As the case of initial curvature estimation, the initial surface matching results are not sufficient for accurate estimation of the dense motion vector field, due to the uniqueness factor as well as the noise factor. Again, under the assumption that the displacement field of the LV wall is sufficiently smooth, any changes of direction and magnitude take place in a gradual manner over space, a regularizing smoothing functional, which includes both an initial estimate adherence term and a local smoothness term, is constructed. It should be clear that strong, unique matches from the bending energy matching process should be preserved, while ambiguous matches should be smoothed over by their neighboring matches. The confidence measures from the initial match are used as weighting coefficients in the smoothing functional.
As the curvature smoothing procedure, our vector field smoothing process is embedded in the irregular triangular grid generated from Delaunay tessellation, instead of the Euclidean space. It is invariant with respect to rotation and translation, and it does not need to construct local Cartesian coordinates, as proposed in our earlier approach.
The functional for smooth motion estimation is given by the following expression, which is similar to the curvature smoothing functional:
In this equation, is the domain of the surface in which the finite element grid from the Delaunay triangulation is embedded, Dis the optimal smoothed motion vector field, Dis the initial motion vector estimate, and is the confidence measure matrix from the initial estimation, which again includes two parts. One weights the overall goodness of the initial match and is inversely proportional to the minimum bending energy measure corresponding to the best match, while the other is a measure of match uniqueness in the search area, and is proportional to the difference between the mean bending energy for all the points and the minimum bending energy for the best match within the window.
The solution of this functional will yield a dense set of smoothed motion vectors that better estimates true LV wall motion. Repeating the process for all the LV wall surfaces, we have a complete set of motion flow fields between pairs of surfaces during the cardiac cycle.