## 1 Introduction  
Ultrasound (US) imaging is a non-invasive technique that allows for the visualization of the morphology and dynamics of body internal tissues in real time, making it a widely used tool in various clinical fields [1]. Nevertheless, the continuous, long-term use of US imaging, which enables the early detection of internal abnormalities [2], remains a significant challenge with existing US devices. This is primarily due to the necessity for the operator to manually hold and press the US probe against the body to ensure proper contact, as any air between the probe and the skin would reflect the US waves and impede the imaging of internal structures [3]. Recently, studies have been conducted into the development of skin-attachable probes with the objective of addressing the aforementioned issues and enabling long-term US monitoring [4]. In some studies, thin and rigid US probes with bio-adhesive hydrogels on their surfaces have been developed, demonstrating the continuous imaging of body internal tissues by attaching the probes to the body surface [5, 6]. Additionally, flexible US probes, comprising US elements arranged on flexible sheets, have been developed [7, 8, 9, 10, 11]. The ability of flexible probes to conform to the body's surface shape allows for monitoring with minimal mechanical stress on the skin. One of the challenges in applying skin-attachable probes to daily life monitoring is the limited field of view (FOV). A conventional one-dimensional (1D) US probe, comprising a single row of US elements, is capable of imaging only a single cross-section. This necessitates precise adjustments in the position and angle of the probe to ensure that the imaging cross-section is aligned with the target cross-section. To illustrate, when employing a 1D probe to image the longitudinal plane of a 6 mm diameter blood vessel, a misalignment of 3 mm in the elevation direction will result in the vessel moving out of the FOV [12]. However, the adjustment of the probe's attachment position and angle requires a comprehensive understanding of the anatomical structures, making it difficult for non-expert users to perform this task in daily life. Furthermore, even when the attachment position and angle are optimal, the relative position between the probe and the imaging target may undergo fluctuations due to the subject movement during the monitoring. A two-dimensional (2D) matrix element array allows the imaging of any cross-section within the three-dimensional field of view (FOV) [13], thereby accommodating positional and angular attachment errors. However, as the distance between elements increases, the quality of the US image deteriorates due to the grating lobe artifact, necessitating the use of a large number of elements for matrix arrays [1]. For example, the conversion of a 128-element 1D array into a 2D array with the same element spacing necessitates the inclusion of \(128\times 128=16,384\) elements. The implementation of such a significant number of elements, cables, and US transmitter/receiver circuits in a thin, skin-attachable device presents a considerable challenge. The need for a significant number of elements in 2D arrays represents a challenge, not only for skin-attachable probes but also for handheld 2D probes used for volumetric imaging [14]. In these cases, sparse configurations have been investigated as a means of suppressing grating lobes through the appropriate arrangement of the elements. For example, [PERSON] et al. developed a random array to reduce the grating lobes [15]. [PERSON] et al. proposed a spiral array configuration that effectively mitigated grating lobes while preserving uniform image quality across the FOV [16]. [PERSON] et al. optimized the element positions by minimizing the number of elements and the side lobe levels simultaneously through the use of a simulated annealing algorithm [17]. [PERSON] et al.