Spatially variant regularization of lateral displacement measurement using variance

The purpose of this work is to confirm the effectiveness of our proposed spatially variant displacement component-dependent regularization for our previously developed ultrasonic two-dimensional (2D) displacement vector measurement methods, i.e., 2D cross-spectrum phase gradient method (CSPGM), 2D autocorrelation method (AM), and 2D Doppler method (DM). Generally, the measurement accuracy of lateral displacement spatially varies and the accuracy is lower than that of axial displacement that is accurate enough. This inaccurate measurement causes an instability in a 2D shear modulus reconstruction. Thus, the spatially variant lateral displacement regularization using the lateral displacement variance will be effective in obtaining an accurate lateral strain measurement and a stable shear modulus reconstruction than a conventional spatially uniform regularization. The effectiveness is verified through agar phantom experiments. The agar phantom 60 mm (height) × 100 mm (lateral width) × 40 mm (elevational width)] that has, at a depth of 10 mm, a circular cylindrical inclusion (dia. = 10 mm) of a higher shear modulus (2.95 and 1.43 × 10⁶ N/m², i.e., relative shear modulus, 2.06) is compressed in the axial direction from the upper surface of the phantom using a commercial linear array type transducer that has a nominal frequency of 7.5-MHz. Because a contrast-to-noise ratio (CNR) expresses the detectability of the inhomogeneous region in the lateral strain image and further has almost the same sense as that of signal-to-noise ratio (SNR) for strain measurement, the obtained results show that the proposed spatially variant lateral displacement regularization yields a more accurate lateral strain measurement as well as a higher detectability in the lateral strain image (e.g., CNRs and SNRs for 2D CSPGM, 2.36 vs 2.27 and 1.74 vs 1.71, respectively). Furthermore, the spatially variant lateral displacement regularization yields a more stable and more accurate 2D shear modulus reconstruction than the uniform regularization (however, for the regularized relative shear modulus reconstructions, slightly accurate, e.g., for 2D CSPGM, 1.51 vs 1.50). These results indicate that the spatially variant displacement component-dependent regularization will enable the 2D shear modulus reconstruction to be used as practical diagnostic and monitoring tools for the effectiveness of various noninvasive therapy techniques of soft tissue diseases (e.g., breast, liver cancers). Application of the regularization to the elevational displacement will also increase the stability of a three-dimensional (3D) reconstruction.