Anisotropic Markov Random Field Research Articles

MRI Reconstruction Using Markov Random Field and Total Variation as Composite Prior.

Reconstruction of magnetic resonance images (MRI) benefits from incorporating a priori knowledge about statistical dependencies among the representation coefficients. Recent results demonstrate that modeling intraband dependencies with Markov Random Field (MRF) models enable superior reconstructions compared to inter-scale models. In this paper, we develop a novel reconstruction method, which includes a composite prior based on an MRF model and Total Variation (TV). We use an anisotropic MRF model and propose an original data-driven method for the adaptive estimation of its parameters. From a Bayesian perspective, we define a new position-dependent type of regularization and derive a compact reconstruction algorithm with a novel soft-thresholding rule. Experimental results show the effectiveness of this method compared to the state of the art in the field.

Hybrid Seismic Inversion Based on Multi-Order Anisotropic Markov Random Field

The prestack seismic inversion effectively transforms reflection data into elastic parameters attributed to rock properties, which has enabled us to explore subsurface in detail. Because the forwarding operator is nonlinear and the observed data are usually insufficient, the inverse problem suffers from ill-posedness. Regularization has proven to be an effective technique making the problem well-posed by adding extra constraints to the estimated model. In particular, the anisotropic Markov random field (AMRF) approach can formulate the prior constraints stabilizing inversion procedure and taking special care of the geological boundaries. However, the standard AMRF only deals with the first-order neighborhood and its capability of revealing geological details is thereby limited. In addition, the auxiliary model, which significantly affects the performance of the AMRFs on correcting anisotropic gradients, is difficult to determine because reliable prior information is usually absent. This articlepresents a hybrid seismic inversion method for estimating the elastic parameters, which is featured by the multi-order AMRF (MAMRF) neighborhoods. In specific, the poststack inverted acoustic impedance result is used as the auxiliary model for the subsequent prestack seismic inversion; the MAMRF-based prior constraint incorporates low- and high-order AMRF neighborhoods during the inversion procedure, whose capability of correcting anisotropic gradients is achieved by the MAMRF coefficients and is enhanced by order-depended and statistically estimated diffusion coefficients. The proposed inversion method is verified by a synthetic test followed by application to inverting field data. Compared with the standard MRF and (first-order) AMRF approaches, the proposed inversion method is applicable for revealing detailed subsurface models, especially under complex geological conditions.

An Improved Anisotropic Markov Random Field Approach for Prestack Seismic Inversion

This letter presents an improved anisotropic Markov random field (IAMRF) approach for prestack seismic inversion. Instead of using multiple potential functions or adding extra weights on clique potentials as done in existing AMRF approaches, IAMRF removes the effects of anisotropic gradients in subsurface models by means of scaling parameters, which directly tune the model gradients right above. In particular, the scaling parameters are not only directionally varied at certain iterations but also updated by a statistical method at every iteration throughout the inversion, thereby generating more accurate and better edge-preserving inverted results. The prestack seismic inversion based on the IAMRF prior constraints is demonstrated on a field data example, which presents encouraging results.

Reservoir Characterization Through Target-Oriented AVA-Petrophysical Inversions with Spatial Constraints

We apply three methods that use different regularization strategies to insert spatial constraints into the seismic-petrophysical inversion. The first method is what we call the structurally constrained inversion (SCI), which directly uses the structural information brought by the seismic stack image to insert geological (structural) constraints into the inversion. The second method is based on anisotropic Markov random field (AMRF) and uses the Huber energy function to reasonably model the spatial heterogeneity of petrophysical reservoir properties. Finally, the last method includes both geostatistical information (describing the lateral variability of reservoir properties) and hard data (i.e. well log data) constraints into the inversion kernel (GHDC inversion). For computationally feasibility, we apply a target-oriented inversion that uses the amplitude versus angle (AVA) responses extracted along the top reservoir reflections to infer the petrophysical properties of interest (i.e. porosity, water saturation and shaliness) for the target layer. In particular, an empirical, linear rock-physics model, properly calibrated for the investigated area, is used to rewrite the P-wave reflectivity equation as a function of the petrophysical contrasts instead of the elastic constants. This reformulation allows for a direct and a simultaneous estimation of petrophysical properties from AVA data. The implemented approaches are tested both on synthetic and field seismic data and compared against the standard method in which each AVA response is inverted independently (laterally unconstrained Bayesian inversion; LUBI). In the case of poor signal-to-noise ratio it turns out that the three considered spatially constrained methods achieve better delineations of reservoir bodies and provide more reliable results than the standard LUBI approach. More in detail, the AMRF recovers sharper geological boundaries than the SCI and GHDC algorithms. The SCI algorithms is more sensitive to data noise, whereas the key advantage of the GHDC consists in analytically providing posterior uncertainties for the model parameters. Finally, for what concerns the computational cost the GHDC and the SCI methods result the most and the least computationally demanding, respectively.

A Hybrid Seismic Inversion Method for VP/VS Ratio and Its Application to Gas Identification

The ratio of compressional wave velocity to shear wave velocity (VP/VS ratio) has established itself as one of the most important parameters in identifying gas reservoirs. However, considering that seismic inversion process is highly non-linear and geological conditions encountered may be complex, a direct estimation of VP/VS ratio from pre-stack seismic data remains a challenging task. In this paper, we propose a hybrid seismic inversion method to estimate VP/VS ratio directly. In this method, post- and pre-stack inversions are combined in which the pre-stack inversion for VP/VS ratio is driven by the post-stack inversion results (i.e., VP and density). In particular, the VP/VS ratio is considered as a model parameter and is directly inverted from the pre-stack inversion based on the exact Zoeppritz equation. Moreover, anisotropic Markov random field is employed in order to regularise the inversion process as well as taking care of geological structures (boundaries) information. Aided by the proposed hybrid inversion strategy, the directional weighting coefficients incorporated in the anisotropic Markov random field neighbourhoods are quantitatively calculated by the anisotropic diffusion method. The synthetic test demonstrates the effectiveness of the proposed inversion method. In particular, given low quality of the pre-stack data and high heterogeneity of the target layers in the field data, the proposed inversion method reveals the detailed model of VP/VS ratio that can successfully identify the gas-bearing zones.

Prestack Seismic Inversion Based on Anisotropic Markov Random Field

Prestack seismic inversion is an ill-posed problem and must be regularized to stabilize the inverted results. In particular, edge-preserving regularization with prior constraints based on Markov random field (MRF) has proved to be an effective technique for reconstructing subsurface models. However, regularized seismic inversion, based on the standard MRF scheme, typically makes use of isotropic MRF neighborhoods, in which the weighting coefficients of the model gradients are equivalent in all directions. Considering real geological conditions, subsurface formations are expected to be laterally continuous and vertically stratified. Therefore, the anisotropic effects caused by model gradients which vary along different directions should not be ignored. In this paper, we proposed a new prestack seismic inversion method based on anisotropic MRF (AMRF). In this method, AMRF coefficients are incorporated into the standard MRF scheme. These coefficients demonstrate directional variations and gradient dependencies, intended to directly correct the errors caused by the anisotropic model gradients on the prior constraints. In particular, we introduced the anisotropic diffusion method to calculate the AMRF coefficients. The proposed inversion method can effectively remove the anisotropic features of the model gradients and significantly improve the inversion results, especially for geologically layered formations. We demonstrated the effectiveness of the inversion method by both 2-D synthetic test and field data example, which presented encouraging results.

A nonlinear method for multiparameter inversion of pre‐stack seismic data based on anisotropic Markov random field

ABSTRACTMultiparameter inversion for pre‐stack seismic data plays a significant role in quantitative estimation of subsurface petrophysical properties. However, it remains a complicated problem due to the non‐unique results and unstable nature of the processing; the pre‐stack seismic inversion problem is ill‐posed and band‐limited. Combining the full Zoeppritz equation and additional assumptions with edge‐preserving regularisation can help to alleviate these problems. To achieve this, we developed an inversion method by constructing a new objective function that includes edge‐preserving regularisation and soft constraints based on anisotropic Markov random fields and is intended especially for layered formations. We applied a fast simulated annealing algorithm to solve the nonlinear optimisation problem. The method directly obtains reflectivity RPP values using the full Zoeppritz equation instead of its approximations and effectively controls the stability of the multiparameter inversion by assuming a sectionally constant S‐ and P‐wave velocity ratio and using the generalised Gardner equation. We substituted the inverted parameters, i.e., the P‐wave velocity, the fitting deviation of S‐wave velocity, and the density were inverted instead of the P‐wave velocity, the S‐wave velocity, and the density, and the generalised Gardner equation was applied as a constraint. Test results on two‐dimensional synthetic data indicated that our substitution obtained improved results for multiparameter inversion. The inverted results could be improved by utilising high‐order anisotropic Markov random field neighbourhoods at early stages and low‐order anisotropic Markov random field neighbourhoods in the later stages. Moreover, for layered formations, using a large horizontal weighting coefficient can preserve the lateral continuity of layers, and using a small vertical weighting coefficient allows for large longitudinal gradients of the interlayers. The inverted results of the field data revealed more detailed information about the layers and matched the logging curves at the wells acceptably over most parts of the curves.

Prestack inversion based on anisotropic Markov random field–maximum posterior probability inversion and its application to identify shale gas sweet spots

Economic shale gas production requires hydraulic fracture stimulation to increase the formation permeability. Hydraulic fracturing strongly depends on geomechanical parameters such as Young’s modulus and Poisson’s ratio. Fracture-prone sweet spots can be predicted by prestack inversion, which is an ill-posed problem; thus, regularization is needed to obtain unique and stable solutions. To characterize gas-bearing shale sedimentary bodies, elastic parameter variations are regarded as an anisotropic Markov random field. Bayesian statistics are adopted for transforming prestack inversion to the maximum posterior probability. Two energy functions for the lateral and vertical directions are used to describe the distribution, and the expectation–maximization algorithm is used to estimate the hyperparameters of the prior probability of elastic parameters. Finally, the inversion yields clear geological boundaries, high vertical resolution, and reasonable lateral continuity using the conjugate gradient method to minimize the objective function. Antinoise and imaging ability of the method were tested using synthetic and real data.

Multimodal brain-tumor segmentation based on Dirichlet process mixture model with anisotropic diffusion and Markov random field prior.

Brain-tumor segmentation is an important clinical requirement for brain-tumor diagnosis and radiotherapy planning. It is well-known that the number of clusters is one of the most important parameters for automatic segmentation. However, it is difficult to define owing to the high diversity in appearance of tumor tissue among different patients and the ambiguous boundaries of lesions. In this study, a nonparametric mixture of Dirichlet process (MDP) model is applied to segment the tumor images, and the MDP segmentation can be performed without the initialization of the number of clusters. Because the classical MDP segmentation cannot be applied for real-time diagnosis, a new nonparametric segmentation algorithm combined with anisotropic diffusion and a Markov random field (MRF) smooth constraint is proposed in this study. Besides the segmentation of single modal brain-tumor images, we developed the algorithm to segment multimodal brain-tumor images by the magnetic resonance (MR) multimodal features and obtain the active tumor and edema in the same time. The proposed algorithm is evaluated using 32 multimodal MR glioma image sequences, and the segmentation results are compared with other approaches. The accuracy and computation time of our algorithm demonstrates very impressive performance and has a great potential for practical real-time clinical use.

Bayesian Image Denoising Using an Anisotropic Markov Random Field Model

This paper presents a Bayesian denoising method based on an anisotropic Markov Random Field (MRF) model in wavelet domain in order to improve the image denoising performance and reduce the computational complexity. The classical single-resolution image restoration method using MRFs and the maximum a posteriori (MAP) estimation is extended to the wavelet domain. To obtain the accurate MAP estimation, a novel anisotropic MRF model is proposed under this framework. As compared to the simple isotropic MRF model, this new model can capture the intrascale dependencies of wavelet coefficients significantly better. Simulation results demonstrate our proposed method has a good denoising performance while reducing the computational complexity.

Segmentation of trabeculated structures using an anisotropic Markov random field: application to the study of the optic nerve head in glaucoma

The study of the architecture of the optic nerve head (ONH) may provide valuable information about the development and progression of glaucoma. To this end, we have generated three-dimensional datasets from monkey eyes under controlled intraocular pressure (IOP). Segmentation of the connective tissues in this area is crucial to obtain an accurate measurement of geometrical parameters and to build mechanical models. However, this segmentation is made difficult by the complicated geometry and the artifacts introduced in the dataset building process. We present a novel segmentation algorithm, based on expectation-maximization, which incorporates an anisotropic Markov random field (MRF) to introduce prior knowledge about the geometry of the structure. The structure tensor is used to characterize the predominant structure direction and the spatial coherence at each point. The algorithm, which has been validated on an artificial validation dataset that mimics our ONH datasets, shows significant improvement over an isotropic MRF. Results on the real datasets demonstrate the ability of the new algorithm to obtain accurate, spatially consistent segmentations of this structure.

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.

Markov Random Fields in Pattern Recognition for Semiconductor Manufacturing

Under the most general conditions of an anisotropic Markov random field, we model the two-dimensional spatial distribution of microchips on a silicon wafer. The proposed model improves on its predecessors as it stipulates the spatial correlation of different strengths in all eight directions. Its canonical parameters represent the intensity of failures, main effects, and interactions of neighboring chips. Explicit forms of conditional distributions are derived, and maximum pseudo-likelihood estimates of canonical parameters are obtained. This numerical characteristic summarizes general patterns of clusters of failing chips on a wafer, capturing their size, shape, direction, density, and thickness. It is used to classify incoming wafers to known root-cause categories by matching them to the closest pattern.