Characterization and applications of linear and nonlinear three-dimensional phase portraits

Abstract

Volumetric phase portraits are mathematical primitives that describe vector field topology in a concise representation surrounding included critical points using a set of coupled differential equations. Phase portraits are classified into one of seven canonical forms depending on the phase portrait eigenvalues, and Jordan form. In addition, the dynamic behavior of these models is defined in terms of their index and signature functions. Relevant volumetric linear and nonlinear phase portrait models for both compressible and incompressible flow are discussed and classified, including their allowable topologies and characteristics. Volumetric phase portrait models are a compact descriptor of smoothly varying vector fields and are used to analyze, compress, and reconstruct vector fields. In addition to their application to vector fields, linear and nonlinear volumetric phase portraits may be used effectively in digital video and volumetric images. Two methods for reconstructing a vector field from its component phase portraits are presented, depending on the complexity of the flow and its boundary behavior. The first method uses a weighted superposition of phase portraits surrounding internal critical points to reconstruct vector fields consisting of non-turbulent, continuous flow and containing a finite number of spatially isolated critical points. For vector fields that violate the necessary assumptions for superposition based reconstruction, a discontinuous block processing method is used. Phase portraits are robust descriptors of field topology and are insensitive to additive noise. Also, an octave tree decomposition and subsequent merge algorithm is presented that models field topology with appropriately scaled phase portrait models. Vector field compression is demonstrated at a compression ratio of 156:1. Other applications include digital video compression, digital video scene and shot transition detection, and volumetric image classifications.