The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

The Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

Stray Current Protection and Monitoring Systems: Characteristic Quantities, Assessment of Performance and Verification

Electrified transportation systems (ETSs) are affected by stray current problems impacting within and outside the right of way on reinforcement, buried metal structures and foundations. Stray current protection systems have recently been integrated in the track structure. Track electrical quantities are, thus, usually measured to assess track insulation and protection efficiency but should be backed up by additional measurements at the affected structures and installations, in order to assess their exposure and risk of corrosion. Ideally, a stray current monitoring system proceeds from the measurement of these quantities, to data collection and archival, to data presentation, analysis and prediction. Feasible sensors and probes, the impact of environmental conditions and uncertainty are considered for the measurement at the physical level. Data analysis is critically reviewed considering the variability of operating conditions and the effectiveness of each quantity as indicator of track insulation and protection efficiency. Given the normal spread of values, for data presentation and interpretation, suitable techniques are considered based on averaging, curve similarity and feature extraction, and also for the task of assessing compliance to limits or reference values and establishing a trend that may drive informed maintenance decision.

Online Signature Verification Based on a Single Template via Elastic Curve Matching

Person verification using online handwritten signatures is one of the most widely researched behavior-biometrics. Many signature verification systems typically require five, ten, or even more signatures for an enrolled user to provide an accurate verification of the claimed identity. To mitigate this drawback, this paper proposes a new elastic curve matching using only one reference signature, which we have named the curve similarity model (CSM). In the CSM, we give a new definition of curve similarity and its calculation method. We use evolutionary computation (EC) to search for the optimal matching between two curves under different similarity transformations, so as to obtain the similarity distance between two curves. Referring to the geometric similarity property, curve similarity can realize translation, stretching and rotation transformation between curves, thus adapting to the inconsistency of signature size, position and rotation angle in signature curves. In the matching process of signature curves, we design a sectional optimal matching algorithm. On this basis, for each section, we develop a new consistent and discriminative fusion feature extraction for identifying the similarity of signature curves. The experimental results show that our system achieves the same performance with five samples assessed with multiple state-of-the-art automatic signature verifiers and multiple datasets. Furthermore, it suggests that our system, with a single reference signature, is capable of achieving a similar performance to other systems with up to five signatures trained.

Curve Similarity Model for Real-Time Gait Phase Detection Based on Ground Contact Forces

This paper proposed a new novel method to adaptively detect gait patterns in real time through the ground contact forces (GCFs) measured by load cell. The curve similarity model (CSM) is used to identify the division of off-ground and on-ground statuses, and differentiate gait patterns based on the detection rules. Traditionally, published threshold-based methods detect gait patterns by means of setting a fixed threshold to divide the GCFs into on-ground and off-ground statuses. However, the threshold-based methods in the literature are neither an adaptive nor a real-time approach. In this paper, the curve is composed of a series of continuous or discrete ordered GCF data points, and the CSM is built offline to obtain a training template. Then, the testing curve is compared with the training template to figure out the degree of similarity. If the computed degree of similarity is less than a given threshold, they are considered to be similar, which would lead to the division of off-ground and on-ground statuses. Finally, gait patterns could be differentiated according to the status division based on the detection rules. In order to test the detection error rate of the proposed method, a method in the literature is introduced as the reference method to obtain comparative results. The experimental results indicated that the proposed method could be used for real-time gait pattern detection, detect the gait patterns adaptively, and obtain a low error rate compared with the reference method.