Wavelet–Based Elemental Spectrum Analysis An Algorithm for Wavelet–Based Elemental Spectrum Analysis Bruce Kessler Western Kentucky University Approximation Theory XIII San Antonio, Texas March 10, 2010 *This work supported by a Kentucky Science and Engineering Grant, KSEF-1653-RDE-011.

Groundwork for the talk: This is a talk on the application of multiwavelets to a real-life problem in physics: the non-invasive deter-mination of the active/inactive status of unexploded ordnance (UXO). Algorithm is designed to remove the “opinion” of an analyst in determining this status. We hope to examine smaller ordnance than usual. The algorithm has wider application in the sciences. The algorithm is currently covered by provisional patent. We will be applying for a full patent over the next month.

Readings are additive – urea (CO(NH2)2) Background on the problem: The Applied Physics Institute at WKU has a neutron generator & detector set-up that they use to identify unknown compounds from a small distance. They count the gamma rays detected at different energy levels. Elements have distinct signatures: carbon silicon Readings are additive – urea (CO(NH2)2) nitrogen oxygen

the actual readings include a large distortion and some noise. The problem is . . . the actual readings include a large distortion and some noise. Some numerical techniques are used, but it is mostly subjective and depends upon an experienced operator.

We had been using a scaling vector developed by Donovan, Geronimo, and Hardin (1999) that generates a space that contains • orthogonal on [n, n+1] • boundary functions • wavelets easily constructed, also with boundary functions

There were issues when the peaks appeared at “corners”.

We now use a scaling vector developed by K (2006) that generates a space that contains • orthogonal on [n, n+1] • has boundary functions • wavelets easily constructed, also with boundary functions • now has a useful prefilter

The method: Step 1: We interpolate the given data with a C1 cubic spline and sample at or above the data rate uniformly over the interval in which our search elements should produce peaks. Step 2: Decompose the signal into d1, . . ., dk, and ck.

The method: Step 3: Do the exact same sampling and decomposition with each of the search element peaks, plus other peaks that belong to contaminants T1, . . ., Tn. Let dE denote the wavelet coefficients in the nth level decomposition of element E from the set {C, N, O, T1, . . ., Tn}. Step 4: Replace d with d*, and reconstruct the signal. We then find nonnegative to minimize the square-error of the reconstructed signal.

Example 1:

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Example 1 (zoom right):

Example 2:

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Example 2 (zoom right):

The method: Step 5: We have a library of (C, N, O) ratios for several known explosives. Treat each as a 3-vector and measure the angle between our resulting 3-vector and the library 3-vectors. If the angle from each of the explosives is sufficiently small, then we deem the UXO “active”. In 5 test-runs before the talk, the algorithm produced 4 right answers.

“And that’s all I have to say about that.” Next steps: • Thorough testing by the API guys (ROC curves, . . .) • Need to look at using longer neutron bombardment time I would like to acknowledge the help of the WKU Applied Physics Institute faculty and staff (Phil Womble, Alex Barzilov, and Joe Howard) for their help in understanding the problem and collecting the data. “And that’s all I have to say about that.” – Forrest Gump Thanks!