Image-based Inverse Problems to Identify Three-dimensional Displacement Field

An identification methodology of three-dimensional displacement within a biological soft tissue is presented in line with non-invasive testing by means of X-ray CT images. The high-compliance and heterogeneity of biological soft tissue induces its complex and nonlinear displacement field hardly measured by conventional methods, although the displacement field is necessary for determinations of its material properties. The full-field digital image correlation method[

1

] with the Levenberg- Marquardt method [

2

] has been established to identify the displacement field by using a pair of twodimensional digital images with crisp contrast in its intensity distribution. The proposed approach is an extension of the method to three-dimensional problems. A pair of three-dimensional images, which are constituted from multi-slice CT images captured with small intervals, is obtained from the deformed and undeformed states of a soft tissue. To identify the displacement field, the undeformed state image is virtually deformed by a tentative displacement field described by the tri-cubic B-Spline basis functions with unknown parameters initially set to tentative values. The error of this identification is evaluated in terms of intensity difference between actually and virtually deformed images. The unknown parameters are successively modified to minimize the error. Primary obstacle against the three-dimensional image correlation is an explosion of computational cost for solving the huge simultaneous equations successively constructed during the error minimization procedure. We reduce the calculation time to a reasonable one by utilizing the numerical symmetry of the coefficient matrix and the locality of the basis functions. Thus, a fast parallel solver for the inverse problem is proposed to minimize the computational time. In this problem, moreover, an indeterminacy of the displacement field arises from noisy images with obscure contrast in its intensity distribution, as is the common case with the X-ray CT images. In this study, therefore, we propose a formulation stabilized by imposing incompressibility of the materials. By using an experimental specimen under compression load, the validity of the algorithm is checked.