Detecting Activations over Graphs using Spanning Tree Wavelet Bases

We consider the detection of activations over graphs under Gaussian noise, where signals are piece-wise constant over the graph. Despite the wide applicability of such a detection algorithm, there has been little success in the development of computationally feasible methods with proveable theoretical guarantees for general graph topologies. We cast this as a hypothesis testing problem, and first provide a universal necessary condition for asymptotic distinguishability of the null and alternative hypotheses. We then introduce the spanning tree wavelet basis over graphs, a localized basis that reflects the topology of the graph, and prove that for any spanning tree, this approach can distinguish null from alternative in a low signal-to-noise regime. Lastly, we improve on this result and show that using the uniform spanning tree in the basis construction yields a randomized test with stronger theoretical guarantees that in many cases matches our necessary conditions. Specifically, we obtain near-optimal performance in edge transitive graphs, $k$-nearest neighbor graphs, and $\epsilon$-graphs

Automated parameters for troubled-cell indicators using outlier detection

In Vuik and Ryan (2014) we studied the use of troubled-cell indicators for discontinuity detection in nonlinear hyperbolic partial differential equations and introduced a new multiwavelet technique to detect troubled cells. We found that these methods perform well as long as a suitable, problem-dependent parameter is chosen. This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically. The choice of the parameter has impact on the approximation: it determines the strictness of the troubled-cell indicator. An inappropriate choice of the parameter will result in detection (and limiting) of too few or too many elements. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In this paper we will see that for each troubled-cell indicator the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting the outliers of a vector (one dimension) or matrix (two dimensions). This is done using Tukey's boxplot approach to detect which coefficients in a vector are straying far beyond others (Tukey, 1977). We provide an algorithm that can be applied to various troubled-cell indication variables. Using this technique the problem-dependent parameter that the original indicator requires is no longer necessary as the parameter will be chosen automatically

Improved Method for Detecting Local Discontinuities in CMB data by Finite Differencing

An unexpected distribution of temperatures in the CMB could be a sign of new physics. In particular, the existence of cosmic defects could be indicated by temperature discontinuities via the Kaiser-Stebbins effect. In this paper, we show how performing finite differences on a CMB map, with the noise regularized in harmonic space, may expose such discontinuities, and we report the results of this process on the 7-year Wilkinson Microwave Anisotropy Probe data.Comment: 5 pages, 6 figures; Text has been edited, in line with the PRD articl

Sensor integration for robotic laser welding processes

The use of robotic laser welding is increasing among industrial applications, because of its ability to weld objects in three dimensions. Robotic laser welding involves three sub-processes: seam detection and tracking, welding process control, and weld seam inspection. Usually, for each sub-process, a separate sensory system is required. The use of separate sensory systems leads to heavy and bulky tools, in contrast to compact and light sensory systems that are needed to reach sufficient accuracy and accessibility. In the solution presented in this paper all three subprocesses are integrated in one compact multipurpose welding head. This multi-purpose tool is under development and consists of a laser welding head, with integrated sensors for seam detection and inspection, while also carrying interfaces for process control. It can provide the relative position of the tool and the work piece in three-dimensional space. Additionally, it can cope with the occurrence of sharp corners along a three-dimensional weld path, which are difficult to detect and weld with conventional equipment due to measurement errors and robot dynamics. In this paper the process of seam detection will be mainly elaborated

Wavelet Analysis and Denoising: New Tools for Economists

This paper surveys the techniques of wavelets analysis and the associated methods of denoising. The Discrete Wavelet Transform and its undecimated version, the Maximum Overlapping Discrete Wavelet Transform, are described. The methods of wavelets analysis can be used to show how the frequency content of the data varies with time. This allows us to pinpoint in time such events as major structural breaks. The sparse nature of the wavelets representation also facilitates the process of noise reduction by nonlinear wavelet shrinkage , which can be used to reveal the underlying trends in economic data. An application of these techniques to the UK real GDP (1873-2001) is described. The purpose of the analysis is to reveal the true structure of the data - including its local irregularities and abrupt changes - and the results are surprising.Wavelets, Denoising, Structural breaks, Trend estimation

Comparing rockfall scar volumes and kinematically detachable rock masses

Scenario-based risk assessment for rockfalls, requires assumptions for different scenarios of magnitude (volume). The magnitude of such instabilities is related to the properties of the jointed rock mass, with the characteristics of the existing unfavourably dipping joint sets playing a major role. The critical factors for the determination of the maximum credible rockfall volume in a study site, the Forat Negre in Andorra, are investigated. The results from two previous analyses for the rockfall size distribution at this site are discussed. The first analysis provides the observed size distribution of the rockfall scars, and it is an empirical evidence of past rockfalls. The second one, calculates the kinematically detachable rock masses, indicating hypothetical rockfalls that might occur in the future. The later gives a maximum rockfall volume, which is one order of magnitude higher, because the persistence of the basal planes is overestimated. The tension cracks and lateral planes interrupt systematically the basal planes, exerting a control over their persistence, and restricting the formation of extensive planes and large rockfall failures. Nonetheless, the formation of basal planes across more than one spacings of tension cracks is possible and small step-path failures have been observed too. Concluding, the key factor for the determination of the maximum credible volume at the study-site is the maximum realistic length of the basal planes, penetrating into the rock mass, their spacing, and, if applied, the contribution of the rock bridges to the overall rock mass resistance.Peer ReviewedPostprint (author's final draft

Detection of Edges in Spectral Data II. Nonlinear Enhancement

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-) \neq 0$. Our approach is based on two main aspects--localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, $K_\epsilon(\cdot)$, depending on the small scale $\epsilon$. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of order ${\cal O}(\epsilon)$) satisfy $K_\epsilon*f(x) = [f](x) +{\cal O}(\epsilon)$, thus recovering both the location and amplitudes of all edges.As an example we consider general concentration kernels of the form $K^\sigma_N(t)=\sum\sigma(k/N)\sin kt$ to detect edges from the first $1/\epsilon=N$ spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, $\sigma^{exp}(\cdot)$, with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where $K_\epsilon*f(x)\sim [f](x) \neq 0$, and the smooth regions where $K_\epsilon*f = {\cal O}(\epsilon) \sim 0$. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors