  Convolutional Neural Networks (CNN) have shown promising results for
displacement estimation in UltraSound Elastography (USE). Many modifications
have been proposed to improve the displacement estimation of CNNs for USE in
the axial direction. However, the lateral strain, which is essential in several
downstream tasks such as the inverse problem of elasticity imaging, remains a
challenge. The lateral strain estimation is complicated since the motion and
the sampling frequency in this direction are substantially lower than the axial
one, and a lack of carrier signal in this direction. In computer vision
applications, the axial and the lateral motions are independent. In contrast,
the tissue motion pattern in USE is governed by laws of physics which link the
axial and lateral displacements. In this paper, inspired by Hooke's law, we
first propose Physically Inspired ConsTraint for Unsupervised Regularized
Elastography (PICTURE), where we impose a constraint on the Effective Poisson's
ratio (EPR) to improve the lateral strain estimation. In the next step, we
propose self-supervised PICTURE (sPICTURE) to further enhance the strain image
estimation. Extensive experiments on simulation, experimental phantom and in
vivo data demonstrate that the proposed methods estimate accurate axial and
lateral strain maps.
