# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 1 Introduction  
High frame rate ultrasound imaging has the potential to broaden the already vast application field of medical ultrasonography [1]. Thus, since the introduction of conventional focused ultrasound imaging, researchers have been working on acquisition schemes that allow for increases in frame rates. Studies of multiline acquisition [2; 3; 4; 5] and wide-beam insonification [6; 7; 8] have converged into three main acquisition schemes that are currently being translated from research to clinical applications: synthetic transmit aperture [9; 10; 11], plane-wave (PW) [12; 13; 14], and diverging-wave (DW) methods [15]. By emitting either PWs or DWs and then reconstructing the full images of the medium, these techniques can increase the frame rate of conventional ultrasound by a factor of hundreds, with the only remaining limiting factor being the pulse-echo propagation time. However, the lack of beam focusing in emission considerably impairs the signal-to-noise ratio (SNR) of the recorded data, and results in decreased quality of the reconstructed images. Thus, spatial compounding has been proposed for each of these techniques [9; 15; 16; 17; 18; 19]. After a succession of insonifications/image reconstructions, a final improved image is obtained by coherently compounding the low-quality images. However, successive emission and reception events result in a drop in the frame rate, which once again yields a trade-off between image quality and frame rate [20]. Parallel with the development of acquisition schemes, the impact of coded excitations on image quality has been studied. It was shown in [21; 22; 23; 24] that excitation signals withhigher time bandwidth products increase penetration depth and SNR, which results in images with better overall quality. In these studies, frequency-modulated signals (i.e., chirps) were emitted and backscattered echoes were pulse compressed using a variety of compression filters. Other studies proposed the use of phase-modulated pseudo-random sequences as excitation signals [21; 25]. Autocorrelation-based filters were then used to compress the echoes. However, all of these techniques consist of modification of the excitation signals and use of post-processing filters to compress the echoes. The frame acquisition rate remains the same, while the image quality is improved. Later, the concept of simultaneous emission of ultrasonic waves by several sources was introduced [26; 27; 28]. The inherent problem of echo separation was solved through the complementarity property of the emission signals (i.e., phase-modulated Golay and Hadamard sequences). As the binary encoded excitation signals carry more energy than classic pulses, the SNR is improved, which again results in better image quality. Simultaneous emission of waves was also addressed by [29], where the total bandwidth of the ultrasound probe was split, with each source emitting in a separate sub-band. However, this technique results in increased SNR without increasing the frame acquisition rate. One of the first coded schemes to increase the frame rate was proposed by [30]. Based on simultaneous emission of phase-modulated pseudo-random sequences, Ref. [30] built a forward model that associated the received signals with the echoes that would be received if each source was emitting alone. By solving the inverse problem, they reconstructed images of the medium at a higher frame rate than for conventional synthetic transmit aperture. However, due to the lack of orthogonal codes, the method is limited to the use of only two simultaneous transmitters, which limits the gain in frame rate. Later, a similar approach was applied to PWs [31; 32]. Once again, the bottleneck was the lack of fully orthogonal codes. The image quality was decreased and the method was limited to two simultaneous PWs. Given that the unknowns in the forward model are the echoes received after each individual source emission, the inverse problem is ill-posed, which leads to decreased image quality without regularization.