акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en

The factor of 12 is necessary because, as discussed in Section 4.2, the (χ , φ0 ) coordinate system

gives a double coverage of the kx, ky space.

This integral gives an expression for the scattered field as a function of the (χ , φ0 ) coordinate system, while the data collected will actually be a function of φ0, the projection angle, and κ, the one-dimensional frequency of the scattered field along the receiver line. To make the final coordinate transformation take the angle χ to be relative to the (κ, γ ) coordinate system shown in Figure 4.4. This is a more natural representation since the data available in a diffraction tomography experiment lies on a semicircle and is therefore available only for 0 ≤ χ ≤ π .

−k0

Using the Fourier diffraction theorem as represented by equation (4.2), the Fourier transform of the object function, O, can be approximated by a simple function of the first-order Born field, uB, at the receiver line. Thus the object function in equation (4.42) can be written as

The coordinates (ξ , η) are illustrated in Figure 4.5. Using the results above, the χ integral is now written as

−k0

Without the extra filter function Gη (ω), the remainder of equation (4.49) would correspond to the filtering operation of the projection data in X-ray tomography. The filtering, as called for by the transfer function Gη (ω), is depth dependent due to the parameter η, which is equal to x cos φ + y sin φ.

In terms of the filtered projections φ (ξ , η) in equation (4.49), the reconstruction integral of equation (4.30) may be expressed as

The computational procedure for reconstructing an image on the basis of equations (4.49) and (4.53) may be presented in the form of the following steps:

Step 1: In accordance with equation (4.49), filter each projection with a separate filter for each depth in the image frame. For example, if only 9 depths are used as shown in Figure 4.5, 9 different filters would need to be applied to the diffracted projection shown there. [In most cases for 128 × 128 reconstruction, the number of discrete depths chosen for filtering the projection will also be around 128. If there are fewer than 128, spatial resolution will be lost.]

Step 2: To each pixel (x, y) in the image frame, in accordance with equation (4.53), allocate a value of the filtered projection that corresponds to the nearest depth line.

Step 3: Repeat the preceding two steps for all projections. As a new projection is taken up, add its contribution to the current sum at pixel (x, y).The depth-dependent filtering in Step 1 makes this algorithm computationally very demanding. For example, if Nη depth values

FFTs compared to 4N FFTs for bilinear interpolation. [For precise comparisons, it must be mentioned that the FFTs for the case of bilinear interpolation are longer due to zero-padding.]

Devaney [20] has also proposed a modified filtered-backpropagation algorithm, in which Gη (ω) is simply replaced by a single Gη0 (ω) where

η0 = x0 cos φ + y0 sin φ, (x0, y0 )

are the coordinates of the point where local accuracy in reconstruction is desired. [The elimination of depth-dependent filtering reduces the number of FFTs to 2Nφ .]

4.3Holography

Holography was invented by Denis Gabor [22] in 1948. It is a three-dimensional imaging system due to the preservation of phase information by adding a reference beam to the object beam. After the arrival of the laser in the 1960s, which provide a coherent light source, holography took off and prompted many researchers to use other forms of radiation to produce a hologram. Acoustical holography was pioneered by Pal Greguss [23] in 1965. In this section we will describe the liquid surface method invented by Mueller and Sheridon in 1966 [24].

4.3.1Liquid Surface Method

The scheme invented by Mueller and Sheridon [24] is shown in Figure 4.6. The object is an acoustic transparency, which is configured so that the interference pattern at the water is the Fourier transform of the object transmission function fs(x, y). The intensity of the water surface in Fresnel’s approximation is

Laser beam

Reconstructed image

Beam splitter

Gas

Liquid

Transducer

Figure 4.6 Liquid surface holography system, generating a Fourier transform hologram at the liquid surface (Mueller [9] © IEEE)

where R is the amplitude of the reference beam, l is the separation between the optical axis of reference and the illumination beams, zs is the distance of the object from the hologram plane, and f˜s is the Fourier transform of the object transmission function:

Disregarding the inconsequential constant R2 and the small second term | f˜s|2, the surface deformation due to the intensity distribution (4.54) is

where K is the impulse response of the liquid surface.

If a plane optical wave u0 is reflected from the deformed surface, one obtains a reflected

The last term of (4.57) follows from the fact that the surface deformation is small compared to the wavelength of the light λL. In order to obtain an optical reconstruction of the object function, fs, we must generate a Fourier transform of the reflected wavefront. The reconstruction can be obtained by shining a laser beam onto the liquid surface, the acoustical hologram. With coherent illumination of the deformed surface, this can be done by a transfer lens. We can then obtain the optical amplitude in its Fourier plane.

Due to the huge difference in magnitude between the optical and acoustical wavelengths of a 10 MHz ultrasound frequency, the size of the image will be reduced by the ratio λs/λL (where λs is the sound wavelength) and the images will have to be viewed through a microscope.

Besides this, image reconstructions obtained with this method are plagued with several aberrations because the reference beam unavoidably deforms the water surface. The blurring of images can only be avoided if the reference beam causes a constant or spherical deformation of the water surface in the hologram area.

Smith and Brenden [25] introduced a method to solve the blurred image problem. Instead of generating a Fourier transform of the object function, they use an acoustic projection system to image the object into the water surface. This image, together with a reference wave, generates a focused image hologram at the water surface (Figure 4.7). It is now the acoustic image, and not its Fourier transform modulated onto the high-frequency carrier, which appears at the surface and is picked up as phase modulation in a light beam reflected from the water surface. Their method achieves a very effective rejection of the aberrations that severely limited the lenseless holographic method. The quality of the experimentally obtained imagery is very good.

Figure 4.8 shows the capability of an experimentally verified acoustic holographic system. Its acceptance and use is, however, very limited in spite of the intriguing potential. With very few exceptions, its use is confined to research rather than diagnostic applications.

Figure 4.10 C-scan transmission image of a full-term stillborn foetus (Havlice and Taenzer [10] © IEEE)

a range-gated receiving system effectively eliminates this problem. The range-gated signal is then processed for display by logarithmic compression and greyscale mapping. Figure 4.10 shows the C-scan transmission image of a full-term, still-born foetus.

The system shown in Figure 4.9 can also be used to obtain a reflection-mode C-scan image by using transducer No. 1 as both transmitter and receiver. The transducer could be scanned mechanically as before to obtain the two-dimensional image. In this case range gating not only removes multipath reverberations but also determines the distance of the image plane from the transducer.

Although the transmission and reflection-mode C-scan imaging methods are similar, the resulting images produced are quite different. The transmission-mode images depend for their contrast primarily on the differential attenuation properties of the medium. The

reflection-mode images on the other hand depend for their contrast primarily on acoustic impedance variations. Reflection-mode images are particularly susceptible to specular reflection effects. Small changes in object orientation often result in significantly different images. Transmission-mode images are independent of specularity but are susceptible to coherent interference effects [26].

The resolution of a C-scan system can be given by the Rayleigh criterion:

where δ = resolution, λ = sound wavelength, F = focal length of the system and D = diameter of the circular entrance pupil.

The effective point response function may or may not be the square of the Airy function, depending on the type of system used. This is in contrast to B-scans where the response function is almost always squared. The bandwidth of the transducer is not a factor in resolution since, in a C-scan image, both dimensions are lateral. As in B-scans, the C-scan resolution suffers when ultrasound passes through the medium due to the frequency-dependent absorption coefficient [27]. Depth of focus is not a major, direct factor in a C-scan resolution but it has some significant indirect effects. For example, out of the focal plane objects may appear as out-of-focus artifacts in the images.

4.4.2B-Scan Method

B-scanning or brightness mode scanning provides a two-dimensional, cross-sectional, frontview, reflection image of the object being scanned. A B-scan image is formed by sweeping a narrow acoustic beam through a plane and positioning the received echoes on a display such that there is a correspondence between the display scan line and the direction of acoustic propagation in the medium. Generally, the same transducer is used to send and receive the acoustic signals. A fundamental feature of a B-scan image is that one of the dimensions is inferred from the arrival time of echoes of a short acoustic pulse as they reflect from structures along a presumed straight-line path. Signals received from structures close to the transducer arrive earlier than signals received from structures far from the transducer [28]. The other dimension (transverse) is obtained by moving the transducer, either physically by mechanical means or electronic means so that a different straight-line path through the object is interrogated by another short acoustic pulse. This process is continued until the entire object region of interest is scanned. Some means of tracking the propagation path through the object is required in order to unambiguously define the image. Figure 4.11 shows the block diagram of a generalized B-scanner. Here an electronic pulser excites a transducer so that a short burst of ultrasound is generated.

Acoustic signals reflected from objects in the acoustic path impinge on the transducer, are converted to electronic signals, and processed for display. Very often the amplifier gain is increased with time in order to partially compensate for the attenuation experienced by signals reflected deeper in the body. This is known as time gain compensation (TGC). The position and angular direction of the ultrasound beam are determined by position-monitoring electronics which keep track of where on the monitor the image signals should be displayed.

SYNC

Pulser

X-axis

Figure 4.11 A block diagram of a simple B-scan system (Havlice and Taenzer [10] © IEEE)

As the echoes are received by the transducer they are amplified, rectified, filtered and the resulting signal is used to brightness modulate the display. One of the most important developments in acoustical imaging was the introduction of a greyscale display [29]. In a greyscale display there are usually 10 or more distinct brightness levels. The imaging system assigns a given brightness level to a small range of echo intensities and distributes the brightness levels such that, for example, strong echoes are displayed at the brightest level and weaker ones at progressively lower levels. This type of display produces B-scan images that are less operator dependent and easier to interpret than the bistable images. Image repeatability also appears to be improved with a greyscale display and grey scale has thus become widely accepted. Colour displays have also been used, with different echo levels being displayed in different colours [30]. However, a colour display provides no more information than a greyscale display.

Figure 4.12 presents various image formats for three scan modalities: linear, sector and arc. Here the transducer size is exaggerated. Typically the transducer diameter is only a small fraction of the scanned dimension. In a linear scan the transducer moves in a straight line, but the field-of-view in this direction is limited by the length of travel of the transducer. However, in the time (or depth) dimension, the field-of-view is limited only by the depth of penetration (i.e. the frequency and attenuation) or the physical size of the object being scanned. One advantage of this technique is that the image may consist of a uniform line density which results in a constant spatial sampling rate of the object and a good display on the monitor. In the sector scan, the transducer position remains fixed at a point on or above the object, but is swept through an angular sector [31]. In this case the field-of-view increases with the depth of penetration; however, the line density diminishes as the field-of-view expands. This type