p-Delay And Sum algorithm

Algorithm

Proposed by [1], it is an extension of the FDMAS algorithm consisting in doing the squared sum of the signed squared-roots vector of delays \(s_{n}(x,z)\). It allows faster computation and also can extend the algorithms to higher degrees. This degree is characterized by the value p, which can be an integer or a float. We can first define \(\tilde{r}(x,z)\), the sum of the scaled squared signed signal along the probe elements:

\[\tilde{r}\left(x,z\right)=\sum_{n=1}^{N}\text{sign}\left(s_{n}\left(x,z\right)\right)\cdot\,{\sqrt[p]{\left|s_{n}\left(x,z\right)\right|}} \tag{1}\]

Then we’ll have the value of the beamformed pixel using the signed p-power:

\[r_{p-DAS}\left(x,z\right)=\text{sign}\left(\tilde{r}\left(x,z\right)\right)\cdot\,\left|\tilde{r}\left(x,z\right)\right|^{p} \tag{2}\]

As for the FDMAS beamformer, we then need to apply a bandpass filter, centered at \(f_{0}\), in order to remove potential harmonics due to non-linear operations, the effect of the filter is shown in the figure 3a.

../_images/pdas_spectra.png

Extension to In-Phase Quadrature data

As for the FDMAS, we can’t operate on the sign of a complex number as it is undefined. Here, we’ve been using the same method as for FDMAS, also defined in [2] and called BB-DMAS-p. It is using the same concept: separating the amplitude and the phase in order to keep the phase unchanged while doing the squared p-th root. It can be formulated as:

\[r_{BB-DMAS-p}=\left(\sum_{n=1}^{N}{\sqrt[p]{a_{n}\left(x,z\right)}}\cdot e^{j\phi_{n}\left(x,z\right)}\right)^{p} \tag{3}\]

Same as for the baseband DMAS version, when we extend it to a degree p, we see on figure 3b that we don’t need any additional filtering. However, since we only do the p-th root of the signal magnitude and not its phase, the frequencies of our signal is only affected by the p-power step, which means that the new central frequency of our beamformed signal becomes \(p.f_{0}\).

An alternative has been proposed by [3], that preserves the central frequency of the beamformed signals to the original \(f_{0}\). This method is the default one, using the same formula as [1], but with the sign of a complex number defined as:

\[\text{sign}(s_n)=\frac{s_n}{|s_n|}=\frac{a_n . e^{j\phi_n}}{a_n}=e^{j\phi_n}\]

  • [1] A Nonlinear Beamformer Based on p-th Root Compression—Application to Plane Wave Ultrasound Imaging, Polichetti & al.

  • [2] Ultrasound Baseband Delay-Multiply-and-Sum (BB-DMAS) nonlinear Beamforming, Shen & al.

  • [3] BB p-DAS, an extension of p-DAS to baseband domain for Doppler imaging, Ecarlat & al.