Enhanced Vector Field Visualization via Lagrangian Accumulation Supplemental Material: Discussion and Additional Results

This document provides additional description to certain con1 tent of the paper and additional results of the A field-based 2 flow visualization. 3 1. Detailed Derivation of Relation between FTLE and Path4 line Orientation Vector 5 Recall that in Section 5.2 of the paper, the Lagrangian accu6 mulation is applied to collect vector-valued properties. Specif7 ically, we use it to accumulate the flow vectors scaled by the 8 integration step size along the pathlines. The resulting A field 9 is a vector field. If we consider a forward accumulation with 10 a time window [t0, t0 +T ], the resulted vector at each sampling 11 point is an orientation vector that points from the starting point 12 to the end point of the integral curve [1] based on vector calcu13 lus. We denote this vector as VSE(x) = φ t0+T t0 (x)−φ t0 t0 (x) based 14 on the notion of flow map [2]. After getting this vector-valued 15 A field, we compute its discrete gradient using finite differ16 ence. For simplicity, we consider a 2D regular grid. The dis17 crete gradient of the vector-valued A field at a sampling point 18 x = (xi,y j) is given as follows 19 F = dVSE (x) dx = V xSE (xi+1, j)−V xSE (xi−1, j) xi+1, j(t0)−xi−1, j(t0) V xSE (xi, j+1)−V xSE (xi, j−1) yi, j+1(t0)−yi, j−1(t0) V ySE (xi+1, j)−V ySE (xi−1, j) xi+1, j(t0)−xi−1, j(t0) V ySE (xi, j+1)−V ySE (xi, j−1) yi, j+1(t0)−yi, j−1(t0) (1)