## Materials and Methods  
### US Experimental Platform  
The new ultrasound platform includes a 1024-element US probe with a specific wiring of the elements and a high-channels density US scanner driven by 4 individual Vantage 256 systems (Verasonics, Kirkland, WA, USA), synchronized together. The development of this US platform was possible thanks to a close collaboration between two laboratories, by sharing together the required materials (the US probe and four US systems) and their competences. The materials were used together for all the experiments presented in this work. **US probe. ** The customized US probe (Vermon, Tours, France) (Figure 1d) for cardiac applications is a 32 \(\times\) 35 matrix array with a 0.3 mm pitch in both \(x\) and \(y\) directions and a square footprint with sides of about 10 mm. In the \(y\) direction, the 9 th, 17 th and 25 th lines are not connected, resulting in a total number of 1024 active elements. The 1024 elements of the probe were carefully spread out across the 4 systems, with 2 connectors per system, each element being physically connected to the same single channel both in transmission and reception (one-element-to-one-channel design). The physical addressing of each US element was  
Figure 1: **(a)** Block diagram of the entre setup including the Master, the 3 Slaves, the synchronization box and the US probe; **(b)** Diagram of the 1024-element distribution across 8 connectors. Each color corresponds to a connector (C); **(c)** Picture of the 4 synchronized Vantage systems and the 3-D US probe connected; **(d)** the customized 3-D US cardiac probe. specifically chosen for allowing various array sparsity combinations while minimizing the number of required driving systems. A detailed diagram illustrating the element distribution is shown in Figure 1b). Indeed, each connector of 128 channels controls 4 entire rows (of 32 elements each) distributed across every 4 rows in the matrix. In this way, for a full active array, 4 systems are required, but regular or irregular sparsity of the US array can be activated using a lower number of US systems. All active channels can be used simultaneously in transmission and reception. Moreover, the connectors incorporate impedance matching circuits for power transmission, if required. The central frequency was chosen as 3 MHz with a 70.9% bandwidth of \(-6\) dB. Therefore, the physical addressing of the US prototype was determined as a compromise between the simplicity of internal connection, flexibility for testing various downsampled configurations and the reduction of the number of systems needed. The interest of several simple sparsity configurations to generate exploitTable 4-D US images was investigated in simulation and then validated experimentally. **Ultrasound systems. ** The Vantage 256 scanners US systems (Verasonics, Kirkland, WA, USA) drive 256 elements each (2 connectors of 128 elements per board in each system) and are a research system entirely programmable (data acquisition and processing platform), designed to allow the development of new advanced US imaging strategies. They have per-channel arbitrary waveform transmit/receive generation capability with full access to RF data from every channel. The systems can be controlled by using Matlab through open user interface, allowing access to RF data of all channels and in particular, to display reconstructed data for ultrasound guidance in real time and offering extreme flexibility and precision. Standard or customized US transducers arrays can be connected to the systems, after declaration of the specifications of the US probe. **Synchronization box. ** A special Multi-System Synchronization Module capable of distributing the Master system clock to up to 8 Vantage systems via common HDMI cables designed and commercialized by Verasonics was used. One of the 4 systems is the \"Master\", providing the 250 MHz clock, trigger and other signals for accurate temporal synchronization.