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Scalar Representation of 2D Steady Vector Fields

Holger Theisel, Michael Motejat, Janos Zimmermann, Christian Rössl

TL;DR

This paper addresses representing a 2D steady vector field ${\mathbf v}$ via two scalars ${a},{b}$ so that stream lines correspond to isolines of ${a}$ and ${b}$ increases with unit speed along integration, enabling local isolation-based visualization.It relaxes the ideal requirement by introducing gradient-preserving cuts and a careful treatment of critical points, allowing a unique, computable solution even when a strict correspondence does not exist.For simple fields, the authors formulate a domain transformation to a constant field and derive boundary- and interior-based procedures to compute ${a}$ and ${b}$, including optimization and minimal-gradient criteria.Results on non-trivial fields show that ${a}$-isocontours align with stream lines while cuts introduce discontinuities in ${a}$ that preserve gradient continuity, indicating practical viability for visualization and connectivity tasks.

Abstract

We introduce a representation of a 2D steady vector field ${\mathbf v}$ by two scalar fields $a$, $b$, such that the isolines of $a$ correspond to stream lines of ${\mathbf v}$, and $b$ increases with constant speed under integration of ${\mathbf v}$. This way, we get a direct encoding of stream lines, i.e., a numerical integration of ${\mathbf v}$ can be replaced by a local isoline extraction of $a$. To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields $a$ and $b$ can be computed. We show several evaluations on non-trivial vector fields.

Scalar Representation of 2D Steady Vector Fields

TL;DR

This paper addresses representing a 2D steady vector field ${\mathbf v}$ via two scalars ${a},{b}$ so that stream lines correspond to isolines of ${a}$ and ${b}$ increases with unit speed along integration, enabling local isolation-based visualization.It relaxes the ideal requirement by introducing gradient-preserving cuts and a careful treatment of critical points, allowing a unique, computable solution even when a strict correspondence does not exist.For simple fields, the authors formulate a domain transformation to a constant field and derive boundary- and interior-based procedures to compute ${a}$ and ${b}$, including optimization and minimal-gradient criteria.Results on non-trivial fields show that ${a}$-isocontours align with stream lines while cuts introduce discontinuities in ${a}$ that preserve gradient continuity, indicating practical viability for visualization and connectivity tasks.

Abstract

We introduce a representation of a 2D steady vector field by two scalar fields , , such that the isolines of correspond to stream lines of , and increases with constant speed under integration of . This way, we get a direct encoding of stream lines, i.e., a numerical integration of can be replaced by a local isoline extraction of . To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields and can be computed. We show several evaluations on non-trivial vector fields.
Paper Structure (11 sections, 1 theorem, 19 equations, 10 figures)

This paper contains 11 sections, 1 theorem, 19 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem description and analysis
  3. Related work
  4. Solution on simple vector fields
  5. Computing $a$, $b$
  6. Gradient-preserving cuts
  7. Treatment of critical points
  8. Placement of the cuts
  9. Discretization and solution
  10. Results
  11. Conclusion

Key Result

Corollary 1

Every 2D linear vector field can be described as co-gradient field of a scalar field $a$ with a gradient-preserving cut.

Figures (10)

  • Figure 1: Finding the fields $a,b$ can be interpreted as searching for a domain transformation of ${\mathbf v}$ to $(0,1)^T$.
  • Figure 2: Definition of $\tau_0({\mathbf x}),\tau_1({\mathbf x})$, ${\mathbf d}_0({\mathbf x}),{\mathbf d}_1({\mathbf x})$.
  • Figure 3: setup of local moving coordinate system.
  • Figure 4: left: 1D field with gradient-preserving cut; right: its derivative.
  • Figure 5: left: 2D field with gradient-preserving cut as height field; right: isocontours
  • ...and 5 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Corollary 1