Cardiac Motion Estimation

0.1

Authors:: Hari Sundar

The problem of motion estimation from cardiac cine images is ill posed, and relying solely on image similarity, even with very accurate similarity measures is not sufficient to capture the true motion of the heart. Current cardiac motion estimation methods rely on image similarity measure to drive the motion estimation, and typically incorporate a regularizer to smooth the deformation field. We propose instead to maximize the similarity between the wavelet attributes, subject to the motion estimate constrained by a mechanical model of the heart.

We present the motion problem as a non-linear pde constrained optimization problem. Most systems can be thought of as having certain control parameters $g$ , which directly or indirectly affect the process' state $\phi$ . An optimization problem can be thought of as one of finding controls $g$ and states $\phi$ such that a cost functional $\mathcal{C}(\phi, g)$ is minimized subject to $\mathcal{F}(\phi, g)=0$ . Here the cost functional $\mathcal{C}(\phi, g)$ is a measure of the how close the current state is to the desired one. $\mathcal{F}(\phi, g)=0$ is a constraint on the relationship between the states and the controls. For the problem of cardiac motion estimation, we solve for the forces $\tau$ in the myocardial fibers and obtain the states $\bf u$ , the displacements produced as a result of these forces. The constraint, $\mathcal{F}({\bf u}, \tau)=0$ is the elastodynamic model of the heart.

This code is built up on the following packages,

PETSc PETSc is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations.
Tao The Toolkit for Advanced Optimization (TAO) is aimed at the solution of large-scale optimization problems on high-performance architectures.

The main optimization loop is written using Tao. The forward problem, i.e., the elastodynamic problem is solved using the Newmark method. PETSc is used for solving the underlying linear equations in a matrix-free, multi-grid manner.

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