Triplet Markov Fields Research Articles

Model-Based Bayesian Compressive Sensing of Non-stationary Images Using a Wavelet-Domain Triplet Markov Fields Model

In this paper, a new model-based Bayesian compressive sensing technique for non-stationary images is proposed. Our algorithm is based on the recently addressed triplet Markov fields (TMF) model. The TMF model is well appropriate for non-stationary image processing, owing to the introduction of a third random field which reflects different non-stationarity of images. Furthermore, TMF can extract the interactions among the neighboring sites of an image in a more complete way than the classic hidden Markov models do. In this paper, the inter-scale dependencies between the wavelet coefficients is exploited explicitly in the proposed TMF model, which results in the wavelet domain TMF model. Our proposed model considers the intra- and inter-scale dependencies and the non-stationarity of images simultaneously. Also, we have developed our proposed algorithm for both Gaussian and non-Gaussian measurement noises, and we have modeled the non-Gaussianity of the noise via Laplace distribution. To approximate the posterior distributions of the hidden variables, we resort to a variational Bayesian expectation–maximization algorithm. Simulation results in both the optical and synthetic aperture radar images show that this model leads to an improvement over state-of-the-art algorithms in terms of the reconstruction error and the structural similarity.

Unsupervised segmentation of hidden Markov fields corrupted by correlated non-Gaussian noise

Pixel labeling problem stands among the most commonly considered topics in image processing. Many statistical approaches have been developed for this purpose, particularly in the frame of hidden Markov random fields. Such models have been extended in many directions to better fit image data. Our contribution falls under such extensions and consists of introducing two new models allowing one to deal with non-Gaussian correlated noise. The first one is purely probabilistic, whereas the second one calls on Dempster–Shafer theory of evidence, both being particular triplet Markov fields. The interest of the proposed models is assessed in unsupervised segmentation of sampled and real images. While both models exhibit significant improvement with respect to classic models, the evidential model turns out to be of particular interest when the hidden label field presents fine details.

Oriented Triplet Markov Fields

Hidden Markov Field modeling is widely used for image segmentation. However, it sometimes lacks power to handle complex situations, e.g. correlated noise, textures or non-stationarities. This is why Pairwise, and then Triplet Markov Fields were introduced to handle in a generic fashion more complex observations. In this paper, we tackle the problem of anisotropic image modeling by introducing an Oriented Triplet Markov Field model, able to explicitly deal with oriented structures. Using oriented features in the framework of Triplet Markov Field modeling, we compare the behavior of this model towards other Markovian modeling on images containing such oriented pattern. We present experiments on synthetic data for segmentation, and application to real data from remote sensing images.

Synthetic aperture radar image segmentation using non‐linear diffusion‐based hierarchical triplet Markov fields model

Triplet Markov fields (TMF) model is widely used to deal with non-stationary synthetic aperture radar (SAR) images. However, its ability to capture global information remains limited due to the non-causal property. A hierarchical TMF model is proposed in this study based on the non-linear diffusion (ND) strategy, which is denoted as ND-hierarchical TMF (HTMF). ND is adopted to generate multiscale decomposition according to local image content, and that is superior to traditional wavelet decomposition in reflecting hierarchical nature of image structure and detailed features. The auxiliary field in ND-HTMF is redefined and initialised on the finest scale to characterise edge information and that enhances the prior modelling ability for non-stationary local image features. The multiscale likelihood and multiscale causal prior energy functions are then defined respectively in bottom-up and top-down procedures to capture local and global information for performing segmentation. Segmentation experiments on simulated and real SAR images demonstrate the effectiveness of ND-HTMF in both edge characterisation accuracy and robustness against speckle noise.

Unsupervised SAR Image Segmentation Using Ambiguity Label Information Fusion in Triplet Markov Fields Model

The recently proposed triplet Markov fields (TMF) model enhances the nonstationary image prior modeling ability by introducing an auxiliary field. Motivated by the TMF model, we propose a generalized TMF model based on ambiguity label information fusion (ALF-TMF) for synthetic aperture radar (SAR) image segmentation. The redefined auxiliary field in ALF-TMF indicates the dominant direction of local image contents and gives explicit nonstationary divisions of SAR images. To reduce the influence of unreliable observations caused by speckle noise, the original label field is adaptively generalized by introducing ambiguity class based on image observation and local nonstationary contextual information. Given the extended label field, prior and likelihood terms are constructed and merged to provide the posterior segmentation decision via the Bayesian fusion rule. Real SAR images are utilized in the experimental analysis, and the effectiveness of the proposed method is validated accordingly.

Unsupervised Segmentation of SAR Images Using Gaussian Mixture-Hidden Evidential Markov Fields

Hidden Markov fields have been extensively applied in the field of synthetic aperture radar (SAR) image processing, mainly for segmentation and change detection. In such models, the hidden process of interest $X$ is assumed to be a Markov field that is to be searched from an observable process $Y$ . The possibility of such estimation lies, however, on several assumptions that turn out to be unsuitable for many natural systems. These models have then been extended in many directions, leading to triplet Markov fields among other extensions. A link has then been established between these models and the theory of evidence, opening new possibilities of uncertainty modeling and information fusion. The aim of this letter is to further generalize the hidden evidential Markov field (EMF) to consider more general forms of noise with application to unsupervised segmentation of SAR images. For parameters estimation, we use iterative conditional estimation, whereas maximization is performed through iterative conditional mode. The performance of the proposed model is assessed against the original EMF on real SAR images.

Dempster–Shafer fusion of evidential pairwise Markov fields

Hidden Markov fields (HMFs) have been successfully used in many areas to take spatial information into account. In such models, the hidden process of interest X is a Markov field, that is to be estimated from an observable process Y. The possibility of such estimation is due to the fact that the conditional distribution of the hidden process with respect to the observed one remains Markovian. The latter property remains valid when the pairwise process (X,Y) is Markov and such models, called pairwise Markov fields (PMFs), have been shown to offer larger modeling capabilities while exhibiting similar processing cost. Further extensions lead to a family of more general models called triplet Markov fields (TMFs) in which the triplet (U,X,Y) is Markov where U is an underlying process that may have different meanings according to the application. A link has also been established between these models and the theory of evidence, opening new possibilities of achieving Dempster–Shafer fusion in Markov fields context. The aim of this paper is to propose a unifying general formalism allowing all conventional modeling and processing possibilities regarding information imprecision, sensor unreliability and data fusion in Markov fields context. The generality of the proposed formalism is shown theoretically through some illustrative examples dealing with image segmentation, and experimentally on hand-drawn and SAR images.

Unsupervised SAR Image Segmentation Using Higher Order Neighborhood-Based Triplet Markov Fields Model

The triplet Markov fields (TMF) model has been successfully applied to statistical segmentation of nonstationary images by introducing the auxiliary field, which represents the different stationarities of images. Commonly, the TMF adopts a four-nearest neighborhood. This limits the modeling ability for complex priors. Therefore, this paper suggests using a higher order neighborhood-based TMF (HN-TMF). In the HN-TMF, the autocovariance analysis is applied to reveal the local fluctuation at each site. The auxiliary field is then redefined based on the local fluctuation information to denote homogeneity or heterogeneity. Based on the auxiliary field, the local energy function in HN-TMF is constructed either in a homogeneous or heterogeneous way, and hence, the local structure can be embedded in the energy function to improve the prior modeling ability. Along with the newly constructed energy function, new initializations of HN-TMF parameters are given to fulfill the physical interpretation of the energy function. The experiments performed on both synthetic and real synthetic aperture radar images demonstrate the effectiveness of the proposed HN-TMF in both speckle noise reduction and heterogeneous region segmentation accuracy.

Unsupervised SAR Image Segmentation Based on Conditional Triplet Markov Fields

Conditional random field (CRF) has been widely used in optical image and remote sensing image segmentation because of the advantage of directly modeling the posterior distribution and capturing arbitrary dependencies among observations. However, for nonstationary SAR images, applications of CRF often fail because of their nonstationary property. The triplet Markov field (TMF) model is well appropriate for nonstationary SAR image processing, owing to the introduction of an auxiliary field which reflects the nonstationarity. Therefore, we introduce an auxiliary field to describe the nonstationarity of the posterior distribution and propose an unsupervised SAR image segmentation algorithm based on a conditional TMF (CTMF) framework which combines the advantages of both CRF and TMF. The proposed CTMF framework explicitly takes into account the nonstationary property of SAR images, directly models the posterior distribution, and considers the interactions among the observed data. Experimental results on real SAR images validate the effectiveness of the algorithm proposed in this letter.

Unsupervised SAR Image Segmentation Based on Triplet Markov Fields With Graph Cuts

The triplet Markov fields (TMF) model is suitable for dealing with nonstationary synthetic aperture radar (SAR) images. Existing optimization approaches for the TMF model cannot balance segmentation accuracy and computational efficiency. Focusing on efficient optimization of the TMF model, we propose an unsupervised SAR image segmentation algorithm based on TMF with graph cuts (GCs) in this letter. Considering the existence of two label fields in the TMF model, an iterative optimization strategy under the criterion of maximum a posteriori is proposed, which iteratively estimates one label field with the other fixed. GCs are is used to find the optimal estimation of each label field. GCs optimization and parameter estimation using iterative conditional estimation perform iteratively, leading to an unsupervised segmentation algorithm. Experiments on simulated and real SAR images demonstrate that the proposed algorithm can obtain accurate segmentation results with reasonable computational cost.

SAR image multiclass segmentation using a multiscale and multidirection triplet Markov fields model in nonsubsampled contourlet transform domain

Triplet Markov fields (TMFs) model recently proposed is to deal with nonstationary image segmentation and has achieved promising results. In this paper, we propose a multiscale and multidirection TMF model for nonstationary synthetic aperture radar (SAR) image multiclass segmentation in nonsubsampled contourlet transform (NSCT) domain, named as NSCT-TMF model. NSCT-TMF model is capable of capturing the contextual information of image content in the spatial and scale spaces effectively by the construction of multiscale energy functions. And the derived multiscale and multidirection likelihoods of NSCT-TMF model can capture the dependencies of NSCT coefficients across scale and directions. In this way, the proposed model is able to achieve multiscale information fusion in terms of image configuration and features in underlying labeling process. Experimental results demonstrate that due to the effective propagation of the contextual information, NSCT-TMF model turns out to be more robust against speckle noise and improves the performance of nonstationary SAR image segmentation.

Triplet Markov fields with edge location for fast unsupervised multi-class segmentation of synthetic aperture radar images

Triplet Markov fields (TMF) model is suitable for dealing with multi-class segmentation of non-stationary synthetic aperture radar (SAR) images. In this study, an algorithm using TMF with edge location for fast unsupervised multi-class segmentation of SAR images is proposed. The new segmentation algorithm can locate edge accurately with reasonable computational cost. First for the statistical characteristics of multiplicative speckle noise in SAR image, an edge strength based on the ratio of exponentially weighted averages operator is introduced into the Turbopixels algorithm to obtain a superpixel graph with accurate edge location in SAR images. To enhance the computational efficiency and suppress the speckle, the TMF model on pixel is generalised to that on the superpixel graph. Then, the new corresponding potential energy function and maximisation of posterior marginal segmentation formula are derived. The experimental results on synthetic and real SAR images show that the proposed algorithm can obtain accurate edge location in multi-class segmentation of SAR images, as well as enhance the computational efficiency. Especially when dealing with SAR images in large size, the proposed algorithm can give a robust and efficient result of segmentation.

Unsupervised multi-class segmentation of SAR images using fuzzy triplet Markov fields model

Triplet Markov fields (TMF) model proposed recently is suitable for nonstationary image segmentation. For synthetic aperture radar (SAR) image segmentation, TMF model can adopt diverse statistical models for SAR data related to diverse radar backscattering sources. However, TMF model does not take into account the inherent imprecision associated with SAR images. In this paper, we propose a statistical fuzzy TMF (FTMF) model, which is a fuzzy clustering type treatment of TMF model, for unsupervised multi-class segmentation of SAR images. This paper contributes to SAR image segmentation in four aspects: (1) Nonstationarity of the statistical distribution of SAR intensity/amplitude data is taken into account to improve the spatial modeling capability of fuzzy TMF model. (2) Mean field theory is generalized to deal with planar variables to derive prior probability in fuzzy TMF model, which resolves the problem in Gibbs sampler in terms of computation cost. (3) A fuzzy objective function with regularization by Kullback–Leibler information of fuzzy TMF model is constructed for SAR image segmentation. The introduction of fuzziness for the belongingness of SAR image pixel makes fuzzy TMF model be able to retain more information from SAR image. (4) Fuzzy iterative conditional estimation (ICE) method, as an extension of the general ICE method is proposed to perform the model parameters estimation. The effectiveness of the proposed algorithm is demonstrated by application to simulated data and real SAR images.

Unsupervised multi-class segmentation of SAR images using triplet Markov fields models based on edge penalty

Non-Gaussian triplet Markov random fields (TMF) model is suitable for dealing with multi-class segmentation of nonstationary and non-Gaussian synthetic aperture radar (SAR) images. However, the segmentation of SAR images utilizing this model still fails to resolve the misclassifications due to the inaccuracy of edge location. In this paper, we propose a new unsupervised multi-class segmentation algorithm by fusing the traditional energy function of TMF model with the principle of edge penalty. Through the introduction of the penalty function based on local edge strength information, the new energy function could prevent segment from smoothing across boundaries. Then we optimize the objective function that stems from the new energy function to obtain an iterative multi-region merging Bayesian maximum posterior mode (MPM) segmentation equation for the new segmentation algorithm. The effectiveness of the proposed algorithm is demonstrated by application to simulated data and real SAR images.

Fast algorithm based on triplet Markov fields for unsupervised multi-class segmentation of SAR images

Non-Gaussian triplet Markov fields (TMF) model is suitable for dealing with multi-class segmentation of non-stationary and non-Gaussian synthetic aperture radar (SAR) images. Considering the complexity of the model and algorithm, as well as the requirement of real-time, and robust and efficient processing of SAR images, a fast algorithm based on TMF for unsupervised multi-class segmentation of SAR images is proposed in this paper. For the speckle noise in SAR images, numerical characteristic, threshold selection and QuadTree decomposition criterion are researched firstly. With the new method, a SAR image can quickly be mapped into an edge-based pixon-representation, which results in a coarse decomposition in smooth regions, and a fine decomposition in edges. Then by combining TMF model with the pixon-representation of SAR image, a new potential energy function of TMF based on pixon-representation is derived. Finally, the segmentation is finished by Bayesian maximum posterior mode (MPM). The effectiveness of the fast TMF algorithm is demonstrated by applying it to simulated data and real SAR images.

Triplet Markov Fields for the Classification of Complex Structure Data

We address the issue of classifying complex data. We focus on three main sources of complexity, namely the high dimensionality of the observed data, the dependencies between these observations and the general nature of the noise model underlying their distribution. We investigate the recent Triplet Markov Fields and propose new models in this class designed for such data and in particular allowing very general noise models. In addition, our models can handle the inclusion of a learning step in a consistent way so that they can be used in a supervised framework. One advantage of our models is that whatever the initial complexity of the noise model, parameter estimation can be carried out using state-of-the-art Bayesian clustering techniques under the usual simplifying assumptions. As generative models, they can be seen as an alternative, in the supervised case, to discriminative Conditional Random Fields. Identifiability issues underlying the models in the non supervised case, are discussed while the models performance is illustrated on simulated and real data exhibiting the mentioned various sources of complexity.

Multisensor triplet Markov fields and theory of evidence

Hidden Markov Fields (HMF) are widely applicable to various problems of image processing. In such models, the hidden process of interest X is a Markov field, which must be estimated from its observable noisy version Y. The success of HMF is due mainly to the fact that X remains Markov conditionally on the observed process, which facilitates different processing strategies such as Bayesian segmentation. Such models have been recently generalized to ‘Pairwise’ Markov fields (PMF), which offer similar processing advantages and superior modeling capabilities. In this generalization, one directly assumes the Markovianity of the pair ( X, Y). Afterwards, ‘Triplet’ Markov fields (TMF) have been proposed, in which the distribution of ( X, Y) is the marginal distribution of a Markov field ( X, U, Y), where U is an auxiliary random field. So U can have different interpretations and, when the set of its values is not too complex, X can still be estimated from Y. The aim of this paper is to show some connections between TMF and the Dempster–Shafer theory of evidence. It is shown that TMF allow one to perform the Dempster–Shafer fusion in different general situations, possibly involving several sensors. As a consequence, Bayesian segmentation strategies remain applicable.

Unsupervised image segmentation using triplet Markov fields

Hidden Markov fields (HMF) models are widely applied to various problems arising in image processing. In these models, the hidden process of interest X is a Markov field and must be estimated from its observable noisy version Y. The success of HMF is mainly due to the fact that the conditional probability distribution of the hidden process with respect to the observed one remains Markovian, which facilitates different processing strategies such as Bayesian restoration. HMF have been recently generalized to “pairwise” Markov fields (PMF), which offer similar processing advantages and superior modeling capabilities. In PMF one directly assumes the Markovianity of the pair ( X, Y). Afterwards, “triplet” Markov fields (TMF), in which the distribution of the pair ( X, Y) is the marginal distribution of a Markov field ( X, U, Y), where U is an auxiliary process, have been proposed and still allow restoration processing. The aim of this paper is to propose a new parameter estimation method adapted to TMF, and to study the corresponding unsupervised image segmentation methods. The latter are validated via experiments and real image processing.