Gradient Covariance Research Articles

Log‐density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods

AbstractA metric tensor for Riemann manifold Monte Carlo particularly suited for nonlinear Bayesian hierarchical models is proposed. The metric tensor is built from symmetric positive semidefinite log‐density gradient covariance (LGC) matrices, which are also proposed and further explored here. The LGCs generalize the Fisher information matrix by measuring the joint information content and dependence structure of both a random variable and the parameters of said variable. Consequently, positive definite Fisher/LGC‐based metric tensors may be constructed not only from the observation likelihoods as is current practice, but also from arbitrarily complicated nonlinear prior/latent variable structures, provided the LGC may be derived for each conditional distribution used to construct said structures. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.

Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance

Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance

A Single-Image Noise Estimation Algorithm Based on Pixel-Level Low-Rank Low-Texture Patch and Principal Component Analysis.

Noise level is an important parameter for image denoising in many image-processing applications. We propose a noise estimation algorithm based on pixel-level low-rank, low-texture subblocks and principal component analysis for white Gaussian noise. First, an adaptive clustering algorithm, based on a dichotomy merge, adaptive pixel-level low-rank matrix construction method and a gradient covariance low-texture subblock selection method, is proposed to construct a pixel-level low-rank, low-texture subblock matrix. The adaptive clustering algorithm can improve the low-rank property of the constructed matrix and reduce the content of the image information in the eigenvalues of the matrix. Then, an eigenvalue selection method is proposed to eliminate matrix eigenvalues representing the image to avoid an inaccurate estimation of the noise level caused by using the minimum eigenvalue. The experimental results show that, compared with existing state-of-the-art methods, our proposed algorithm has, in most cases, the highest accuracy and robustness of noise level estimation for various scenarios with different noise levels, especially when the noise is high.

Low-Dose CT Image Denoising Model Based on Sparse Representation by Stationarily Classified Sub-Dictionaries

Low-dose computed tomography (LDCT) technique is an important imaging modality, but LDCT images are always severely degraded by mottle noise and streak artifacts. The recently proposed nonlocally centralized sparse representation (NCSR) algorithm has good performance in natural image denoising, but it suffers from residual streak artifacts and can't preserve edges structure information well when implemented in LDCT image denoising. In addition, it has high computational complexity. To address this problem, in this paper, we propose an improved model, i.e. SNCSR model, based on the stationary PCA sub-dictionaries, nonlocally centralized sparse representation and relative total variation. In the SNCSR model, in order to learn more accurate sub-dictionaries, the LDCT image is preprocessed by the improved total variation (ITV) model in which the weighted coefficient of the regularization term is constructed depending on a clipped and normalized local activity. In addition, the maximum eigenvalue of the gradient covariance matrix of the image patch is used to distinguish edge structure information from background region so that the restored image can be represented more sparsely. Moreover, unlike the NCSR model that needs to learn sub-dictionaries in each outer loop, the proposed model learns stationary sub-dictionaries only once before iteration starts, which shorten the computation time significantly. At last, the relative total variation (RTV) algorithm is applied to further reduce the residual artifacts in the recovered image more thoroughly. The experiments are performed on the simulated pelvis phantom, the actual thoracic phantom and the clinical abdominal data. Compared with several other competitive denoising algorithms, both subjective visual effect and objective evaluation criteria show that the proposed SNCSR model has lower computational complexity and can improve LDCT images quality more effectively.

Understanding the seismic disorder attribute and its applications

While reflections associated with conformal sedimentary layers are usually coherent and continuous, other reflections, such as mass-transport complexes, karst collapse, and salt, may appear to be quite chaotic, without any specific orientation. We also may see chaotic events that have little to do with the target geology but rather are artifacts due to variations in the overburden and surface or budget limitations resulting in a suboptimum acquisition program. While some of these artifact issues can be handled at the time of processing, a certain level of randomness remains in most seismic data volumes. Geologic features of interpretational interest, such as fault damage zones, unconformities, and gas chimneys, often have randomness associated with them. The published work on randomness describes techniques such as cross correlation, which measures randomness associated with noise, the chaos attribute based on eigen-analysis of gradient covariance, and the seismic disorder attribute based on application of second derivative operators in the three axial directions in a 3D seismic volume. We apply the seismic disorder attribute to two different data sets and find that it is a useful attribute for assessing the signal-to-noise ratio and data quality, in addition to helping delineate damage zones associated with large faults and the interior of salt dome structures.

3D unconstrained and geologically constrained stochastic inversion of airborne vertical gravity gradient data

We present an inversion tool for airborne gravity gradient data that yields a 3D density model using stochastic methods i.e. cokriging and conditional simulation. This method uses geostatistical properties of the measured gravity gradient to estimate a 3D density model whose gravity response fits the measured gravity gradient anomaly. Linearity between gravity gradient data and density allows estimation of the model (density) covariance using observed data, i.e. we adjust iteratively the density covariance matrix by fitting experimental and theoretical gravity gradient covariance matrices. Inversion can be constrained by including densities known at some locations. In addition we can explore various reasonable solutions that honour both the estimated density covariance model and the gravity gradient data using geostatistical simulation.The proposed method is first tested with two synthetic datasets generated from a sharp-boundary model and a smooth stochastic model respectively. The results show the method to be capable of retrieving models compatible with the true models; it also allows the integration of complex a priori information. The technique is then applied to gravity gradient survey data collected for the Geological Survey of Canada in the area of McFaulds Lake (Ontario, Canada) using the Falcon airborne gravity system. Unconstrained inversion returns a density model that is geologically plausible and the computed response exactly fits the observed gravity gradient anomaly.

색상과 에지 공분산 특징을 이용한 변화영역 검출

Abstract This paper proposes a change detection method based on the covariance matrices of color and edge gradient in a color video. The YCbCr color format was used instead of RGB. The color covariance matrix was calculated from the CbCr-channels and the edge gradient covariance matrix was calculated from the Y-channels. The covariance matrices were effectively calculated at each pixel by calculating the sum, squared sum, and sum of two values' multiplication of a rectangle area using the integral images from a background image. The background image was updated by a running the average between the background image and a current frame. The change areas in a current frame image against the background were detected using the Mahalanobis distance, which is a measure of the statistical distance using covariance matrices. The experimental results of an expressway color video showed that the proposed approach can effectively detect change regions for color and edge gradients against the background.

Role of Convective Structures and Background Turbulence in the Dry Convective Boundary Layer

We quantify the role of the convective buoyant structures and the remainder turbulence, here called background turbulence, in the convective atmospheric boundary layer in horizontally homogeneous, dry and barotropic conditions. Three filtering methods to separate the structures and the background turbulence are first evaluated. These are: short-time averaging, Fourier filtering and proper orthogonal decomposition. The Fourier method turns out to be the most appropriate for the present purpose. The decomposition is applied to two cases: one with no mean flow and another with moderate mean wind speed. It is shown that roughly 85 % of the vertical flux of the potential temperature and about 72 % of the kinetic energy is carried by the structures in the mixed layer in both cases. The corresponding percentage for the potential temperature variance is 81 % in the zero mean-wind case and 76 % in the moderate mean-wind case. The structures are responsible for as much as 94 % of the momentum flux in the mixed layer of the moderate mean-wind case. In the surface layer the background turbulence is generally more important than the structure contribution in both cases. The budget of the potential temperature flux is analyzed in detail and it is shown that its turbulent transport term is mostly built up by the structures but also the interaction between the structures and the background turbulence plays a significant role. The other important budget terms are shown to be dominated by the structures except for the pressure–temperature gradient covariance.

Learning sparse gradients for variable selection and dimension reduction

Variable selection and dimension reduction are two commonly adopted approaches for high-dimensional data analysis, but have traditionally been treated separately. Here we propose an integrated approach, called sparse gradient learning (SGL), for variable selection and dimension reduction via learning the gradients of the prediction function directly from samples. By imposing a sparsity constraint on the gradients, variable selection is achieved by selecting variables corresponding to non-zero partial derivatives, and effective dimensions are extracted based on the eigenvectors of the derived sparse empirical gradient covariance matrix. An error analysis is given for the convergence of the estimated gradients to the true ones in both the Euclidean and the manifold setting. We also develop an efficient forward-backward splitting algorithm to solve the SGL problem, making the framework practically scalable for medium or large datasets. The utility of SGL for variable selection and feature extraction is explicitly given and illustrated on artificial data as well as real-world examples. The main advantages of our method include variable selection for both linear and nonlinear predictions, effective dimension reduction with sparse loadings, and an efficient algorithm for large p, small n problems.

Effects of axisymmetric strain on a passive scalar field: modelling and experiment

Homogeneous, approximately isotropic turbulence at two Taylor-scale Reynolds numbers,Rλ=50, 190, with a mean transverse temperature gradient is passed through an axisymmetric contraction. The effects of the straining on the velocity field, and on the passive scalar field, are investigated within the contraction as are the effects of releasing the strain in the post-contraction region. Components of the fluctuating velocity and scalar gradient covariance are measured in order to understand their relation to the large-scale anisotropy of the flow. The scale-dependent spectral evolution of the scalar is also determined. A tensor model is constructed to predict the evolution of the fluctuating scalar gradient covariance. The model constants are determined in the post-contraction relaxation region, where the flow geometry does not vary. The model is shown to perform well throughout the flow, even in the contraction in which the geometry varies. Rapid distortion theory is applied to the scalar field in the contraction, and its solutions are compared to the experimental results.

Image field categorization and edge/corner detection from gradient covariance

Edges, corners, and vertices in an image correspond to 1D (one-dimensional) and 2D discontinuities in the intensity surface of the underlying scene. Ridges and peaks correspond to 1D and 2D extrema in it. All of them can be characterized by the distribution of gradients, particularly by the dimensionality of it. The approach to image field categorization here is to construct a covariance matrix of the gradient vector in each small window and apply the canonical correlation analysis to it. Schwarz's inequality on the matrix determinant and the related differential equation is the key to this analysis. We obtain two operators P/sub EG/ and Q/sub EG/ to categorize the image field into a unidirectionally varying region (UNIVAR), an omidirectionally varying region (OMNIVAR), and a nonvarying region. We investigate the conditions under which their absolute maximum response, i.e. P/sub EG/=1 and Q/sub EG/=1, occurs in the small window and show that they are, respectively, the desired 1D and 2D discontinuities/extrema and OMNIVAR, is in many cases, a 1D pattern in polar coordinates. This leads to an algorithm to obtain further classification and accurate localization of them into edges, ridges, peaks, corners, and vertices through detailed analysis in the informative (varying) axis of them. We examined and compared the performance of the operators and the localization algorithm on various types of images and various noise levels. The results indicate that the proposed method is superior with respect to stability, localization, and resolution.