Robust curve reconstruction with k-order α-shapes

Abstract

We combine classical concepts from different disciplines -those of α-hull and α-shape from computational geometry, splitting data into training and test sets from artificial intelligence, density-based spatial clustering from data mining, and moving average from time series analysis -to develop a robust algorithm for reconstructing the shape of a curve from noisy samples.The novelty of our approach is two-fold. First, we introduce the notion of k-order α-hull and α-shape -generalizations of α-hull and α-shape. Second, we use white noise to "train" our k-order α-shaper, i.e., to choose the right values of α and k.The difference of the k-order α-hull and α-shape from the α-hull and α-shape is also two-fold. First, k-order α-hull and α-shape provide a robust estimate of the shape by ignoring outliers. Second, it reconstructs the "inner" shape, with the amount of "digging" into the data controlled by k. Index Terms: I.5.2 [Pattern Recognition]: Design MethodologyPattern analysis; J.2 [Physical Sciences and Engineering]: Earth and atmospheric sciences- α -HULL AND α -SHAPELet P ⊂ R 2 be a finite planar set. Classical tools, defining "shape" of the set, are those of the α-hull and α-shape of P [1]. Adapting the notions from [1] to make our exposition simpler, we make the following definitions (cf. Definitions 2, 3, 4 in [1]):Definition 1. Let α > 0. An α-ball is an (open) disc of radius α. An empty α-ball is an α-ball containing no points of P. The α-hull of P is the complement of the union of all empty α-balls. Two points p, q ∈ P are α-neighbors if there exists an empty α-ball having both p and q on its boundary. The α-shape of P is the straightline graph, whose vertices are points in P and whose edges connect α-neighbors.For an example of α-shape see Figure 1, (a) and (b). HANDLING OUTLIERS WITH k-ORDER α -HULL AND α -SHAPEIt is easy to imagine situations in which several outliers will severely harm the shape produced by the α-hull or α-shape (see Fig. 1, (a) and (b)). To reduce the impact of outliers on shape reconstruction, we introduce a generalization of the α-hull and α-shape, the k-order α-hull and k-order α-shape:Definition 2. Let α > 0, k ≥ 0. An α-ball is an (open) disc of radius α. An empty α-ball is an α-ball containing exactly k points of P. The k-order α-hull of P is the complement of the union of all * empty α-balls. Two points p, q ∈ P are called α-neighbors if there exists an empty α-ball having both p and q on its boundary. The k-order α-shape of P is the straight-line graph, whose vertices are points in P and whose edges connect a-neighbors.Of course, α-hull and α-shape are just 0-order α-hull and α-shape. With k ≥ 1 we allow the k-order α-hull and α-shape to ignore some amount of outliers (Fig. 1). RECONSTRUCTING INNER SHAPE WITH k-ORDER α -HULL AND α -SHAPEα-hull and α-shape, being generalizations of the convex hull, capture the "outer" shape of P. At the same time, in many applications it is of interest to infer the "inner" shape of the set. (For a polygonal domain its inner shape may be represe...

An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes

“…Most of these reconstruction approaches first perform noise removal through clustering, thinning, or averaging, which often leads to a significant blunting of features. Robustness to outliers has also beento a lesser extent-investigated, with approaches ranging from data clustering [Son10] and robust statistics [FCOS05], to k-order alpha shapes [KP08], spectral methods [KSO04] and 1 -minimization [ASGCO10]-but often at the cost of a Simplification. Polygonal curve simplification has received attention from many fields, including cartography, computer graphics, computer vision and other medical and scientific imaging applications.…”

Visual Analytics for Spatial Clustering: Using a Heuristic Approach for Guided Exploration

“…In order to overcome these difficulties, we use a new data structure that generalizes the α-shape. Introduced by Krasnoshchekov and Polishchuk [30], it was termed k-order α-shape. Its main motivation is to exclude outliers from the shape.…”

“…Since real data is usually noisy, we employ a recently developed generalization of the α-shape, called k-order α-shape [30]. The k-order α-shape is a generalization of both the α-shape and of the k-hull [9]a statistical data depth measuring tool.…”

“…The idea behind the k-order α-shape is that the disk defining the shape may contain up to k data points; thus the ordinary α-hull is obtained by setting k = 0. Using k > 0 allows one to remove the noise locally by digging deeper into the data; the amount digging is controlled by k. Formal definitions of the k-order α-shape and α-hull and their combinatorial and algorithmic properties are given in [30]; here we illustrate the concept in Figure 5 (in the full version of the paper we will discuss noise reduction in much greater length). The left column in the figure shows the α-shapes for different α's on the same dataset.…”