Laplacian Smoothing Research Articles

A Bayesian Method for Incorporating Self‐Similarity Into Earthquake Slip Inversions

AbstractDistributions of coseismic slip help illuminate many properties of earthquakes, including fault geometry, stress changes, frictional properties, and potential future hazard. Slip inversions take observations and calculate slip at depth, but there are a number of commonly adopted assumptions such as minimizing the second spatial derivative of slip (the Laplacian) that have little physical basis and potentially bias the result. In light of growing evidence that fault slip shows fractal properties, we suggest that this information should be incorporated into slip inversions as a form of regularization, instead of Laplacian smoothing. We have developed a Bayesian approach to efficiently solve for slip incorporating von Karman regularization. In synthetic tests, our approach retrieves fractal slip better than Laplacian regularization, as expected, but even performs comparably, or better, when the input slip is not fractal. We apply this to the 2014 Mw 6.0 Napa Valley earthquake on a two‐segment fault using Interferometric Synthetic Aperture Radar (InSAR) and Global Positioning System (GPS) data. We find the von Karman and Laplacian inversions give similar slip magnitude but in different locations, and the von Karman solution has much tighter confidence bounds on slip than the Laplacian solution. Differences in earthquake slip due to the regularization technique could have important implications for the interpretation and modeling of stress changes on the causative and neighboring faults. We therefore recommend that choice of regularization method should be routinely made explicit and justified and that von Karman regularization is a better default than Laplacian.

Structured sparse K-means clustering via Laplacian smoothing

We propose a structured sparse K-means clustering algorithm that learns the cluster assignments and feature weights simultaneously. Compared to previous approaches, including K-means in MacQueen [28] and sparse K-means in Witten and Tibshirani [46], our method exploits the correlation information among features via the Laplacian smoothing technique, so as to achieve superior clustering accuracy. At the same time, the relevant features learned by our method are more structured, hence have better interpretability. The practical benefits of our method are demonstrated through extensive experiments on gene expression data and face images.

GETOpt mesh smoothing: Putting GETMe in the framework of global optimization-based schemes

We present a novel approach to mesh smoothing, called GETOpt, by using a variation of the Geometric Element Transformation Method (GETMe) which falls within the framework of global optimization-based schemes and mesh quality improvement methods. We introduce a global mesh quality measure, which is inspired by rigorous convergence results concerning the size of the geometric elements (non-degeneracy and comparable sizing) and whose gradient flow yields a combination of GETMe and Laplace smoothing. Under this scheme, it can be shown rigorously that the tetrahedral mesh never degenerates. Furthermore, numerical tests show that the sizing of the mesh elements become more comparable and that the FEM gives better approximations in case the PDE problem being solved demonstrates isotropic physical behavior. More specifically, we present in this article an example where the mesh is used to solve Poisson's equation using the standard Galerkin finite element method with piecewise linear and order 2 functions on the low-order mesh.

Deeper Insights Into Graph Convolutional Networks for Semi-Supervised Learning

Many interesting problems in machine learning are being revisited with new deep learning tools. For graph-based semi-supervised learning, a recent important development is graph convolutional networks (GCNs), which nicely integrate local vertex features and graph topology in the convolutional layers. Although the GCN model compares favorably with other state-of-the-art methods, its mechanisms are not clear and it still requires considerable amount of labeled data for validation and model selection. In this paper, we develop deeper insights into the GCN model and address its fundamental limits. First, we show that the graph convolution of the GCN model is actually a special form of Laplacian smoothing, which is the key reason why GCNs work, but it also brings potential concerns of over-smoothing with many convolutional layers. Second, to overcome the limits of the GCN model with shallow architectures, we propose both co-training and self-training approaches to train GCNs. Our approaches significantly improve GCNs in learning with very few labels, and exempt them from requiring additional labels for validation. Extensive experiments on benchmarks have verified our theory and proposals.

Design optimization and additive manufacturing of nodes in gridshell structures

In this paper, two new design approaches are proposed to design structural nodes of complex shapes for gridshell structures, seeking improved structural performance and design efficiency by using the transitional section method and the bi-directional evolutionary structural optimization (BESO) method, respectively. Detailed design methodologies are introduced and compared with two types of conventionally designed structural nodes in real use, i.e., Seele node and Sun Valley node. Finite element analyses are conducted to evaluate the stress distribution, maximum stress and mean compliance of different structural nodes. The results show that, although the BESO node has the highest stiffness and the lowest structural volume, it seems to have a higher maximum stress level. Multiple load cases are considered in the analysis and design optimization processes. Moreover, the application of Laplacian smoothing in structural node design effectively reduces the stress concentration. Prototypes of the newly designed structural nodes are fabricated by additive manufacturing, which enables the rapid and precise manufacturing of customized structural nodes optimized for specific loading and boundary conditions.

Adaptive <italic>h</italic>-Refinement for the RWG-Based EFIE

An adaptive h -refinement algorithm is investigated for the electric field integral equation applied to a conducting missile target. The process incorporates the advancing front Delaunay algorithm to refine the triangular-cell surface mesh in conjunction with Laplacian smoothing to maintain the mesh quality. The performance of several error estimators used to guide the refinement process is compared. Two of the estimators used within the adaptive refinement procedure work well to improve the accuracy of the computed radar cross section.

Features Fusion based on the Fisherface and Simplification of the Laplacian Smoothing Transform

Features Fusion based on the Fisherface and Simplification of the Laplacian Smoothing Transform

Contradictory of the Laplacian Smoothing Transform and Linear Discriminant Analysis Modeling to Extract the Face Image Features

Laplacian smoothing transform uses the negative diagonal element to generate the new space. The negative diagonal elements will deliver the negative new spaces. The negative new spaces will cause decreasing of the dominant characteristics. Laplacian smoothing transform usually singular matrix, such that the matrix cannot be solved to obtain the ordered-eigenvalues and corresponding eigenvectors. In this research, we propose a modeling to generate the positive diagonal elements to obtain the positive new spaces. The secondly, we propose approach to overcome singularity matrix to found eigenvalues and eigenvectors. Firstly, the method is started to calculate contradictory of the laplacian smoothing matrix. Secondly, we calculate the new space modeling on the contradictory of the laplacian smoothing. Moreover, we calculate eigenvectors of the discriminant analysis. Fourth, we calculate the new space modeling on the discriminant analysis, select and merge features. The proposed method has been tested by using four databases, i.e. ORL, YALE, UoB, and local database (CAI-UTM). Overall, the results indicate that the proposed method can overcome two problems and deliver higher accuracy than similar methods.

Multi-body separation simulation with an improved general mesh deformation method

This paper presents an improved general mesh deformation method and its application in multi-body separation simulation. Intelligent Laplacian smoothing algorithm is combined with previously developed radial basis function (RBF) mesh deform technology to avoid mesh quality decreasing and topology damaging in successive large mesh deformation. In the intelligent Laplacian smoothing algorithm, mesh qualities of tetrahedron, triangular prism, hexahedron and pyramid are evaluated by a unified standard which is used to setup the criterion of the smoothing onset. This improved mesh deformation methodology is integrated into a solver of Navier–Stokes equations coupled with six degrees of freedom (6-DOF) moving equations. The store separation trajectory of a generic wing, pylon, and store configuration is successfully simulated to validate the ability of improved mesh deformation methodology in dealing with large magnitude of relative motion among multiple moving boundaries. Numerical simulation results agree well with experimental data, which states that the accuracy and robustness of the above numerical methods can reach the requirement of describing the traces of multi-body separation.

Graph Regularized and Locality-Constrained Coding for Robust Visual Tracking

Visual tracking is complicated due to factors, such as occlusion, background clutter, abrupt target motion, and illumination variations, among others. In recent years, subspace representation and sparse coding techniques have demonstrated significant improvements in tracking. However, performance gain in tracking has been at the expense of losing locality and similarity attributes among the instances to be encoded. In this paper, a graph regularized and locality-constrained coding (GRLC) technique that encapsulates local manifold structure of the data in order to preserve locality and similarity information among instances is proposed. The GRLC methodology incorporates a similarity-preserving term within the objective function of the locality-constrained linear coding model, thereby overcoming some of the inherent instability issues common to such coding methods. In the proposed GRLC scheme, a graph Laplacian regularizer is chosen as a smoothing operator to learn both the representation dictionary and the coefficients by preserving the local structure of the data. This graph Laplacian smoothing operator ensures that the representations vary smoothly along the geodesics of the data manifold. Thus, by deriving the objective function of the GRLC method, a discriminative dictionary of instances can be iteratively obtained and the corresponding coefficients for each candidate can be computed using this learned dictionary. Finally, an effective observation likelihood function based on reconstruction error and a simple dictionary update scheme for visual target tracking are also proposed. Experimental results on the CVPR2013 visual tracker benchmark have demonstrated a favorable performance of the proposed technique both in terms of accuracy and robustness.

Quality improvement of surface triangular mesh using a modified Laplacian smoothing approach avoiding intersection.

We present a systematic procedure to improve the qualities of triangular molecular surface meshes and at the same time preserve the manifoldness. The procedure utilizes an algorithm to remove redundant points having three or four valences and another algorithm to smooth the mesh using a modified version of Laplacian method without causing intersecting triangles. This approach can be effectively applied to any manifold surface meshes with arbitrary complex geometry. In this paper, the tested meshes are biomolecular surface meshes exhibiting typically highly irregular geometry. The results show that the qualities of the surface meshes are greatly improved and the manifoldness of the surface meshes are preserved. Compared with the original meshes, these improved molecular surface meshes can be directly applied to boundary element simulations and generation of body-fitted volume meshes using Tetgen. The procedure has been incorporated into our triangular molecular surface mesh generator, TMSmesh 2.0. It can be also used as a standalone program and works together with any other surface triangular mesh generator to obtain qualified manifold mesh. The package is downloadable at https://doi.org/10.6084/m9.figshare.5346169.v1 and can be run online at http://www.xyzgate.com.

An Improved Tomography Approach Based on Adaptive Smoothing and Ground Meteorological Observations

Using the Global Navigation Satellite System (GNSS) to sense three-dimensional water vapor (WV) has been intensively investigated. However, this technique still heavily relies on the a priori information. In this study, we propose an improved tomography approach based on adaptive Laplacian smoothing (ALS) and ground meteorological observations. By using the proposed approach, the troposphere tomography is less dependent on a priori information and the ALS constraints match better with the actual situation than the constant constraints. Tomography experiments in Hong Kong during a heavy rainy period and a rainless period show that the ALS method gets superior results compared with the constant Laplacian smoothing (CLS) method. By validation with radiosonde and European Centre for Medium-Range Weather Forecasts (ECMWF) data, we found that the introduction of ground meteorological observations into tomography can solve the perennial problem of resolving the wet refractivity in the lower troposphere and thus significantly improve the tomography results. However, bad data quality and incompatibility of the ground meteorological observations may introduce errors into tomography results.

Ternary Sparse Matrix Representation for Volumetric Mesh Subdivision and Processing on GPUs

AbstractIn this paper, we present a novel volumetric mesh representation suited for parallel computing on modern GPU architectures. The data structure is based on a compact, ternary sparse matrix storage of boundary operators. Boundary operators correspond to the first‐order top‐down relations of k‐faces to their (k − 1)‐face facets. The compact, ternary matrix storage format is based on compressed sparse row matrices with signed indices and allows for efficient parallel computation of indirect and bottom‐up relations. This representation is then used in the implementation of several parallel volumetric mesh algorithms including Laplacian smoothing and volumetric Catmull‐Clark subdivision. We compare these algorithms with their counterparts based on OpenVolumeMesh and achieve speedups from 3× to 531×, for sufficiently large meshes, while reducing memory consumption by up to 36%.

An Introduction to Laplacian Spectral Distances and Kernels: Theory, Computation, and Applications

Abstract In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Within this context, this book is intended to provide a common background on the definition and computation of the Laplacian spectral kernels and distances for geometry processing and shape analysis. To this end, we define a unified representation of the isotropic and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and their discretization in terms of the Laplacian spectrum. As main applications, we discuss the design of smooth functions and the Laplacian smoothing of noi...

Hierarchical probabilistic models for multiple gene/variant associations based on next-generation sequencing data.

MotivationThe identification of genetic variants influencing gene expression (known as expression quantitative trait loci or eQTLs) is important in unravelling the genetic basis of complex traits. Detecting multiple eQTLs simultaneously in a population based on paired DNA-seq and RNA-seq assays employs two competing types of models: models which rely on appropriate transformations of RNA-seq data (and are powered by a mature mathematical theory), or count-based models, which represent digital gene expression explicitly, thus rendering such transformations unnecessary. The latter constitutes an immensely popular methodology, which is however plagued by mathematical intractability.ResultsWe develop tractable count-based models, which are amenable to efficient estimation through the introduction of latent variables and the appropriate application of recent statistical theory in a sparse Bayesian modelling framework. Furthermore, we examine several transformation methods for RNA-seq read counts and we introduce arcsin, logit and Laplace smoothing as preprocessing steps for transformation-based models. Using natural and carefully simulated data from the 1000 Genomes and gEUVADIS projects, we benchmark both approaches under a variety of scenarios, including the presence of noise and violation of basic model assumptions. We demonstrate that an arcsin transformation of Laplace-smoothed data is at least as good as state-of-the-art models, particularly at small samples. Furthermore, we show that an over-dispersed Poisson model is comparable to the celebrated Negative Binomial, but much easier to estimate. These results provide strong support for transformation-based versus count-based (particularly Negative-Binomial-based) models for eQTL mapping.Availability and implementationAll methods are implemented in the free software eQTLseq: https://github.com/dvav/eQTLseqSupplementary information Supplementary data are available at Bioinformatics online.

Phantom experiments using soft-prior regularization EIT for breast cancer imaging

Objective. A soft-prior regularization (SR) electrical impedance tomography (EIT) technique for breast cancer imaging is described, which shows an ability to accurately reconstruct tumor/inclusion conductivity values within a dense breast model investigated using a cylindrical and a breast-shaped tank. Approach. The SR-EIT method relies on knowing the spatial location of a suspicious lesion initially detected from a second imaging modality. Standard approaches (using Laplace smoothing and total variation regularization) without prior structural information are unable to accurately reconstruct or detect the tumors. The soft-prior approach represents a very significant improvement to these standard approaches, and has the potential to improve conventional imaging techniques, such as automated whole breast ultrasound (AWB-US), by providing electrical property information of suspicious lesions to improve AWB-US’s ability to discriminate benign from cancerous lesions. Main results. Specifically, the best soft-regularization technique found average absolute tumor/inclusion errors of 0.015 S m−1 for the cylindrical test and 0.055 S m−1 and 0.080 S m−1 for the breast-shaped tank for 1.8 cm and 2.5 cm inclusions, respectively. The standard approaches were statistically unable to distinguish the tumor from the mammary gland tissue. An analysis of false tumors (benign suspicious lesions) provides extra insight into the potential and challenges EIT has for providing clinically relevant information. Significance. The ability to obtain accurate conductivity values of a suspicious lesion (>1.8 cm) detected from another modality (e.g. AWB-US) could significantly reduce false positives and result in a clinically important technology.

New Weights in Laplacian Smoothing on Triangular Mesh

Mesh smoothing is one of the basic procedures for improvement of mesh quality. Most smoothing techniques move vertices of the mesh without changing topology of the connectivity. Laplace smoothing is one of the simplest and efficient algorithm, where in each step vertex of the mesh is move to the barycenter of its neighbors. The only problem with Laplacian smoothing is surface shrinking when it is performed iteratively. In this paper, three relatively simple weights proposed in Laplacian smoothing which has less surface shrinking from the previous weights. The performance and effectiveness of the presented weights aredemonstrated on two knowing simply and doubly connected geometrical shapes respectively.

Variable-scale maps in real-time generalisation using a quadtree data structure and space deforming algorithms

ABSTRACTVariable-scale maps have been advocated by several authors in the context of mobile cartography. In the literature on real-time map generalisation, however, corresponding methods that resolve cartographic conflicts by deformation of the underlying map space together with the map foreground, are underrepresented. This paper demonstrates how the concept of a malleable space can be applied as a part of the generalisation process and incorporated into the overall methodology of point generalisation. Two different algorithms are used, a density-equalising cartogram algorithm and Laplacian smoothing. Both methods work in real-time and are datadriven. In addition, they allow for a parameterisation in combi-nation with a quadtree data structure, as well as a combination with 'classic' generalisation operators (e.g. selection, aggregation, displacement) based on the quadtree. The quadtree serves both as a spatial index for fast retrieval and search of points, and as a density estimator to inform generalisation operators. The use of the quadtree as a common spatial index provides a tool to combine variable-scale maps with classic generalisation. A combination of the two allows, at small map scales, the maintenance of detail in dense areas and data reduction in sparse areas. Additionally, it facilitates building a modular workflow for real-time map generalisation.

Implication of adaptive smoothness constraint and Helmert variance component estimation in seismic slip inversion

When ill-posed problems are inverted, the regularization process is equivalent to adding constraint equations or prior information from a Bayesian perspective. The veracity of the constraints (or the regularization matrix R) significantly affects the solution, and a smoothness constraint is usually added in seismic slip inversions. In this paper, an adaptive smoothness constraint (ASC) based on the classic Laplacian smoothness constraint (LSC) is proposed. The ASC not only improves the smoothness constraint, but also helps constrain the slip direction. A series of experiments are conducted in which different magnitudes of noise are imposed and different densities of observation are assumed, and the results indicated that the ASC was superior to the LSC. Using the proposed ASC, the Helmert variance component estimation method is highlighted as the best for selecting the regularization parameter compared with other methods, such as generalized cross-validation or the mean squared error criterion method. The ASC may also benefit other ill-posed problems in which a smoothness constraint is required.

Interactive reconstructions of cranial 3D implants under MeVisLab as an alternative to commercial planning software.

In this publication, the interactive planning and reconstruction of cranial 3D Implants under the medical prototyping platform MeVisLab as alternative to commercial planning software is introduced. In doing so, a MeVisLab prototype consisting of a customized data-flow network and an own C++ module was set up. As a result, the Computer-Aided Design (CAD) software prototype guides a user through the whole workflow to generate an implant. Therefore, the workflow begins with loading and mirroring the patients head for an initial curvature of the implant. Then, the user can perform an additional Laplacian smoothing, followed by a Delaunay triangulation. The result is an aesthetic looking and well-fitting 3D implant, which can be stored in a CAD file format, e.g. STereoLithography (STL), for 3D printing. The 3D printed implant can finally be used for an in-depth pre-surgical evaluation or even as a real implant for the patient. In a nutshell, our research and development shows that a customized MeVisLab software prototype can be used as an alternative to complex commercial planning software, which may also not be available in every clinic. Finally, not to conform ourselves directly to available commercial software and look for other options that might improve the workflow.