Speckle reduction techniques

By translating the k-space window developed by the imaging system, more information can be gathered from the target spatial frequency space. If such translation is used to develop multiple windows in image k-space, the correlation among these windows is proportional to their geometric overlap. The averaging of the detected speckle patterns from such shifted windows in k-space is a means of reducing speckle noise, and is achieved through spatial or frequency compounding. The correlation coefficient among the k-space windows will tell us if compounding will reduce the speckle. (If the original spectra are in phase, the correlation coefficient is proportional to overlap. This is the case under ideal conditions, i.e. at the focus and with no aberration.)

Spatial compounding

The transducer-target geometry affects the spatial frequencies in the target function that are interrogated by the imaging system. The effect of movement of either relative to the other is shown in Figure 8.1. Spatial compounding is achieved by the translation of the aperture, as shown in Figure 8.2. The detected signals acquired at different aperture positions are averaged together.

Figure 8.1: If you move a single element with respect to a fixed group of scatterers, you sweep out an arc in k-space.

Figure 8.2: Under spatial compounding, the aperture is translated laterally (left) between interrogations of the same region of interest. This shifts the system's region of support in k-space, reflecting decorrelation of the response, and hence of the observed speckle patterns. The region of overlap (shaded) is proportional to the correlation between the two speckle patterns.

In a classic paper on speckle second-order statistics by Wagner, et al., we read that the lateral spatial correlation of speckle is equal to the autocorrelation of a triangle function[28]. Translation of one aperture length takes the cross correlation curve to 0, and hence produces statistically independent speckle patterns, but even translation of only 1/4 aperture length reduces the cross correlation function to a very low value, as shown in Figure 8.3.

Figure 8.3: As the sub-apertures of a 2:1 spatial compounding system are separated in space, the normalized correlation of the signals received by them changes as a function of this separation.

Frequency compounding

Frequency compounding adds together detected data from different frequency bands. These frequency bands have different regions of support in k-space, as shown in Figure 8.4.

Figure 8.4: Under frequency compounding, the temporal frequency band used in imaging is altered among different interrogations of the same region of interest (left). This shifts the system's region of support along the $k_{z}$ dimension in k-space, reflecting decorrelation of the response, and hence of the observed speckle patterns. The region of overlap is proportional to the correlation between the two speckle patterns.

A simple frequency compounding method is to transmit a single broad bandwidth pulse and to then select different receive frequency ``sub-bands'' through the application of filters, as shown in Figure 8.5. The axial resolution is now determined by the smaller bandwidth of these compounding filters. This represents a trade-off between speckle SNR and axial resolution.

Again, the correlation between sub-bands is a function of their overlap. Completely discrete bands have no correlation. The signal-to-noise ratio will be different in each band, thus ``pre-whitening'' filters may be used to level off each band, although doing so amplifies portions of the noise spectrum.

Figure 8.5: Under frequency compounding, a broad temporal frequency band is used on transmit. The receive signal is filtered to produce several receive sub-bands (3, in the diagram above.) Frequency compounding relies on detection of these sub-bands before summation, which averages the partially-uncorrelated speckle patterns they produce. The region of sub-band overlap is proportional to the correlation between the speckle patterns.

Each sub-band typically has a Gaussian profile axially. Therefore as we change center frequencies of the compounding bands and calculate their cross correlation, we are calculating the autocorrelation function of a Gaussian. For a frequency compounding system, the correlation coefficient between sub-band speckle patterns as a function of the separation of sub-bands is thus the autocorrelation function of a Gaussian.

Figure 8.6: As the bands of two receive filters are separated (top), the normalized correlation of the signals received by them changes as a function of the separation in center frequency (bottom).

Note that frequency compounding only works on detected data. Averaging RF signals is simply superposition, which recreates the original bandwidth system but achieves no compounding.

Trahey and Smith investigated the speckle reduction vs. spatial resolution trade-off using an expression for lesion detectability based on in contrast-detail study. This work suggested that it is better to use the entire bandwidth coherently.

On the other hand, there is reason to believe commercial scanners are routinely using frequency compounding. This reflects the fact that the axial resolution of a typical scanner's display monitor is not capable of exhibiting the true axial resolution of the system. Given this ``excess'' of axial resolution, frequency compounding can be applied without an apparent penalty in axial resolution. This cannot be done when the scanner is operated in a ``zoom'' mode.