The Wavelet Packets Equalization Technique: Applications on LASCO Images M.Mierla, R. Schwenn, G....
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The Wavelet Packets The Wavelet Packets Equalization Equalization Technique: Technique: Applications on LASCO Applications on LASCO Images Images M.Mierla, R. Schwenn, G. M.Mierla, R. Schwenn, G. Stenborg Stenborg
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Transcript of The Wavelet Packets Equalization Technique: Applications on LASCO Images M.Mierla, R. Schwenn, G....
The Wavelet Packets The Wavelet Packets Equalization Technique: Equalization Technique: Applications on LASCO Applications on LASCO
ImagesImages
M.Mierla, R. Schwenn, G. StenborgM.Mierla, R. Schwenn, G. Stenborg
ContentsContents
1. Motivation1. Motivation
2. Objectives2. Objectives
3. The Data 3. The Data
4. The Wavelet Packets Equalization Technique4. The Wavelet Packets Equalization Technique
5. Applications on LASCO Images5. Applications on LASCO Images
6. Conclusions and Perspectives6. Conclusions and Perspectives
What are we looking for?What are we looking for? Observational signatures that Observational signatures that
would allow us to quantify the would allow us to quantify the coronal outflow from regions coronal outflow from regions close to the limb up to larger close to the limb up to larger distances distances
MotivationMotivation
Studying the near-sun Studying the near-sun solar wind; sources solar wind; sources and topologyand topology
Focusing on the slow Focusing on the slow solar wind since the solar wind since the fast solar wind is much fast solar wind is much better knownbetter known
Unambiguous tracking of the motion of the Unambiguous tracking of the motion of the coronal material coronal material
Quantification of such motionsQuantification of such motions
ObjectivesObjectives
Approach
• Selective contrast-enhancement of internal structures of both close-to limb features and streamers
• Temporal correlation of isolated features (HT diagrams)
LASCO/SOHOLASCO/SOHO
LASCO = Large Angle and Spectrometric COronagraph
SOHO = The Solar and Heliospheric Observatory
The DataThe Data
C1 ( LASCO/SOHO)C1 ( LASCO/SOHO)
T~2*10T~2*106 6 KK
C2 (LASCO/SOHO) white light
C3 (LASCO/SOHO) white light
Fe XIV
The Wavelet Packets Equalization The Wavelet Packets Equalization TechniqueTechnique
The techniqueThe technique consists in consists in decomposingdecomposing a given image in the so called a given image in the so called wavelet scales wavelet scales oror
wavelet planes, wavelet planes, the the first scalesfirst scales containing containing the higher (spatial) frequency componentsthe higher (spatial) frequency components and and the the
last oneslast ones containing the containing the lower (spatial) frequency signatureslower (spatial) frequency signatures. .
Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent
scales.scales.
By assigning different weight to these levels and subsequently recombining them (plus a 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.smoothed array, called continuum), a very good contrast enhanced image can be obtained.
Stenborg & Cobelli, A&A, 2003, in press
Wavelet Transform
MW: B3-spline (1D)
The 1D “à trous” algorithm
Bn-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 ScalesStarck J.-L. et al., ApJ, 1997
The 2D “à trous” algorithm WeightWeight
00 1111 5522 0033 0044 0055 00
WeightWeight00 1111 5522 5533 5544 5555 55
WeightWeight00 1111 0022 5533 0044 0055 00
WeightWeight00 1111 0022 0033 5544 0055 00
WeightWeight00 1111 0022 0033 0044 0055 55
original
The Wavelet Packets Equalization The Wavelet Packets Equalization TechniqueTechnique
The techniqueThe technique consists in consists in decomposingdecomposing a given image in the so called a given image in the so called wavelet scales wavelet scales oror
wavelet planes, wavelet planes, the first scales containing the higher (spatial) frequency components and the the first scales containing the higher (spatial) frequency components and the
last ones containing the lower (spatial) frequency signatures. last ones containing the lower (spatial) frequency signatures.
Wavelet Transform properties allow Wavelet Transform properties allow further decompositionfurther decomposition of each wavelet scale in subsequent of each wavelet scale in subsequent
scales (wavelet packets).scales (wavelet packets).
By assigning different weight to these levels and subsequently recombining them (plus a 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.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:
w0(0)
w1(0)
w2(0)
wp1(0)
...
...
wk(0) w0
(0,k)
w1(0,k)
w2(0,k)
wp2(0,k)
...
...
wm(0,k)
w0(0,k,m)
w1(0,k,m)
w2(0,k,m)
wp3(0,k,m)
...
...
wk(0,k,m)
I(x,y)
1
0
2
0
3
0
),,0(),(p
i
p
j
p
h
ijhwyxI
Reconstruction:
Wickerhauser, 1991
1-D variant Fligge & Solanki, 1997(Noise reduction in astronomical spectra)
Stenborg & Cobelli, A&A, 2003
The Wavelet Packets Equalization The Wavelet Packets Equalization TechniqueTechnique
The techniqueThe technique consists in consists in decomposingdecomposing a given image in the so called a given image in the so called wavelet scales wavelet scales oror wavelet wavelet
planes, planes, the first scales containing the higher (spatial) frequency components and the last ones the first scales containing the higher (spatial) frequency components and the last ones
containing the lower (spatial) frequency signatures. containing the lower (spatial) frequency signatures.
Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent scales.Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent scales.
AfterAfter noise filteringnoise filtering in the wavelet domain, and in the wavelet domain, and assigning different weightsassigning different weights to the last level wavelet to the last level wavelet
scales (including the “continuum”) a reconstructed image is obtained, showing selectively scales (including the “continuum”) a reconstructed image is obtained, showing selectively contrast- contrast-
enhancedenhanced features. features.
Stenborg & Cobelli, A&A, 2003, in press
),( if ),(
),( if 0),(
(...)(...)(...)
(...)(...)
(...)
yxkwyxw
yxkwyxW
hhh
hh
h
),(ˆ),( (...)(...) yxyx Ihh Noise Progressionin Wavelet Space:
The Reconstruction Strategy
1
0
2
0
3
0
),,0(,,),(
p
i
p
j
p
h
ijhhji WyxI Weighted Reconstruction:
3333
03030202 0101000000
2323
1313121212121010 11
535352525151505055
434342424141414144
32323131303033
22222121202022
33221100
51 p
32 p
Stenborg & Cobelli, A&A, 2003
00 11 …… 44
00 11 11 11 11
11 44 44 44 44
22 44 44 44 44
33 44 44 44 44
44 44 44 44 44
…… 00 00 00 00
88 00 00 00 00
00 11 …… 4400 11 11 11 11
11 44 44 44 44
22 44 44 44 44
33 44 44 44 44
44 44 44 44 44
…… 44 44 44 44
88 44 44 44 44
00 11 …… 4400 11 11 11 11
11 00 00 00 00
22 44 1010 77 44
33 44 44 77 44
44 44 44 44 44
…… 00 00 00 00
88 00 00 00 00
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, 2002. 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.
00 11 22 33
00 00 11 11 11
11 00 55 55 55
22 00 55 55 55
33 00 55 55 55
44 00 55 55 55
…… 00 55 55 55
88 00 55 55 55
00 11 22 33
00 00 00 00 00
11 88 88 88 88
22 88 88 88 88
33 44 44 44 44
44 44 44 44 44
…… 44 44 44 44
88 44 44 44 44
Original image
00 11 22 33
00 00 11 11 11
11 00 55 00 00
22 00 88 00 00
33 00 88 00 00
44 00 88 00 00
…… 00 88 00 00
88 00 88 00 00
00 11 22 3300 11 11 11 11
11 11 11 11 11
22 11 11 11 11
33 1010 1010 1010 1010
44 55 55 55 55
…… 55 55 55 55
1010 55 55 55 55
14:41
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 UT 15:41 UT 16:41 UT
14:42 UT 15:41 UT 16:41 UT
The corona observed by LASCO-C2 coronagraph on August 12-13, 2002. 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.
00 11 22 3300 11 11 11 1111 55 55 55 55
22 55 55 55 55
33 55 55 55 55
44 55 55 55 55
…… 55 55 55 55
1010 55 55 55 55
00 11 22 3300 11 11 11 1111 00 1010 1010 1010
22 00 1515 1515 1515
33 00 1515 1515 1515
44 00 1515 1515 1515
…… 00 1010 1010 1010
1010 00 1010 1010 1010
Original images
ConclusionsConclusions
By applying the wavelet packet equalization technique to By applying the wavelet packet equalization technique to LASCO images:LASCO images:
- diffuse close-to-limb magnetic field structures are - diffuse close-to-limb magnetic field structures are better discerned,better discerned,
- faint, small structures, hidden in the background - faint, small structures, hidden in the background can be revealed, can be revealed,
- the unseen internal details of coronal transients are - the unseen internal details of coronal transients are revealed.revealed.
What do we need this for?What do we need this for?
LASCOimages
time
time
I
time
R
PA
R
PA
m
Perspectives
C2/LASCO, 3 June 1998, 23:57 UT
Polar coordinates
100
200
300
123.4
2
4
1
3
5
(Sol
ar r
adii)
3.9 solar radii
Rad
ial
dist
ance
Angular distance from the equator in west limb
END END
The 2D “à trous” algorithm
WeightWeight
00 11
11 00
22 55
33 00
44 00
55 00
The 2D “à trous” algorithm
WeightWeight
00 11
11 00
22 00
33 55
44 00
55 00
The 2D “à trous” algorithm
WeightWeight
00 11
11 00
22 00
33 00
44 55
55 00
WeightWeight
00 11
11 00
22 00
33 00
44 00
55 55
The 2D “à trous” algorithm
The 2D “à trous” algorithm
ReconstructionWeightWeight
00 11
11 55
22 55
33 55
44 55
55 55
00 11 22 33
00 11 11 11 11
11 00 55 00 00
22 00 55 00 00
33 00 55 00 00
44 00 55 00 00
55 00 55 00 00
00 11 22 33
00 11 11 11 11
11 00 00 55 00
22 00 00 55 00
33 00 00 55 00
44 00 00 55 00
55 00 00 55 00
00 11 22 33
00 11 11 11 11
11 55 00 00 00
22 55 00 00 00
33 55 00 00 00
44 55 00 00 00
55 55 00 00 00
00 11 22 33
00 11 11 11 11
11 55 1515 1010 77
22 33 1010 77 55
33 11 77 55 33
44 11 55 33 11
55 00 33 11 11
00 11 22 33
00 11 11 11 11
11 00 00 11 33
22 11 11 33 55
33 11 33 55 77
44 33 55 77 1010
55 55 77 1010 1515
- 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
- B3 minimum curvature property.
Why B3 splines as mother wavelets?
A succession of LASCO-C3 images recorded on June 2nd, 1998. The first column corresponds to the LASCO-C3 raw images with the background removed and the last column corresponds to:
00 11 22 3300 11 11 11 11
11 11 11 11 11
22 11 11 11 11
33 1010 1010 1010 1010
44 55 55 55 55
…… 55 55 55 55
1010 55 55 55 55
