The discussion is separated into two separate sections, on ``first'' and ``second'' order aberrators. In this terminology, first order refers to geometric focal errors due to the difference between the actual mean sound speed in the tissue and that assumed by the ultrasound scanner beamformer. Second order aberration refers to the effects of the inhomogeneity of the tissue that still remain after the first order focal error is removed. Both types of aberration result in blurring of the point spread function and consequently the reduction of spatial and contrast resolution.
A first order aberrator arises from a gross sound speed error in the ultrasound system beamformer relative to the actual tissue, and is usually caused by the application of a single assumed tissue sound speed to all patients. For example, the first order aberration resulting from a 5% overestimation of sound speed is shown in Fig. 9.1. The impact of this aberrator on the system point spread function is shown in Fig. 9.3. In this example, the aberrator was applied to the element delays of a typical array at the f/1 focal range (7.5 MHz, 60% relative bandwidth, 128 elements of width equal to the 1540 m/s wavelength, f/8 in elevation) in an acoustic field simulation program. The aberration leads to an 85% reduction in the PSF energy and a 75% reduction in its peak amplitude. The lateral resolution has been degraded by a factor of 3.9.
An example of a second order aberrator is shown in Fig. 9.2. The impact of this aberrator on the system point spread function is shown in in Fig. 9.4. Once again, this aberrator was applied to the element delays of a typical array (7.5 MHz, 60% relative bandwidth, 128 elements of width equal to the wavelength, f/8 in elevation) in an acoustic field simulation program. The aberration produces an 95% reduction in the PSF energy and a 91% reduction in its peak amplitude. The lateral resolution in this case has been degraded by a factor of 2.3.
A variety of techniques have been proposed to compensate for the effects of phase aberration[35,36,37,38,39,40,41,42,43,44,45]. Ideally, the estimated aberrator should be applied on both transmit and receive, as the phase screen affects wave propagation on both transmit and receive.
Speckle has a finite correlation length due to aperture-scatterer path length differences that distinguish one aperture position from another. A point target as infinite correlation length due to the lack of such path length differences.
The Van Cittert-Zernicke theorem states that for an incoherent, diffuse source, the receive spatial correlation function is equal to the Fourier transform of the intensity pattern of the incoherent source. Speckle is a truly incoherent source. The location of scatterers on one line is uncorrelated with that of those in the next.
Cross-correlation methods are affected by VCZ. The VCZ curve falls off even faster in the presence of a transmit phase aberrator due to be broadening of the beam, which scales in k-space to a narrower intensity pattern, thereby producing a steeper correlation curve.
Deconvolution or ``super-resolution'' methods rely mainly on two approximations. The first of these is known as the Born approximation, which disregards echoes reduced by multiple scattering from targets. The second is that the point spread function is accurately known throughout the region of interest.
Under ideal conditions, deconvolution can work very well. This implies that the noise is very low and that the number of targets is relatively small. Deconvolution does not work well for larger numbers of targets under conditions of low signal-to-noise ratio, especially away from the focus and/or in the presence of phase aberration or attenuation. There performance is also degraded when physical conditions reduce the stationarity of the system's point spread function.
The simplest form of detection is AM demodulation using a full wave rectifier followed by a low pass filter. This is shown schematically in Figure 9.7, using the squaring operation to represent rectification.
A more sophisticated demodulation technique is to mix the signal with that from a local oscillator. This produces base-band and double-frequency components, the latter of which are typically filtered out using a low pass filter. An extension of this concept, known as quadrature demodulation, is to mix the original signal with two local oscillators having the same frequency but shifted by 90 in phase relative to one another. The two baseband signals that remain after low-pass filtering are often referred to as the I (for ``in-phase'') and Q (for ``quadrature'') signals. Generating both I and Q signals preserves the phase information in the original signal.
Given the trigonometric identities,
Neither of these techniques achieve ``ideal'' demodulation. While the level of energy at zero Hertz (DC) is preserved, the base-band signal away from DC contains the integration across of the product of frequencies at shift :
The most accurate demodulation technique is to use the Hilbert operator or the Hilbert transform to generate the quadrature signal at the RF center frequency. This operation does not assume any particular center frequency, and is thus immune to discrepancies between the local oscillator's frequency and that of the echo signal. The echo envelope is then found by calculating the magnitude of the complex signal formed by combining the original RF signal with its synthesized quadrature signal. With this method, no filtering is necessary.
Various methods for creating elastic waves or motion in a tissue are under investigation, including an internally or externally applied vibrator, an externally-applied compression paddle, or pumping waves using low frequency sound. Motion under vibration is imaged using Doppler techniques or speckle-tracking techniques. Standing waves are a problem, especially with strong vibration necessary for the targets.
Compression based techniques typically used some kind of correlation search before and after compression. Some error is produced in these techniques by the decorrelation of speckle during compression. Decorrelation during compression results in part from scatterers moving into or out of the resolution volume, and partly from changes in the geometric relationships among particles within the resolution volume[50,51].
Despite these limitations, we hope this text has demonstrated the utility of a k-space approach in understanding and analyzing many aspects of ultrasound imaging systems and beamforming techniques.