## 4 Conclusions  
This paper proposes a nonlinear beamformer, \(p\)-DAS, to enhance image quality. This beamformer is tested here in simulation and also under experimental conditions in the context of ultrafast plane wave imaging. The main benefits of this method are to improve lateral resolution, to better reject side lobes, and, as a consequence, to improve the contrast ratio, depending on the \(p\)-value used. However, \(p\)-DAS tends to distort the speckle statistics as the nonlinear operations increase the gap between the coherent and incoherent targets that compose the speckle, which leads to decreased CNR.  
Another interesting aspect is that the proposed method is an extended version of an already well-known nonlinear beamformer: FDMAS. The signed \(p\)-power allows the removal of the PSF flat  
Figure 7: Comparison of PSFs obtained with 2-DAS and FDMAS. The scatterer was placed at (\(x_{0}=0\) mm, \(z_{0}=35\) mm) on the numerical phantom. B-mode log-compressed images are shown over a 40-dB dynamic range for 2-DAS (**a**) and FDMAS (**b**). Their respective normalized lateral profiles are shown in (**c**).  
main lobe in FDMAS and better side-lobe rejection. Finally, the \(p\)-value can be tuned to find the best trade-off between CR and CNR, as required by the user.  
Several applications are expected for this \(p\)-DAS beamforming. First, as \(p\)-DAS highlights punctual targets in noisy environments, it appears to be a very promising tool for bubble localization. For instance, [PERSON] et al. reconstructed high-resolution vascular maps using ultrafast imaging of microbubbles [22]. The efficiency of the method relied on the assumption that the bubbles are punctual and separable sources, and then on the detection of a large number of them in B-mode images. \(p\)-DAS also represents a solution to increase the number of detected bubbles, as when the echo signals are weak due to ultrasound attenuation (e.g., deep tissues, through the skull, high frequency ultrasound). Another application might be imaging of cavitation for therapeutic monitoring. In [23], [PERSON] et al. proposed enhancing the resolution of passive imaging of cavitation with the adaptive phase coherence factor based beamformer [10]. \(p\)-DAS might also be an appropriate tool to reject interference between the punctual bubbles that compose the cavitation cloud, which generate strong artifacts on cavitation maps. Moreover, the performance of \(p\)-DAS might be investigated with other high frame rate imaging strategies. In the case of multi-line transmit [4; 12], \(p\)-DAS might be a good candidate to reject the cross-talk for reception.  
For the methodology, further studies can be conducted on the function used to compress the adaptive weights. Indeed, the coupling of '\(p^{th}\)-root and \(p\)-power' has been used in this paper, while it might be interesting to use other couplings of functions, such as 'exponential and logarithm', to understand their impact on image quality.  
The authors would like to thank [PERSON] and [PERSON] for their valuable discussions, and for sharing their experience of FDMAS. This work was supported by LABEX CELYA (ANR-10-LABX-0060) and was performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of the Universite de Lyon, within the programme _'Investissements d'Avenir'_ (ANR-11-IDEX-0007), operated by the French National Research Agency (ANR).  
[PERSON], [PERSON] and [PERSON] developed the methodology, analyzed the data, and wrote the paper. [PERSON] and [PERSON] are the leaders of the project, and contributed with reading and improving the manuscript.  
The authors declare that they have no conflicts of interest.