A “Peak” at the Algorithm Behind “Peaklet Analysis” Software Bruce Kessler Western Kentucky University for KY MAA Annual Meeting March 26, 2011 * This work was partially supported by KSEF grant KSEF-1653-RDE-011, and is currently supported by KCF grant COMMFUND-1280-RFP-011. * Patent pending on the algorithm, application #12/756,991.

Wavelet Basics Wavelets are signal processing and analysis tools, analogous to trigonometric polynomials in the Fourier transform. Wavelet bases are local, with coefficients representing different translates of the basis. Thus, coefficients reflect not just if a feature occurs, but give some idea of where a feature occurs. Wavelet transforms ignore polynomial components of the signal up to the approximation order of the basis. There are a wide selection of wavelet bases that can be used, depending on the needs or application of the user. Multiwavelets can be simultaneously orthogonal (quick decompositions) and have symmetry properties. There is only one single wavelet construction that can have both.

Wavelet Basics

Then we may decompose a parent space into a smoother space and all of the included wavelet spaces. This type of decomposition has numerous applications in denoising, edge detection, data/image compression, etc., and now spectral analysis!

Example with r = 1 (Haar) What is approximation order? If a scaling vector  has approximation order k, then all of the spaces V j generated by  will contain polynomials of degree k – 1. Approximation order 1 basis – just showed it to you.

Approximation order 2 bases (Hardin, Geronimo, Massopust) orthogonal to integer translates Another approximation order 2 basis (Daubechies) Also orthogonal to its integer translates, but not when restricted to a shorter interval with integer endpoints.

Approximation order 3 bases (Donovan, Hardin, Massopust)

Approximation order 4 bases (DHM)

Approximation order 4 bases (with one derivative, K.)

When to use (multi)wavelets – Quantitative data. Background noise. Wavelet decompositions ignore signal components up to the polynomial approximation order of the basis, so a wavelet analysis to look for patterns over the top of background noise. Random noise. Gaussian noise appears as wavelet coefficients that are very close to 0, so they have little effect (and can be threshholded out if we desire). Non-sparse data. Multiwavelets allow us to build boundary basis functions, allowing us to ignore data outside our analysis window.

Background on the problem: There are several different ways of interrogating material that generate counts at different energy- levels or frequencies, with peak locations indicative of different elements.

The problem is... the actual readings never look that clean. Background subtraction is typical. Some other numerical techniques may be used, but results are sometimes subjective.

Wavelet-Based Algorithm  The user inputs peak information (center, standard deviation) for each element for which they are searching, along with other peaks in the same horizontal range. The user has the option of combining area under multiple peaks into one combined measure, and of setting relative areas.  The spectral data is input by the user. The peak centers may be adjusted slightly, within a user-set tolerance, to match the apparent calibration of the spectral data.  Wavelet decompositions are generated, to a level specified by the user, for each of the user-defined Gaussian peaks, using the adjusted centers.

Wavelet-Based Algorithm  The wavelet decomposition of the spectral data is calculated to the user-defined level, and the wavelets are replaced with a variable linear combination of the wavelets from the elemental decompositions. The spectral data is then reconstructed.  The variables are set to amounts that give a least- squares fit to the original data.  (Optional) The user may input a library of elemental ratios for substances for which they are searching, and an angle will be calculated from the fit vector and each library vector, or combination of vectors.

Example 1:

 If relative heights are given, then the n-vector of amounts is compared to the n-vectors of substances in a user- input library.

Example 1:

Next steps: The algorithm is partially coded into JAVA, but we may be starting over in a partnership with a software company, to provide a “try-before-you-buy” webpage. We have our first potential customers waiting to test the algorithm on their data. “And that’s all I have to say about that.” – Forrest Gump Thanks! works.bepress.com/bruce_kessler Mission Critical: I would like to thank you for your kind attention today. Licensing? Nobel?