Adaptive multiscale detection of filamentary structures in a background of uniform random points

We are given a set of $n$ points that might be uniformly distributed in the unit square $[0,1]^2$. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with $C^{\alpha}$-norm bounded by $\beta$. An asymptotic detection threshold exists in this problem; for a constant $T_-(\alpha,\beta)>0$, if the number of points sampled from the curve is smaller than $T_-(\alpha,\beta)n^{1/(1+\alpha)}$, reliable detection is not possible for large $n$. We describe a multiscale significant-runs algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information $\alpha$ or $\beta$ in advance, provided that the number of points on the curve exceeds $T_*(\alpha,\beta)n^{1/(1+\alpha)}$. This algorithm therefore has an optimal detection threshold, up to a factor $T_*/T_-$. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will have area $2/n$ and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices all are strips of the same anisotropy and whose edges link such strips to their ``good continuations.'' The point-cloud data are reduced to counts that measure membership in strips. Each anisotropy graph is reduced to a subgraph that consist of strips with significant counts. The algorithm rejects $\mathbf{H}_0$ whenever some such subgraph contains a path that connects many consecutive significant counts.Comment: Published at http://dx.doi.org/10.1214/009053605000000787 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Virtual restoration of the Ghent altarpiece using crack detection and inpainting

In this paper, we present a new method for virtual restoration of digitized paintings, with the special focus on the Ghent Altarpiece (1432), one of Belgium's greatest masterpieces. The goal of the work is to remove cracks from the digitized painting thereby approximating how the painting looked like before ageing for nearly 600 years and aiding art historical and palaeographical analysis. For crack detection, we employ a multiscale morphological approach, which can cope with greatly varying thickness of the cracks as well as with their varying intensities (from dark to the light ones). Due to the content of the painting (with extremely many fine details) and complex type of cracks (including inconsistent whitish clouds around them), the available inpainting methods do not provide satisfactory results on many parts of the painting. We show that patch-based methods outperform pixel-based ones, but leaving still much room for improvements in this application. We propose a new method for candidate patch selection, which can be combined with different patch-based inpainting methods to improve their performance in crack removal. The results demonstrate improved performance, with less artefacts and better preserved fine details

Verification of Unstructured Grid Adaptation Components

Adaptive unstructured grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Known differences and previously unknown differences in grid adaptation components and their integrated processes are identified here for study. Unstructured grid adaptation tools are verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. Laminar flow over a delta wing and turbulent flow over an ONERA M6 wing are verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and a moment. The scalar problems illustrate known differences in a grid adaptation component implementation and a previously unknown interaction between components. The wing adaptation cases in the current study document a clear improvement to existing grid adaptation procedures. The stage is set for the infusion of verified grid adaptation into production fluid flow simulations

A machine learning approach for efficient uncertainty quantification using multiscale methods

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.Comment: Journal of Computational Physics (2017

Generalised additive multiscale wavelet models constructed using particle swarm optimisation and mutual information for spatio-temporal evolutionary system representation

A new class of generalised additive multiscale wavelet models (GAMWMs) is introduced for high dimensional spatio-temporal evolutionary (STE) system identification. A novel two-stage hybrid learning scheme is developed for constructing such an additive wavelet model. In the first stage, a new orthogonal projection pursuit (OPP) method, implemented using a particle swarm optimisation(PSO) algorithm, is proposed for successively augmenting an initial coarse wavelet model, where relevant parameters of the associated wavelets are optimised using a particle swarm optimiser. The resultant network model, obtained in the first stage, may however be a redundant model. In the second stage, a forward orthogonal regression (FOR) algorithm, implemented using a mutual information method, is then applied to refine and improve the initially constructed wavelet model. The proposed two-stage hybrid method can generally produce a parsimonious wavelet model, where a ranked list of wavelet functions, according to the capability of each wavelet to represent the total variance in the desired system output signal is produced. The proposed new modelling framework is applied to real observed images, relative to a chemical reaction exhibiting a spatio-temporal evolutionary behaviour, and the associated identification results show that the new modelling framework is applicable and effective for handling high dimensional identification problems of spatio-temporal evolution sytems