subsequently recombining them (plus a smoothed array, called continuum), a very good contrast enhanced image can be obtained. Stenborg & Cobelli, A&A, 2003, in press

Wavelet Transform MW: B 3 -spline (1D) The 1D “à trous” algorithm B n -splines (1D) Mother Wavelets Analysis produces a set of resolution- related views of the original signal, called scales. Scaling is achieved by dilating and contracting the basic wavelet to form a set of wavelet functions. Wavelet Scales Starck J.-L. et al., ApJ, 1997

The 2D “à trous” algorithmWeight Weight Weight Weight Weight original

The Wavelet Packets Equalization Technique  The technique consists in decomposing a given image in the so called wavelet scales or wavelet planes, the first scales containing the higher (spatial) frequency components and the last ones containing the lower (spatial) frequency signatures.  Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent scales (wavelet packets).  By assigning different weight to these levels and subsequently recombining them (plus a smoothed array, called continuum), a very good contrast enhanced image can be obtained. Stenborg & Cobelli, A&A, 2003, in press

The splitting algorithm A Multiple level wavelet decomposition: w 0 (0) w 1 (0) w 2 (0) w p1 (0)... w k (0) w 0 (0,k) w 1 (0,k) w 2 (0,k) w p2 (0,k)... w m (0,k) w 0 (0,k,m) w 1 (0,k,m) w 2 (0,k,m) w p3 (0,k,m)... w k (0,k,m) I(x,y) Reconstruction: Wickerhauser, D variant  Fligge & Solanki, 1997 (Noise reduction in astronomical spectra) Stenborg & Cobelli, A&A, 2003

The Wavelet Packets Equalization Technique  The technique consists in decomposing a given image in the so called wavelet scales or wavelet planes, the first scales containing the higher (spatial) frequency components and the last ones containing the lower (spatial) frequency signatures.  Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent scales.  After noise filtering in the wavelet domain, and assigning different weights to the last level wavelet scales (including the “continuum”) a reconstructed image is obtained, showing selectively contrast- enhanced features. Stenborg & Cobelli, A&A, 2003, in press

Noise Progression in Wavelet Space: The Reconstruction Strategy Weighted Reconstruction:  33  03  02  01  00 0  23  13  12  10 1  53  52  51  50 5  43  42  41 4  32  31  30 3  22  21  Stenborg & Cobelli, A&A, 2003

01… … … … … … Original image Fe XIV green line loops in the inner corona as seen by LASCO/C1 on June 01, 1998 at 04:12 UT (upper left). The other frames show three different reconstruction schemes based on an 8 first-level scales plus continuum, each scale further subdivided in 4 scales plus continuum

A CME observed by LASCO-C2 coronagraph on August 13, The upper left image corresponds to the LASCO-C2 raw image with the background removed and the other 3 images correspond to different restoration processes based on an 8 first-level scales plus continuum, each scale further subdivided in 3 scales plus continuum … … Original image …

… :41 15:41 LASCO-C3 image recorded on June 2nd, 1998, 15:41 UT. The image corresponds to the LASCO-C3 raw image with the background removed The reconstructed image

14:42 UT15:41 UT16:41 UT 14:42 UT15:41 UT16:41 UT

The corona observed by LASCO-C2 coronagraph on August 12-13, The first movie corresponds to the LASCO-C2 raw images with the background removed and the other 2 movies correspond to different restoration processes based on an 10 first-level scales plus continuum, each scale further subdivided in 3 scales plus continuum … … Original images

Conclusions  By applying the wavelet packet equalization technique to LASCO images: - diffuse close-to-limb magnetic field structures are better discerned, - diffuse close-to-limb magnetic field structures are better discerned, - faint, small structures, hidden in the background can be revealed, - faint, small structures, hidden in the background can be revealed, - the unseen internal details of coronal transients are revealed. - the unseen internal details of coronal transients are revealed. What do we need this for?

LASCO images time I R PA R m Perspectives

C2/LASCO, 3 June 1998, 23:57 UT Polar coordinates (Solar radii) 3.9 solar radii Radial distance Angular distance from the equator in west limb

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The 2D “à trous” algorithmWeight

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- Splines: piecewise polynomials - Spline degree n: each segment is a polynomial of degree n (n+1 coef needed). Additional smoothness constraint: continuity of the spline and derivatives until order n-1. - B splines: basic atoms by which splines are constructed - B 3 minimum curvature property. Why B 3 splines as mother wavelets?

A succession of LASCO-C3 images recorded on June 2 nd, The first column corresponds to the LASCO-C3 raw images with the background removed and the last column corresponds to: …