Statistical approach to X-ray CT imaging and its applications in image analysis. II. A new stochastic model-based image segmentation technique for X-ray CT image

For pt.I, see ibid., vol.11, no.1, p.53.61 (1992). Based on the statistical properties of X-ray CT imaging given in pt.I, an unsupervised stochastic model-based image segmentation technique for X-ray CT images is presented. This technique utilizes the finite normal mixture distribution and the underlying Gaussian random field (GRF) as the stochastic image model. The number of image classes in the observed image is detected by information theoretical criteria (AIC or MDL). The parameters of the model are estimated by expectation-maximization (EM) and classification-maximization (CM) algorithms. Image segmentation is performed by a Bayesian classifier. Results from the use of simulated and real X-ray computerized tomography (CT) image data are presented to demonstrate the promise and effectiveness of the proposed technique.     

The EM/MPM algorithm for segmentation of textured images: analysisand further experimental results

In this paper we present new results relative to the "expectation-maximization/maximization of the posterior marginals" (EM/MPM) algorithm for simultaneous parameter estimation and segmentation of textured images. The EM/MPM algorithm uses a Markov random field model for the pixel class labels and alternately approximates the MPM estimate of the pixel class labels and estimates parameters of the observed image model. The goal of the EM/MPM algorithm is to minimize the expected value of the number of misclassified pixels. We present new theoretical results in this paper which show that the algorithm can be expected to achieve this goal, to the extent that the EM estimates of the model parameters are close to the true values of the model parameters. We also present new experimental results demonstrating the performance of the EM/MPM algorithm.     

Estimation of 2-D noisy fractional Brownian motion and itsapplications using wavelets

The two-dimensional (2-D) fractional Brownian motion (fBm) model is useful in describing natural scenes and textures. Most fractal estimation algorithms for 2-D isotropic fBm images are simple extensions of the one-dimensional (1-D) fBm estimation method. This method does not perform well when the image size is small (say, 32x32). We propose a new algorithm that estimates the fractal parameter from the decay of the variance of the wavelet coefficients across scales. Our method places no restriction on the wavelets. Also, it provides a robust parameter estimation for small noisy fractal images. For image denoising, a Wiener filter is constructed by our algorithm using the estimated parameters and is then applied to the noisy wavelet coefficients at each scale. We show that the averaged power spectrum of the denoised image is isotropic and is a nearly 1/f process. The performance of our algorithm is shown by numerical simulation for both the fractal parameter and the image estimation. Applications to coastline detection and texture segmentation in a noisy environment are also demonstrated.     

EM image segmentation algorithm based on an inhomogeneous hidden MRF model

This paper introduces a Bayesian image segmentation algorithm that considers the label scale variability of images. An inhomogeneous hidden Markov random field is adopted in this algorithm to model the label scale variability as prior probabilities. An EM algorithm is developed to estimate parameters of the prior probabilities and likelihood probabilities. The image segmentation is established by using a MAP estimator. Different images are tested to verify the algorithm and comparisons with other segmentation algorithms are carried out. The segmentation results show the proposed algorithm has better performance than others.     

ML parameter estimation for Markov random fields with applicationsto Bayesian tomography

Markov random fields (MRFs) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: (i) direct parameter estimation for MRFs is known to be mathematically and numerically challenging; (ii) parameters can not be directly estimated because the true image cross section is unavailable. We propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as generalized Gaussian MRF (GGMRF). We derive methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, sigma, has a simple closed-form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an off-line numerical computation of the dependence of the partition function on p. We present a fast algorithm for computing ML parameter estimates when the true image is unavailable. To do this, we use the expectation maximization (EM) algorithm. We develop a fast simulation method to replace the E-step, and a method to improve the parameter estimates when the simulations are terminated prior to convergence. Experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets.     

Design and Analysis of an Efficient Evolutionary Image Segmentation Algorithm

Evolutionary image segmentation algorithms have a number of advantages such as continuous contour, non-oversegmentation, and non-thresholds. However, most of the evolutionary image segmentation algorithms suffer from long computation time because the number of encoding parameters is large. In this paper, design and analysis of an efficient evolutionary image segmentation algorithm EISA are proposed. EISA uses a K-means algorithm to split an image into many homogeneous regions, and then uses an intelligent genetic algorithm IGA associated with an effective chromosome encoding method to merge the regions automatically such that the objective of the desired segmentation can be effectively achieved, where IGA is superior to conventional genetic algorithms in solving large parameter optimization problems. High performance of EISA is illustrated in terms of both the evaluation performance and computation time, compared with some current segmentation methods. It is empirically shown that EISA is robust and efficient using nature images with various characteristics.     

Precise segmentation of multimodal images.

We propose new techniques for unsupervised segmentation of multimodal grayscale images such that each region-of-interest relates to a single dominant mode of the empirical marginal probability distribution of grey levels. We follow the most conventional approaches in that initial images and desired maps of regions are described by a joint Markov-Gibbs random field (MGRF) model of independent image signals and interdependent region labels. However, our focus is on more accurate model identification. To better specify region borders, each empirical distribution of image signals is precisely approximated by a linear combination of Gaussians (LCG) with positive and negative components. We modify an expectation-maximization (EM) algorithm to deal with the LCGs and also propose a novel EM-based sequential technique to get a close initial LCG approximation with which the modified EM algorithm should start. The proposed technique identifies individual LCG models in a mixed empirical distribution, including the number of positive and negative Gaussians. Initial segmentation based on the LCG models is then iteratively refined by using the MGRF with analytically estimated potentials. The convergence of the overall segmentation algorithm at each stage is discussed. Experiments show that the developed techniques segment different types of complex multimodal medical images more accurately than other known algorithms.     

Estimation of generalized mixtures and its application in imagesegmentation

We introduce the notion of a generalized mixture and propose some methods for estimating it, along with applications to unsupervised statistical image segmentation. A distribution mixture is said to be "generalized" when the exact nature of the components is not known, but each belongs to a finite known set of families of distributions. For instance, we can consider a mixture of three distributions, each being exponential or Gaussian. The problem of estimating such a mixture contains thus a new difficulty: we have to label each of three components (there are eight possibilities). We show that the classical mixture estimation algorithms-expectation-maximization (EM), stochastic EM (SEM), and iterative conditional estimation (ICE)-can be adapted to such situations once as we dispose of a method of recognition of each component separately. That is, when we know that a sample proceeds from one family of the set considered, we have a decision rule for what family it belongs to. Considering the Pearson system, which is a set of eight families, the decision rule above is defined by the use of "skewness" and "kurtosis". The different algorithms so obtained are then applied to the problem of unsupervised Bayesian image segmentation, We propose the adaptive versions of SEM, EM, and ICE in the case of "blind", i.e., "pixel by pixel", segmentation. "Global" segmentation methods require modeling by hidden random Markov fields, and we propose adaptations of two traditional parameter estimation algorithms: Gibbsian EM (GEM) and ICE allowing the estimation of generalized mixtures corresponding to Pearson's system. The efficiency of different methods is compared via numerical studies, and the results of unsupervised segmentation of three real radar images by different methods are presented.     

Bayesian pixel classification using spatially variant finitemixtures and the generalized EM algorithm

A spatially variant finite mixture model is proposed for pixel labeling and image segmentation. For the case of spatially varying mixtures of Gaussian density functions with unknown means and variances, an expectation-maximization (EM) algorithm is derived for maximum likelihood estimation of the pixel labels and the parameters of the mixture densities, An a priori density function is formulated for the spatially variant mixture weights. A generalized EM algorithm for maximum a posteriori estimation of the pixel labels based upon these prior densities is derived. This algorithm incorporates a variation of gradient projection in the maximization step and the resulting algorithm takes the form of grouped coordinate ascent. Gaussian densities have been used for simplicity, but the algorithm can easily be modified to incorporate other appropriate models for the mixture model component densities. The accuracy of the algorithm is quantitatively evaluated through Monte Carlo simulation, and its performance is qualitatively assessed via experimental images from computerized tomography (CT) and magnetic resonance imaging (MRI).     

Nonlinear estimation of scene parameters from digital images usingzero-hit-length statistics

A zero-hit run-length probability model for image statistics is derived. The statistics are based on the lengths of runs of pixels that do not include any part of objects that define a scene model. The statistics are used to estimate the density and size of the discrete objects (modeled as disks) from images when the image pixel size is significant relative to the object size. Using different combinations of disk size, density, and image resolution (pixel size) in simulated images, parameter estimation may be used to investigate the essential invertibility of object size and density. Analysis of the relative errors and 95% confidence intervals indicates the accuracy and reliability of the estimates. An integrated parameter r, reveals relationships between errors and the combinations of the three basic parameters of object size, density, and pixel size. The method may be used to analyze real remotely sensed images if simplifying assumptions are relaxed to include the greater complexity found in real data     

结合EM/MPM算法和Voronoi划分的图像分割方法

为了实现在模型参数先验分布知识未知情况下进行基于区域和统计方法的图像分割,同时获取更精确的模型参数估计结果,提出了一种结合Voronoi划分技术、最大期望值(Expectation Maximization, EM)和最大边缘概率(Maximization of the Posterior Marginal, MPM)算法的图像分割方法。该方法利用Voronoi划分技术将图像域划分为若干子区域,待分割图像中的同质区域可以由一组子区域拟合而成,并假定同一同质区域内像素强度服从同一独立的正态分布,从而建立图像模型,然后结合EM/MPM算法进行图像分割和模型参数估计,其中,MPM算法用于实现面向同质区域的图像分割,EM算法用于估计图像模型参数。为了验证本文图像分割方法,分别对合成图像和真实图像进行了分割实验,测试结果的定性和定量分析表明了该方法的有效性和准确性。      

A unified framework for MAP estimation in remote sensing image segmentation

A complete framework is proposed for applying the maximum a posteriori (MAP) estimation principle in remote sensing image segmentation. The MAP principle provides an estimate for the segmented image by maximizing the posterior probabilities of the classes defined in the image. The posterior probability can be represented as the product of the class conditional probability (CCP) and the class prior probability (CPP). In this paper, novel supervised algorithms for the CCP and the CPP estimations are proposed which are appropriate for remote sensing images where the estimation process might to be done in high-dimensional spaces. For the CCP, a supervised algorithm which uses the support vector machines (SVM) density estimation approach is proposed. This algorithm uses a novel learning procedure, derived from the main field theory, which avoids the (hard) quadratic optimization problem arising from the traditional formulation of the SVM density estimation. For the CPP estimation, Markov random field (MRF) is a common choice which incorporates contextual and geometrical information in the estimation process. Instead of using predefined values for the parameters of the MRF, an analytical algorithm is proposed which automatically identifies the values of the MRF parameters. The proposed framework is built in an iterative setup which refines the estimated image to get the optimum solution. Experiments using both synthetic and real remote sensing data (multispectral and hyperspectral) show the powerful performance of the proposed framework. The results show that the proposed density estimation algorithm outperforms other algorithms for remote sensing data over a wide range of spectral dimensions. The MRF modeling raises the segmentation accuracy by up to 10% in remote sensing images.     

Sonar image segmentation using an unsupervised hierarchical MRFmodel

This paper is concerned with hierarchical Markov random field (MRP) models and their application to sonar image segmentation. We present an original hierarchical segmentation procedure devoted to images given by a high-resolution sonar. The sonar image is segmented into two kinds of regions: shadow (corresponding to a lack of acoustic reverberation behind each object lying on the sea-bed) and sea-bottom reverberation. The proposed unsupervised scheme takes into account the variety of the laws in the distribution mixture of a sonar image, and it estimates both the parameters of noise distributions and the parameters of the Markovian prior. For the estimation step, we use an iterative technique which combines a maximum likelihood approach (for noise model parameters) with a least-squares method (for MRF-based prior). In order to model more precisely the local and global characteristics of image content at different scales, we introduce a hierarchical model involving a pyramidal label field. It combines coarse-to-fine causal interactions with a spatial neighborhood structure. This new method of segmentation, called the scale causal multigrid (SCM) algorithm, has been successfully applied to real sonar images and seems to be well suited to the segmentation of very noisy images. The experiments reported in this paper demonstrate that the discussed method performs better than other hierarchical schemes for sonar image segmentation.     

Optimal segmentation of cell images

An optimal segmentation algorithm for light microscopic cell images is presented. The image segmentation is performed by thresholding a parametric image approximating the original image. Using the mean squared error between the original and the constructed image as the cost function, the segmentation problem is transformed into an optimisation process where parametric parameters are determined that minimise the defined cost function. The cost function is iteratively minimised using an unsupervised learning rule to adjust the parameters, and a parametric image is constructed at each iteration, based on the obtained parameters. The cell region is extracted by thresholding the final parametric image, where the threshold is one of the image parameters. Application results to real cervical images are provided to show the performance of the proposed segmentation approach. Experimental segmentation results are presented for the proposed optimal algorithm for synthetic cell images corrupted by variant levels of noise; these results are compared with the K-means clustering method and Bayes classifier in terms of classification errors     

烟花算法在SAR图像属性散射中心参数估计中的应用

针对合成孔径雷达（SAR）属性散射中心估计问题,提出基于烟花算法的方法。首先,在图像域对SAR图像中高能量区域进行分割解耦,获得单个独立散射中心在图像域的表现形式。在此基础上,以属性散射中心参数化模型为基础,构建优化问题,对分离出来的单个散射中心进行最优参数的搜索。在此阶段,引入烟花算法进行参数寻优。该算法具有强大的全局和局部搜索能力,在保证优化精度的条件下避免陷入局部最优,从而保证散射中心参数估计的可靠性。在原始图像中剔除求解后的单个散射中心,对残余图像进行高能量区域分割,序惯估计下一个散射中心的属性参数。最终,获取输入SAR图像上所有散射中心的参数集。实验中,首先基于MSTAR数据集中的SAR图像进行参数估计验证,通过参数估计结果与原始图像的对比以及基于估计参数集对原始图像进行重构,反映了提出算法的有效性。此外,实验还基于估计得到的属性参数进行SAR目标识别算法验证,通过与其他参数估计算法在相同条件下进行识别性能的对比,进一步体现了提出方法在属性散射中心参数估计上的性能优势。     

On-line estimation of hidden Markov model parameters based on theKullback-Leibler information measure

Sequential or online hidden Markov model (HMM) signal processing schemes are derived, and their performance is illustrated by simulation. The online algorithms are sequential expectation maximization (EM) schemes and are derived by using stochastic approximations to maximize the Kullback-Leibler information measure. The schemes can be implemented either as filters or fixed-lag or sawtooth-lag smoothers. They yield estimates of the HMM parameters including transition probabilities, Markov state levels, and noise variance. In contrast to the offline EM algorithm (Baum-Welch scheme), which uses the fixed-interval forward-backward scheme, the online schemes have significantly reduced memory requirements and improved convergence, and they can estimate HMM parameters that vary slowly with time or undergo infrequent jump changes. Similar techniques are used to derive online schemes for extracting finite-state Markov chains imbedded in a mixture of white Gaussian noise (WGN) and deterministic signals of known functional form with unknown parameters     

A multiscale random field model for Bayesian image segmentation

Many approaches to Bayesian image segmentation have used maximum a posteriori (MAP) estimation in conjunction with Markov random fields (MRF). Although this approach performs well, it has a number of disadvantages. In particular, exact MAP estimates cannot be computed, approximate MAP estimates are computationally expensive to compute, and unsupervised parameter estimation of the MRF is difficult. The authors propose a new approach to Bayesian image segmentation that directly addresses these problems. The new method replaces the MRF model with a novel multiscale random field (MSRF) and replaces the MAP estimator with a sequential MAP (SMAP) estimator derived from a novel estimation criteria. Together, the proposed estimator and model result in a segmentation algorithm that is not iterative and can be computed in time proportional to MN where M is the number of classes and N is the number of pixels. The also develop a computationally efficient method for unsupervised estimation of model parameters. Simulations on synthetic images indicate that the new algorithm performs better and requires much less computation than MAP estimation using simulated annealing. The algorithm is also found to improve classification accuracy when applied to the segmentation of multispectral remotely sensed images with ground truth data.     

Medical image segmentation based on non-parametric mixture models with spatial information

Because of too much dependence on prior assumptions, parametric estimation methods using finite mixture models are sensitive to noise in image segmentation. In this study, we developed a new medical image segmentation method based on non-parametric mixture models with spatial information. First, we designed the non-parametric image mixture models based on the cosine orthogonal sequence and defined the spatial information functions to obtain the spatial neighborhood information. Second, we calculated the orthogonal polynomial coefficients and the mixing ratio of the models using expectation-maximization (EM) algorithm, to classify the images by Bayesian Principle. This method can effectively overcome the problem of model mismatch, restrain noise, and keep the edge property well. In comparison with other methods, our method appears to have a better performance in the segmentation of simulated brain images and computed tomography (CT) images.     

Multivariate Gaussian MRF for multispectral scene segmentation andanomaly detection

Gaussian Markov random field texture models and multivariate parametric clustering algorithms have been applied extensively for segmentation, restoration, and anomaly detection of single-band and multispectral imagery, respectively. The present work extends and combines these previous efforts to demonstrate joint spatial-spectral modeling of multispectral imagery, a multivariate (vector observations) GMRF texture model is employed. Algorithms for parameter estimation and image segmentation are discussed, and a new anomaly detection technique is developed. The model is applied to imagery from the Daedalus sensor. Image segmentation results from test images are discussed and compared to spectral clustering results. The test images are collages, with known texture boundaries constructed from larger data cubes. Anomaly detection results for two Daedalus images are also presented, assessed using receiver operating characteristic (ROC) performance curves, and compared to spectral clustering models. It is demonstrated that even the simplest first-order isotropic texture models provide significant improvement in image segmentation and anomaly detection over pure spectral clustering for the data sets examined. The sensitivity of anomaly detection performance to the choice of parameter estimation method and to the number of texture segments is examined for one example data set     

EM algorithm for image segmentation initialized by a tree structurescheme

In this correspondence, the objective is to segment vector images, which are modeled as multivariate finite mixtures. The underlying images are characterized by Markov random fields (MRFs), and the applied segmentation procedure is based on the expectation-maximization (EM) technique. We propose an initialization procedure that does not require any prior information and yet provides excellent initial estimates for the EM method. The performance of the overall segmentation is demonstrated by segmentation of simulated one-dimensional (1D) and multidimensional magnetic resonance (MR) brain images.