Deep Null Space Learning Improves Dataset Recovery for High Frame Rate Synthetic Transmit Aperture Imaging

Synthetic transmit aperture (STA) imaging benefits from the two-way dynamic focusing to achieve optimal lateral resolution and contrast resolution in the full field of view, at the cost of low frame rate (FR) and low signal-to-noise ratio (SNR). In our previous studies, compressed sensing-based STA (CS-STA) and minimal
${l}_{{2}}$
-norm least squares (LS-STA) methods were proposed to recover the complete STA dataset from fewer Hadamard-encoded (HE) plane wave (PW) transmissions. Results demonstrated that, compared with STA imaging, CS/LS-STA can maintain the high resolution of STA in the full field of view and improve the contrast in the deep region with increased FR. However, these methods would introduce errors to the recovered STA datasets and subsequently produce severe artifacts to the beamformed images, especially in the shallow region. Recently, we discovered that the theoretical explanation for the error introduced in the LS-STA-based recovery is that the LS-STA method neglects the null space component of the real STA dataset. To deal with this problem, we propose to train a convolutional neural network under the null space learning framework (CNN-Null) to estimate the missing null space component) for high-accuracy recovery of the STA dataset from fewer HE PW transmissions. The mapping between the low-quality STA dataset (i.e., the range space component of the real STA dataset recovered using the LS-STA method) and the missing null space component of the real STA dataset was learned by the network with the high-quality STA dataset (obtained using full HE STA (HE-STA) imaging) as training labels. The performance of the proposed CNN-Null method was compared with the baseline LS-STA, conventional STA, and HE-STA methods, in terms of the visual quality, the normalized root mean square error (NRMSE), the generalized contrast-to-noise ratio (gCNR), and the lateral full-width at half-maximum (FWHM). The results demonstrate that the proposed method can greatly improve the recovery accuracy of the STA datasets (lower NRMSE) and, therefore, effectively suppress the artifacts presented in the images (especially in the shallow region) obtained using the LS-STA method (with a gCNR improvement of 0.4 in the cross-sectional carotid artery images). In addition, the proposed method can maintain the high lateral resolution of STA with fewer (as low as 16) PW transmissions, as LS-STA does.