Showing papers on "Markov random field published in 2021"

Quantifying the Reliability and Replicability of Psychopathology Network Characteristics.

Abstract: Pairwise Markov random field networks—including Gaussian graphical models (GGMs) and Ising models—have become the “state-of-the-art” method for psychopathology network analyses. Recent research has...

Generative Adversarial Networks and Markov Random Fields for oversampling very small training sets

Abstract: In this work, we propose a new method for oversampling the training set of a classifier, in a scenario of extreme scarcity of training data. It is based on two concepts: Generative Adversarial Networks (GAN) and vector Markov Random Field (vMRF). Thus, the generative block of GAN uses the vMRF model to synthesize surrogates by the Graph Fourier Transform. Then, the discriminative block implements a linear discriminant on features measuring clique similarities between the synthesized and the original instances. Both blocks iterate until the linear discriminant cannot discriminate the synthetic from the original instances. We have assessed the new method, called Generative Adversarial Network Synthesis for Oversampling (GANSO), with both simulated and real data in experiments where the classifier is to be trained with just 3 or 5 instances. The applications consisted of classification of stages of neuropsychological tests using electroencephalographic (EEG) and functional magnetic resonance imaging (fMRI) data and classification of sleep stages using electrocardiographic (ECG) data. We have verified that GANSO can effectively improve the classifier performance, while the benchmark method SMOTE is not appropriate to deal with such a small size of the training set.

An Enhanced 3-D Discrete Wavelet Transform for Hyperspectral Image Classification

Abstract: In the classification of hyperspectral image (HSI), there exists a common issue that the collected HSI data set is always contaminated by various noise (e.g., Gaussian, stripe, and deadline), degrading the classification results. To tackle this issue, we modify the 3-dimensional discrete wavelet transform (3DDWT) method by considering the noise effect on feature quality and propose an enhanced 3DDWT (E-3DDWT) approach to extract the feature and meanwhile alleviate the noise. Specifically, the proposed E-3DDWT method first applies classical 3DDWT method to the HSI data cube and thus can generate eight subcubes in each level. Then, the stripe noise is concentrated into several subcubes due to its spatial vertical property. Finally, we abandon these subcubes and obtain the feature cube by stacking the remaining ones. After acquiring the feature, we then adopt the convolutional neural network (CNN) model with an active learning strategy for classification since CNN has been verified to be a state-of-the-art feature extraction method for HSI classification, and active learning strategy can alleviate the insufficient labeled sample issue to some extent. In addition, we apply the Markov random field to enhance the final categorized results. Experiments on two synthetically striped data sets show that our proposed approach achieves better categorized results than other advanced methods.

Real-Time Object Detection for AUVs Using Self-Cascaded Convolutional Neural Networks

Abstract: This article presents an automatic real-time object detection method using sidescan sonar (SSS) and an onboard graphics processing unit (GPU). The detection method is based on a modified convolutional neural network (CNN), which is referred to as self-cascaded CNN (SC-CNN). The SC-CNN model segments SSS images into object-highlight, object-shadow, and seafloor areas, and it is robust to speckle noise and intensity inhomogeneity. Compared with typical CNN, SC-CNN utilizes crop layers which enable the network to use local and global features simultaneously without adding convolution parameters. Moreover, to take the local dependencies of class labels into consideration, the results of SC-CNN are postprocessed using Markov random field. Furthermore, the sea trial for real-time object detection via the presented method was implemented on our autonomous underwater vehicle (AUV) named SAILFISH via its GPU module at Jiaozhou Bay Bridge, Qingdao, China. The results show that the presented method for SSS image segmentation has obvious advantages when compared with the typical CNN and unsupervised segmentation methods, and is applicable in real-time object detection task.

Fine Building Segmentation in High-Resolution SAR Images Via Selective Pyramid Dilated Network

Abstract: The building extraction from synthetic aperture radar (SAR) images has always been a challenging research topic. Recently, the deep convolution neural network brings excellent improvements in SAR segmentation. The fully convolutional network and other variants are widely transferred to the SAR studies because of their high precision in optical images. They are still limited by their processing in terms of the geometric distortion of buildings, the variability of building structures, and scattering interference between adjacent targets in the SAR images. In this article, a unified framework called selective spatial pyramid dilated (SSPD) network is proposed for the fine building segmentation in SAR images. First, we propose a novel encoder–decoder structure for the fine building feature reconstruction. The enhanced encoder and the dual-stage decoder, composed of the CBM and the SSPD module, extract and recover the crucial multiscale information better. Second, we design the multilayer SSPD module based on the selective spatial attention. The multiscale building information with different attention on multiple branches is combined, optimized, and adaptively selected for adaptive filtering and extracting features of complex multiscale building targets in SAR images. Third, according to the building features and SAR imaging mechanism, a new loss function called L-shape weighting loss (LWloss) is proposed to heighten the attention on the L-shape footprint characteristics of the buildings and reduce the missing detection of line buildings. Besides, LWloss can also alleviate the class imbalance problem in the optimization stage. Finally, the experiments on a large-scene SAR image dataset demonstrate the effectiveness of the proposed method and verify its superiority over other approaches, such as the region-based Markov random field, U-net, and DeepLabv3+.

A Comprehensive Review of Markov Random Field and Conditional Random Field Approaches in Pathology Image Analysis

Abstract: Pathology image analysis is an essential procedure for clinical diagnosis of numerous diseases. To boost the accuracy and objectivity of the diagnosis, nowadays, an increasing number of intelligent systems are proposed. Among these methods, random field models play an indispensable role in improving the investigation performance. In this review, we present a comprehensive overview of pathology image analysis based on the Markov Random Fields (MRFs) and Conditional Random Fields (CRFs), which are two popular random field models. First of all, we introduce the framework of two random field models along with pathology images. Secondly, we summarize their analytical operation principle and optimization methods. Then, a thorough review of the recent articles based on MRFs and CRFs in the field of pathology is presented. Finally, we investigate the most commonly used methodologies from the related works and discuss the method migration in computer vision.

Sparse to Dense Scale Prediction for Crowd Couting in High Density Crowds

Abstract: Head detection-based crowd counting is of great importance and serves as a preprocessing step in many visual applications, for example, counting, tracking, and crowd dynamics understanding. Despite significant importance, limited amount of work is reported in the literature to detect human heads in high-density crowds. The problem of detecting heads in crowded scenes is challenging due to significant scale variations in the scene. In this paper, we tackle this problem by exploiting contextual constraints offer by the crowded scenes. For this purpose, we propose two networks, i.e., sparse-scale convolutional neural network (SS-CNN) and dense-scale convolutional neural network (DS-CNN). SS-CNN detects human heads with coarse information about the scales in the image. DS-CNN utilizes detection obtained from SS-CNN and generates dense scalemap by globally reasoning the coarse scales of detections obtained from SS-CNN via Markov Random Field (MRF). The dense scalemap has unique property that it captures all scale variations in image and provides an aid in generating scale-aware proposals. We evaluated our framework on three challenging state-of-the-art datasets, i.e., UCF-QNRF, WorldExpo’10, and UCF_CC_50. Experiment results show that proposed framework outperforms existing state-of-the-art methods.

A Fast Point Cloud Ground Segmentation Approach Based on Coarse-To-Fine Markov Random Field

Abstract: Ground segmentation is an important preprocessing task for autonomous vehicles (AVs) with 3D LiDARs. However, the existing ground segmentation methods are very difficult to balance accuracy and computational complexity. This paper proposes a fast point cloud ground segmentation approach based on a coarse-to-fine Markov random field (MRF) method. The method uses the coarse segmentation result of an improved local feature extraction algorithm instead of prior knowledge to initialize an MRF model. It provides an initial value for the fine segmentation and dramatically reduces the computational complexity. The graph cut method is then used to minimize the proposed model to achieve fine segmentation. Experiments on two public datasets and field tests show that our approach is more accurate than both methods based on features and MRF and faster than graph-based methods. It can process Velodyne HDL-64E data frames in real-time (24.86 ms, on average) with only one thread of the I7-8700 CPU. Compared with methods based on deep learning, it has better environmental adaptability.

Prestack Seismic Inversion With Data-Driven MRF-Based Regularization

Abstract: Regularization is effective in mitigating the ill-condition existing in inverse problems. With respect to the ill-conditioned prestack seismic inversion, regularization aims to stabilize the multiple inverted results and, essentially, reconstruct structural features of subsurface parameter (model) as realistic as possible. Among variants of regularization method, Markov random field (MRF) is an effective approach in formulating prior constraint. However, standard MRF-based or other methods often require prior knowledge of the structural features of desired models, e.g., smoothness, blockiness, and sparsity, after which the prior constraint is formulated. Therefore, such a model-driven regularization method lacks applicability to the cases with geological complexity or limited prior knowledge. In this article, we propose a data-driven regularization scheme for prestack seismic inversion. The MRF-based constraints formulated by multiple orders are quantitatively integrated into the inversion procedure driven by seismic data. In order to endow the method with high adaptation to geological complexity, we iteratively adjust the regularization parameters of multiple orders via the maximum likelihood estimator. Besides, we incorporate the multivariate Gaussian distribution among elastic parameters into the model update/perturbation in fast simulated annealing, by which the objective function is optimized while the multiple results are correlated and stabilized. Synthetic tests indicate that the proposed method is capable of revealing structural details and achieving multiple results with less uncertainty. Field application provides further validation, wherein the results distinctly reveal structural details within the target formation.

A Local Region-Based Level Set Method With Markov Random Field for Side-Scan Sonar Image Multi-Level Segmentation

Abstract: Side-scan sonar is widely utilized in the field of underwater exploration. Accurate segmentation of the side-scan sonar images is an essential part of sonar image processing. To improve the accuracy of image segmentation and reduce the occupancy of computing resources, a new active contour model for image segmentation is proposed in this article. Firstly, we make the energy function of each pixel determined by itself and its neighbors. This article embeds the local texture neighborhood region system and define its structure to against the noise and object boundary pollution of the image. Further, we introduce a Bayesian framework that embeds the Markov Random Field model and local texture information to counteract the problem of intensity inhomogeneity, improve the accuracy of segmentation, and reduce computational cost. Finally, according to this model, we customize a new multi-class energy function which applied to the level set function. By minimizing this energy equation, we can divide the image into three classes: object, shadow, and seabed background. The experimental results show that this method has a good effect on the synthesized high-noise sonar images and real sonar images, and also have significantly increased the accurate recognition of underwater targets.

A novel unsupervised forest change detection method based on the integration of a multiresolution singular value decomposition fusion and an edge-aware Markov Random Field algorithm

Abstract: As a leading natural wealth, forests play an essential role in the development and prosperity of countries. Hence, monitoring their changes can lead to proper management and planning in conserving ...

Maximum Response Deep Learning Using Markov, Retinal & Primitive Patch Binding With GoogLeNet & VGG-19 for Large Image Retrieval

Abstract: Smart and productive image retrieval from flexible image datasets is an unavoidable necessity of the current period. Crude picture marks are imperative to mirror the visual ascribes for content-based image retrieval (CBIR). Algorithmically enlightened and recognized visual substance structure image marks to accurately file and recover comparative outcomes. Consequently, highlighted vectors ought to contain adequate image data with color, shape, objects, spatial data viewpoints to recognize image class as a qualifying applicant. This article presents the maximum response of visual features of an image over profound convolutional neural networks in blend with an innovative content-based image retrieval plan to recover phenomenally precise outcomes. For this determination, a serial fusion of GoogLeNet and VGG-19 based generated signatures are formulated with visual features including texture, color and shape. Initially, the maximum response is calculated for texture pattern by using Markov Random Field (MRF) classifier. Thereafter, cascaded samples are passed through a human retinal system like descriptor named Fast Retina Keypoint (FREAK) for corresponding fundamental points through the image. GoogLeNet and VGG-19 are applied to extract deep features of an image; hence color components are obtained using a correlogram. Finally, all the image signatures are combined and passed through the BoW scheme. The proposed method is applied experimentally to challenging datasets, including Caltech-256, ALOT (250), Corel 1000, Cifar-100, and Cifar 10. Remarkable precision,Recall and F-score results obtained.The texture dataset ALOT (250) with the uppermost precision rate 0.99 for a maximum of its categories, whereas Caltech-256 gives 0.66 precision, and Corel 1000 0.99 for VGG-19 and 0.95 for GoogLeNet. Recall, F-score, ARR and ARP rates shows the significant rates in most of the image categories.

A Spatially Coupled Data-Driven Approach for Lithology/Fluid Prediction

Abstract: Prediction of lithology/fluid (LF) characteristics is always the bottleneck problem and difficulty of reservoir characterization. Deep-learning-based data-driven methods can review data and find specific trends and patterns that would not be apparent to humans, and have been successfully used in many geophysical applications including LF prediction (LFP). However, the above methods mostly predict LF point-by-point, which means that the spatial correlation of LF is not considered. When the predicted LF results are combined to form a 2-D/3-D image, the resulting image will be noisy or even geologically unreliable. To overcome these issues, we proposed a spatially coupled data-driven (convolutional neural network, CNN) approach for LFP from the poststack seismic data and well observations. Here, the vertical couplings of the LF are modeled by a Markov chain (MC) prior and the lateral continuity of the LF is further defined by a Markov random field (MRF) prior. We also proposed to perform spectral decomposition via inversion strategies (ISD) to get a time–frequency (TF) spectrum as the input of CNN. ISD helps make full use of the information hidden in the frequency domain of the poststack seismic data. Well-logs and poststack seismic data are integrated in a consistent manner to obtain predictions of the LF classes with the associated uncertainty statements. The LFP results of the proposed approach are more laterally continuous and geologically reliable than the LFP results of the point-by-point. We determined the effectiveness of this methodology on a 2-D synthetic model and a 3-D field seismic data set.

Unsupervised Bayesian Subpixel Mapping of Hyperspectral Imagery Based on Band-Weighted Discrete Spectral Mixture Model and Markov Random Field

Abstract: Although accurate training and initialization information is difficult to acquire, unsupervised hyperspectral subpixel mapping (SPM) without relying on this predefined information is an insufficiently addressed research issue. This letter presents a novel Bayesian approach for unsupervised SPM of hyperspectral imagery (HSI) based on the Markov random field (MRF) and a band-weighted discrete spectral mixture model (BDSMM), with the following key characteristics. First, this is an unsupervised approach that allows adjustment of abundance and endmember information adaptively for less relying on algorithm initialization. Second, this approach consists of the BDSMM for accommodating the noise heterogeneity and the hidden label field of subpixels in HSI. The BDSMM also integrates SPM into the spectral mixture analysis and allows enhanced SPM by fully exploring the endmember-abundance patterns in HSI. Third, the MRF and BDSMM are integrated into a Bayesian framework to use both the spatial and spectral information efficiently, and an expectation-maximization (EM) approach is designed to solve the model by iteratively estimating the endmembers and the label field. Experiments on both simulated and real HSI demonstrate that the proposed algorithm can yield better performance than traditional methods.

A Unifying Generative Model for Graph Learning Algorithms: Label Propagation, Graph Convolutions, and Combinations

Abstract: Semi-supervised learning on graphs is a widely applicable problem in network science and machine learning. Two standard algorithms -- label propagation and graph neural networks -- both operate by repeatedly passing information along edges, the former by passing labels and the latter by passing node features, modulated by neural networks. These two types of algorithms have largely developed separately, and there is little understanding about the structure of network data that would make one of these approaches work particularly well compared to the other or when the approaches can be meaningfully combined. Here, we develop a Markov random field model for the data generation process of node attributes, based on correlations of attributes on and between vertices, that motivates and unifies these algorithmic approaches. We show that label propagation, a linearized graph convolutional network, and their combination can all be derived as conditional expectations under our model, when conditioning on different attributes. In addition, the data model highlights deficiencies in existing graph neural networks (while producing new algorithmic solutions), serves as a rigorous statistical framework for understanding graph learning issues such as over-smoothing, creates a testbed for evaluating inductive learning performance, and provides a way to sample graphs attributes that resemble empirical data. We also find that a new algorithm derived from our data generation model, which we call a Linear Graph Convolution, performs extremely well in practice on empirical data, and provide theoretical justification for why this is the case.

End-to-end learning of multiple sequence alignments with differentiable Smith-Waterman

Abstract: Multiple Sequence Alignments (MSAs) of homologous sequences contain information on structural and functional constraints and their evolutionary histories. Despite their importance for many downstream tasks, such as structure prediction, MSA generation is often treated as a separate pre-processing step, without any guidance from the application it will be used for. Here, we implement a smooth and differentiable version of the Smith-Waterman pairwise alignment algorithm that enables jointly learning an MSA and a downstream machine learning system in an end-to-end fashion. To demonstrate its utility, we introduce SMURF (Smooth Markov Unaligned Random Field), a new method that jointly learns an alignment and the parameters of a Markov Random Field for unsupervised contact prediction. We find that SMURF mildly improves contact prediction on a diverse set of protein and RNA families. As a proof of concept, we demonstrate that by connecting our differentiable alignment module to AlphaFold2 and maximizing the predicted confidence metric, we can learn MSAs that improve structure predictions over the initial MSAs. This work highlights the potential of differentiable dynamic programming to improve neural network pipelines that rely on an alignment.

Random Fields in Physics, Biology and Data Science

Abstract: A random field is the representation of the joint probability distribution for a set of random variables. Markov fields, in particular, have a long standing tradition as the theoretical foundation of many applications in statistical physics and probability. For strictly positive probability densities, a Markov random field is also a Gibbs field, i.e. a random field supplemented with a measure that implies the existence of a regular conditional distribution. Markov random fields have been used in statistical physics, dating back as far as the Ehrenfests. However, their measure theoretical foundations were developed much later by Dobruschin, Lanford and Ruelle, as well as by Hammersley and Clifford. Aside from its enormous theoretical relevance, due to its generality and simplicity, Markov random fields have been used in a broad range of applications in equilibrium and non-equilibrium statistical physics, in non-linear dynamics and ergodic theory. Also in computational molecular biology, ecology, structural biology, computer vision, control theory, complex networks and data science, to name but a few. Often these applications have been inspired by the original statistical physics approaches. Here, we will briefly present a modern introduction to the theory of random fields, later we will explore and discuss some of the recent applications of random fields in physics, biology and data science. Our aim is to highlight the relevance of this powerful theoretical aspect of statistical physics and its relation to the broad success of its many interdisciplinary applications.

Cauchy Markov Random Field Priors for Bayesian Inversion.

Abstract: The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo (MCMC) methods are usually required In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison We firstly propose a one-dimensional second order Cauchy difference prior, and construct new first and second order two-dimensional isotropic Cauchy difference priors Another new Cauchy prior is based on the stochastic partial differential equation approach, derived from Matern type Gaussian presentation The comparison also includes Cauchy sheets Our numerical computations are based on both maximum a posteriori and conditional mean estimationWe exploit state-of-the-art MCMC methodologies such as Metropolis-within-Gibbs, Repelling-Attracting Metropolis, and No-U-Turn sampler variant of Hamiltonian Monte Carlo We demonstrate the models and methods constructed for one-dimensional and two-dimensional deconvolution problems Thorough MCMC statistics are provided for all test cases, including potential scale reduction factors

Iterated graph cut method for automatic and accurate segmentation of finger-vein images

Abstract: Recent advances in computer vision and machine intelligence have facilitated biometric technologies, which increasingly rely on image data in security practices. As an important biometric identifier, the near-infrared (NIR) finger-vein pattern is favoured by non-contact, high accuracy, and enhanced security systems. However, large stacks of low-contrast and complex finger-vein images present barriers to manual image segmentation, which locates the objects of interest. Although some headway in computer-aided segmentation has been made, state-of-the-art approaches often require user interaction or prior training, which are tedious, time-consuming and prone to operator bias. Recognizing this deficiency, the present study exploits structure-specific contextual clues and proposes an iterated graph cut (IGC) method for automatic and accurate segmentation of finger-vein images. To this end, the geometric structures of the image-acquisition system and the fingers provide the hard (centreline along the finger) and shape (rectangle around the finger) constraints. A node-merging scheme is applied to reduce the computational burden. The Gaussian probability model determines the initial labels. Finally, the maximum a posteriori Markov random field (MAP-MRF) framework is tasked with iteratively updating the data models of the object and the background. Our approach was extensively evaluated on 4 finger-vein databases and compared with some benchmark methods. The experimental results indicate that the proposed IGC method outperforms the state-of-the-practice approaches in finger-vein image segmentation. Specifically, the IGC method, relative to its level set deep learning (LSDL) counterpart, can increase the average F-measure value by 5.03%, 6.56%, 49.91%, and 22.89% when segmenting images from four different finger-vein databases. Therefore, this work can provide a feasible path towards fully automatic image segmentation.

Boundary-Aware and Semiautomatic Segmentation of 3-D Object in Point Clouds

Abstract: Scene understanding in 3-D point clouds requires to annotate points manually at the model training stage. To reduce manual efforts in labeling points, this letter focuses on proposing an efficient method to implement semiautomatic segmentation of 3-D objects in 3-D point clouds. Specifically, to handle point clouds with high point density, supervoxels are treated as basic operating units during the object segmentation procedure. To obtain the valuable boundaries for guiding 3-D object segmentation, we propose to filter meaningless boundaries obtained by a traditional boundary detection method. Once valuable boundaries are obtained, we propose a boundary-aware Markov random field (MRF) model to consider the object-boundary constraint into generating the boundary-preserved segmentation results. Extensive experiments on two data sets show the effectiveness of our proposed framework on segmenting 3-D objects from point cloud scenes.

Multigranularity Multiclass-Layer Markov Random Field Model for Semantic Segmentation of Remote Sensing Images

Abstract: Semantic segmentation is one of the most important tasks in remote sensing. However, as spatial resolution increases, distinguishing the homogeneity of each land class and the heterogeneity between different land classes are challenging. The Markov random field model (MRF) is a widely used method for semantic segmentation due to its effective spatial context description. To improve segmentation accuracy, some MRF-based methods extract more image information by constructing the probability graph with pixel or object granularity units, and some other methods interpret the image from different semantic perspectives by building multilayer semantic classes. However, these MRF-based methods fail to capture the relationship between different granularity features extracted from the image and hierarchical semantic classes that need to be interpreted. In this article, a new MRF-based method is proposed to incorporate the multigranularity information and the multilayer semantic classes together for semantic segmentation of remote sensing images. The proposed method develops a framework that builds a hybrid probability graph on both pixel and object granularities and defines a multiclass-layer label field with hierarchical semantic over the hybrid probability graph. A generative alternating granularity inference is suggested to provide the result by iteratively passing and updating information between different granularities and hierarchical semantics. The proposed method is tested on texture images, different remote sensing images obtained by the SPOT5, Gaofen-2, GeoEye, and aerial sensors, and Pavia University hyperspectral image. Experiments demonstrate that the proposed method shows a better segmentation performance than other state-of-the-art methods.

Ground Segmentation Algorithm for Sloped Terrain and Sparse LiDAR Point Cloud

Abstract: Distinguishing obstacles from ground is an essential step for common perception tasks such as object detection-and-tracking or occupancy grid maps. Typical approaches rely on plane fitting or local geometric features, but their performance is reduced in situations with sloped terrain or sparse data. Some works address these issues using Markov Random Fields and Belief Propagation, but these rely on local geometric features uniquely. This article presents a strategy for ground segmentation in LiDAR point clouds composed by two main steps: (i) First, an initial classification is performed dividing the points in small groups and analyzing geometric features between them. (ii) Then, this initial classification is used to model the surrounding ground height as a Markov Random Field, which is solved using the Loopy Belief Propagation algorithm. Points are finally classified comparing their height with the estimated ground height map. On one hand, using an initial estimation to model the Markov Random Field provides a better description of the scene than local geometric features commonly used alone. On the other hand, using a graph-based approach with message passing achieves better results than simpler filtering or enhancement techniques, since data propagation compensates sparse distributions of LiDAR point clouds. Experiments are conducted with two different sources of information: nuScenes’s public dataset and an autonomous vehicle prototype. The estimation results are analyzed with respect to other methods, showing a good performance in a variety of situations.

Masked Autoencoder for Distribution Estimation on Small Structured Data Sets

Abstract: Autoregressive models are among the most successful neural network methods for estimating a distribution from a set of samples. However, these models, such as other neural methods, need large data sets to provide good estimations. We believe that knowing structural information about the data can improve their performance on small data sets. Masked autoencoder for distribution estimation (MADE) is a well-structured density estimator, which alters a simple autoencoder by setting a set of masks on its connections to satisfy the autoregressive condition. Nevertheless, this model does not benefit from extra information that we might know about the structure of the data. This information can especially be advantageous in case of training on small data sets. In this article, we propose two autoencoders for estimating the density of a small set of observations, where the data have a known Markov random field (MRF) structure. These methods modify the masking process of MADE, according to conditional dependencies inferred from the MRF structure, to reduce either the model complexity or the problem complexity. We compare the proposed methods with some related binary, discrete, and continuous density estimators on MNIST, binarized MNIST, OCR-letters, and two synthetic data sets.