Vector Field Analysis and Synthesis Using Three-Dimensional Phase Portraits

Abstract

Tools from dynamical systems theory are used to decompose 3-D vector fields into simpler building blocks consisting of critical points and phase portraits. A robust critical point detector is developed for three dimensions. Samples from the vector field surrounding each critical point are then used to estimate the associated linear phase portrait, which is written as a 3 × 3 matrix. The estimated matrix may be categorized into one of seven canonical forms by its eigenvalues, which remain consistent under an arbitrary differentiable mapping of the region. The original vector field behavior is estimated using two methods. In one technique, the global behavior is reconstructed using a weighted superposition of phase portraits. For more complex field patterns, a regular partition is imposed prior to phase portrait representation, and each individual partition is decomposed into a separate phase portrait. These methods provide a means of extracting the relevant features and information from the vector field in the form of a higher level descriptor and provide a means of reconstructing the field qualitatively from those descriptors. The method is demonstrated on fluid flow data.