Recently, the hidden Markov model (HMM) with student’s t-mixture model (SMM), called student’s t-HMM (SHMM) for short, has received much attention in unsupervised learning of sequential data. However, the current existing SHMMs fail to take into consideration of the relevant features embedded in local subspaces, thus influencing their performances in clustering. To address the problem, a novel SHMM is proposed by combining the measure of localized feature saliency (LFS) with SMM and utilizing two student’s t-distributions as subcomponents to respectively describe the distributions of useful features and non-salient “features,” with the purpose of accurately modeling the hidden state observation emission distributions of SHMM. Moreover, we exploit the variational Bayesian learning technique to simultaneously estimate the LFS, the number of components and other parameters of the herein proposed SHMM. Experimental results on both synthetic and real data sets demonstrate the improved robustness, effectiveness, and accuracy of our model.

As a significant tool, finite mixture models (FMMs) have been widely used for image segmentation. However, there are two problems with standard FMMs: first, the conditional probability is sensitive to outliers. Second, the robustness to image noise is inadequate. In this study, the authors present a novel hierarchical Student's‐ t MM (HSMM), which includes standard FMMs as a sub‐problem. Additionally, to incorporate more image spatial information, they apply a mean template not only to the prior/posterior probability, but also to the sub‐conditional distribution. Thus, their HSMM is more robust to outliers and image noise owing to the spatial constraints from the mean template. In the standard SMM, a t ‐distribution is used to calculate the conditional probability. In this study, the authors present a novel hierarchical student's‐ t mixture model (HSMM), which includes the standard FMM as a sub‐problem. Finally, though they use Student's‐ t ‐distribution to solve the image segment problems of this study, their HSMM achieves excellent performance, is elastic and can encompass any other model that is based on FMMs. Experimental results demonstrate that their proposed method is robust and effective.

Owing to the existence of noise and intensity inhomogeneity in brain magnetic resonance (MR) images, the existing segmentation algorithms are hard to find satisfied results. In this study, the authors propose an improved fuzzy C ‐mean clustering method (FCM) to obtain more accurate results. First, the authors modify the traditional regularisation smoothing term by using the non‐local information to reduce the effect of the noise. Second, inspired by the mechanism of the Gaussian mixture model, the distance function of FCM is defined by using the form of certain exponential function consisting of not only the distance but also the covariance and the prior probability to improve the robustness. Meanwhile, the bias field is modelled by using orthogonal basis functions to reduce the effect of intensity inhomogeneity. Finally, they use the hierarchical strategy to construct a more flexibility function, which considers the improved distance function itself as a sub‐FCM, to make the method more robust and accurate. Compared with the state‐of‐the‐art methods, experiment results based on synthetic and real MR images demonstrate its accuracy and robustness.

A mixture model based on the symmetric Gaussian distribution that simultaneously treats the feature selection, and the model detection has recently received great attention for pattern recognition problems. However, in many applications, the distribution of the data has a non-Gaussian and nonsymmetric form. This brief presents a new asymmetric mixture model for model detection and model selection. In this brief, the proposed asymmetric distribution is modeled with multiple student's- t distributions, which are heavily tailed and more robust than Gaussian distributions. Our method has the flexibility to fit different shapes of observed data, such as non-Gaussian and nonsymmetric. Another advantage is that the proposed algorithm, which is based on the variational Bayesian learning, can simultaneously optimize over the number of the student's- t distribution that is used to model each asymmetric distribution, the number of components, and the saliency of the features. Numerical experiments on both synthetic and real-world datasets are conducted. The performance of the proposed model is compared with other mixture models, demonstrating the robustness, accuracy, and effectiveness of our method.

The Dirichlet process mixture model (DPMM) with spatial constraints – e.g. hidden Markov random field (HMRF) model – has been considered as an effective algorithm for image processing application. However, the HMRF model is complex and time‐consuming for implementation. A new DPMM has been introduced, where a generalised mean (GDM) is selected as the spatial constraints function. The GDM is applied not only on prior probability (and posterior probability) to incorporate local spatial information and component information, but also on conditional probability to incorporate local spatial information and observation information. The purpose of the HMRF model and GDM are the same for incorporating some spatial constraints into the system. However, compared to HMRF, GDM is easier, faster and simpler to implement. Finally, a variational Bayesian approach has been adopted for parameters estimation and model selection. Experimental results on image segmentation application demonstrate the improved performance of the proposed approach.

Fuzzy c‐means (FCMs) with spatial constraints have been considered as an effective algorithm for image segmentation. The well‐known Gaussian mixture model (GMM) has also been regarded as a useful tool in several image segmentation applications. In this study, the authors propose a new algorithm to incorporate the merits of these two approaches and reveal some intrinsic relationships between them. In the authors model, the new objective function pays more attention on spatial constraints and adopts Gaussian distribution as the distance function. Thus, their model can degrade to the standard GMM as a special case. Our algorithm is fully free of the empirically pre‐defined parameters that are used in traditional FCM methods to balance between robustness to noise and effectiveness of preserving the image sharpness and details. Furthermore, in their algorithm, the prior probability of an image pixel is influenced by the fuzzy memberships of pixels in its immediate neighbourhood to incorporate the local spatial information and intensity information. Finally, they utilise the mean template instead of the traditional hidden Markov random field (HMRF) model for estimation of prior probability. The mean template is considered as a spatial constraint for collecting more image spatial information. Compared with HMRF, their method is simple, easy and fast to implement. The performance of their proposed algorithm, compared with state‐of‐the‐art technologies including extensions of possibilistic fuzzy c‐means (PFCM), GMM, FCM, HMRF and their hybrid models, demonstrates its improved robustness and effectiveness.

Synthetic aperture radar (SAR) data are often affected by speckle noise, which originates in the SAR system's coherent nature. In this paper, we introduce a simple and effective algorithm to make the traditional Student's t-mixture model (SMM) more robust to noise. The proposed new modified SMM (MSMM) is applied for SAR image segmentation. SMM has come to be regarded as an alternative to the Gaussian mixture model (GMM) as it is heavy tailed and more robust to outliers. However, a major shortcoming of this method is that it does not take into account the spatial dependencies in the image. Although some existing methods incorporate the spatial relationship between neighboring pixels, they are still not robust enough to noise. The advantages of our method are as follows. First, we introduce MSMM to incorporate the local spatial information and pixel intensity value by considering the conditional probability of an image pixel influenced by the probabilities of pixels in its immediate neighborhood. Furthermore, we introduce the additional parameter α to control the extent of this influence. The larger α indicates the heavier extent of influence in the neighborhoods. Second, the prior probability of an image pixel is influenced by the probabilities of pixels in its immediate neighborhood, which incorporates local spatial and component information. Third, our model is based on the finite mixture model (FMM); it is simple and easy to implement, and the expectation maximization algorithm can be applied for estimation of optimal parameters. Finally, the traditional SMM can be considered as a special case of our model. Thus, our method is general enough for FMM-based techniques. Experimental results on both simulated and real SAR images demonstrate the improved robustness and effectiveness of our approach.

Segmentation of a medical image based on the modeling and estimation of the tissue intensity probability density functions via a Gaussian mixture model has recently received great attention. However, the Gaussian distribution is unbounded and symmetrical around its mean. This study presents a new bounded asymmetric mixture model for analyzing both univariate and multivariate data. The advantage of the proposed model is that it has the flexibility to fit different shapes of observed data such as non-Gaussian, nonsymmetric, and bounded support data. Another advantage is that each component of the proposed model has the ability to model the observed data with different bounded support regions, which is suitable for application on image segmentation. Our method is intuitively appealing, simple, and easy to implement. We also propose a new method to estimate the model parameters in order to minimize the higher bound on the data negative log-likelihood function. Numerical experiments are presented where the proposed model is tested in various images from simulated to real 3- D medical ones.

The Gaussian Mixture Model (GMM) with a spatial constraint, e.g. a Hidden Markov Random Field (HMRF), has been proven effective for image segmentation. However, the determination of parameter β in the HMRF model is, in fact, noise dependent to some degree. In this paper, we propose a simple and effective algorithm to make the traditional Gaussian Mixture Model more robust to noise, with consideration of the relationship between the local spatial information and the pixel intensity value information. The conditional probability of an image pixel is influenced by the probabilities of pixels in its immediate neighborhood to incorporate the spatial and intensity information. In this case, the parameter β can be assigned to a small value to preserve image sharpness and detail in non-noise images. At the same time, the neighborhood window is used to tolerate the noise for heavy-noised images. Thus, the parameter β is independent of image noise degree in our model. Finally, our algorithm is not limited to GMM-it is general enough so that it can be applied to other distributions based on the construction of the Finite Mixture Model (FMM) technique.

Segmentation of medical image based on the modeling and estimation of the tissue intensity probability density functions via Gaussian mixture model (GMM) has recently received great attention. However, Gaussian distribution is unbounded and symmetrical around its mean. This study presents a new bounded asymmetric mixture model for analyzing both univariate and multivariate data. The advantage of the proposed model is that it has the flexibility to fit different shapes of observed data such as non-Gaussian, non-symmetric, and bounded support data. Another advantage is that each component of the proposed model has the ability to model the observed data with different bounded support regions, which is suitable for application on image segmentation. Our method is intuitively appealing, simple, and easy to implement. We also propose a new method to estimate the model parameters in order to minimize the higher bound on the data negative log-likelihood function. Numerical experiments are presented where the proposed model is tested in various images from simulated to real 3D medical ones.

This paper presents a new finite mixture model for image segmentation. First, in order to take into account the spatial dependencies in an image, existing mixture models use a constant temperature parameter (β) throughout the image for every label. The constant value of β reduces the impact of noise in homogeneous regions but negatively affects segmentation along the border of two regions. We propose a new way to use a different value of β throughout the image. Secondly, in order to incorporate the correlation between each centre pixel and its neighboring pixels, existing mixture model gives the same importance to all pixels in a neighborhood window. We assign different weights to different pixels appearing in the window, which is based on the fact that the clique strength should be reduced with distance. Thirdly, our model is based on the Student's-t distribution, which is heavily tailed and more robust than Gaussian. We exploit Dirichlet distribution and Dirichlet law to incorporate the spatial relationships between pixels in an image. Finally, expectation maximization (EM) algorithm is adopted to maximize the data log-likelihood and to optimize the parameters. The performance is compared to other existing models based on the model-based techniques, demonstrating superiority of the proposed model for image segmentation.

Fuzzy c-means (FCM) with spatial constraints has been considered as an effective algorithm for image segmentation. In this paper, we propose a new algorithm to incorporate the local spatial information with the consideration of mean template. Our algorithm is fully free of the empirically predefined parameters that are used in other FCM methods to balance between robustness to noise and effectiveness of preserving the image sharpness and details. Furthermore, in our algorithm, the prior probability of an image pixel is influenced by the fuzzy memberships of pixels in its immediate neighborhood to incorporate the local spatial information and intensity information. Finally, we utilize the mean template instead of the traditional hidden Markov random field (HMRF) model for estimation of prior probability. Compared to HMRF, our method is simple, easy and fast to implement.

Fuzzy c-means (FCM) has been considered as an effective algorithm for image segmentation. However, it lacks of sufficient robustness to image noise. In this paper, we propose a simple and effective method to make the traditional FCM more robust to noise, with the help of generalized mean. Traditional FCM can be considered as a linear combination of membership and distance (function) from the expression of its mathematical formula. The proposed generalized FCM (GFCM) is generated by applying generalized mean on these two items. We impose generalized mean on membership to incorporate local spatial information and cluster information, and on distance function to incorporate local spatial information and observation information (image intensity value). Thus, our GFCM is more robust to image noise with the spatial constraints: the generalized mean. The performance of our proposed algorithm, compared with state-of-the-art technologies including modified FCM, HMRF and their hybrid models, demonstrates its improved robustness and effectiveness.

Wireless sensor networks are deployed in areas we are interested in to help users to retrieve raw data. Due to the limited power, it is desired that data retrieving routing protocols are energy efficient, and energy consumption can be balanced among all nodes so that lifetime of WSNs can be prolonged. In this paper, we propose a centralized clustering geographic energy aware routing (GEAR-CC) for wireless sensor networks. It has the dominance of both hierarchical routing and geographic energy aware routing. As a centralized algorithm, it takes full advantage of base station's sufficient power to conduct every transmission in the wireless sensor network. In GEAR-CC, the base station can easily formulate the optimal transmitting schemes for all sensor nodes based on the global information of topology and energy. The optimization is achieved by making trade-off between energy cost and node's residual power. Simulation results show that GEAR-CC can dramatically balance residual energy of each node to avoid hot-spot phenomenon, and the time the first cluster dies in network using GEAR-CC is about 1.57 times of clustering direct transmission routing and 1.32 times of cluster-head-to-cluster-head (CH-to-CH) routing.

In this study, the authors introduce a new weighted Student's t ‐mixture model (WSMM) for image segmentation. Gaussian distribution and Student's t ‐distribution are the two commonly used probabilities in the finite mixture model (FMM). The Student's t ‐mixture model has come to be regarded as an alternative to Gaussian mixture models, as it is heavily tailed and more robust for outliers. Moreover, the pixels are considered independent of each other in the FMM. Although some existing methods incorporate the spatial relationship between neighbouring pixels, they do not consider the relationship between spatial information and clustering information, thus those reported methods remain sensitive to noise. The advantages of the authors method are as follows: first, the authors introduce WSMM to incorporate the local spatial information, pixel intensity value and clustering information in an image. Second, the authors model is simple, easy to implement and has a good balance between noise insensitiveness and image detail preservation. Third, they adopt the gradient method and expectation maximisation algorithm, which allow for simultaneous estimation of optimal parameters. Finally, the most useful statistical tool for image segmentation, the well‐known hidden Markov random field model, is a special case of their model. Thus, their method is general enough for model‐based techniques construction. Experimental results on synthetic and real images demonstrate the improved robustness and effectiveness of their approach.

The Gaussian hidden Markov model has been successfully used in pattern recognition and classification applications; however, recently the Student's t ‐mixture model is regarded as an alternative to Gaussian mixture models, as it is more robust for outliers. The model using Student's t ‐mixture distribution as its hidden state is the Student's t ‐hidden Markov model (SHMM). The authors propose a novel Student's t ‐hidden Markov model, which considers the relationship among Markov states, latent components and observations by introducing a regularising scalar exponent in the component densities of the model's emission densities. Moreover, the standard SHMM can be considered as a special case of the modified SHMM with the selection of proper parameter values. Finally, the authors adopt the gradient method to estimate optimal weight parameters. Simultaneously, the expectation–maximisation algorithm is used to fit the modified SHMM. Thus, our model is simple and easy to implement. The experimental results using synthetic and real data demonstrate the improved robustness of the proposed approach.

In this paper, we propose a novel algorithm for feature selection and model detection using Student's t-distribution based on the variational Bayesian (VB) approach. First, our method is based on the Student's t-mixture model (SMM) which has heavier tail than the Gaussian distribution and is therefore less sensitive to small numbers of data points and consequent precision-estimates of the components number. Second, the number of components, the local feature saliency and the parameters of the mixture model are simultaneously estimated by Bayesian variational learning. Experimental results using synthetic and real data demonstrate the improved robustness of our approach.

Fuzzy c-means (FCM) with spatial constraints has been considered as an effective algorithm for image segmentation. Student's t-distribution has come to be regarded as an alternative to Gaussian distribution, as it is heavily tailed and more robust for outliers. In this letter, we propose a new algorithm to incorporate the merits of these two approaches. The advantages of our method are as follows: First, we incorporate the local spatial information and pixel intensity value by considering the labeling of an image pixel influenced by the labels in its immediate neighborhood. Second, we introduce additional parameterato control the extent of this influence. The largeraindicates heavier extent of influence in the neighborhoods. Finally, we utilize a mean template instead of the traditional hidden Markov random field (HMRF) model for estimation of prior probability. Compared with HMRF, our method is simple, easy and fast to implement. Experimental results on synthetic and real images demonstrate the improved robustness and effectiveness of our approach.

The well-known finite mixture model (FMM) has been regarded as a useful tool for image segmentation application. However, the pixels in FMM are considered independent of each other and the spatial relationship between neighboring pixels is not taken into account. These limitations make the FMM more sensitive to noise. In this brief, we propose a simple and effective method to make the traditional FMM more robust to noise with the help of a mean template. FMM can be considered a linear combination of prior and conditional probability from the expression of its mathematical formula. We calculate these probabilities with two mean templates: a weighted arithmetic mean template and a weighted geometric mean template. Thus, in our model, the prior probability (or conditional probability) of an image pixel is influenced by the probabilities of pixels in its immediate neighborhood to incorporate the local spatial and intensity information for eliminating the noise. Finally, our algorithm is general enough and can be extended to any other FMM-based models to achieve super performance. Experimental results demonstrate the improved robustness and effectiveness of our approach.

Affine moment invariants are important shape descriptors in pattern recognition and computer vision. Existing affine invariants methods are based on geometric and complex moments. In this paper, we propose a set of affine invariants extracted from Legendre moments. These invariants are derived by the relationship between the Legendre moment of the affine transformed image and that of the original image. The performance of the proposed descriptor is evaluated with a set of binary and gray images. Experimental results show that the proposed method behaves better than existing methods in terms of pattern recognition accuracy.

This paper presents two new sets of nonseparable discrete orthogonal Charlier and Meixner moments describing the images with noise and that are noise-free. The basis functions used by the proposed nonseparable moments are bivariate Charlier or Meixner polynomials introduced by Tratnik et al. This study discusses the computational aspects of discrete orthogonal Charlier and Meixner polynomials, including the recurrence relations with respect to variable x and order n. The purpose is to avoid large variation in the dynamic range of polynomial values for higher order moments. The implementation of nonseparable Charlier and Meixner moments does not involve any numerical approximation, since the basis function of the proposed moments is orthogonal in the image coordinate space. The performances of Charlier and Meixner moments in describing images were investigated in terms of the image reconstruction error, and the results of the experiments on the noise sensitivity are given.

The derivation of moment invariants has been extensively investigated in the past decades. In this paper, we construct a set of invariants derived from Zernike moments which is simultaneously invariant to similarity transformation and to convolution with circularly symmetric point spread function (PSF). Two main contributions are provided: the theoretical framework for deriving the Zernike moments of a blurred image and the way to construct the combined geometric-blur invariants. The performance of the proposed descriptors is evaluated with various PSFs and similarity transformations. The comparison of the proposed method with the existing ones is also provided in terms of pattern recognition accuracy, template matching and robustness to noise. Experimental results show that the proposed descriptors perform on the overall better.

Geometric distortions are generally simple and effective attacks for many watermarking methods. They can make detection and extraction of the embedded watermark difficult or even impossible by destroying the synchronization between the watermark reader and the embedded watermark. In this paper, we propose a new watermarking approach which allows watermark detection and extraction under affine transformation attacks. The novelty of our approach stands on a set of affine invariants we derived from Legendre moments. Watermark embedding and detection are directly performed on this set of invariants. We also show how these moments can be exploited for estimating the geometric distortion parameters in order to permit watermark extraction. Experimental results show that the proposed watermarking scheme is robust to a wide range of attacks: geometric distortion, filtering, compression, and additive noise.

The completeness property of the invariant descriptors, which is of fundamental importance from the theoretical as well as the practical points of views, has been investigated by several research groups. In this paper, we propose a new approach to derive a complete set of pseudo-Zernike moment invariants. We first establish a relationship between the pseudo-Zernike moments of the original image and those of the image having the same shape but distinct orientation and scale. Based on this relationship, a complete set of scale and rotation invariants is derived. Experimental results show that the proposed method has better performance in pattern recognition compared to existing method.

In this paper, we proposed a set of translation and rotation invariants extracted from Tchebichef moments. A set of Tchebichef moment invariants is derived from the relationship between Tchebichef moments of the original image and those of the transformed image. These invariants are then used for symmetric image recognition. Contrarily to the methods based on the complex moments in symmetric image analysis, our method does not need the pre-selection of moment values. Experimental results show that the proposed method achieves better performance compared to the existing methods.

Moments and moment invariants are useful tool in pattern recognition and image analysis. Conventional methods to deal with color images are based on RGB decomposition or graying. In this paper, by using the theory of quaternions, we introduce a set of quaternion Zernike moments (QZMs) for color images in a holistic manner. It is shown that the QZMs can be obtained via the conventional Zernike moments of each channel. We also construct a set of combined invariants to rotation and translation (RT) using the modulus of central QZMs. Experimental results show that the proposed descriptors are more efficient than the existing ones.

In this paper, we introduce a new algorithm to register images with rotation and image blurring. The characteristic of this approach is that a set of Legendre moment invariants are used to establish the correspondence of corner points between the reference image and the distorted image, after these points are extracted by Harris corner detector. Transformation parameters can be estimated from those matched points.

Discrete orthogonal moments have been recently introduced in the field of image analysis. It was shown that they have better image representation capability than the continuous orthogonal moments. One problem concerning the use of moments as feature descriptors is the high computational cost, which may limit their application to the problems where the online computation is required. In this paper, we present a new approach for fast computation of the 2-D Tchebichef moments. By deriving some properties of Tchebichef polynomials, and using the image block representation for binary images and intensity slice representation for grayscale images, a fast algorithm is proposed for computing the moments of binary and grayscale images. The theoretical analysis shows that the computational complexity of the proposed method depends upon the number of blocks of the image, thus, it can speed up the computational efficiency as far as the number of blocks is smaller than the image size.

Processing blurred images is a key problem in many image applications. Existing methods to obtain blur invariants which are invariant with respect to centrally symmetric blur are based on geometric moments or complex moments. In this paper, we propose a new method to construct a set of blur invariants using the orthogonal Legendre moments. Some important properties of Legendre moments for the blurred image are presented and proved. The performance of the proposed descriptors is evaluated with various point-spread functions and different image noises. The comparison of the present approach with previous methods in terms of pattern recognition accuracy is also provided. The experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments.