УДК 620.179.16
MATCHING PURSUIT-BASED ADAPTIVE WAVELET PACKET ATOMIC DECOMPOSITION APPLIED IN ULTRASONIC
INSPECTION
Yang Guang, Zhang Qi, Que Pei-wen Institute of automatic Detection, Shanghai Jiaotong University Shanghai, 200240, P.R. China
АДАПТИВНАЯ ВЕЙВЛЕТ-ПАКЕТНАЯ ДЕКОМПОЗИЦИЯ, ОСНОВАННАЯ НА АЛГОРИТМЕ СОГЛАСОВАННОГО ПРЕСЛЕДОВАНИЯ И ЕЕ ПРИМЕНЕНИЕ В УЛЬТРАЗВУКОВОМ
КОНТРОЛЕ
Янь Гуанъ, Жень Ки, Ке Пей-вен Институт автоматического обнаружения, Шанхайский Джиатонь университет Шанхай, 200240, КНР
Новый метод частотно-временного анализа, представляющий собой вейвлет-пакет-ную "атомную" декомпозицию, основанную на алгоритме Matching Pursuit (MP согласованного преследования), предложен для улучшения обнаружения дефектов при ультразвуковом неразрушающем контроле. Алгоритм MP используется, чтобы разложить нелинейный и нестационарный ультразвуковой сигнал по заданным базисным функциям ("атомам") из полной ортонормированной системы базисных функций ("словаря"). Вейв-лет-пакетный "словарь" построен на функциях Добечи, которые похожи на наблюдаемые ультразвуковые эхосигналы. Подбираются адаптивно оптимальные "атомы", чтобы восстановить сигнал и получить образ, приближающийся к исходному сигналу и эффективно улучшить воспроизведение сигнала на фоне шума. Компьютерное моделирование и экспериментальные результаты подтвердили существенное улучшение отношения сигнал—шум для ультразвуковых эхосигналов.
Abstract. A novel method of time-frequency amalysis, wavelet packet atomic decomposition based on Matching Pursuit (MP) algorithm, is proposed to improve ultrasonic flaw detection in ultrasonic NDT. MP algorithm is used to decompose the nonlinear and non-stationary signal into given atoms in an over-complete wavelet packet dictionary. The wavelet packet dictionary with Daubechies wavelet function is selected, which well match the observed ultrasonic flaw echoes. Selecting adaptively the optimum atoms to reconstruct signal can obtain spares approximations of the original signal with the less complexity and efficiently improve signal corrupted by noise. Computer simulation and experimental results have verified distinct SNR enhancement for ultrasonic echo in flaw detection.
Key words: Matchning Pursuit; wavelet packet dictionary; atomic decomposition; denoising; ultrasonic; NDT.
I. INTRODUCTION
The ultrasonic pulse echoes reflected from crack, flaws or other defects contain a large amount of information of the reflectors. So, ultrasonic techniques have been widely utilized to detect damages and assure the performance of materials. In practice, the reflected ultrasonic signals are usually corrupted by outer or inner noise. The outer noise is mainly induced by detection equipments and the circuits. The grain or structure noise of tested materials is called inner noise. The noise embedded in useful signal is time-invatiant and contaminates the echoes from the flaw to be detected. Therefore, it is difficult to extract useful information directly from the measured echoes. Different noise restricts the detection quality and makes a fundamental limit on the accuracy of measurement.
Various signal processing techniques have been used for enhancing Signal-to-Noise Ratio (SNR) in ultrasonic flaw detection. The Split Spectrum Processing (SSP) and wavelet transform-based threshold method have been utilized to denoise ultrasonic signal. Time-frequency analysis by basis pursuit
(BP) is a recently applied technique for decomposing a signal into an optimal superposition of functions in an over-complete dictionary [1]. This decomposition method has been used to denoise ultrasonic echoes embedded in grain noise of highly scattering materials [1]. However, the high computational cost is its main drawback.
Matching Pursuit (MP) algorithm [2] decomposes any given signal into a set of waveforms or atoms each providing compact time-frequency description of original signal according to its correlative frequency, time and amplitude properties. This algorithm has been applied for ultrasonic signal denoising, compression and approximation [3—6]. Jin-Chul Hong [4, 5] adopted a two-stage MP strategy based on Gabor dicrionary and Chirp function dictionary to extract useful waves out of noisy signal in guide-wave damage inspection, respectively. Ramazan Demirli [6] presented a TF representation for ultrasonis signal based on MP algorithm by enabling the addition of the TF distribution of composing Gabor functions. Ruiz-Reyes [3] utilized the discrete Morlet function as dictionary elements by MP technique to improve SNR in ultrasonic NDT.
In this paper, a novel MP-based method for SNR enhancement in ultrasonic flaw detection is proposed with the less computational complexity than BP algorithm. The proposed method adopts wavelet packet dictionary with Daubechies function, the over-complete dictionary composed of atoms that are well localized in time-frequency plane and properly mimic the flaw waves. So, wavelet packet atoms are suitable to detect non-stationary ultrasonic signals buried in a noisy background. The proposed method has been successfully tested by considerable computer simulation and experimental noisy signals, even when SNR is rather low.
II. MATCHING PURSUIT ALGORITHM
MP is an iterative algorithm introduced by Mallat and Zhang [2]. Let H be a Hilbert space. The over-complete dicrionary T is defined as a family T = i = 0, 1, ..., L) of functions in H, such asllgJI = 1, where L is the dicrionary number of elements. MP offers a suboptimal solution for decomposing a signal x(ri) in terms of unit-norm expansion vectors g.fji) chosen from a given dictionary, where P norm is used as the approximation metric due to its mathematical convenience. When a well-designed dictionary is adopted by MP, the analyzed signals are able to be compactly and adaptively decomposed.
At each step of iteration, the atom g.(n) owning the largest inner product with analyzed signal is chosen. The contribution of this atom is then subtracted from the signal, and the process is repeated on the residual. At the mth iterationm the residue is
{x\n\ m = 0
mf 1 r 1 -t- r\> (1)
where a.(m) is the weight associated to optimum element gi(m)["] at the mth iteration.
The weight a™ assosiated to each atom at the mth iteration is computed in terms of the inner products with the residual г™[п\:
= <&■[/.],r>]> = <&[nlr-[n]> ^ w я[ ] (2)
<*[»UM> На ИII
6 Дефектоскопия, № 1. 2007
At the mth iteration, the optimum atom can be performed as:
rn] = argmin ||/-m+1[«] J" = argmax ||a,m||" =argmax ||af||. (3)
g/eT g/eT p/er
The correlation updating procedure [2] is obtained as follows:
[«], g,[«]> = g,[n]> - aKm)<g.[n], gim[n]>. (4)
One dictionary Y has been determined, the correlations <g,[/i], gi(m)[/i]> can be pre-calculated and stored. Therefore, it is only required to compute <g.[n], at the first iteration, according to (4).
III. WAVELET PACKET DICTIONARY
If a dictionary T is defined as a family of parameterised waveforms \j/, (\j/, yeT), a collection of the wavelet atoms is a wavelet dictionary [7]. A wavelet packet dictionary is a family of orthonormal bases composed of elements well localized in time-frequency. For signals of N samples, each vector of a wavelet packet dictionary (a wavelet packet aton ij/y) is indexed by y = (/, P, k), with 0 < j < \og2(N), 0 <p< 2~'N, 0<k< 2J. Such an atom has similar time-frequency localization properties as a discrete window function, dilated by 1>, centered at 2!(p + 1/2), and modulated by a sinusoidal wave of frequency 2n2~J(k + 1/2) [2].
Presently, various orthogonal wavelet functions have been available in the wavelet packet dictionary, including the family of Haar, Daubechies, coiflets, and symlets [7]. If the choosing atoms are very similar to the echoes coming from the measured flaw, MP algorithm can efficiently extract flaw echoes from the noisy signals. On the other hand, the complexity needs to be relatively simple in terms of its on the computational cost.
In this paper, Daubechies wavelet function is selected, owing to the similarity of its waveforms to ultrasonic flaw echo. The Daubechies family is composed of diverse wavelet functions, Daubechies 10 appearing to most closely match flaw echo. Therefore, Daubechies 10 is chosen to generate the quadrature mirror filters in wavelet packet decomposition.
A wavelet packet dictionary with Daubechies 10 is well-fitted to analyze ultrasonic flaw signal due to the following reasons: (1) it owns time — frequency properties and can capture time frequency atoms with high-energy levels. (2) The atoms in the wavelet packet dictionary can well mimic ultrasonic waveform. The more similar the wavelet packet atoms are to the echoes, the less the number of coefficients is needed to approximate the flaw signal [7]. (3) A redundant wavelet packet dictionary is able to represent signal compactly and offers the freedom to describe a signal in various ways.
IV. ADAPTIVE WAVELET PACKET ATOMIC DECOMPOSITION BY MP
In the ultrasonic NDT, the measured ultrasonic signal x{t) can be expressed as the sum of two components [1]:
x(t)=y(t) + n(t). (5)
Here, y(t) is the reflected signal of the detected flaw and buried in the noise n(t). Noise elimination is to obtain a signal x'(t) as close as possible to y(t), thus minimizing the effect of n(t).
According to the principle of atomic decomposition, a signal can be decomposed into elementary atoms (waveforms) and be sparsely approximated with atoms when a dictionary composed of vectors adapted to the time — frequency behavior of the signal is defined [8]. Therefore, the ultrasonic signal is able to be
decomposed as a linear expansion on the basis of vectors g. chosen from a given dictionary. Using MP algorithm, wavelet packet atomic decomposition can describe the observed ultrasonic signal x(t) with adaptive approximation:
m
x(i)=(6) 1=1
Here, y is an index of an over-complete dictionary T, which is the wavelet packet dictionary with Dau
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