An image-space Morse decomposition for 2D vector fields

Morse decompositions have been proposed to compute and represent the topological structure of steady vector fields. Compared to the conventional differential topology, Morse decomposition and the resulting Morse Connection Graph (MCG) is numerically stable. However, the granularity of the original Morse decomposition is constrained by the resolution of the underlying spatial discretization, which typically results in non-smooth representation. In this work, an Image-Space Morse decomposition (ISMD) framework is proposed to address this issue. Compared to the original method, ISMD first projects the original vector field onto an image plane, then computes the Morse decomposition based on the projected field with pixels as the smallest elements. Thus, pixel-level accuracy can be achieved. This ISMD framework has been applied to a number of synthetic and real-world steady vector fields to demonstrate its utility. The performance of the ISMD is carefully studied and reported. Finally, with ISMD an ensemble Morse decomposition can be studied and visualized, which is shown useful for visualizing the stability of the Morse sets with respect to the error introduced in the numerical computation and the perturbation to the input vector fields.