Gradient-Preserving Cuts for Scalar Representations of Vector Fields

我们提出了一种方法来表示向量场（通常由流动模拟或流量测量方法产生）作为标量场的（共）梯度。由于已知对于光滑标量场一般来说这是不可能的，我们在标量场中引入了保持梯度切割的概念。我们给出了一个精确的定义，并研究了它的性质。这样，向量场可以精确地表示为标量场的（共）梯度（2D），或两个标量场梯度的叉乘（3D）。在此基础上，我们探索是否可以建立一种比传统集成技术更快、更准确的流线集成替代方法。如果成功，这将影响流可视化中的许多标准方法。我们的目标是通过引入新的基于纹理的流可视化技术和精确计算三维无发散流的Clebsch地图的新方法来证明这一点。

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.