Interpolation from sparse imaging data is typically required to achieve dense, three-dimensional quantification of left ventricular function. Although the heart muscle is known to be incompressible, this fact is ignored by most previous approaches that address this problem. In this paper, we present a method to reconstruct a dense representation of the three-dimensional, incompressible deformation of the left ventricle from tagged MR images acquired in both short-axis and long axis orientations. The approach applies a smoothing, divergence-free, vector spline to interpolate velocity fields at intermediate discrete times such that the collection of velocity fields integrate over time to match the observed displacement components. Through this process, the method yields a dense estimate of a displacement field that matches our observations and also corresponds to an incompressible motion.
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