Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data A wavelet packets equalization technique to reveal the multiple spatial-scale nature of coronal structures Guillermo A. Stenborg The Catholic University of America & NASA Goddard Space Flight Center

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data Objective 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 ? Approach Selective contrast enhancement of boundaries and internal details of coronal features More reliable identification of coronal structures to help in the process of automatic recognition of coronal events

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

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data the first scales the higher (spatial) frequency componentsthe last ones the lower (spatial) frequency signatures 1) The technique consists in decomposing a given signal 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. 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 Technique

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 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 Wavelet Transform MW: B 3 -spline (1D)

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The 2D “à trous” algorithm Weight

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The 2D “à trous” algorithm Weight

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The 2D “à trous” algorithm Weight

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The 2D “à trous” algorithm Weight

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data Weight The 2D “à trous” algorithm

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The 2D “à trous” algorithm 1) Reconstruction of original image 4) Weighted Reconstruction: The Figure depicts the wavelet scales 1 to 4 (W i ) and the smoothed image (W o ), i.e., continuum, of I (x,y) = EIT Fe IX/X ( 171 Å) image =  =1  =[1,1,1,1] k=0 For comparison, continuum corresponding to decomposition based on 50 scales when 2D B 3 -spline : : 2) Local standard deviation of Noise at scale j Local standard deviation of noise in original image (first scale) Noise progression in wavelet space 3) Noise filtering: (hard thresholding as in Donoho & Johnstone, 1994)

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The 2D “à trous” algorithm Reconstruction Weight

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

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data The alternative: The Wavelet Packets -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).

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 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) 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) 2) 3)  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  In matrix form the weighting coefficients can be depicted as (for 2 levels): 4) Briefly, the first level decomposition of the given image in p 1 scales gives rise to the wavelet transform set {w i (0) }, i=1...p 1, i=0 corresponding to the continuum component. Afterward, further decomposition is applied to each wavelet plane.

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

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

 = 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 K = 0 EIT Fe XIV image reconstructed with:

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

1/5 LASCO-C2: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 2/5 LASCO-C2: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 3/5 LASCO-C2: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 4/5 LASCO-C2: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 5/5 LASCO-C2: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 1/2 LASCO-C3: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 2/2 LASCO-C3: April 21, 2002

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data /14 LASCO-C2 Level 0.5

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data LASCO-C :06 – :06 Original

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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 00:06 UT) Original Processed 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

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 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. Original Processed (Continuum component removed) Position Angle Radial distance (pxls) Time (hours since January 8, 2001 at 00:06 UT) Time (hours since January 8, 2001 at 00:06 UT)

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data Position Angle Radial distance (pxls) Time (hours since January 21, 2001 at 00:06 UT) Original Processed (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.

Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data 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 Radial distance (pxls) Angular distance

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

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Pattern Recognition Techniques Applied to Solar Image Data SIRW October 23-24, 2003 Pattern Recognition Techniques Applied to Solar Image Data - 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? 2D B 3 -spline

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

(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. (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 (4) Example: Original imageOriginal image + white noise (gaussian) comparable to that of the original signal After filtering by application of the multiresolution approach

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

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

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

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

 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 methods