Sparse arrays give rise to the possibility of miniaturization of sensor arrays for ultrasound imaging. The design of sparse arrays that preserves the full-array beampattern has received much attention in the recent past. Based on the transmit/receive pair array synthesis model proposed by Hoctor and Kassam, issues related to producing effective apertures twice their corresponding physical arrays have been addressed. We consider two sparse array designs — SCOBA and SCOBAR — proposed recently and investigate the sparsity of their acquisition apertures and sub-apertures. We constructively formulate the problem of designing sparse arrays using fundamental properties from Minkowski algebra. Our investigation leads to two novel super-sparse array designs, which are sparser than SCOBA and SCOBAR and computationally more efficient for performing convolutional beamforming. They also have identical beampatterns and consequently similar lateral resolution as SCOBA and SCOBAR. We substantiate our findings using Field-II simulations.

中文翻译：

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用于超分辨率超声成像的基于 Minkowski 代数的超稀疏阵列设计

稀疏阵列产生了用于超声成像的传感器阵列小型化的可能性。保留全阵列波束图的稀疏阵列设计在最近受到了很多关注。基于 Hoctor 和 Kassam 提出的发射/接收对阵列合成模型，与产生两倍于其相应物理阵列的有效孔径相关的问题已得到解决。我们考虑最近提出的两种稀疏阵列设计——SCOBA 和 SCOBAR，并研究它们的采集孔径和子孔径的稀疏性。我们使用闵可夫斯基代数的基本性质建设性地表述了设计稀疏数组的问题。我们的研究导致了两种新颖的超稀疏阵列设计，它们比 SCOBA 和 SCOBAR 更稀疏，并且在执行卷积波束成形时计算效率更高。它们还具有相同的波束图，因此具有与 SCOBA 和 SCOBAR 相似的横向分辨率。我们使用 Field-II 模拟证实了我们的发现。