Multidimensional Inverse Problems in Ultrasonic Imaging

Len J. Sciacca; Robin J. Evans    Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Australia 3052

I INTRODUCTION

In this chapter the reconstruction of reflectivity profiles from ultrasonic back-scattered echoes is treated as an inverse problem involving incomplete and noisy measurements. Given pulse-echo measurements from a synthetically created one-dimensional or two-dimensional array of sensors, we show that the maximum likelihood estimate of the reflectivity profile requires the pseudoinverse of the array point spread function (PSF). As a consequence of the physical properties of our imaging system, the point spread function is shown to be a highly structured block Toeplitz linear operator and is a function of the sensor and array characteristics. The near singularity of the PSF, the continuous-discrete model approximation, and measurement noise, mean that inversion of this linear operator is ill-posed and hence direct inversion will yield unsatisfactory estimates of the image. Regularisation in the Tikhonov sense and subspace rank-reducing techniques based on singular value decomposition are utilised to yield stable inversion algorithms and are shown to provide meaningful image estimates. Singular value decomposition inversion is shown to yield computational advantages and is particularly suited to large scale multidimensional linear deconvolution and regularisation problems.

In order to apply these procedures, regularising parameters in the case of Tikhonov regularisation and a level of rank reduction for truncated singular value inversion must be determined. If the signal-to-noise ratio of the imaging system is known a priori then this is a straightforward procedure. However it is often difficult to estimate the noise or signal variances and furthermore, Gaussian assumptions may not be appropriate. The problem of choosing regularisation parameters is resolved in this case using a technique known as generalised cross-validation (GCV). The advantage of this approach is that no a priori knowledge of the noise or its distribution is required and such a procedure may therefore be used as a good "first guess". Generalised cross-validation is also useful in partial blind deconvolution problems where both the PSF model and regularising parameters must be estimated simultaneously.

In Section II, we motivate this study which grew out of an industrial problem. In Section III, we review the literature in the field of ultrasonic digital signal processing and image reconstruction. Section IV provides an imaging model based on the physical structure of the imaging system and the discretised problem is stated in terms of a linear discrete multidimensional convolution. The inverse problem is posed in Section V and issues such as ill-posedness, ill-conditioned systems and regularisation are treated in a practical way. In Section VI we derive image reconstruction estimation procedures using singular value decomposition. Optimal regularisation is treated in Section VIII. Section IX briefly introduces the N-dimensional inverse problem. In Section X partial blind deconvolution is discussed in the context of our application. Recursive solutions that allow strip imaging are then derived in Section XI. In Section XII, results using both synthesised and real experimental data are presented for many of the techniques discussed in this chapter. Finally in Section XIV we provide some concluding remarks.

II THE PIPE-LINE PROJECT

Gas authorities around the world have for many years engaged in the development of automated crack detection devices to inspect gas pipelines. In the past, the detection of cracks in pipe walls has primarily been performed by human operators manually inspecting pipes using ultrasonic hand-probes and portable flaw detectors. This practice is dangerous, expensive, and slow. To remedy this situation gas authorities have developed automated testing equipment built into pipeline crawlers [1]. Such devices make use of a range of sensing techniques including X-rays, neutron bombardment, and ultrasonics. The benefits of an automated imaging system are very high, lengthening the life of in-service pipelines and preventing catastrophic failures. Our work is aimed at providing practical techniques for multidimensional image reconstruction through high resolution three-dimensional imaging of submillimeter cracks.

III REVIEW

Acoustic imaging has its origins in early sound navigation ranging (sonar) technology developed by the French scientist Paul Langevin during World War I. This technique gave the distance to an object in water by timing the delay between transmission of an acoustic pulse and the received echo. The technique is widely used in a diverse range of applications and is the basis of many procedures employed in nondestructive testing.

Nondestructive evaluation (NDE) using ultrasonics typically uses ultrasonic frequencies between 1 MHz and 100 MHz. Ultrasound above 1GHz is becoming more common in microelectronic circuit imaging and in medical acoustic-microscope applications. There are three common examination procedures used in nondestructive testing. These are usually called A, B and C scanning.

A-scan ultrasonic material evaluation utilises a high frequency broad-band pulse that is transmitted into the material. Should defects be present they will present a different acoustic impedance to the wave resulting in refracted and reflected waves [2]. A-scans are a simple one-dimensional analysis procedure where the time-of-arrival of an echo is used in order to estimate the depth of the defect. Experienced operators are able to characterise defects from A-scan data.

B-scan is a technique involving a linear array of sensors or, in its simplest form, a single sensor moved in a linear motion. As with the A-scan, a broad-band pulse is emitted by an ultrasound sensor. In so doing, a two-dimensional image can be obtained. To a limited extent this image can also give information about the three-dimensional nature of the defects. Once again this requires experience and knowledge of the response of ultrasound to defects.

C-scan imaging constructs a three-dimensional image of a region utilising a two-dimensional array of sensors. This removes the ambiguities present in the linear array. As with B-scan imaging, the array may be synthesised by taking measurements with a single sensor in a uniform grid arrangement, transmitting a broad-band pulse at each grid position.

Perhaps the simplest processing technique employed in modern NDE is to use the A-scan procedure and display the rectified received echo envelope on an oscilloscope. This limits discrimination of defects to approximately the length of the transmitted pulse. The size of defects is often estimated by the intensity of the returned echo. The time-of-arrival of the echo gives the distance of the defect to the sensor as this is related by the velocity of sound in the material being tested. Many practical systems [4] use correlation to improve the system resolution or detection capabilities. Correlation is a form of matched filtering where the echo signal is convolved with the delayed time-reversed version of the original transmitted pulse [5],

^t=∫st−t′zt′dt′

where s (t) is the transmitted pulse, y (t) is the measured noisy signal and ^t is the estimated reflectivity profile.

Twomey and Phillips studied practical inverse problems during the mid 1960's [6][7][8]. Twomey's research was motivated by atmospheric physics but the problems were framed as discretised versions of a general linear convolution problem and reduced to the problem of solving overdetermined or underdetermined sets of linear equations. Twomey incorporated constrained solutions and the separation of signal and noise subspaces in an intuitive way to solve discretised inverse problems. It is well known that many of the convolution problems encountered in physics can be expressed as a Fredholm equation of the first kind,

t=∫wt−t′xt′dt′

where w (t) describes the sensing system and x (t) is the image we are seeking to reconstruct. By discretising this equation and expressing it as a sampled convolution problem, a linear algebraic expression can be derived. Inverse filtering can then be used to obtain an estimate of x,

^=W−1y

where the operator W , is a function of w (t) in Eq. (2).

The use of inverse filtering to perform deconvolution in ultrasonic inspection appeared to coincide with the advances in computer technology in the late 1970's. Hundt and Trantenberg in [9] provide a review of the history of digital deconvolution in the ultrasonic field. It is evident from this and other literature that early processing techniques were crude and only considered the one-dimensional A-scan case with no regard for noise.

Early attempts to improve resolution were motivated by seismic applications which operate at much lower acoustic frequencies than used for ultrasonic inspection and therefore require less computing power for real-time...