Laplacian Term Research Articles

Salient object detection based on super-pixel clustering and unified low-rank representation

In this paper, we present a novel salient object detection method, efficiently combining Laplacian sparse subspace clustering (LSSC) and unified low-rank representation (ULRR). Unlike traditional low-rank matrix recovery (LRMR) based saliency detection methods which mainly extract saliency from pixels or super-pixels, our method advocates the saliency detection on the super-pixel clusters generated by LSSC. By doing so, our method succeeds in extracting large-size salient objects from cluttered backgrounds, against the detection of small-size salient objects from simple backgrounds obtained by most existing work. The entire algorithm is carried out in two stages: region clustering and cluster saliency detection. In the first stage, the input image is segmented into many super-pixels, and on top of it, they are further grouped into different clusters by using LSSC. Each cluster contains multiple super-pixels having similar features (e.g., colors and intensities), and may correspond to a part of a salient object in the foreground or a local region in the background. In the second stage, we formulate the saliency detection of each super-pixel cluster as a unified low-rankness and sparsity pursuit problem using a ULRR model, which integrates a Laplacian regularization term with respect to the sparse error matrix into the traditional low-rank representation (LRR) model. The whole model is based on a sensible cluster-consistency assumption that the spatially adjacent super-pixels within the same cluster should have similar saliency values, similar representation coefficients as well as similar reconstruction errors. In addition, we construct a primitive dictionary for the ULRR model in terms of the local-global color contrast of each super-pixel. On top of it, a global saliency measure covering the representation coefficients and a local saliency measure considering the sparse reconstruction errors are jointly employed to define the final saliency measure. Comprehensive experiments over diverse publicly available benchmark data sets demonstrate the validity of the proposed method.

PCA Based on Graph Laplacian Regularization and P-Norm for Gene Selection and Clustering

In modern molecular biology, the hotspots and difficulties of this field are identifying characteristic genes from gene expression data. Traditional reconstruction-error-minimization model principal component analysis (PCA) as a matrix decomposition method uses quadratic error function, which is known sensitive to outliers and noise. Hence, it is necessary to learn a good PCA method when outliers and noise exist. In this paper, we develop a novel PCA method enforcing P-norm on error function and graph-Laplacian regularization term for matrix decomposition problem, which is called as PgLPCA. The heart of the method designing for reducing outliers and noise is a new error function based on non-convex proximal P-norm. Besides, Laplacian regularization term is used to find the internal geometric structure in the data representation. To solve the minimization problem, we develop an efficient optimization algorithm based on the augmented Lagrange multiplier method. This method is used to select characteristic genes and cluster the samples from explosive biological data, which has higher accuracy than compared methods.

An Organelle Correlation-Guided Feature Selection Approach for Classifying Multi-Label Subcellular Bio-Images.

Nowadays, with the advances in microscopic imaging, accurate classification of bioimage-based protein subcellular location pattern has attracted as much attention as ever. One of the basic challenging problems is how to select the useful feature components among thousands of potential features to describe the images. This is not an easy task especially considering there is a high ratio of multi-location proteins. Existing feature selection methods seldom take the correlation among different cellular compartments into consideration, and thus may miss some features that will be co-important for several subcellular locations. To deal with this problem, we make use of the important structural correlation among different cellular compartments and propose an organelle structural correlation regularized feature selection method CSF (Common-Sets of Features) in this paper. We formulate the multi-label classification problem by adopting a group-sparsity regularizer to select common subsets of relevant features from different cellular compartments. In addition, we also add a cell structural correlation regularized Laplacian term, which utilizes the prior biological structural information to capture the intrinsic dependency among different cellular compartments. The CSF provides a new feature selection strategy for multi-label bio-image subcellular pattern classifications, and the experimental results also show its superiority when comparing with several existingalgorithms.

Estimates of eigenvalues of the Laplacian by a reduced number of subsets

Chung–Grigor’yan–Yau’s inequality describes upper bounds of eigenvalues of the Laplacian in terms of subsets (“input”) and their volumes. In this paper we will show that we can reduce “input” in Chung–Grigor’yan–Yau’s inequality in the setting of Alexandrov spaces satisfying CD(0,∞). We will also discuss a related conjecture for some universal inequality among eigenvalues of the Laplacian.

RGBD Salient Object Detection by Structured Low-Rank Matrix Recovery and Laplacian Constraint

A structured low-rank matrix recovery model for RGBD salient object detection is proposed. Firstly, the problem is described by a low-rank matrix recovery, and the hierarchical structure of RGB image is added to the sparsity term. Secondly, the depth information is fused into the model by a Laplacian regularization term to ensure that the image regions which share similar depth value will be allocated to similar saliency value. Thirdly, a variation of alternating direction method is proposed to solve the proposed model. Finally, both quantitative and qualitative experimental results on NLPR1000 and NJU400 show the advantage of the proposed RGBD salient object detection model.

Estimating a smooth function on a large graph by Bayesian Laplacian regularisation

We study a Bayesian approach to estimating a smooth function in the context of regression or classification problems on large graphs. We derive theoretical results that show how asymptotically optimal Bayesian regularisation can be achieved under an asymptotic shape assumption on the underlying graph and a smoothness condition on the target function, both formulated in terms of the graph Laplacian. The priors we study are randomly scaled Gaussians with precision operators involving the Laplacian of the graph.

Decomposition of the Laplacian and pluriharmonic Bloch functions

We decompose the invariant Laplacian of the deleted unit complex ball by two directional Laplacians, tangential one and radial one. We give a characterization of pluriharmonic Bloch function in terms of the growth of these Laplacians.

Iteratively Reweighted Blind Deconvolution With Adaptive Regularization Parameter Estimation

In many realistic image processing applications, the acquired images often suffer from mixed noises and blurring, which greatly degrade the image quality. In this paper, we propose an iteratively reweighted blind deconvolution method with robust regression for obtaining high quality images with mixed noises present. First, we construct a variational regularization model, including a robust regression data term with an adaptive reweighted least square criterion, which is robust to the mixed noises. To preserve the sharp edges and suppress the noise, a total variation-based regularization term for the image is incorporated into the model. Moreover, a Laplacian regularization term is imposed on the point spread function (PSF) for better smoothness. The subsequent optimization problems for the image and the PSF are solved using the limited-memory BFGS-B algorithm suitable for the large-scale problems. In addition, to improve the practicality of the method, a variant of the generalized cross validation method is derived and adopted to automatically estimate the regularization parameters for the image and the PSF. Experiments on simulated and real images demonstrate that the proposed method is superior to the state-of-the-art methods in terms of both subjective measure and visual quality.

Bilevel Model-Based Discriminative Dictionary Learning for Recognition.

Most supervised dictionary learning methods optimize the combinations of reconstruction error, sparsity prior, and discriminative terms. Thus, the learnt dictionaries may not be optimal for recognition tasks. Also, the sparse codes learning models in the training and the testing phases are inconsistent. Besides, without utilizing the intrinsic data structure, many dictionary learning methods only employ the l0 or l1 norm to encode each datum independently, limiting the performance of the learnt dictionaries. We present a novel bilevel model-based discriminative dictionary learning method for recognition tasks. The upper level directly minimizes the classification error, while the lower level uses the sparsity term and the Laplacian term to characterize the intrinsic data structure. The lower level is subordinate to the upper level. Therefore, our model achieves an overall optimality for recognition in that the learnt dictionary is directly tailored for recognition. Moreover, the sparse codes learning models in the training and the testing phases can be the same. We further propose a novel method to solve our bilevel optimization problem. It first replaces the lower level with its Karush-Kuhn-Tucker conditions and then applies the alternating direction method of multipliers to solve the equivalent problem. Extensive experiments demonstrate the effectiveness and robustness of our method.

Computational magneto‐hydrodynamic modeling of hypersonic flows with resolved shock wave diffusion

AbstractThe present model describes magneto‐hydrodynamic (MHD) effects on diffusing shock waves in compressible flows. The accurateness and the robustness of these new density‐based numerical algorithm in his ideal form was tested on a series of benchmark numerical experiments like the present MHD shock tube [1]. Interpolation schemes are used for interpolation of density, temperature, velocity and the magnetic field. For the discretization of laplacian terms with diffusion coefficient for polyhedral meshes, we make use of the Gauss linear uncorrected schemes to deal with the case that the consecutive cell faces are non‐orthogonal. The approach described above and developed in the present project has been coupled with semi‐discrete central schemes (e.g. Greenshields et al. [2]). All computational results are validated with verified data and analytical results. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Existence of a positive solution for problems with ( p , q ) $(p,q)$ -Laplacian and convection term in R N $\mathbb{R}^{N}$

This paper provides a positive solution for the $(p,q)$ -Laplace equation in $\mathbb{R}^{N}$ with a nonlinear term depending on the gradient. The solution is constructed as the limit of positive solutions in bounded domains. Strengthening the growth condition, it is shown that the solution is also bounded. The positivity of the solution is obtained through a new comparison principle. Finally, under a stronger growth condition, we show the existence of a vanishing at infinity solution.

Some remarks on the homogeneous Boltzmann equation with the fractional Laplacian term

We study the homogeneous Boltzmann equation with the fractional Laplacian term. Working on the Fourier side we solve the resulting integral equation, and improve a previous result by Y.-K. Cho. We replace the initial data space with a certain space $\mathcal{M}^{\alpha}$ introduced by Morimoto, Wang, and Yang. This space precisely captures the Fourier image of probability measures with bounded fractional moments, providing a more natural initial condition. We show existence of a unique global solution, in addition to the expected maximal growth estimates and stability estimates. As a consequence we obtain a continuous density solution of the original equation.

Renormalized solutions to a reaction-diffusion system applied to image denoising

This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.

Thermal rectification based on phonon hydrodynamics and thermomass theory

Abstract The thermal diode is the fundamental device for phononics. There are various mechanisms for thermal rectification, e.g. different temperature dependent thermal conductivity of two ends, asymmetric interfacial resistance, and nonlocal behavior of phonon transport in asymmetric structures. The phonon hydrodynamics and thermomass theory treat the heat conduction in a fluidic viewpoint. The phonon gas flowing through the media is characterized by the balance equation of momentum, like the Navier-Stokes equation for fluid mechanics. Generalized heat conduction law thereby contains the spatial acceleration (convection) term and the viscous (Laplacian) term. The viscous term predicts the size dependent thermal conductivity. Rectification appears due to the MFP supersession of phonons. The convection term also predicts rectification because of the inertia effect, like a gas passing through a nozzle or diffuser.

Supervised Matrix Factorization Hashing for Cross-Modal Retrieval

The target of cross-modal hashing is to embed heterogeneous multimedia data into a common low-dimensional Hamming space, which plays a pivotal part in multimedia retrieval due to the emergence of big multimodal data. Recently, matrix factorization has achieved great success in cross-modal hashing. However, how to effectively use label information and local geometric structure is still a challenging problem for these approaches. To address this issue, we propose a cross-modal hashing method based on collective matrix factorization, which considers both the label consistency across different modalities and the local geometric consistency in each modality. These two elements are formulated as a graph Laplacian term in the objective function, leading to a substantial improvement on the discriminative power of latent semantic features obtained by collective matrix factorization. Moreover, the proposed method learns unified hash codes for different modalities of an instance to facilitate cross-modal search, and the objective function is solved using an iterative strategy. The experimental results on two benchmark data sets show the effectiveness of the proposed method and its superiority over state-of-the-art cross-modal hashing methods.

Fokker–Planck equations for stochastic dynamical systems with symmetric Lévy motions

The Fokker–Planck equations for stochastic dynamical systems, with non-Gaussian α-stable symmetric Lévy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker–Planck equations on either a bounded or infinite domain. Under a specified condition, the scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein–Uhlenbeck system and the double-well system are shown.

Identification of Mixed Causal-Noncausal Models in Finite Samples

Gourieroux and Zakoian (2013) propose to use noncausal models to parsimoniously capture nonlinear features often observed in financial time series and in particular bubble phenomena. In order to distinguish causal autoregressive processes from purely noncausal or mixed causal-noncausal ones, one has to depart from the Gaussianity assumption on the error distribution. Financial (and to a large extent macroeconomic) data are characterized by large and sudden changes that cannot be captured by the Normal distribution, which explains why leptokurtic error distributions are often considered in empirical finance. By means of Monte Carlo simulations, this paper investigates the identication of mixed causal-noncausal models in finite samples for different values of the excess kurtosis of the error process. We compare the performance of the MLE, assuming a t-distribution, with that of the LAD estimator that we propose in this paper. Similar to Davis, Knight and Liu (1992) we find that for infinite variance autoregressive processes both the MLE and LAD estimator converge faster. We further specify the general asymptotic normality results obtained in Andrews, Breidt and Davis (2006) for the case of t-distributed and Laplacian distributed error terms. We first illustrate our analysis by estimating mixed causal-noncausal autoregressions to model the demand for solar panels in Belgium over the last decade. Then we look at the presence of potential noncausal components in daily realized volatility measures for 21 equity indexes. The presence of a noncausal component is confirmed in both empirical illustrations.

An improved KGF‐SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows

SummaryThe kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF‐SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF‐SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF‐SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF‐SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two‐dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF‐SPH and improved KGF‐SPH. The numerical results show that the improved KGF‐SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF‐SPH. Copyright © 2015 John Wiley & Sons, Ltd.

Spectral Determinants on Mandelstam Diagrams

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $${|\omega|^2}$$ , where $${\omega}$$ is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form $${\omega}$$ . As an important intermediate result we prove a decomposition formula of the type of Burghelea–Friedlander–Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.

On some semilinear equation in $R^4$ containing a Laplacian term and involving nonlinearity with exponential growth

In this paper, we prove a multiplicity result for some semilinear elliptic equation of biharmoninc type in $R^4$ containing a Laplacian term. The nonlinear term exhibits an exponential growth.