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Spatially Constrained Student's T-Distribution Based Mixture Model J Math Imaging Vis (2018) 60:355–381 https://doi.org/10.1007/s10851-017-0759-8 Spatially Constrained Student’s t-Distribution Based Mixture Model for Robust Image Segmentation Abhirup Banerjee1 · Pradipta Maji1 Received: 8 January 2016 / Accepted: 5 September 2017 / Published online: 21 September 2017 © Springer Science+Business Media, LLC 2017 Abstract The finite Gaussian mixture model is one of the Keywords Segmentation · Student’s t-distribution · most popular frameworks to model classes for probabilistic Expectation–maximization · Hidden Markov random field model-based image segmentation. However, the tails of the Gaussian distribution are often shorter than that required to model an image class. Also, the estimates of the class param- 1 Introduction eters in this model are affected by the pixels that are atypical of the components of the fitted Gaussian mixture model. In In image processing, segmentation refers to the process this regard, the paper presents a novel way to model the image of partitioning an image space into some non-overlapping as a mixture of finite number of Student’s t-distributions for meaningful homogeneous regions. It is an indispensable step image segmentation problem. The Student’s t-distribution for many image processing problems, particularly for med- provides a longer tailed alternative to the Gaussian distri- ical images. Segmentation of brain images into three main bution and gives reduced weight to the outlier observations tissue classes, namely white matter (WM), gray matter (GM), during the parameter estimation step in finite mixture model. and cerebro-spinal fluid (CSF), is important for many diag- Incorporating the merits of Student’s t-distribution into the nostic studies. For example, in multiple sclerosis diseases, hidden Markov random field framework, a novel image seg- accurate quantification of WM lesions is necessary for drug mentation algorithm is proposed for robust and automatic treatment assessment, while in schizophrenia and epilepsy, image segmentation, and the performance is demonstrated volumetric analysis of GM, WM, and CSF is required to char- on a set of HEp-2 cell and natural images. Integrating the acterize the morphological differences between subjects. In bias field correction step within the proposed framework, a a similar way, automatic segmentation of human epithelial novel simultaneous segmentation and bias field correction type 2 (HEp-2) cells from indirect immunofluorescent (IIF) algorithm has also been proposed for segmentation of mag- images is a necessary step of the IIF antinuclear antibody test netic resonance (MR) images. The efficacy of the proposed for the diagnosis of connective tissue diseases. approach, along with a comparison with related algorithms, is Many image processing techniques for automatic image demonstrated on a set of real and simulated brain MR images segmentation exist throughout the literature [10,53]. Among both qualitatively and quantitatively. them, the thresholding methods [39,52,59] segment the scaler images by using one or more thresholds. These meth- ods are usually very simple, fast, and work reasonably well for images with very good contrast between distinctive sub- B Pradipta Maji regions. But they do not consider the spatial characteristics [email protected] of an image, which makes them sensitive to noise and out- Abhirup Banerjee liers. Li et al. [33] tried to remove this noise sensitivity of [email protected] the thresholding algorithms by incorporating the local inten- sity information to the framework. Lee et al. [29]alsosolved 1 Biomedical Imaging and Bioinformatics Lab, Machine Intelligence Unit, Indian Statistical Institute, 203 B. T. Road, the similar problem by finding a boundary between two sub- Kolkata, West Bengal 700108, India regions using the path connection algorithm and changing 123 356 J Math Imaging Vis (2018) 60:355–381 the threshold adaptively. Region growing is another popular dependent on its neighboring pixels. Liu and Zhang [38] inte- technique for image segmentation [45], which is generally grated the level set method with FGM framework for image applied for delineation of small, simple structures. However, segmentation in the presence of intensity inhomogeneity and it also gets affected by the presence of noise and outliers. noise. Mangin et al. [44] solved this problem by applying a homo- However, none of the works reported above, based on topic region growing algorithm, which is able to preserve either FCM or FM, considers spatial information for seg- the topology between initial and extracted regions. Artificial mentation. In this regard, spatial information of neighboring neural network-based algorithms have also been investigated pixels has been incorporated into the FCM framework to in the image segmentation problems [12], either applied in make it robust from the effect of noise and outliers [2, supervised fashion [24] or unsupervised fashion [10,57]. 13,15,21,28,36,55,64,66,70,71], while the Markov random Variational models [32,60] such as snake model [47] and field (MRF) model is integrated within the FM model-based geodesic active contour model [56] have been applied to seg- probabilistic framework [19,25,50,73]. Incorporating spa- ment the complex-shaped image structures in an efficient and tial information of the pixels with their intensity distribution, automated manner. Among other techniques, self-organizing Zhang et al. [73] introduced the hidden Markov random maps [34], wavelet transform [6,42,43], k-means [1], fuzzy field (HMRF) model and proposed a joint EM-HMRF frame- connectedness [17,58,74], optimum-path forest [7], support work to achieve robust segmentation in noisy environment. vector machine [18], level set [37], and graph-cut [14,16,30]- In another work, Held et al. [25] developed an MRF seg- based approaches are applied in the segmentation of various mentation algorithm, based on the adaptive segmentation image structures. algorithm of Wells et al. [65]. Their algorithm incorpo- One of the major problems in image segmentation is rated three features, namely nonparametric distributions of uncertainty. Imprecision in computations and vagueness in tissue intensities, neighborhood correlations, and signal inho- class definitions often cause this uncertainty. In this back- mogeneities, that are of special importance for an image. ground, fuzzy c-means (FCM) algorithm is considered as Diplaros et al. [19] introduced a generative model with the one of the most popular techniques in modeling uncertainty assumption that the hidden class labels of the pixels are gen- of image segmentation problems [11,24,31,67]. The proba- erated by prior distributions that share similar parameters bilistic model [4,5] is another popular framework to model for neighboring pixels. The same EM algorithm was applied image classes for segmentation. This model generally applies with a smoothing step interleaved between the E- and the M- the expectation-maximization (EM) algorithm that labels the steps that couple the posteriors of neighboring pixels in each pixels according to their probability values, calculated based iteration. In another work, Nguyen and Wu [50] proposed a on the intensity distribution of the image. Using a suitable new way to incorporate spatial relationships among neigh- assumption about the intensity distribution, the probabilis- boring pixels using a simple metric in the joint FGM-MRF tic approaches attempt to estimate the associated class label, framework. given only the intensity of each pixel. Such an estimation Although the aforementioned FM- or MRF-based seg- problem is necessarily formulated using maximum a poste- mentation methods provide different ways to achieve robust riori (MAP) or maximum likelihood (ML) principles. In this image segmentation, all of them assume the underlying regard, the finite mixture (FM) model, more specifically, the intensity distribution of each image class as a Gaussian distri- finite Gaussian mixture (FGM) model, is one of the most bution. Gaussian distribution is a unimodal distribution that popular models for image segmentation [22,35,38,51,65]. attains highest probability density value at a single point in Wells et al. [65] applied the EM framework in the FGM its range of variation, which is mean, and probability den- model to achieve optimal segmentation performance, while sity decreases symmetrically as one traverses away from the simultaneously reducing the shading artifacts in brain MR mean. This assumption is particularly very useful to model images. Liang et al. [35] developed a statistical method to an image class, as an image class, in general, also has a ten- simultaneously classify the pixels and estimate the parame- dency to have high concentration of intensity values around ters of the intensity distribution of each class in multispectral the mean and the concentration decreases as one deviates images. Their EM framework assumed the distribution of further from the mean. However, the tails of the Gaussian image intensities as a mixture of finite number of multivariate distribution are often shorter than that required to model an normal distributions and the prior distribution of each class as image class. There generally exist outliers, that is, observa- a Markov random field (MRF). In another work, Greenspan tions with large deviation from the class mean, in an image et al. [22] modeled each image class by large number of class, which are very difficult to model and have the ability Gaussian components in the same FGM framework. Nguyen to degrade the performance of the segmentation algorithm et al. [51] proposed an extension of the standard Gaussian if Gaussian distribution is fitted to model the image classes. mixture model. Their proposed framework assumed the prior Also, the estimates of the class parameters in this model get distribution of each class to be different for each pixel and 123 J Math Imaging Vis (2018) 60:355–381 357 corrupted by the pixels that are atypical of the components portions, while the second model applies Gamma prior on of the fitted Gaussian mixture model. the Student’s t-distributed local differences of the contextual The Student’s t-distribution, in this regard, provides a mixing proportions.
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