Reconstructing the Deflector Images From the Ultrasonic Echo Signals by the Maximum Entropy Method

This article proposes using the maximum entropy method (MEM) for processing the ultrasonic echo signals in order to restore the deflectors images with high signal/noise ratio and small lateral petals of the point scattering function. When processing the ultrasonic echo signals, it is possible to take into consideration the pulses propagation paths, together with both their deflections from uneven boundary of inspected object and the respective wave type transformation. In model experiments the images of deflectors were received in two registering modes: in the combined mode by using the conventional single element probe; in double and triple scan mode by using the antenna array. The deflection from uneven surface during the echo signals acquisition had been taken into account in both modes. The restored images had a resolution exceeding the one required by Rayleigh criterion. MEM allows obtaining the defect images of low-level lateral petals by processing less than 10 % of echo signals full set.


Introduction
Getting the information on the internal structure of industrial facilities is an actual problem belonging to a class of problems inverse to that one of scattering, which is to determine the quantitative characteristics of unknown flaws on the basis of observing the scattered irradiating field.The most important problem of nondestructive testing is the found deflectors classifying and determining their sizes.This information may be used by experts on strength calculations to assess the operating life of the inspected objects.
In practice, coherent systems of ultrasonic nondestructive testing are being widely used.These systems register the echo signals by means of several single-plate probes moving over the surface of the inspected object.To restore the flaw images from the acquired echo signals, the algorithms based on the Born approximated solution of reverse scalar problem of scattering are normally used.Here are some of them: Synthetic Aperture Focusing Technique (SAFT) method (Hall, Doctor, Reid, Littlefield, & Gilbert, 1987;Erhard, Lucht, Schulz, Montag, Wüstenberg, & Beine, 2000;Osetrov, 1991), the method of angular spectrums (Goodman, 1968) and the method of Projection in the Spectral Space (PSS) (Mayer, Markelein, Langenberg, & Kreutter, 1990;Badalyan & Bazulin, 1988).Practical realization of the last method is especially effective in 3D-option due to high speed of getting the image.SAFT method provides images of alike quality.But in this case data processing takes place not in a spectral space with its fast Fourier transformation, but in time sphere and, therefore, takes more time.However, SAFT method enables to consider the multiple deflections of ultrasonic pulse from uneven boundary of the inspected object and, in the same time, the effect of wave type transformation, and thus has a very important advantage over PSS method, especially today that rapid development of computer engineering allows restoring deflector images with a frequency of more than 10 Hz.
Recently the equipment exploiting the antenna arrays (AA), or matrixes, for emitting and receiving the ultrasound has become actively used in practice of ultrasonic testing.As a rule these devices operate in the mode of phased antenna arrays (PA) providing instrumental (hardware) focusing the ultrasound due to multichannel emitting and receiving the pulses with timed delays.However, PA-flaw detectors have a number of disadvantages (Bazulin, 2013), one of them being the defocusing and the shifting of deflectors glares from their actual locations proportionally to the distance from the focusing line.The increasing the antenna array or matrix aperture leads, on the one hand, to the improving of the image quality in the focusing area, and, on the other hand, to decreasing of dimensions of this very area.This problem is particularly acute in inspecting the thick-walled objects with help of 32-element PA (Braconnier, Okuda, & Dao, 2009).A technology of dynamic depth focusing (DDF) (Olympus, 2007) was developed in order to partially eliminate such a negative effect.
One of the ways to obtain the deflectors images of higher quality is to initially receive the echo signals by antenna array operating in the dual scan mode (Bazulin, Golubev, & Kokolev, 2009) in which the echo signals are being registrated by all pairs of array elements.In the article (Chatillon, Fidahoussen, Iakovleva, & Calmon, 2009) such a mode of echo signals acquisition is called the Full Matrix Capture (FMC).As the dimensions of antenna matrix elements are commensurable with the wave length, each element forms a field of wide directional diagram (Danilov, Samokrutov, & Lyutkevich, 2003) that enables acquisition of lots of pulses at longitudinal and transverse waves running from the deflector either directly or after they have been deflected by the borders of the inspected object.At the second stage the deflectors images are being restored by combined SAFT (C-SAFT) (Kovalev, Kozlov, Samokrutov, Shevaldykin, & Yakovlev, 1990) method using the echo signals measured in dual scan mode.In article (Bazulin, 2011) this method, designated as TS-M-C-SAFT, is generalized for cases of multiple deflections from uneven boundary of the inspected object; the method also takes into account the transformation of wave type, it is applicable to the mode of triple scan when antenna array operating in the mode of dual scan moves along the inspected object.The C-SAFT method is also referred to as Total Focusing Method (TFM) (Holmes, Drinkwater, & Wilcox, 2005).
The dark side of the simplicity of the algorithms being used is that the quality of flaws images is not always sufficiently high.They oftenly have many glares formed by re-scattered pulses and by pulses appeared in result of wave type transformation during the re-scattering on heterogeneities.Besides this, it is not always possible to make definite conclusion on the deflector's form, as it is the restored image of only the part of the deflector's edge which actually had reflected the pulses received in acquisition area.Therefore, there must, with not claim of completeness, be mentioned the existing methods of getting scatterers images based on a more precise solution of the inverse problem of scattering.Here are some of them: the MUltiple SIgnal Classification (MUSIC) (Devaney, 2000), the iterative algorithm for solving the inverse boundary problem of ultrasound scattering on the cavity in an isotropic solid body (Buro & Prudnikova, 1999), Novikov-Henkin algorithm (Novikov, & Henkin,1987;Novikov, 1986;Burov, Vecherin, Morozov, & Rumyantsev, 2010), methods based on the acquisition of echo signals appearing due to nonlinear effects of second and third order (Burov, Gurinovitch, Rudenko, & Tagunov, 1994;Burov & Shmelev, 2009).It is worth mentioning a very promising direction for solving inverse problems, which becomes in more and more demand as far as the computing power grows.The solution of the inverse coefficient problem for the scalar wave equation (Goncharsky & Romanov, 2011) is based on the possibility of the direct calculation of the gradient of the function closure via the solution of the so-called conjugate problem for partial derivatives equation by using the finite difference method.By turning to the vector version of the wave equation, it seems possible that all effects of propagation and scattering of sound would be taken into account.The appearance of run-round impulse, multiple re-scattering and wave type transformation of waves types will give information, which, at unilateral access, will allow to restore the field of density and elastic coefficients values in the inspected object, by which it will be possible to determine not only the entire boundary of the deflector, but also the properties of its material.The algorithms mentioned are quite complex in practical implementation, but, with the increasing of computing power, they will be actively used in the systems of mass ultrasound expert testing.
There is a number of mathematical methods not associated with solving acoustic inverse problem but enabling to improve the quality of deflectors images.Thus, to produce images of super resolution (exceeding the Rayleigh limit), the mathematical algorithms based on the extrapolation of temporal and spatial spectra of signals are used.For example, there is the iterative Gershberg-Papoulis algorithm (Gershberg, 1974;Lasaygues & Lefebvre, 1998;Papoulis, & Chamras, 1979;Bazulin, 1991;Pickalov, & Kazantsev, 2008), or algorithm of signal spectrum extrapolation of echo based on building its AR-model (Marple, 1987;Faur, Morisseau, & Poradis, 1998;Box, Jenkins, & Reinsel, 1994); Bazulin, 1993).Gershberh-Papoulis algorithm is used for both echo signals spectra extrapolating, which leads to an increase in the longitudinal image resolution, and for extrapolating the complex spectrum of the image, which leads to the both longitudinal and transverse resolution enhance.As the use of the Gershberg-Papoulis (Papoulis, & Chamras, 1979) method assumes the use of image-cutting-off operation at about 30 % of the maximum value, this leads to a loss of information on the small amplitude scatterers.Extrapolation of echo signal spectrum in course of its the constructing of its AR-model leads to an increasing mainly the longitudinal, but not transverse resolution of the restored image.In article (Wan, Balasundar, & Mandayam, 2003) a method for processing the echo signals using wavelet technology to achieve super-resolution is offered.The mentioned methods belong to the field of signal and image processing without taking into account specific features of nondestructive ultrasonic testing.
The maximum entropy method (MEM) is an outstanding one among the methods of solving inverse problems.In article (Freiden, 1972) the possibility is shown of achieving super-resolution in the images obtaining system using entropy as a stabilizing factor in the Tikhonov regularization method context (Tikhonov & Arsenin, 1986).Investigations have confirmed the effectiveness of MEM practical application in restoring images tomography, radio astronomy (Wernecke, & D`Addario, 1977;Baykova, 2008), nuclear magnetic resonance (NMR) (Hore, 1991), as well as in ultrasonic testing (Battle, 1999;Bazulin, & Bazulin, 2005).In article (Bazulin, 2010) MEM is used for solving the inverse scattering problem in single-dimensional option selected in order to account for re-scattering pulses at pointlike discontinuities.This article explores the possibility of using MEM in ultrasonic nondestructive testing for image reconstructing from the measured set of deflectors echo signals, considering their deflecting from uneven boundaries of the inspected object.In this paper, the second section is proposed to use MEM for reconstruction the image reflectors during of ultrasonic testing.In the third section describes the method of calculation of the ultrasonic field from a pointlike scatterer with account of multiple reflected pulse from the boundaries of the inspected object.The fourth section presents the results of the application of MEM to reconstruct the image reflectors and for compression of complex signals.

Maximum Entropy Method (MEM)
The solving the inverse scattering problem means finding the function describing the deflective properties of inhomogeneity within the area, by using both the known field sources located within the area and the scattered field measured over the area.This reverse problem is nonlinear one (Gorjunov & Saskovets, 1989), since, besides the unknown parameters of inhomogeneity, the unknown total field in the entire region of interest (ROI) has to be determined.In the practice of ultrasonic testing the Born approximation is commonly used where the scattered field amplitude is assumed much smaller than that one of the incident field, i.e.
, which, from a strictly mathematical point of view, is oftenly not true.
Let the solution of the direct problem, i.e. the calculating the scattered field by the known and functions be formally written down as follows , (1) -the function describing the deflective properties, -the known field sources, -the scattered field.As the direct problem is linear or, as in the case of the Born approximation, can be linearized, the equation (1) can be written in the following matrix form: where matrix describes the propagation of ultrasonic waves from their sources belonging to area to the pointlike deflector and then to the acquisition area, the vector being the noise of measurements.Since matrix is usually poor-conditioned, there are, besides simple matrix inversion, some other solution options.One of them is finding an estimate as the solution of unconstrained optimization, a criterion of the quality of the reconstructed image being the square of discrepancy solution (3) where symbol denotes the transpose operation.Estimate can be written as (4) The solution of the inverse problem in the form of ( 4) is called the method of least squares (Helstrom, 1967).To solve the problem of finding out the (4) minimum with the maximum speed, it seems better to use the second-order methods (Letova & Panteleev, 1998).Their application requires a knowledge of the gradient and Hessian of the formula (3), which is calculated by the following formula , ( 5) In terms of article (Tikhonov & Arsenin, 1986), the solving the degenerated system of linear algebraic equations (4) with respect to , provides a minimum 2 χ discrepancy and is called pseudosolution.There may be the infinite number of pseudosolutions, and their parameters such as resolution, speckle noise level and etc. in general case may appear to be far from ideal.In article (Lingvall, Olofsson, Wennerström, & Stepinski, 2004) the LSM method was applied to restore the images of the scatterers by their echo signals in the numerical experiment, and that was called Extended Synthetic Aperture Focusing Technique (ESAFT).
The clear regularized solution of equation ( 4) does exist, the one valid for the case of non-square matrixes being , where is the diagonal matrix, is spectral density of the noise energy.This formula is structurally equivalent to the formula describing Wiener filter (Vasilenko, 1986) in the frequency domain.Using (7) allows to obtain high-quality images, but in cases of low level of noise.In matrix form the evaluation of function can be represented analogously with a correlation formula . (8) The inverse problem is considered incorrect in the Hadamard sense (Tikhonov & Arsenin, 1986), unless all of the three conditions are not fulfilled: for any element there is a solution , the solution is determined unequivocally and the problem is stable within the spaces .Unfortunately, the problem ( 2) is incorrect (Devaney, 2000).To solve alike problems, Tikhonov (Tikhonov & Arsenin, 1986) has developed a method of regularization which justifies the replacement of the problem in the form (2) by the optimization problem stable to small changes in input data (9) where is the square of solution discrepancies in the metric defined by the specifics of the problem, is a stabilizing function.The sense of using the stabilizing functions lays in possibility to consider any a priori information about the solution and thereby to narrow the area of searching for solutions.
The entropy of function can be used as the stabilizing function .Initially statistical physics concept of entropy, with symbol especially assigned for it, was introduced in 1865 by R. Clausius to denote a function describing the state of a thermodynamic system.In 1948, Shannon (Shannon, 1984) used the notion of entropy to assess the information volume of messages consisting of a finite set of characters of a certain alphabet.The text message of finite length contains the maximum amount of information, provided any sign of the message is followed by any alphabet character in arbitrary way and with equal probability.From a combinatorial point of view, such a message allows to create the maximum number of different character combinations with a limited set of alphabetic symbols.From the point of view of increasing the images resolution, using the entropy function as regularizing one allows to transfer the search for solutions on a set of images, in which the count with any value may be with any count, that is, the limitation on the solution front steepness is removed.Thus, restoring the image by linear method using the formula (4), the resolution remains practically unchanged, i.e. letter «z» will never be followed by letter «x», but after regularization the front steepness can rise -this means that next to the letter «x» there may appear any letter of the alphabet.In case the entropy is applied as a stabilizing function, formula (9) can be rewritten as , where is the entropy of a set of discrete independent random variables.This is defined for the case of real, non-negative as where is the number of points in the reconstructed image along the axis and axis .In practice, the so-called cross-entropy (Kullback, 19668)   Suppose that for the emitting and receiving the ultrasonic waves an antenna array mounted on a prism is used which has a tilt angle and is made of the material featuring velocities of longitudinal and transverse waves and a density of (Figure 1).The antenna array consists of elements of size located at a distance from each other.Let us designate sound velocity and density of the inspected object as .The location of transmitters and receivers relative to the center of the prism front edge are described by and vectors shown in Figure 1 with blue arrows.The prism front edge is shifted from the center of the coordinate system by the value of .The antenna array with a single element can be regarded as a single element transducer.
To determine the elements of matrix in the formula (2), it is necessary to calculate the field in the area of acquisition for the case when the pointlike deflector is placed at any arbitrary point of ROI and the transmitter is put in area, ROI being shown with pink rectangle in Figure 1.
The trajectory of pulse movement in case of emission (shown in Figure 1 with red arrows) can be described as a sequence of vectors , where is the number of deflections from the boundaries of the specimen, in case of acquisition -as a sequence , where is the number of deflections from the boundaries of the specimen in course of acquisition (in Figure 1 the trajectory is shown with green arrows).The first vector of such a sequence always corresponds to the path of sound propagation in the prism.In case of emission the velocity of pulse propagation long the path is defined by the list , and in case of acquisition -by list .The first element in these lists is in all cases -the longitudinal wave velocity in the prism.The remaining elements can take a value of either longitudinal or transverse velocity of sound in the inspected object, thus allowing taking into account the effect of the transformation of the wave type once it is reflected from the boundary.The minimally possible rates list for the case or looks like or and corresponds to the option of working with a transverse wave in a direct beam.
For describing the acoustic schemes in which the reflection of ultrasonic pulses from both the bottom and surface of the inspected object take place, we will use the following designations: such events as surface refraction and surface deflection will be designated by letter T, and the deflection from the bottom will be denoted by letter B. Type of waves after the event is occurred will be denoted by the letters L (longitudinal) and S (shearing).Expression T(S)-T(L) denotes a common direct beam on a transverse wave in case of emission and the direct beam but on the longitudinal wave ( , ) in case of acquisition.Acoustic scheme TB(LL)-TBT(SSS) describes a situation where, in case of emission, only the radiation beams once reflected on the longitudinal wave are considered, and, in case of acquisition -only the ones doubly reflected on the transverse wave ( , ); acoustic scheme T(L)-TB(SL) corresponds to that one called «self tandem» (Ermolov & Lange, 2004) ( , ).The beam shown with the solid line corresponds to transverse wave propagation, and the dashed line longitudinal wave propagation.Figure 1 represents a scheme of the rays propagation in correspondence with acoustic scheme T(*)-TB(**).
To build either a function for reflections from the inspected object boundary at emission or function for deflections at the acquisition, one may use the geometrical optics approximation (Kravtsov & Orlov, 1980;Gengembre, 2003).Since the sizes of the antenna elements are comparable to the wave length, they may be considered to be the pointlike emitters -receivers with the directivity diagram in prism, where is beam angle to the normal of piezoelectric element of the emitter or receiver.Having taken into account these approximations, the functions and can be written as   minimum period of time.This variational approach makes it easy to take into account effects such as multiple reflections from uneven boundary of the inspected object and transformation of wave types in course of reflecting and refracting at the boundaries of different environments.However, it is usually implemented by means of an iterative procedure, which slows down the calculations rate.
To reduce the amount of echo signals to be processed, let us introduce the notion of switching matrix of dimensions.The expression means that it is the antenna array element numbering that is radiating, while the element numbering is receiving.For reducing the volume of echo can be measured only the upper (lower) triangle of matrix (Samokrutov & Shevaldykin, 2011).The characteristic form of the switching matrix for 32-element antenna array at a random choice of 70 pairs is shown in Figure 3.

Model experiments
It should be noted that the more accurately echo signals are calculated by a given estimate of the function , the better convergents the solution of the problem (9).Errors caused by inaccuracies of matrix calculation by formula ( 14) are called an operating noise.In this paper for calculating the matrix a simplified version of formula ( 14) was used in which the reference signal of constant amplitude was shifted for the time , and for acoustic schemes with an odd number of deflections T(S)-TB(SS) and TB(SS)-TBT( SSS) the reference signal was assumed to be equal to (Samokrutov & Shevaldykin, 2011).

Specimen With Six Inclined Saw Cuts (Frontal Resolution for Inclined Probe Testing)
To register echo signals from the tops of six crack models each of 15 mm height, 0.05 mm opening and inclined at 60º angle within a duralumin block, there was used a sloping single element probe operating on the longitudinal wave, with beam width of about 40 degrees and the central frequency of 2.5 MHz.The scan area is shown schematically in Figure 4 with several probe images.

Figure 3 .
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