#### Technical Report USC-SIPI-158

“Computation of 3-D Velocity Fields from 3-D Cine CT Images of a Human Heart”

by Samuel M. Song and Richard M. Leahy

June 1990

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 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.

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 AX = 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.

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