Decomposition Of The Time Reversal Operator Method Research Articles

A time-reversal imaging system for buried objects in layered media using complex images Green’s functions

In this paper, the three-dimensional problem of detection and localization of buried metallic objects inside a multilayered medium is studied. Decomposition of the time-reversal operator (DORT) is used as the inversion algorithm in the imaging method. In this approach, one has to analyze the time-reversal (TR) operator obtained by using the multistatic data matrix (MDM) of a TR array. To attain MDM in the forward problem, full-wave simulation results from a commercial simulation software are used. For synthetic retransmission of signals in this inverse problem, determination of the appropriate medium Green’s function is required. Here, for the first time the complex images technique is employed in calculating the required Green’s functions. The complex images technique gives a closed-form Green's function, instead of evaluating time-consuming numerical integrals resulting in a fast and accurate Green’s function calculation. Using singular value decomposition of the MDM at a frequency range of 1–3 GHz and synthesized DORT method, images are constructed. Numerical results show the accuracy of the proposed approach in the detection and localization of buried objects. Furthermore, noise effect and sensitivity analysis on layers’ permittivity values are studied. The results confirm the robustness of the method against noise and uncertain layers parameters.

Nonlinear Lamb wave based DORT method for detection of fatigue cracks

This research reports a selective detecting and focusing approach for fatigue cracks through use of decomposition of the time reversal operator (DORT) in combination with nonlinear Lamb waves. In the proposed approach, the time reversal operator (TRO) is only decomposed at the second-harmonic frequency, which has been widely utilized in recent years to evaluate micro-damage, rather than the fundamental frequency in the traditional DORT method. Further, an imaging algorithm based on the total focus method is introduced to locate nonlinear scatterers. The proposed approach is experimentally investigated to locate two fatigue cracks with different fatigue cycles in an aluminum plate with a set of surface bounded piezoelectric (PZT) transducers. The results show that the significant eigenvalues are related to the fatigue cycles of fatigue cracks when the TRO is decomposed at the second-harmonic frequency, and the number of significant eigenvalues is exactly equal to the number of fatigue cracks. The two fatigue cracks are selectively located accurately and are also distinguished from linear scatterer by use of the proposed method.

Ultrasonic imaging of defects in coarse-grained steels with the decomposition of the time reversal operator.

In the present work, the Synthetic Transmit Aperture (STA) imaging is combined with the Decomposition of the Time Reversal Operator (DORT) method to image a coarse grained austenitic-ferritic steel using a contact transducer array. The highly heterogeneous structure of this material produces a strong scattering noise in ultrasound images. Furthermore, the surface waves guided along the array interfere with the bulk waves backscattered by defects. In order to overcome these problems, the DORT method is applied before calculating images with the STA algorithm. The method consists in analyzing in the frequency domain the singular values and singular vectors of the full array transfer matrix. This paper first presents an analysis of the singular values of different waves contained in the data acquisition, which facilitates the identification of the subspace associated with the surface guided waves for filtering operations. Then, a filtered matrix is defined where the contribution of structural noise and guided waves are reduced. Finally, in the time domain, the STA algorithm is applied to this matrix in order to calculate an image with reduced structural noise. Experiments demonstrate that this filtering improves the signal-to-noise ratio by more than 12 dB in comparison with the STA image before filtering.

An efficient technique based on DORT method to locate multiple soft faults in wiring networks

Decomposition of the time reversal operator (DORT) recently adapted to wiring fault detection and location presented effectual results when dealing with a single soft fault along with complex network configurations. On the other hand, it failed to handle the task of locating multiple faults within even simple ones. In this article, we propose an enhanced version of the standard DORT technique (EDORT) based on a complementary procedure that enables the accurate and selective location of multiple soft faults in various wiring systems.

Lamb wave-based subwavelength damage imaging using the DORT-MUSIC technique in metallic plates

A Lamb wave-based, subwavelength imaging algorithm is developed for damage imaging in a metallic plate based on a decomposition of the time-reversal operator method together with a multiple signal classification imaging condition in the space-frequency domain. In this study, a hybrid non-contact inspection system was proposed to image damage in an aluminum plate using a piezoelectric linear array for actuation and a laser Doppler vibrometer line-scan perpendicular to the piezoelectric array for sensing. The physics of incident waves, reflection, and reflected waves that underlie the transfer matrix in the decomposition of the time-reversal operator method is mathematically formulated in the context of guided waves based on the first-order Born approximation. Singular value decomposition is then employed to decompose the experimentally measured transfer matrix into three matrices, detailing the incident wave propagation from the linear actuator array, reflection from the damage, and followed by reflected waves toward the linear sensing array for each small damage. The singular value decomposition and multiple signal classification imaging condition enable providing insight on both the damage “reflectivity” and detecting damage smaller than a wavelength (subwavelength). With the flexibility of this inspection system, a considerably large area can be imaged using lower frequency Lamb waves with rapid line-scans. The experimental results showed that the hardware system with a signal processing tools such as the decomposition of the time-reversal operator with multiple signal classification (time-reversal with multiple signal classification) imaging technique can provide robust, highly accurate imaging results as well as providing damage reflectivity estimation with unknown material properties.

A comparison of methods for focusing the field of a HIFU array transducer through human ribs

A forward model, which predicts the scattering by human ribs of a multi-element high-intensity focused ultrasound transducer, was used to investigate the efficacy of a range of focusing approaches described in the literature. This forward model is based on the boundary element method and was described by Gélat et al (2011 Phys. Med. Biol. 56 5553–81; 2012 Phys. Med. Biol. 57 8471–97). The model has since been improved and features a complex surface impedance condition at the surface of the ribs. The inverse problem of focusing through the ribs was implemented on six transducer array-rib topologies and five methods of focusing were investigated, including spherical focusing, binarized apodization based on geometric ray tracing, phase conjugation and the decomposition of the time-reversal operator method. The excitation frequency was 1 MHz and the array was of spherical-section type. Both human and idealized rib topologies were considered. The merit of each method of focusing was examined. It was concluded that the constrained optimization approach offers greater potential than the other focusing methods in terms of maximizing the ratio of acoustic pressure magnitudes at the focus to those on the surface of the ribs whilst taking full advantage of the dynamic range of the phased array.

Time-reversal ESPRIT imaging method for the detection of single target

Estimation of signal parameters via rotational invariance techniques (ESPRIT) is employed to time-reversal (TR) imaging for single target in this paper. By the precondition of ESPRIT, the signal subspace is divided into two parts, which are connected with each other by a transfer matrix. This transfer matrix can be used to get the TR-ESPRIT imaging pseudo-spectrum. Two examples corresponding to the noise-free and noise case are considered. The last simulated results show that the proposed TR-ESPRIT method has higher resolution than the conventional decomposition of the time-reversal operator method, and it can keep the efficiency of time reversal operator-based imaging by the signal subspace.

Lamb wave damage detection using time reversal DORT method

Among structural health monitoring methods, Lamb waves are frequently used as a diagnostic tool to detect damage in plate-like structures. The decomposition of the time reversal operator (DORT) method is a selective detection and focusing technique using an array of transmit–receive transducers. Unlike body waves, the propagation of Lamb waves is complicated due to their dispersive and multimode characteristics. So it is hard to locate different scatterers on the plate directly using the DORT method because of the asymmetry of the time reversal transfer matrix. In the present paper a damage detection technique to make the transfer matrix symmetric for the Lamb wave DORT method is proposed. It is found that the number of significant eigenvalues is not exactly equal to the number of damage targets and two significant eigenvalues are associated with one damage target. An experiment is carried out to distinguish and locate two simulated points of damage on an aluminum plate. The behavior of reflected multimode waves is analyzed by studying the eigenvalues and eigenvectors of the time reversal operator matrix. This method, which uses a PZT array and operates under multimode pulse-echo mode, can estimate the position of the damage zone in the 2D image by numerically backpropagating a selective eigenvector using the A0 and/or S0 Lamb wave propagation analytical solution. The experimental results illustrate the proposed method.

Characterization of circumferential guided waves in a cylindrical cortical bone-mimicking phantom

The femoral neck cortical shell was recently demonstrated to act like a waveguide for circumferential waves. Femoral neck assessment with ultrasound could be enhanced by guided waves measurement. In this study, the decomposition of the time reversal operator (DORT) method is used to measure the phase velocities of circumferential guided modes in a circular tube with dimensions characteristic of femoral neck. The tube is made of a bone-mimicking material. Five guided modes are obtained and compared to theoretical predictions. The work substantiates the feasibility of measuring guided waves in a relatively thick tube of attenuating material with the DORT method.

Time reversal DORT method applied to nonlinear elastic wave scattering

The decomposition of the time reversal operator (DORT) method is a selective detection and focusing technique widely used in acoustics. The background theory is based on the time reversal (TR) where a signal collected from an array of transducers is time reversed and then back-propagated into the medium to focus on selected targets. The DORT method was developed to detect damages and inhomogeneities which induce linear changes of the elastic moduli. However, material may experience some type of damage (cracks, voids) which may affect the nonlinear elastic wave propagation behaviour with a small undetectable changes of the linear elastic moduli. This paper presents an extension of this method for the detection of multiple linear and nonlinear scatterers. In the presence of nonlinear elastic scatterers, forcing the solid with a harmonic excitation, the time reversal operator can be obtained not only at the fundamental excitation frequency but also at the odd harmonics. At the fundamental harmonic, inhomogeneities or linear damages can be individually selected, but using the odd harmonics nonlinear elastic damages can be detected. Thus, by decomposing the operator at the 3rd harmonic is possible to focus on nonlinear scatterers and to distinguish them from the linear inhomogeneities. The Preisach–Mayergoz (PM) material constitutive model was used for modeling the presence of nonlinear elastic hysteretic scatterers. This paper presents a complete mathematical analysis of this method for 1 and 2D structures. The paper includes numerical simulations applied to 1 and 2 dimensional structures showing the capability of the method to focus selectively on linear and nonlinear inhomogeneous medium.

Focusing with the decomposition of the time reversal operator method in noisy environments

This paper presents a detailed analysis of the effects of noise (reverberation) on the focusing performance of decomposition of the time reversal operator (DORT) in a noise-limited case and in a reverberation-limited case, respectively. Quantitative results obtained from simulations and experiments are presented. The results show the DORT method can be efficiently applied to target detection with enough source level to yield significant backscatter. For a target placed on the bottom, the influence of the reverberation on the focusing performance is slight. However, distinguishing between a target and constant backscattering returning from strong local clutter on the bottom (false alarms) needs further research.

Characterization of an elastic target in a shallow water waveguide by decomposition of the time-reversal operator

This paper reports the results of an investigation into extracting of the backscattered frequency signature of a target in a waveguide. Retrieving the target signature is difficult because it is blurred by waveguide reflections and modal interference. It is shown that the decomposition of the time-reversal operator method provides a solution to this problem. Using a modal theory, this paper shows that the first singular value associated with a target is proportional to the backscattering form function. It is linked to the waveguide geometry through a factor that weakly depends on frequency as long as the target is far from the boundaries. Using the same approach, the second singular value is shown to be proportional to the second derivative of the angular form function which is a relevant parameter for target identification. Within this framework the coupling between two targets is considered. Small scale experimental studies are performed in the 3.5 MHz frequency range for 3 mm spheres in a 28 mm deep and 570 mm long waveguide and confirm the theoretical results.

Time reversal operator decomposition with focused transmission and robustness to speckle noise: Application to microcalcification detection

The decomposition of the time reversal operator (DORT) is a detection and focusing technique using an array of transmit-receive transducers. In the absence of noise and under certain conditions, the eigenvectors of the time reversal operator contain the focal laws to focus ideally on well-resolved scatterers even in the presence of strong aberration. This paper describes a new algorithm, FDORT, which uses focused transmission schemes to acquire the operator. It can be performed from medical scanner data. A mathematical derivation of this algorithm is given and it is compared with the conventional algorithm, both theoretically and with numerical experiments. In the presence of strong speckle signals, the DORT method usually fails. The influence of the speckle noise is explained and a solution based on FDORT is presented, that enables detection of targets in complex media. Finally, an algorithm for microcalcification detection is proposed. In-vivo results show the potential of these techniques.

DORT method as applied to ultrawideband signals for detection of buried objects

The decomposition of the time reversal operator (DORT) method, originally developed in acoustics, allows detection of scatterers embedded in the probed domain and provides some very robust means for focusing an incident wave onto a given scatterer. Hence, this method is very helpful for clutter reduction. Here, it is applied to the detection of buried cylindrical objects with the help of electromagnetic ultrawideband signals. It is shown that when the set of antennas is located on a piece of line above an interface, the use of the DORT method remains simple, whatever the polarization, provided the contribution from the target can be separated from that of the interface. Using wideband signals also permits one to excite natural resonances of the buried scatterer, which can easily be extracted from the eigenvalues of the time reversal operator. Numerical examples based on a finite-difference time-domain algorithm are given.

Imaging with the DORT method

The decomposition of the time reversal operator (DORT) method applies to active detection and focusing of acoustic waves using arrays of transmitters and receivers. This method allows detection and selective focusing on scatterers through complex media. It consists in the construction of the invariants of the time reversal process, that are also the singular vectors of the array response matrix. The DORT method is particularly interesting for detection in inhomogeneous media when the acoustic properties are poorly known. However, if an estimate of the medium’s Green function is available, images of the medium can be formed by backpropagation of the dominant eigenvectors. For the detection of defect in scattering media like titanium, the method is useful to separate the echo of a defect from the microstructure contribution and thus to reduce speckle noise. We will also see that the time reversal operator can be interpreted as a pseudocovariance matrix like those encountered in classical array signal processing. And as shown by Anthony J. Devaney [J. Acoust. Soc. Am. 110, 2617 (2001)], nonlinear estimators like MUSIC can be applied to achieve high resolution. These different points will be illustrated through several experimental results.

Ultrasonic nondestructive testing of scattering media using the decomposition of the time-reversal operator

In ultrasonic nondestructive testing, the iterative time-reversal process is an adaptive technique that can be used to detect flaws in complex samples with a large array of transducers. The decomposition of the time-reversal operator (DORT) method is a detection technique that is derived from the mathematical analysis of the iterative time-reversal process. Contrary to time-reversal techniques, the DORT method does not require programmable generators, and it allows the simultaneous detection and separation of several defects. In this paper, the method is applied to a Ti6-4 titanium cylindrical sample to separate the echo of a defect from the speckle due to microstructure contribution. The grain structure of this titanium alloy makes detection very difficult and, for large depths, conventional techniques do not allow the detection of small flaws with a satisfactory signal-to-noise ratio. The efficiency of the DORT method to detect a flat bottom hole with a diameter of 0.4 mm located at a depth of 140 mm in a titanium alloy sample is shown.

Detection and Selective Focusing on Scatterers Using Decomposition of Time Reversal Operator Method in Pekeris Waveguide Model

The decomposition of the time reversal operator (DORT) method is a selective detection and focusing technique for the pulse-echo mode that uses an array of transmit-receive transducers. Because it is an application of the acoustic time reversal mirror, the irregular shape of the array and the aberration of the propagation medium are compensated and the selective focusing on targets is made possible. In this paper, we examined the application of this method to the underwater acoustics and the application to active sonar and communication are made possible. To clarify the relationship between the focusing effect of the DORT method and the sound propagation property in shallow water, we examined the selective focusing effect with respect to the propagation distance by using the cylindrical spread model without the sea surface and bottom and the Pekeris model, which is typical of shallow water models. There are few influences of the propagation distance on the focusing effect in shallow water model as a result, and it is shown that the focusing effect improves with the increase of the sound velocity of the bottom.

Decomposition of the time reversal operator: Detection and selective focusing on two scatterers

The decomposition of the time reversal operator (DORT) method is a selective detection and focusing technique using an array of transmit–receive transducers. It relies on the theory of iterative time reversal mirrors which was presented by Prada et al. [C. Prada, J. L. Thomas, and M. Fink, J. Acoust. Soc. Am. 97, 62–71 (1995)]. The time reversal operator was defined as K*(ω)K(ω), where ω is the frequency, * means complex conjugate, and K(ω) is the transfer matrix of the array of L transducers insonifying a time invariant scattering medium. It was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities, each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers. The DORT method consists in determining these eigenvectors and using them for the selective focusing. This paper presents a complete analysis of this method in the case of two scatterers. The mathematical expressions of the eigenvectors are given and several experimental results are described. In particular, the effectiveness of the method to focus selectively through an inhomogeneous medium is established.