## 2 Methods and Materials
### Methods
#### 2.1.3 Proposed Method: \(p\)-DAS Beamforming  
The proposed method is referred to as \(p\)-DAS. A block diagram is given in Figure 1 to describe how the value of a pixel \(r_{p\text{-}\text{DAS}}(x,z)\) is computed from the corresponding compounded delayed samples \(s_{n}(x,z)\) with the \(p\)-DAS beamformer. After being correctly delayed and compounded, the \(s_{n}(x,z)\) amplitudes are compressed through the application of the signed \(p^{th}\) root:  
\[\tilde{s}_{n}(x,z)=sign(s_{n}(x,z))\cdot|s_{n}(x,z)|^{\frac{1}{p}}. \tag{8}\]  
Note that the \(p\)-value can be an integer or a float. Then, the \(\tilde{s}_{n}(x,z)\) are summed on the element dimension:\[\bar{r}_{p\text{-DAS}}(x,z)=\sum_{n=1}^{N}s_{n}(x,z). \tag{9}\]  
With the original dimensionality of \(s_{n}(x,z)\) in volts, this implies that \(\bar{r}_{p\text{-DAS}}(x,z)\) is homogeneous to volt\({}^{1/p}\). Thus, \(p\)-powering is necessary to recover the conventional image dimensionality in volts, as for DAS. Note that the signed \(p\)-power is used to keep the polarity of the samples. The value of the pixel obtained is then:  
\[r_{p\text{-DAS}}(x,z)=sign(\bar{r}_{p\text{-DAS}}(x,z))\cdot|\bar{r}_{p\text{- DAS}}(x,z)|^{p}. \tag{10}\]  
Note that while \(s_{n}(x,z)\) oscillates at \(f_{0}\) along the depth dimension, the nonlinear operations (such as signed \(p^{th}\) root and signed \(p\)-power) distort the shape of oscillations, and so generate artificial harmonics on \(r_{p\text{-DAS}}(x,z)\), which have no acoustic meaning. The reconstructed image must be band-pass filtered to remove these artificial harmonics. The central frequency of this band-pass filter along the \(z\)-dimension is \(f_{0}\). Note also that the frequency sampling has to be high enough (e.g., \(\geq\)8\(f_{0}\) ) to ignore the potential aliasing of artificial harmonics.  
As for FDMAS, \(p\)-DAS is a coherence-based nonlinear beamformer, but it has the advantage that it is tunable with the \(p\)-value to balance the effects of the beamformer on the images. Moreover, \(p\)-DAS preserves the sign of the oscillations through the reconstruction process thanks to the use of the signed \(p^{th}\) root and the signed \(p\)-power. Such difference and the impact on the images are investigated in Section 3.3.  
In actuality, \(p\)-DAS can be viewed as DAS applied to the signed \(p^{th}\) root of the raw signals, followed by the signed \(p\)-power, in order to re-establish the original dimension of the image. In this way, the effects of the coherent summation are reinforced. The principle of the proposed method is illustrated in Section 3.1, considering \(p\)-DAS as a DAS beamformer with adaptive weighting, as in [6]. Indeed, Equation (9) can be rearranged as:  
\[\bar{r}_{p\text{-DAS}}(x,z)=\sum_{n=1}^{N}w_{n}(x,z)\cdot s_{n}(x,z), \tag{11}\]  
where \(w_{n}(x,z)\) are the adaptive weights that depend on the \(s_{n}(x,z)\) amplitudes:  
\[w_{n}(x,z)=\frac{1}{|s_{n}(x,z)|^{\frac{p-1}{p}}}. \tag{12}\]  
Figure 1: Block diagram describing how the value of a pixel \(r_{p\text{-DAS}}(x,z)\) is obtained from the \(N\times M\) raw echo signals recorded after \(M\) plane wave transmissions, using the \(N\) channels of the probe. Once all the pixel values have been computed in this way, a band-pass filter (not represented here) centered at \(f_{0}\) is applied along the \(z\)-dimension of the complete image, in order to remove potential artificial harmonics due to nonlinear operations.