Pattern Search in Flows based on Similarity of Stream Line Segments Additional Material

As a supplement of the paper, we give further details and analysis of the globally consistent stream line segmentation algorithm. Moreover, a more detailed description of the intrinsic similarity measure for comparing the resulting segments is also provided. Notation. We make use of the following formal concepts: let v(x) denote steady differentiable vector fields with associated over two (d = 2) and three-dimensional (d = 3) flow domains D⊂ Rd with x ∈D. Parametric stream lines c(t) = x0 + ∫ t 0 v(c(u))du are curves defined through integration along v starting from a seed point x0 for an integration time t. We partition stream lines c into disjoint stream line segments si(t) by splitting c at integration times ti such that the points of c and si coincide for t ∈ [ti, ti+1]. For a stream line integrated from t0 to tn, a segmentation is defined by the sequence [t0, . . . ti, . . . tn] of segment boundaries ti. We denote the length of si by li. 1. Globally Consistent Segmentation of Stream Lines 1.1. Curve Curvatures. In this work, we demonstrate that all curve segmentation requirements can be addressed by considering intrinsic curve properties only. In particular, the requirements can be expressed only in terms of curve curvature: feature locations along a curve are identified with high curvature values, whereas low curvature points along the curve are possible segmentation points. Also, segmentation consistency can be achieved using a curvature-based normalization. Let ċ(t) := d dt c = v and c̈(t) := d dt2 c = (∇ċ) ċ denote the first two stream line derivatives in terms of vector field quantities (cf. [WT02]). Then stream line curvatures κd(t) are given (a) (b) Figure 1: Segment Merge Criteria. Pre-merge segment boundaries are colored (•), and two different average segment orientations are colored (•) and (•). (a) A pair of segments is mergeable if they both have similar average orientations. (b) A triplet of segments is mergeable if the center segment (•) has a low average total curvature compared to its neighboring segments, which have similar average orientations. in two and three dimensions d by the well-known expressions κ2 = det([ċ, c̈] ) ||ċ|| κ3 = ||ċ× c̈|| ||ċ|| . (1) The signed curvature κ2 defines the geometry of 2D curves up to rigid transformations (see, e.g., [dC76]), which allows us to use κ2 to represent their intrinsic geometry as a basis for segmentation. Although unsigned curvature κ3 does not fully define 3D curves up to rigid transformations, we will show that for segmentation it is sufficient to only consider κ3 for 3D curves. By only relying on curvature estimates, our segmentation scheme supports 2D and 3D curves in a unified way. Stream line segmentation proceeds in two phase, curvature-based splitting and subsequent segment merging, and we continue to describe both in more detail. c © The Eurographics Association 2014. Z. Wang, J. Martinez Esturo, H.-P. Seidel, T. Weinkauf / Pattern Search in Flows based on Similarity of Stream Line Segments Additional Material