Markov Random Field Prior Model Research Articles

A one-step Bayesian inversion framework for 3D reservoir characterization based on a Gaussian mixture model — A Norwegian Sea demonstration

We have developed a one-step approach for Bayesian prediction and uncertainty quantification of lithology/fluid classes, petrophysical properties, and elastic attributes conditional on prestack 3D seismic amplitude-variation-with-offset data. A 3D Markov random field prior model is assumed for the lithology/fluid classes to ensure spatially coupled lithology/fluid class predictions in the lateral and vertical directions. Conditional on the lithology/fluid classes, we consider Gauss-linear petrophysical and rock-physics models including depth trends. Then, the marginal prior models for the petrophysical properties and elastic attributes are multivariate Gaussian mixture models. The likelihood model is assumed to be Gauss-linear to allow for analytic computation. A recursive algorithm that translates the Gibbs formulation of the Markov random field into a set of vertical Markov chains is proposed. This algorithm provides a proposal density in a Markov chain Monte Carlo algorithm such that efficient simulation from the posterior model of interest in three dimensions is feasible. The model is demonstrated on real data from a Norwegian Sea gas reservoir. We evaluate the model at the location of a blind well, and we compare results from the proposed model with results from a set of 1D models in which each vertical trace is inverted independently. At the blind well location, we obtain at most a 60% reduction in the root-mean-square error for the proposed 3D model compared to the model without lateral spatial coupling.

A Bayesian hidden Potts mixture model for analyzing lung cancer pathology images.

Digital pathology imaging of tumor tissues, which captures histological details in high resolution, is fast becoming a routine clinical procedure. Recent developments in deep-learning methods have enabled the identification, characterization, and classification of individual cells from pathology images analysis at a large scale. This creates new opportunities to study the spatial patterns of and interactions among different types of cells. Reliable statistical approaches to modeling such spatial patterns and interactions can provide insight into tumor progression and shed light on the biological mechanisms of cancer. In this article, we consider the problem of modeling a pathology image with irregular locations of three different types of cells: lymphocyte, stromal, and tumor cells. We propose a novel Bayesian hierarchical model, which incorporates a hidden Potts model to project the irregularly distributed cells to a square lattice and a Markov random field prior model to identify regions in a heterogeneous pathology image. The model allows us to quantify the interactions between different types of cells, some of which are clinically meaningful. We use Markov chain Monte Carlo sampling techniques, combined with a double Metropolis-Hastings algorithm, in order to simulate samples approximately from a distribution with an intractable normalizing constant. The proposed model was applied to the pathology images of $205$ lung cancer patients from the National Lung Screening trial, and the results show that the interaction strength between tumor and stromal cells predicts patient prognosis (P = $0.005$). This statistical methodology provides a new perspective for understanding the role of cell-cell interactions in cancer progression.

Joint probabilistic petrophysics-seismic inversion based on Gaussian mixture and Markov chain prior models

Seismic reservoir characterization focuses on the prediction of reservoir properties based on the available geophysical and petrophysical data. The inverse problem generally includes continuous properties, such as petrophysical and elastic attributes, and discrete properties, such as lithology/fluid classes. We have developed a joint probabilistic inversion methodology for the prediction of petrophysical and elastic properties and lithology/fluid classes that combined statistical rock physics and Bayesian seismic inversion. The elastic attributes depend on continuous petrophysical variables, such as porosity and clay content, and discrete lithology/fluid classes, through a nonlinear rock-physics relationship together. The seismic model relates the elastic attributes, such as velocities and density, to their seismic response (reflectivity, traveltime, and amplitudes). The advantage of our integrated approach is that the inversion method accounts for the uncertainty associated to each step of the modeling workflow. The lithology/fluid classes are assigned by a Markov random field prior model to capture vertical continuity and vertical sorting of the lithology/fluid classes. Because rock and fluid properties are in general not Gaussian, a spatially coupled Gaussian mixture prior model based on the lithology/fluid classes is constructed. The forward geophysical operator includes a lithology-/fluid-dependent rock physics model and a linearized seismic model based on the convolution of the seismic wavelet with the reflectivity coefficient series. The solution of the inverse problem consists of the posterior distributions of petrophysical and elastic properties and lithology/fluid classes. We proposed an efficient Markov chain Monte Carlo algorithm to sample from the posterior models and assess the uncertainty. Our methodology is demonstrated on a seismic cross section from a survey in the Norwegian Sea, and it shows promising results consistent with well-log data measured at the well location as well as reliable prediction uncertainties.

Sparse Recovery in Magnetic Resonance Imaging With a Markov Random Field Prior.

Recent research in compressed sensing of magnetic resonance imaging (CS-MRI) emphasizes the importance of modeling structured sparsity, either in the acquisition or in the reconstruction stages. Subband coefficients of typical images show certain structural patterns, which can be viewed in terms of fixed groups (like wavelet trees) or statistically (certain configurations are more likely than others). Wavelet tree models have already demonstrated excellent performance in MRI recovery from partial data. However, much less attention has been given in CS-MRI to modeling statistically spatial clustering of subband data, although the potentials of such models have been indicated. In this paper, we propose a practical CS-MRI reconstruction algorithm making use of a Markov random field prior model for spatial clustering of subband coefficients and an efficient optimization approach based on proximal splitting. The results demonstrate an improved reconstruction performance compared with both the standard CS-MRI methods and the recent related methods.

Multi-channel beam-scanning imaging at kHz frame rates by Lissajous trajectory microscopy.

A beam-scanning microscope based on Lissajous trajectory imaging is described for achieving streaming 2D imaging with continuous frame rates up to 1.4 kHz. The microscope utilizes two fast-scan resonant mirrors to direct the optical beam on a circuitous trajectory through the field of view. By separating the full Lissajous trajectory time-domain data into sub-trajectories (partial, undersampled trajectories) effective frame-rates much higher than the repeat time of the Lissajous trajectory are achieved with many unsampled pixels present. A model-based image reconstruction (MBIR) 3D in-painting algorithm is then used to interpolate the missing data for the unsampled pixels to recover full images. The MBIR algorithm uses a maximum a posteriori estimation with a generalized Gaussian Markov random field prior model for image interpolation. Because images are acquired using photomultiplier tubes or photodiodes, parallelization for multi-channel imaging is straightforward. Preliminary results show that when combined with the MBIR in-painting algorithm, this technique has the ability to generate kHz frame rate images across 6 total dimensions of space, time, and polarization for SHG, TPEF, and confocal reflective birefringence data on a multimodal imaging platform for biomedical imaging. The use of a multi-channel data acquisition card allows for multimodal imaging with perfect image overlay. Image blur due to sample motion was also reduced by using higher frame rates.

Construction of Binary Multi-grid Markov Random Field Prior Models from Training Images

Bayesian modeling requires the specification of prior and likelihood models. In reservoir characterization, it is common practice to estimate the prior from a training image. This paper considers a multi-grid approach for the construction of prior models for binary variables. On each grid level we adopt a Markov random field (MRF) conditioned on values in previous levels. Parameter estimation in MRFs is complicated by a computationally intractable normalizing constant. To cope with this problem, we generate a partially ordered Markov model (POMM) approximation to the MRF and use this in the model fitting procedure. Approximate unconditional simulation from the fitted model can easily be done by again adopting the POMM approximation to the fitted MRF. Approximate conditional simulation, for a given and easy to compute likelihood function, can also be performed either by the Metropolis–Hastings algorithm based on an approximation to the fitted MRF or by constructing a new POMM approximation to this approximate conditional distribution. The proposed methods are illustrated using three frequently used binary training images.

Bayesian lithology/fluid prediction constrained by spatial couplings and rock physics depth trends

Elastic properties of rocks are strongly influenced by geological trends, and the uncertainties in the lithology/fluids prediction can be reduced if these trends are included in the inversion model. In this study lithology/fluid classification constrained by depth trends (Avseth et al., 2003) is considered in a spatial setting with a Markov random field prior model for spatial coupling of the discrete lithology/fluids classes.

A joint inter- and intrascale statistical model for Bayesian wavelet based image denoising

This paper presents a new wavelet-based image denoising method, which extends a "geometrical" Bayesian framework. The new method combines three criteria for distinguishing supposedly useful coefficients from noise: coefficient magnitudes, their evolution across scales and spatial clustering of large coefficients near image edges. These three criteria are combined in a Bayesian framework. The spatial clustering properties are expressed in a prior model. The statistical properties concerning coefficient magnitudes and their evolution across scales are expressed in a joint conditional model. The three main novelties with respect to related approaches are (1) the interscale-ratios of wavelet coefficients are statistically characterized and different local criteria for distinguishing useful coefficients from noise are evaluated, (2) a joint conditional model is introduced, and (3) a novel anisotropic Markov random field prior model is proposed. The results demonstrate an improved denoising performance over related earlier techniques.

Endocardial Boundary E timation and Tracking in Echocardiographic Images using Deformable Template and Markov Random Fields

We present a new approach to shape-based segmentation and tracking of deformable anatomical structures in medical images, and validate this approach by detecting and tracking the endocardial contour in an echocardiographic image sequence. To this end, some global prior shape knowledge of the endocardial boundary is captured by a prototype template with a set of predefined global and local deformations to take into account its inherent natural variability over time. In this deformable model-based Bayesian segmentation, the data likelihood model relies on an accurate statistical modelling of the grey level distribution of each class present in the ultrasound image. The parameters of this distribution mixture are given by a preliminary iterative estimation step. This estimation scheme relies on a Markov Random Field prior model, and takes into account the imaging process as well as the distribution shape of each class present in the image. Then the detection and the tracking problem is stated in a Bayesian framework, where it ends up as a cost function minimisation problem for each image of the sequence. In our application, this energy optimisation problem is efficiently solved by a genetic algorithm combined with a steepest ascent procedure. This technique has been successfully applied on synthetic images, and on a real echocardiographic image sequence.

Author Index for Volume 76

This paper presents an original method for analyzing, in an unsupervised way, images supplied by high resolution sonar. We aim at segmenting the sonar image into three kinds of regions: echo areas (due to the reflection of the acoustic wave on the object), shadow areas (corresponding to a lack of acoustic reverberation behind an object lying on the sea-bed), and sea-bottom reverberation areas. This unsupervised method estimates the parameters of noise distributions, modeled by a Weibull probability density function (PDF), and the label field parameters, modeled by a Markov random field (MRF). For the estimation step, we adopt a maximum likelihood technique for the noise model parameters and a least-squares method to estimate the MRF prior model. Then, in order to obtain an accurate segmentation map, we have designed a two-step process that finds the shadow and the echo regions separately, using the previously estimated parameters. First, we introduce a scale-causal and spatial model called SCM (scale causal multigrid), based on a multigrid energy minimization strategy, to find the shadow class. Second, we propose a MRF monoscale model using a priori information (at different level of knowledge) based on physical properties of each region, which allows us to distinguish echo areas from sea-bottom reverberation. This technique has been successfully applied to real sonar images and is compatible with automatic processing of massive amounts of data.

Three-Class Markovian Segmentation of High-Resolution Sonar Images

This paper presents an original method for analyzing, in an unsupervised way, images supplied by high resolution sonar. We aim at segmenting the sonar image into three kinds of regions: echo areas (due to the reflection of the acoustic wave on the object), shadow areas (corresponding to a lack of acoustic reverberation behind an object lying on the sea-bed), and sea-bottom reverberation areas. This unsupervised method estimates the parameters of noise distributions, modeled by a Weibull probability density function (PDF), and the label field parameters, modeled by a Markov random field (MRF). For the estimation step, we adopt a maximum likelihood technique for the noise model parameters and a least-squares method to estimate the MRF prior model. Then, in order to obtain an accurate segmentation map, we have designed a two-step process that finds the shadow and the echo regions separately, using the previously estimated parameters. First, we introduce a scale-causal and spatial model called SCM (scale causal multigrid), based on a multigrid energy minimization strategy, to find the shadow class. Second, we propose a MRF monoscale model using a priori information (at different level of knowledge) based on physical properties of each region, which allows us to distinguish echo areas from sea-bottom reverberation. This technique has been successfully applied to real sonar images and is compatible with automatic processing of massive amounts of data.

Adaptive quadrature filters and the recovery of phase from fringe pattern images

A principled approach, based on Bayesian estimation theory and complex-valued Markov random-field prior models, is introduced for the design of a new class of adaptive quadrature filters. These filters are capable of adapting their tuning frequency to the local dominant spatial frequency of the input image while maintaining an arbitrarily narrow local frequency response; therefore they may be effectively used for the accurate recovery of the phase of broadband spatial-carrier fringe patterns, even when they are corrupted by a significant amount of noise. Also, by constraining the spatial variation of the adaptive frequency to be smooth, they permit the completely automatic recovery of local phase from single closed fringe pattern images, since the spurious discontinuities and sign reversals that one obtains from the classical Fourier-based methods are avoided in this case. Although the applications discussed here come from fringe pattern analysis in optics, these filters may also be useful in the solution of other problems, such as texture characterization and segmentation and the recovery of depth from stereoscopic pairs of images.

Tomographic image sequence reconstruction by edge-preserving interslice MAP methods

We consider a new problem in tomographic imaging. In a sequence of transverse images, adjacent slices often look similar to each other. We investigate the possibility of using this correlation to help reconstruct the sequence with 2-D in-plane data. The problem is formulated as a maximum a posteriori (MAP) estimation problem, using an edge-preserving Markov random field prior model. Both Gaussian and Poisson data are considered. Some improvements over the single-slice MAP reconstructions are observed.