An image-space Morse decomposition for 2D vector fields

Abstract

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.

Morse Decomposition of 3D Piecewise Linear Vector Fields

“…In the second stage, we sub-divide the voxels that fall in the resulting Morse sets from the first stage and re-compute the local Morse decomposition within each Morse set region with a higher grid resolution. This enables us to partially improve the smoothness of the Morse set boundaries for visualization, which is in a much similar spirit of the image-space Morse decomposition framework [5]. In particular, our refinement framework can be described as follows.…”

“…This integration time will be increased gradually at regions that may contain flow recurrent dynamics to refine the obtained results. In addition, our refinement framework borrows ideas from the recently introduced image-space Morse decomposition for 2D vector fields [5], and procedurally sub-divides the 3D cells to refine the boundary of the obtained Morse sets. Since the input vector fields are defined on regular grids, a CUDA implementation of our framework is possible, which largely improves the computation performance of our framework.…”