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Linear

(a)

Sector

(b)

Arc

(c)

Figure 4.12 Mechanical motion and image format for (a) linear, (b) sector and (c) arc B-scans (Havlice and Taenzer [10] © IEEE)

of scan is particularly well suited for imaging through narrow apertures, such as imaging the heart through the ribs. In an arc scan a transducer is moved along the arc of a circle, which gives rise to an image format that is the inverse of the sector scan [32]. Here the field-of-view is largest near the transducer and decreases with depth of penetration. The arc scan is most often encountered in manual scans of the abdomen, the surface of which resembles the arc of a circle.

The compound scan [33] is a combination of the sector scan and either an arc scan or a linear scan, as illustrated in Figure 4.13. For illustrative purposes, only two positions in the linear travel and the respective sectors are shown. The sector is usually much smaller in compound scanning than in simple sector scanning, where angles as large as ±45◦ are used. Note that in compound scanning, object points are imaged by more than one acoustic pulse along different ray paths. Compound scanning is used to overcome a major problem in B-scan imaging, namely the difficulty of imaging specular reflectors and objects lying behind specular reflectors. A specular reflector reflects sound towards a direction that is dependent

Compound

linear

Partial rays only

Figure 4.13 Mechanical motion and image format for a compound linear scan (only two positions shown). Note that in the compound region, object points are imaged with more than one acoustic ray (Havlice and Taenzer [10] © IEEE)

on its orientation to the transducer. Hence it is possible for an incident sound beam to reflect from a specular reflector in a direction that does not return the sound beam to the transducer. The images of cystic structures within the liver obtained with a manual compound contact B-scanner is shown in Figure 4.14.

There are two types of resolution in a B-scan system:

1.Lateral or transverse resolution, which is resolution in the direction of transducer motion.

2.Axial resolution, which is resolution in the direction of acoustic pulse propagation.

Figure 4.14 These two images were made with a manual compound contact B-scanner showing cystic structures within the liver (Havlice and Taenzer [10] © IEEE)

The definition of the lateral resolution is due to Lord Rayleigh and is given in equation (4.58). Lord Rayleigh derived this resolution criterion in order to predict the ability of an optical system to distinguish two self-luminous incoherent point sources. His optical system was operating in a receiver-only mode, whereas an ultrasonic B-scan system operates in a transmit/receive mode. This means that the effective spatial response of the ultrasonic system to a point source reflector is the product of the transmitter field pattern and the receiver field pattern. Because the same transducer is employed as a transmitter and a receiver, the effective spatial response pattern for a B-scan system is not an Airy pattern but the square of the Airy pattern.

The axial resolution of a B-scan is inferred from the arrival time of sequentially reflected acoustic pulses. The resolution in the axial dimension is relatively unaffected by the presence or absence of focusing elements but is determined principally by the bandwidth of the transducer [34]. The larger the bandwidth the shorter the acoustic pulse that can be generated and received, and the finer the definition along the axis of propagation. For a typical 2.25 MHz commercial medical transducer, it is possible to attain 70% bandwidth to provide a resolution of about 2 mm.

There is another factor that negatively affects resolution. Most B-scanners achieve transverse resolution with fixed focus elements. Thus the resolution is better for structures both nearer to and farther from the transducer than the focal length of the fixed focus element. In addition, the design of a transducer will also meet the following compromise: resolution at the focal depth may be improved by increasing the aperture D in (4.58). However, the depth-of-focus – that region over which the optimum resolution is obtained – becomes smaller. This is a bad compromise since the resolution improves only as the first power of the aperture whereas the depth-of-focus becomes smaller as the square of the aperture [35]. In other words, one loses depth-of-focus much more quickly than one gains resolution. To minimize this effect, only weak focusing is generally used in diagnostic instruments. Even so, the resolution is noticeably poorer for points far from the focal distance when fixed focus elements are used. There are acoustic focusing elements that are not fixed in their focal distance, but are electronically variable [36].

4.5Acoustic Microscopy

Acoustic microscopy uses high-frequency ultrasound for imaging. The frequencies cover the range 5 MHz to 3 GHz. The concept of acoustic microscopy, first proposed by Sokolov in 1936 [37], was expected to produce magnified views of structures. The ultrasound frequency used was as high as 3 GHz and was known as the Sokolov tube. However, due to technological limitations at that time, no such instrument was constructed. It was not until 1959 that Dunn and Fry [38] performed the first acoustic microscopy experiments, though at much lower frequencies. There had been little progress in acoustic microscopy until around 1970 when two groups – Lemons and Quate [39] and Korpel, Kessler and Palermo [40] – emerged. Earlier efforts in the development of an operational acoustic microscope focused on the highfrequency adaptation of low-frequency ultrasonic visualization methods [41, 42]. In 1970, the Korpel et al. group began to pursue a scanning laser detection system for acoustic microscopy. In 1974, the activity was shifted to another organization under Kessler where practical aspects of the instrument were developed. This instrument, the scanning laser acoustic microscope (SLAM) was made commercially available in 1975. In 1973, the Quate group at Stamford University began the development of its present instrument concept.

Due to the advancement in electronic circuitry and digital signal processing techniques, various forms of acoustic microscopy have been developed during the past decades. Higherfrequency acoustic microscopy will enable higher resolutions to be attained. For instance, frequencies of 2 to 3 GHz will enable a resolution of 1 to 2 microns, equivalent to that of the optical microscope. However, there is a trade-off due to absorption and attenuation in the material, resulting in lesser penetration depth.

Acoustic microscopy has also opened up several areas of application: to the nondestructive evaluation in the semiconductor industries, the aerospace industries, the automotive industries, the oil and gas industries, and to biomedical areas for the detection of cancerous tissues and cells. The V(z) curve technique of acoustic microscopy will also enable the quantitative nondestructive evaluation of the quantitative characterization of defects and damages by providing measured values of elastic modulus.

A more detailed treatment on acoustic microscopy will be given in a later chapter on high-frequency imaging.

References

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[6]Greenleaf, J.F. et al. (1974) Algebraic reconstruction of spatial distributions of acoustic absorption within tissue from their two-dimensional acoustic projections, in Acoustical Holography, vol. 5 (ed. P.S. Green), Plenum Press, New York, pp. 591–603.

[7]Greenleaf, J.F., Johnson, S.A., Samayoa, W.F. and Duck, F.A. (1975) Algebraic reconstruction of spatial distributions of acoustic velocities in tissue from their time-of-flight profiles, in Acoustical Holography, vol. 6, Plenum Press, New York, pp. 71–90.

[8]Mueller, R.K., Kaveh, M. and Iverson, R.D. (1978) A new approach ot acoustic tomography using diffraction techniques, in Acoustical Holography, vol. 8 (ed. A. Metherell), Plenum Press, New York; and Mueller, R.K., Kaveh, M. and Wade, G. (1979). Reconstructive tomography and applications to ultrasonics. Proc. IEEE 67, 567–587.

[9]Mueller, R.K. (1979) Acoustic holography, in Modern Acoustical Imaging (eds H. Lee and G. Wade), IEEE Press, New York, pp. 73–89.

[10]Havlice, J.F. and Taenzer, J.C. (1979) Medical ultrasonic imaging: an overview of principles and instrumentation, in Modern Acoustical Imaging (eds H. Lee and G. Wade), IEEE Press, New York, pp. 7–28.

[11]Morse, P.M. and Feshbach, H. (1953) Methods of Theroretical Physics, McGraw-Hill Book Company, New York.

[12]Ishimaru, A. (1978) Wave Propagation and Scattering in Random Media, Academic Press, New York.

[13]Kak, A.C. (1984) Tomographic imaging with diffracting and non-diffracting sources, in Array Processing Systems (ed. S. Haykin), Prentice Hall, New Jersey.

[14]Kak, A.C. and Slaney, M. (1988) Principles of Computerized Tomographic Imaging. IEEE, New York.

[15]Wolf, E. (1969) Three-dimensional structure determination of semi-transparent objects from holographic data. Opt. Commun., 1, 153–156.

[16]Nahamoo, D., Pan, S.X. and Kak, A.C. (1984) Synthesis aperture diffraction tomography and its interpolationfree computer implementation. IEEE Trans. Sonics Ultrason., SU-31, 218–229.

[17]Devaney, A.J. (1984) Geophysical diffraction tomography. IEEE Trans. Geosci. Remote Sens., GE-22, 3–13.

[18]Slaney, M. and Kak, A.C. (1985) Imaging with diffraction tomography, Preprint, TR-EE 85-5, February, Pur due University, USA.

[19]Soumekh, M., Kaveh, M. and Mueller, R.K. (1984) Fourier domain reconstruction methods with application to diffraction tomography. Acoust. Imaging, 13, 17–30.

[20]Devaney, A.J. (1982) A filtered backpropagation algorithm for diffraction tomography. Ultrasonic Imaging, 4, 336–350.

[21]Kaveh, M., Soumekh, M. and Mueller, R.K. (1982) Tomographic imaging via wave equation inversion. ICASSP, 82, 1553–1556.

[22]Gabor, D. (1948) A new microscope principle. Nature, 161, 777.

[23]Greguss, P. (1965) Research Film, 5, 330.

[24]Mueller, R.K. and Sheridon, N.K. (1966) Sound holograms and optical reconstruction. Appl. Phys. Lett., 9, 328.

[25]Smith, R.B. and Brenden, B.B. (1969) Refinements and variations in liquid surface and scanned ultrasound holography. IEEE Trans. Sonics Ultrason. (1968 IEEE Ultrasonics Symposium Dig.), SU-16, 29.

[26]Korpel, A. et al. (1974) Elimination of spurious detail in acoustic images, in Acoustical Holography, vol. 5, Plenum Press, New York, pp. 373–390.

[27]Dunn, F. et al. (1969) Absorption and dispersion of ultrasound in biological media, in Biological Engineering (Inter-University Electronic Series, vol. 9), McGraw-Hill, New York.

[28]Erikson, K.R. et al. (1974) Ultrasound in medicine – A review. IEEE Trans. Sonics Ultrason., SU-21(3).

[29]Kossoff, G. (1972) Improved techniques in ultrasonic cross-sectional echography. Ultrasonics, 10, 221.

[30]Yokoi, H. et al. (1973) Quantized colour ultrasonotomography. Excerpta Medica, 277(103).

[31]Tucker, D.G. et al. (1958) Electronic sector scanning. Brit. Inst. Radio Eng., 26, 465.

[32]Baum, G. (1978) The current status of ultrasound mammography, in Ultrasound in Medicine, vol. 4, Plenum Press, New York.

[33]Fleming, J.E. and Hall, A.J. (1968) Two dimensional compound scanning-effects of maladjustment and calibration. Ultrasonics, 6, 160–166.

[34]Papadakis, E.P. and Fowler, K.A. (1969) Broad-band transducers: radiation field and selected applications. J. Acoust. Soc. Amer., 50, 729–745.

[35]Papoulis, A. (1968) Systems and Transforms with Applications in Optics, McGraw-Hill, New York.

[36]Halice, J.F. et al. (1974) An electronically focused acoustic imaging device, in Acoustical Holography, vol. 5, Plenum Press, New York, pp. 317–334.

[37]Sokolov, S. (1936) USSR Patent no. 49 (Aug. 31, 1936), British Patent no. 477 139 (1937) and US Patent no. 21 64 125 (1939).

[38]Dunn, F. and Fry, W.J. (1959) Ultrasonic absorption microscope. J. Acoust. Soc. Amer., 31, 632–633.

[39]Lemons, R.A. and Quate, C.F. (1973) A scanning acoustic microscope. Proc. IEEE Ultrason. Symp. (ed. J. de Klerk) pp. 18–21. Catalog 73CHO 807-8 SU.

[40]Korpel, A., Kessler, L.W. and Palermo, P.R. (1971) Acoustic microscope operating at 100 MHz. Nature, 232(5306), 110–111.

[41]Havlice, J., Quate, C.F. and Richardson, B. (1968) Visualization of sound beams in quartz and sapphire near 1 GHz, paper 1-4 presented to the 1967. IEEE Ultrason. Symp., IEEE Trans. Sonics Ultrason., SU-15, 68.

[42]Korpel, A. (1966) Visualization of the cross section of a sound beam by Bragg diffraction of light. Appl. Phys. Lett., 9, 425.

5

Time-Reversal Acoustics

and Superresolution

5.1Introduction

The new field of time-reversal acoustics was pioneered by Fink in 1992 [1]. It is based on the time-reversal invariance of the acoustic wave equation. It has the characteristics of working well in heterogeneous media where multiple scatterings take place instead of in ordered or homogeneous media. Hence it enables sharp focusing in heterogeneous media. Since most solid media are inhomogeneous in nature, this technique has great potential in acoustical imaging and even shows the possibility of superresolution – that is, defeating the diffraction limit.

5.2Theory of Time-Reversal Acoustics

Time-reversal acoustics is based on the principle of time-reversal invariance of the acoustic wave equation in lossless media [1–4]. It means that, if ϕ(r, t ) is an acoustic field and thus is a solution of the wave equation, then ϕ(r, −t ) is another solution and thus a possible acoustic field. In particular, if ϕ(r, t ) is a wave diverging from a point, ϕ(r, −t ) must be focused on this same location. This gives rise to the novel idea of focusing procedure in two steps. First, an acoustic source is installed and its emitted field is measured by means of a closed receiving surface around the medium. We assume that each point of this surface is able to record the wave as a function of time and then to re-emit in order to generate the time-reversal solution. This time-reversed wave is backpropagated through the medium and finally focuses on the locations of the initial source.

However, the concept of a closed time-reversed cavity (CTRC) is difficult to realize [3] and is usually replaced by a time-reversed medium of finite size which shows a performance comparable to the cavity in spite of the loss of information [4]. We will consider the case of a solid–fluid interface which is applicable to situations of nondestructive testing (NDT) and medical imaging.

Draeger et al. [5] consider a point-like source of elastic waves located inside a solid half space at the origin x = y = z = 0 (Figure 5.1(a)). The plane solid–fluid interface is at z = h > 0 and

Acoustical Imaging: Techniques and Applications for Engineers, First Edition. Woon Siong Gan. © 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

Figure 5.1 Time reversal behaves as a two-step process. (a) Emission of a short pulse of longitudinal and transverse waves by an active source in the solid, yielding two pressure wavefronts in the fluid; recording by the TRM. (b) Re-emission of the time-reversed fields into the fluid by the TRM; backpropagation of the two wavefronts in the fluid, yielding four wavefronts in the solid. The desired waves arrive simultaneously at the initial source location, the undesired ones arrive before and afterwards (Draeger et al. [5] © Acoustical Society of America)

the time-reversed medium is located in the fluid at z = Z > H. The source emits a short pulse of longitudinal and transverse waves (or P and S waves) which are partially terminated at the interface into the fluid and are thus both converted into pressure waves. They also consider that all waves emitted or reflected into the negative z-directions are lost. SH waves, in particular, are totally reflected and therefore it is impossible to apply the time-reversed process to this polarization of transverse waves. This results in a limitation of the time-reversed medium device and is why the P and SV wave components yield two wavefronts in the fluid.

The time-reversed medium records them both and is able to distinguish between the two wavefronts if their interval time is sufficiently distinct. In this case, they can choose to backpropagate only the wavefronts corresponding to the P waves or the SV waves, or both. When the backpropagating wavefronts arrive at the fluid–solid interface, each of them creates two wavefronts in the solid, one corresponding to the original type of wave (wanted) and one of the other type of wave (unwanted) (see Figure 5.1(b)). Also the unwanted SV wave, created by the returned P wavefront, and the unwanted P wave, due to the returned SV wavefront, yield a low-level noise and as they are not focused they arrive at different times. On the contrary, the two wanted wavefronts focus at the same place – that is, the location of the initial source – which now remains passive.

Their derivations here are for the P and SV waves propagating in the positive z-direction [5]. They describe the displacement vector u of the incident elastic field as a function of

the scalar and vector potentials φ and ψ of the P and SV waves propagation in the positive z-direction [5]:

The initial field can be for a point-like source or extended source. The only condition is the existence of the 2D Fourier transform over x and y in the frequency domain of the potential(kx, ky, z, ω) and (kx, ky, z, ω): φ is the scalar potential and gives rise to the longitudinal wavefield

Referring to the acoustic wave equation (5.1), the dependence on z can be written explicitly as

where va is a function of kr = kx2 + ky2 and the longitudinal wave speed α, and is defined by

If vα is real, that is, if kr ≤ ωα , it can be considered as the z component kz of the wave vector

= ( x, ky, kz ) of a propagative wave. k k

If vα is imaginary, the wave is called an evanescent wave and is nonpropagative. Its amplitude decreases exponentially with depth in the z-direction.

is the vector potential that gives rise to the shear wave. Following same procedure as for the longitudinal wave one obtains:

At the interface, each wave type in the solid generates a wavefront in the fluid. Describing the transmitted sound wave by its pressure field P, they can split it formally into the part created by the P wave Pp and the one from the SV wave Ps :

where Tp f , Ts f = transmission coefficients.

66 Acoustical Imaging: Techniques and Applications for Engineers

The derivation of the transmission coefficients is as follows:

The incident plane sound waves are on a solid–fluid or fluid–solid interface. For simplicity, the interface is at z = 0 and ky = 0; kx = kr . Two cases are considered: (1) an incident P wave,

(2) an incident SV wave. Referring to equation (5.1), the expression of the trial wavefields are as follows:

Eliminating the reflection coefficients for each case described earlier, the transmission coefficients are obtained as