Gradient-Preserving Cuts for Scalar Representations of Vector Fields

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We propose an approach to represent vector fields (usually resulting from flow simulation or flow measurement approaches) as the (co-)gradient of a scalar field. Since it is known that in general this is impossible for smooth scalar fields, we introduce the concept of gradient-preserving cuts in scalar fields. We give an exact definition and study its properties. With this, vector fields can be exactly represented as (co-)gradients of scalar fields (2D), or as cross product of two scalar field gradients (3D). We explore whether based on this we can establish an alternative approach to stream line integration being both faster and more accurate than traditional integration techniques. If successful, this influences a number of standard approaches in Flow Visualization. We aim in demonstrating this by introducing new texture based Flow Visualization techniques and new methods for exact Clebsch map computation for 3D divergence-free flows.