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“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.