Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data A wavelet.

SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image DataPattern Recognition Techniques Applied to Solar Image Data

A wavelet packets A wavelet packets equalization technique equalization technique to reveal the multiple to reveal the multiple spatial-scale nature spatial-scale nature of coronal structuresof coronal structures

Guillermo A. Stenborg Guillermo A. Stenborg

The Catholic University of America&

NASA Goddard Space Flight Center


ObjectiveObjective

• More accurate tracking of coronal events

• More accurate determination of onset times

• Tracking of continuous coronal outflow (slow solar wind ?) seen in LASCO-C2 and -C3 images ?

ApproachApproach

Selective contrast enhancement of boundaries and internal details of

coronal features

More reliable identification of coronal structures to More reliable identification of coronal structures to help in the process of automatic recognition of help in the process of automatic recognition of coronal eventscoronal events


WTs (Wavelet Transforms)

• Transforms data to time-scale domain

• Use of “mother wavelets”

How do we analyse signals?

• Dilations and compressions

• Traslations over the signal´s domain

Analyzing wavelet adapted to frequency

Spatialy Localized

Additional capabilities & features

• Infinite set of possible basis functions

• Quantitative measure of information

Adapted wavelets

Time-scale based methodsTime-scale based methods


1) The technique consists in decomposing a given signal in the so-called wavelet scales or wavelet planes, the the

first scalesfirst scales containing the higher (spatial) frequency componentsthe higher (spatial) frequency components and the last onesthe last ones containing the lower (spatial) the lower (spatial)

frequency signaturesfrequency signatures.

2) Wavelet Transform properties allow further decomposition of each wavelet scale in subsequent scales.

3) After noise filtering in the wavelet domain, and assigning different weights to the wavelet scales (including a smoothed array

called “continuum”) a reconstructed image is obtained, showing selectively contrast- enhanced features (in a way

resembling the technique known as „unsharp masking“).

The Wavelet-based Equalization TechniqueThe Wavelet-based Equalization Technique


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

Wavelet Transform

MW: B3-spline (1D)



Weight

0 1

1 1

2 0

3 0

4 0

5 0



Weight

0 1

1 0

2 1

3 0

4 0

5 0



Weight

0 1

1 0

2 0

3 1

4 0

5 0



Weight

0 1

1 0

2 0

3 0

4 1

5 0


Weight

0 1

1 0

2 0

3 0

4 0

5 1




p

jj yxwyxwyxI

10 ),(),(),(1) Reconstruction

of original image

),(1 yxw ),(2 yxw ),(3 yxw ),(4 yxw

),(0 yxw

),( yxI

p

jjj yxWyxWyxI

10 ),(),(),(' 4) Weighted Reconstruction:

The Figure depicts the wavelet scales 1 to 4 (Wi) and the smoothed image (Wo), i.e., continuum, of I (x,y)

= EIT Fe IX/X ( 171 Å) image = =1 =[1,1,1,1]k=0

),(' yxI

For comparison, continuum corresponding to decomposition

based on 50 scales

when

2DB3-spline

:

:

),(ˆ),( yxyx Ijj 2) Local standard deviation of Noise at scale j

Local standard deviation of noise in original image (first scale)Noise progression in wavelet space

),(),( if ),(

),(),( if 0),(

yxkyxwyxw

yxkyxwyxW

jjj

jj

j

3) Noise filtering: (hard thresholding as in Donoho & Johnstone, 1994)



ReconstructionWeight

0 1

1 5

2 5

3 5

4 5

5 5


• The technique shown so far involves decomposing a given image in wavelet planes (i.e., spatial frequency bands), the finer scales containing the higher frequency components and the coarser ones the lowest frequency signatures.

• For non-orthogonal wavelets (as for the “à trous” algorithm) the Signal to Noise Ratio (SNR) increases toward coarser scales.

• Straightforward filtering of wavelet coeficients at this stage produces rejection of signal along with noise.

Comments on the 2D “à trous” algorithm

A better alternative is a technique allowing a finer analysis of the frequency content of the signal


The alternative:

The Wavelet The Wavelet PacketsPackets -based Equalization Technique -based Equalization Technique • The splitting algorithm of wavelet packets on non-orthogonal wavelets allows much

better frequency localization. That is achieved by recursively decomposing (transforming) the wavelet scales obtained with the “à trous” algorithm (thanks to the fact that wavelet transform is not its own inverse).

• 1-D variant of the algorithm was first implemented for an astronomical aplication by Fligge & Solanki, 1997 to reduce noise in astronomical spectra.

• 2-D variant of the algorithm was first implemented for an astronomical application by Stenborg & Cobelli, 2003 to reveal the multiple spatial-scale nature of coronal structures (hereafter SC2003).


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)

The technique

Multiple-level decomposition scheme:

3-level decomposition tree. For clarity only one branch is shown at each decomposition level, but it is assumed that when computing a new level all coefficients of the previous one are decomposed

1

0

2

0

3

0

),,0(),(p

i

p

j

p

h

ijhwyxI

),( if ),(

),( if 0),(

,(...)(...)

,(...)

(...)

yxkwyxw

yxkwyxW

lkhhh

lkhh

h

),(ˆ),( (...)(...) yxyx Ihh

1)

2)

1

0

2

0

3

0

),,0(,,),('

p

i

p

j

p

h

ijhhji WyxI

3)

33

0302 01000

23

13121210 1

535251505

434241414

3231303

2221202

3210

51 p

32 pIn matrix form the weighting coefficients can be depicted as (for 2 levels):

4)

Briefly, the first level decomposition of the given image in p1 scales gives rise to the wavelet transform set {wi

(0)}, i=1...p1, i=0 corresponding to the continuum component. Afterward, further decomposition is applied to each wavelet plane.


0 1 2 3

0 1 1 1 1

1 0 1 0 0

2 0 1 0 0

3 0 1 0 0

4 0 1 0 0

5 0 1 0 0


0 1 2 3

0 1 1 1 1

1 0 0 1 0

2 0 0 1 0

3 0 0 1 0

4 0 0 1 0

5 0 0 1 0


0 1 2 3

0 1 1 1 1

1 1 0 0 0

2 1 0 0 0

3 1 0 0 0

4 1 0 0 0

5 1 0 0 0


0 1 2 3

0 1 1 1 1

1 5 15 10 7

2 3 10 7 5

3 1 7 5 3

4 1 5 3 1

5 0 3 1 1

0 1 2 3

0 1 1 1 1

1 0 0 1 3

2 1 1 3 5

3 1 3 5 7

4 3 5 7 10

5 5 7 10 15


0 1 2 3

0 0 0 0 0

1 1 0 0 1

2 1 0 0 1

3 1 0 0 1

4 1 0 0 1

5 1 0 0 1

0 1 2 3

0 0 0 0 0

1 0 1 1 0

2 0 1 1 0

3 0 1 1 0

4 0 1 1 0

5 0 1 1 0


= 1 = [0,10,1,1,1,1,1,1]K = 0

EIT Fe XIV image reconstructed with:

= 1 0,10,0,0 1, 1,1,1 1, 1,1,1 = 1, 1,1,1 1, 1,1,1 1, 1,1,1 1, 1,1,1 1, 1,1,1K = 0

EIT Fe XIV image reconstructed with:




1/5LASCO-C2: April 21, 2002














1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

01/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

02/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

03/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

04/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

05/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

06/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

07/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

08/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

09/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

10/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

11/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

12/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

13/14

LASCO-C2 Level 0.5


1998.06.02

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 1 1 1 1

1 8 8 8 8

2 8 8 8 8

... 8 8 8 8

7 8 8 8 8

8 8 8 8 8

0 1 2 3

0 1 1 1 1

1 8 18 18 18

2 8 18 18 18

... 8 18 18 18

7 8 18 18 18

8 8 18 18 18

0 1 2 3

0 1 1 1 1

1 18 8 8 8

2 18 8 8 8

... 18 8 8 8

7 18 8 8 8

8 18 8 8 8

0 1 2 3

0 0 0 0 0

1 0 1 1 1

2 0 1 1 1

... 0 1 1 1

7 0 1 1 1

8 0 1 1 1

14/14

LASCO-C2 Level 0.5


LASCO-C22002.08.12 00:06 – 2002.08.14 02:06

Original

0 1 2 3

0 1 1 1 1

1 0 5 5 5

2 0 5 5 5

... 0 5 5 5

7 0 5 5 5

8 0 5 5 5

0 1 2 3

0 0 0 0 0

1 1 1 1 1

2 1 1 1 1

... 1 1 1 1

7 1 1 1 1

8 1 1 1 1

0 1 2 3

0 0 0 0 0

1 0 3 3 3

2 0 3 3 3

... 0 0 0 0

7 0 0 0 0

8 0 0 0 0

1 2

3 4


1 2 3 4


Highlights

• Typical coronal images show coexistent structures exhibiting high and low intensities, i.e., a wide dynamic range.

Method´s property of being highly localized (depending upon the value of N, i.e., size of the mother wavelet, relative to the image size) allows to treat them on the same ground and without affecting each other.

Radial distance from the border of the occulter (in pixels)

C2 image: Polar representation(01/01/2001 @ 00:06 UT)

Original

Processed

Angular distance (0 at equator, West limb)

Angular distance (0 at equator, West limb)

As shown with the examples, the SC2003 technique is suitable for the selective enhancement of specific spatial scales composing any 2D

image.

An example showing the application of the SC2003 technique to a polar representation of a LASCO C2 image


Towards automatic tracking of dynamical events

The temporal evolution of dynamical events can be seen in a single 2D image by stacking one image on top of the other and obtaining the intensity profile along the time axis i) at a given position angle for all radial distances (Heigth - Time maps), or ii) at a given radial distance for all position angles (Position angle - Time maps), so that the SC2003 technique can be applied. Two examples using LASCO-C2 data follows .

OriginalProcessed

(Continuum component removed)

Posit

ion

An

gle

Rad

ial d

ista

nce (

pxls

)

Time (hours since January 8, 2001 at 00:06 UT)0 25 50 75 100 125 Time (hours since January 8, 2001 at 00:06 UT)

0 25 50 75 100 125

0

50

100

150

90

270

110

130

150

170

190

210

230

250


270

300

330

0

30

60

90

Posit

ion

An

gle

Rad

ial d

ista

nce (

pxls

) 150

100

50

00 25 50 75 100 125

0 25 50 75 100 125 Time (hours since January 21, 2001 at 00:06 UT)

0 25 50 75 100 125 0 25 50 75 100 125

OriginalProcessed

(Continuum component removed)CMEs

Streamers

Top: Position Angle - Time map Bottom: Corresponding Height-Time Map for a radial cut at P=98° -Example 1- and

P=285° -Example 2- (solid white line in Position Angle - Time map). The dashed white line depicts the border of the occulter.


Future Prospects...Future Prospects...

Time-lapse sequences of LASCO-C2, and C3 show a continuous outflow resembling the flow of the slow solar wind. However, the small inhomogeneities forming the flow cannot be distinguished from noise when observing individual images.

This side effect can be used for good to enhance the inhomogeneities forming the outward flow. As the object to be characterize is a flow, static images will not reveal anything unless the dynamic is in the image itself (e.g., Carrington maps, or Height-Time maps).

If there is no small-scale inhomogeneities moving outward the Heigth-Time image will exhibit just white noise. Otherwise, the background will exhibit a preferential direction (noise correlated in time).

Under way... The SC2003 technique to enhance such inhomogeneities to help quantify the slow solar wind speed...

Without proper noise removal, the technique developed also enhances the noise

Note the inclination of the pattern !!!!!

Time Time

Rad

ial d

ista

nce (

pxls

)

An

gu

lar

dis

tan

ce


END


1/4


2/4


3/4


4/4


- 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 BWhy B33 splines as mother wavelets? splines as mother wavelets?

2DB3-spline



(1) The arrival of photons, and their expression by electron counts on CCD detectors may be modeled by a Poisson distribution. If the noise in data I(x,y) is Poisson, the Anscombe transformation acts as if the data arose from Gaussian white noise model.

Determination of the Noise

(3) Noise Progression in wavelet scales: by simulating an image containing Gaussian noise with a standard deviation equal to 1, and taking the same WT applied to the original image to this sintetic image. is the standard deviation of each wavelet scale.

j

(2) Calculation of local standard deviation: For a fixed pixel position, say ( k,h ), the local standard deviation is calculated by taking its N x N neighbouring pixels given by the cartesian product [ k,k+N ] x [ h,h+N ] and computing their standard deviation. This value is stored in an array at its corresponding position, i.e., ( k,h ). The operation is extended to cover all pixels, the resulting array being the local standard deviation

),( yxI

(4) Example:

Original image

Original image + white noise (gaussian)

comparable to that of the original signal

After filtering by application of the multiresolution

approach


0 1 2 3

0 1 1 1 1

1 5 0 0 0

2 5 0 0 0

3 5 0 0 0

4 5 0 0 0

5 5 0 0 0

0 1 2 3

0 1 1 1 1

1 5 0 0 0

2 5 0 0 0

... 5 0 0 0

14 5 0 0 0

15 5 0 0 0


0 1 2 3

0 1 1 1 1

1 0 0 0 0

2 0 0 0 0

3 5 5 5 5

... 5 5 5 5

15 5 5 5 5

Original


Original

0 1 2 3

0 1 1 1 1

1 5 0 0 0

2 5 5 5 5

3 5 5 5 5

4 5 5 5 5

5 5 5 5 5



FTs (Fourier Transforms):

- Transforms data from time to frequency domain

- Functions as superpositions of sin and cos

Non-localized

WFTs (Windowed Fourier Transforms):

- Signal is chopped into sections for separate analysis

- Windowing via weight functions

- Gives information both in time and frequency domain

Weight functions are translated

but window size remains constant, i.e.,

Time-widths are not adapted to frequency

Spatially

Localized

Frequency based methodsFrequency based methods

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data A wavelet.

Documents

pattern recognition

solar image data sirw

solar image data objective

solar image data t

wavelet domain

reconstructed image

wavelet planes

basic wavelet