MixTURE: L1-Norm-Based Mixed Second-Order Continuity in Strain Tensor Ultrasound Elastography

Energy-based displacement tracking of ultrasound images can be implemented by optimizing a cost function consisting of a data term, a mechanical congruency term, and first- and second-order continuity terms. This approach recently provided a promising solution to 2-D axial and lateral displacement tracking in ultrasound strain elastography. However, the associated second-order regularizer only considers the unmixed second derivatives and disregards the mixed derivatives, thereby providing suboptimal noise suppression and limiting possibilities for total strain tensor imaging. We propose to improve axial, lateral, axial shear, and lateral shear strain estimation quality by formulating and optimizing a novel -norm-based second-order regularizer that penalizes both mixed and unmixed displacement derivatives. We name the proposed technique -MixTURE, which stands for -norm Mixed derivative for Total UltRasound Elastography. When compared with simulated ground-truth results, the mean structural similarity (MSSIM) obtained with -MixTURE ranged 0.53-0.86 and the mean absolute error (MAE) ranged 0.00053-0.005. In addition, the mean elastographic signal-to-noise ratio (SNR) achieved with simulated, experimental phantom, and in vivo breast datasets ranged 1.87-52.98, and the mean elastographic contrast-to-noise ratio (CNR) ranged 7.40-24.53. When compared with a closely related existing technique that does not consider the mixed derivatives, -MixTURE generally outperformed the MSSIM, MAE, SNR, and CNR by up to 37.96%, 67.82%, and 25.53% in the simulated, experimental phantom, and in vivo datasets, respectively. These results collectively highlight the ability of -MixTURE to deliver highly accurate axial, lateral, axial shear, and lateral shear strain estimates and advance the state-of-the-art in elastography-guided diagnostic and interventional decisions.