Directional Analysis of 2D Cardiac Motion Slices Using the Discrete Helmholtz Hodge Decomposition

The 2D Discrete Helmholtz Hodge Decomposition (DHHD) is a candidate for decomposition of cardiac motion vector field (MVFs) from modalities such as echocardiography, allowing analyses of rotational, radial and harmonic MVFs separately and the tracking of centers of flow phenomena. However, the literature contains differing opinions about how boundary conditions (BCs) should be applied. In the proposed study, we investigate two sets of BCs, found in the literature, using synthetic motion vector fields: (i) zero BCs for all nodes along the domain boundary; (ii) a single zero BC for the first node. We found that (i) produces a decomposition with Normalised Root Mean Square Error (NRMSE) below 1%, while (ii) produces NRMSE greater than 200% when the input field contains a harmonic component. Further, we demonstrate the decomposition of a synthetic motion field from the Extended Cardiac Torso Phantom v2 into radial, rotational and harmonic components.