The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-SIPI-193

“Recovery of 3-D Motion from 3-D Density Images”

by Samuel Moon-Ho Song

December 1991

The motion of a deforming body is completely characterized by the velocity field (with initial position) generated by the motion. A method of computing the 3-D velocity field from 3-D cine CTs of a beating heart is proposed.

Continuum theory provides two constraints on the velocity field generated by a deforming body. Assuming that (1) the image intensity is proportional to some conserved quantity and (2) the imaged medium is incompressible, the velocity field must satisfy the divergence-free constraint and the incompressibility constraint. Computation of the velocity field using these two constraints is an ill-posed problem which may be regularized using a smoothness term. We define a penalty function as a weighted sum of the two constraining terms and the smoothness term. Minimization of this function yields the desired velocity field. It is shown that, under certain conditions on the image, a unique minimizer of the penalty exists.

Via variational calculus, it can be shown that the solution minimizing the penalty satisfies the Euler-Lagrange equations for this problem. The solution of the Euler-Lagrange equation is a set of coupled elliptic partial differential equations (PDEs). For numerical implementation, the PDEs obtained are discretized resulting in a system of linear equations A x = b where x is the solution velocity field. The matrix equation is solved using the conjugate gradient algorithm. Examples of experiments using simulated images and a cine CT of a beating heart are presented.

The traditional regularization method does not provide a rigorous approach for obtaining the so-called regularization parameters. For this reason, we reformulate the problem as a constrained minimization. Here, instead of the regularization parameters, we require knowledge of the mean-squared errors of the constraints, which is physically and intuitively more appealing.A solution (and the numerical algorithm) is obtained by the dual space method.

To download the report in PDF format click here: USC-SIPI-193.pdf (4.4Mb)