Bayesian multiscale analysis of images modeled as Gaussian Markov random fields

P-spline smoothing for spatial data collected worldwide

Image Classification Based on Bayes Discriminant Functions

Circular data can be found across many areas of science, for instance meteorology (e.g., wind directions), ecology (e.g., animal movement directions), or medicine (e.g., seasonality in disease onset). The special nature of these data means that conventional methods for non-periodic data are no longer valid. In this paper, we consider wrapped Gaussian processes and introduce a spatial model for circular data that allow for non-stationarity in the mean and the covariance structure of Gaussian random fields. We use the empirical equivalence between Gaussian random fields and Gaussian Markov random fields which allows us to considerably reduce computational complexity by exploiting the sparseness of the precision matrix of the associated Gaussian Markov random field. Furthermore, we develop tunable priors, inspired by the penalized complexity prior framework, that shrink the model toward a less flexible base model with stationary mean and covariance function. Posterior estimation is done via Markov chain Monte Carlo simulation. The performance of the model is evaluated in a simulation study. Finally, the model is applied to analyzing wind directions in Germany.

A powerful modelling tool for spatial data is the framework of Gaussian Markov random fields (GMRFs), which are discrete domain Gaussian random fields equipped with a Markov property. GMRFs allow us to combine the analytical results for the Gaussian distribution as well as Markov properties, thus allow for the development of computationally efficient algorithms. Here we briefly review popular spatial GMRFs, show how to construct them, and outline their recent developments and possible future work.

Exospheric tomography is a computational 3-D imaging technique that provides the estimates of the neutral density distributions of the terrestrial exosphere from space-based ultraviolet (UV) measurements. Variability of neutral densities during geomagnetically active conditions has been previously reported, motivating the development of time-dependent tomographic techniques that can characterize both the spatial and temporal scales of densities during these events. However, solving the dynamic exospheric tomography problem can be challenging owing to its ill-posedness. In this letter, we introduce a novel algorithm for 4-D exospheric tomography based on optimal interpolation (OI) and Gaussian Markov random field (GMRF) theory. The OI analysis enables iterative reconstructions of the exosphere when a statistical background field is provided. Its mean is selected from previous knowledge of the exosphere, and its covariance matrix is estimated using GMRF. To validate the performance, we apply our proposed methodology to six days of UV data acquired by National Aeronautics and Space Administration (NASA) two-wide angle imaging neutral-atom spectrometer (TWINS) mission during the geomagnetic storm that occurred on June 15, 2008.

Assessing the impact of uncertainties in ice-sheet models is a major and challenging issue that needs to be faced by the ice-sheet community to provide more robust and reliable model-based projections of ice-sheet mass balance. In recent years, uncertainty quantification (UQ) has been increasingly used to characterize and explore uncertainty in ice-sheet models and improve the robustness of their projections. A typical UQ analysis involves first the (probabilistic) characterization of the sources of uncertainty followed by the propagation and sensitivity analysis of these sources of uncertainty. Previous studies concerned with UQ in ice-sheet models have generally focused on the last two steps but paid relatively little attention to the preliminary and critical step of the characterization of uncertainty. Sources of uncertainty in ice-sheet models, like uncertainties in ice-sheet geometry or surface mass balance, typically vary in space and potentially in time. For that reason, they are more adequately described as spatio(-temporal) random fields, which account naturally for spatial (and temporal) correlation. As a means of improving the characterization of the sources of uncertainties in ice-sheet models, we propose in this paper to represent them as Gaussian random fields with Matérn covariance function. The class of Matérn covariance functions provides a flexible model able to capture statistical dependence between locations with different degrees of spatial correlation or smoothness properties. Samples from a Gaussian random field with Matérn covariance function can be generated efficiently by solving a certain stochastic partial differential equation. Discretization of this stochastic partial differential equation by the finite element method results in a sparse approximation known as a Gaussian Markov random field. We solve this equation efficiently using the finite element method within the Ice-sheet and Sea-level System Model (ISSM). In addition, spatio-temporal samples can be generated by combining an autoregressive temporal model and the Matérn field. The implementation is tested on a set of synthetic experiments to verify that it captures well the desired spatial and temporal correlations. Finally, we demonstrate the interest of this sampling capability in an illustration concerned with assessing the impact of various sources of uncertainties on the Pine Island Glacier, West Antarctica. We find that both larger spatial and temporal correlations lengths will likely result in increased uncertainty in the projections.

This paper proposes a GMRF (Gaussian Markov random field)-model based restoration method of missing areas in still images. The GMRF model used in the proposed method is realized by a new assumption that reasonably holds for an image source. This model can express important image features such as edges because of the use of the new assumption. Therefore, the proposed method restores the missing areas using the modified GMRF model and can correctly reconstruct the missing edges. Consequently, the proposed method achieves more accurate restoration than those of the traditional methods on both objective and subjective measures. Extensive experimental results demonstrate the improvement of the proposed method over previous methods.

In this paper, we propose a new estimation procedure for discovering the structure of Gaussian Markov random fields (MRFs) with false discovery rate (FDR) control, making use of the sorted l1-norm (SL1) regularization. A Gaussian MRF is an acyclic graph representing a multivariate Gaussian distribution, where nodes are random variables and edges represent the conditional dependence between the connected nodes. Since it is possible to learn the edge structure of Gaussian MRFs directly from data, Gaussian MRFs provide an excellent way to understand complex data by revealing the dependence structure among many inputs features, such as genes, sensors, users, documents, etc. In learning the graphical structure of Gaussian MRFs, it is desired to discover the actual edges of the underlying but unknown probabilistic graphical model-it becomes more complicated when the number of random variables (features) p increases, compared to the number of data points n. In particular, when p >> n, it is statistically unavoidable for any estimation procedure to include false edges. Therefore, there have been many trials to reduce the false detection of edges, in particular, using different types of regularization on the learning parameters. Our method makes use of the SL1 regularization, introduced recently for model selection in linear regression. We focus on the benefit of SL1 regularization that it can be used to control the FDR of detecting important random variables. Adapting SL1 for probabilistic graphical models, we show that SL1 can be used for the structure learning of Gaussian MRFs using our suggested procedure nsSLOPE (neighborhood selection Sorted L-One Penalized Estimation), controlling the FDR of detecting edges.

Unsupervised segmentation of noisy and textured images using Markov random fields

On Digital Mammogram Segmentation and Microcalcification Detection Using Multiresolution Wavelet Analysis

A novel multiresolution wavelet analysis (MWA) and non-stationary Gaussian Markov random field (GMRF) technique is introduced for the identification of microcalcifications with high accuracy. The hierarchical multiresolution wavelet information in conjunction with the contextual information of the images extracted from GMRF provides a highly efficient technique for microcalcification detection. A Bayesian learning paradigm realized via the expectation maximization (EM) algorithm was also introduced for edge detection or segmentation of larger lesions recorded on the mammograms. The effectiveness of the approach has been extensively tested with a number of mammographic images provided by a local hospital.

Modeling textures using Gaussian Markov random fields (GMRF) has been successfully used in classifying textures. However, these models do not perform well for self-similar textures such as those generated from fractional Brownian motion. The authors show that by using the difference images at different scales instead of the original image, one can significantly increase the performance of classifying self-similar texture patterns using GMRF models. >

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

Almost sure limit theorem for stationary Gaussian random fields

In recent years most of researcher’s has done tremendous work in the field of medical image applications such as Magnetic Resonance Imaging (MRI), Ultra Sound, CT scan but still there are many research and experiments in medical imaging field and diagnosing of human health by Health Care Institutes. There is a growing interest for medical imaging de-noising as a hot area of research and also imaging equipment as a device. It is used for better image processing and highlighting the important features. These images are affected with random noise during acquisition, analyzing and Transmission process. This results in blurry image visible in low contrast. Wavelet transforms have effective method to separate the noise from the original medical image by using threshold techniques without affecting the important data of an image. Wavelet transform enables us to use the forward wavelet transform to represent sub-band of the original image in decomposition process then reconstructing this sub band coefficients to original image using inverse wavelet transform. In this work, the quality of medical image has been evaluated using filter assessment parameters like Variance, standard deviation, the squared difference error between original medical image & de-noised image (MSE) and the ratio between original image & noisy image. From numerical results, we can see that the algorithm is efficient de-noising of noisy medical image. When, investigating with Baye’s threshold techniques it achieved the Best value of peak signal to noise ratio (PSNR). For best medical image de-noising, the wavelet based de-noising algorithm has been investigated and results of Baye’s techniques and hard & soft threshold methods have been compared.