Identification of Region of Interest

The skull stripping is an important preprocessing phase in brain imaging application, which involves removal of non-cerebral tissues like skull, scalp, and dura from brain MR images. It reduces misclassification during segmentation as well as minimizes the execution time of segmentation algorithm by eliminating the objects for non-cerebral tissues [41].

Although many skull stripping techniques have been studied, most of them are based on either region growing techniques [42, 43] or mathematical morphology [44–46]. The success of skull removing methods using region growing approaches relies on the seed selection algorithm, which automatically selects seeds corresponding to the brain and non-brain regions. However, most of the brain MR image segmentation methods use morphological operations to separate the brain tissues from the surrounding tissues without holes, although morphology requires a prior binarization of the image and thresholding may be used to create binary images from gray level ones.

Since the morphology operations need binarization of the image, selection of a threshold value is crucial to generate the mask for brain tissues, in turns, to ensure the accurateness of the segmented results. The thresholding method used in this work is based on the mean image intensity value. On the other hand, the morphology operations use an octagonal shaped structuring element to produce the skull-stripped image. In the proposed brain MR image segmentation methodology, the skull stripping technique includes the following steps:

1. Apply median filtering with a window of size 3×3 to the input image.
2. Compute the initial mean intensity value T_i of the image.
3. Identify the top, bottom, left, and right pixel locations, from where brain skull starts in the image, considering gray values of the skull are greater than T_i.
4. Form a rectangle using the top, bottom, left, and right pixel locations.
5. Compute the final mean value T_f of the brain using the pixels located within the rectangle.
6. Approximate the region of brain membrane or meninges that envelop the brain, based on the assumption that the intensity of skull is more than T_f and that of membrane is less than T_f.
7. Set the average intensity value of membrane as the threshold value T.
8. Convert the given input image into binary image using the threshold T.
9. Apply a 13×13 opening morphological operation to the binary image in order to separate the skull from the brain completely.
10. Find the largest connected component and consider it as brain.
11. Finally, apply a 21×21 closing morphological operation to fill the gaps within and along the periphery of the intracranial region.

Feature Extraction Using Multiresolution Wavelet Analysis

Wavelets mean small waves, that is, short duration finite energy functions. Wavelet function must be chosen from the space of all measurable functions that are absolutely and square integrable, that is, L¹(ℝ)∩L²(ℝ). Being in this space a mother wavelet function ψ(t) must satisfy the conditions of zero mean and square norm, that is, (1) Mathematically, a wavelet is defined as (2) where b is the location parameter and a is the scaling parameter. The continuous wavelet transform is defined using mother wavelet ψ(t) as (3)

According to (3), for every (a, b), a wavelet transform coefficient is obtained, representing how much the scaled wavelet is similar to the function at location t = (b/a). The mother wavelet ψ(t) has to satisfy the admissibility condition (4) In (4), Ψ(w) denotes Fourier transform of ψ(t). The admissibility condition assures that the function f(t) can be reconstructed with the knowledge of its wavelet transform by the following inverse transform relationship (5)

The multiresolution analysis is designed to provide good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies [16, 19, 20]. In multiresolution analysis, a scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 from its nearest neighboring approximation. Additional functions, called wavelets, are then used to encode the difference in information between adjacent approximations. Scaling function ϕ(t) and wavelet function ψ(t) are defined as follows: (6) (7) for all j, k ∈ ℤ. Here k determines the position along x-axis; and j determines function’s width, that is, how broad or narrow it is along x-axis. The scaling function ϕ(t) and wavelet function ψ(t) are chosen appropriately to satisfy the orthonormality condition. The wavelet subspaces W_js form an orthogonal decomposition of L²(ℝ) function space and hence they are related to nested subspaces V_js as follows: where V_j is the function space spanned by ϕ_{j, k}(t) over k and W_j is the function space spanned by ψ_{j, k}(t).

If a function f(t) ∈ L²(ℝ), its wavelet transform is (8) (9) where j₀ is an arbitrary starting scale and N is the number of samples taken from the signal. The W_ϕ(j₀, k)’s are called approximation or scaling coefficients and W_ψ(j, k)’s are referred to as detail or wavelet coefficients. In fast wavelet transform, at scale j, the approximation and detail coefficients, W_ϕ(j, k) and W_ψ(j, k), respectively, can be computed by convolving the scale j+1 approximation coefficients, W_ϕ(j+1, k), with the time-reversed scaling and wavelet vectors, h_ϕ(−n) and h_ψ(−n), respectively, and subsampling the results.

Like one dimensional wavelet transform, the two dimensional wavelet transform can be implemented using the separable two dimensional scaling and wavelet functions. It is done by taking one dimensional wavelet transform of the rows of two dimensional function or image, followed by one dimensional wavelet transform of the resulting columns. Hence, it generates four subbands at each level, namely, approximation, horizontal, vertical, and diagonal subbands. The approximation part is iteratively decomposed as the decomposition level is increased in case of standard wavelet transform. Hence, if an input image is decomposed upto lth level, total d = 3l+1 number of subbands are generated.

The classical wavelet transform includes downsampling operations by a factor that cause wavelet expansions to be shift-variant. But, overcomplete representation of wavelets overcomes the shift-varying nature of classical wavelet expansion. Additionally, the overcomplete wavelet transform is convenient over the subsampled methods as downsampling decreases the size of the subbands at each increasing level of decomposition and thus may bias the decomposition at higher levels. Hence, the proposed methodology includes feature-extraction scheme that uses multiresolution dyadic wavelet filtering without downsampling.

Generation of Skull Stripped Feature Vectors

After generating d subbands or features for a given image I, the mask is applied to each subband to generate the feature vectors for the region of interest only. Hence, a reduced set X = {x₁, ⋯, x_i, ⋯, x_n} is generated from $\acute{X}$ consisting of pixels within the region of interest. Each pixel or object x_i ∈ X is represented by d-dimensions, each dimension corresponding to each subband generated from wavelet decomposition. Next section presents an unsupervised feature selection algorithm, which is used to select m relevant and significant subbands or features from the whole set of d features for efficient segmentation.

Unsupervised Feature Selection

In wavelet-based image segmentation method, a number of insignificant and irrelevant features may be generated. The presence of such features may lead to a reduction in the valuable information of segmentation. The objective of the feature selection is to identify a reduced feature subset with optimum salient characteristics of the image. It not only decreases the processing time, but also leads to more compactness and better generalization. The selected features should have high relevance and high significance in the feature set. In effect, these features will be able to predict the belongingness of the objects or samples in different segmented regions. However, as the segmentation problem is unsupervised in nature, the feature selection method should be unsupervised. Accordingly, a measure is required that can efficiently assess the effectiveness of feature set in unsupervised manner.

There exist many approaches to select a reduced set of relevant subbands for texture segmentation [47–51]. While Coifman and Wickerhauser [47] proposed entropy for the selection of best subbands, Saito et al. [48] used empirical probability density estimation and a local basis library for the extraction of discriminant features. On the other hand, Huang and Aviyente used mutual information in [50] for computing dependence among subbands to discard redundant or insignificant features, while both dependence and energy measures are used in [52] to select relevant and nonredundant subbands for texture classification. In the proposed feature selection algorithm, the energy measure is used to identify the relevant and significant features. The energy content of a feature set 𝕊 is defined as follows: (10) as $\overline{x}$ is the mean of feature vectors of all objects and is given by (11) where n is the number of objects and 𝕊 is the feature set. Hence, the energy content provides higher values for the features having high frequency components, indicating that these features contain more information. On the other hand, the energy content measure yields low energy values for the smoothed subbands indicating textural uniformity. Also, this measure is unsupervised in nature as it does not require any class information. Hence, this measure can be used to select potential wavelet features for image segmentation.

Let ℂ = {𝒜₁, ⋯, 𝒜_i, ⋯, 𝒜_j, ⋯, 𝒜_d} be the set of d features and 𝕊 is the set of selected features. The relevance γ_𝒜i of the feature 𝒜_i is defined as (12) The significance σ_𝒜j({𝒜_i, 𝒜_j}) of the feature 𝒜_j with respect to the feature set {𝒜_i, 𝒜_j} defines the extent to which the feature 𝒜_j is contributing in the energy estimation computed using (10). The change in energy estimation when a feature is removed from the feature set, is the measure of the significance of the feature, and is given as (13) The higher the change in energy estimation, the more significant the feature is. If the significance is 0, then the feature is dispensable.

Therefore, the problem of selecting a set 𝕊 of relevant and significant features from the whole set ℂ of d features is equivalent to maximizing both the total relevance of all selected features and the total significance among the selected features. To solve the above problem satisfying maximum relevance-maximum significance (MRMS) criterion [53], the following greedy algorithm can be used:

1. Initialize ℂ ← {𝒜₁, ⋯, 𝒜_i, ⋯, 𝒜_d} and 𝕊 ← ∅.
2. Calculate the relevance γ_𝒜i of each feature 𝒜_i ∈ ℂ.
3. Select the feature 𝒜_i as the most relevant feature that has the highest relevance value γ_𝒜i. In effect, 𝕊 ← 𝕊∪{𝒜_i} and ℂ ← ℂ\{𝒜_i}.
4. Repeat the following two steps until the desired number of features m is selected.
5. Calculate the significance of each of the remaining features of ℂ with respect to the already selected features of 𝕊 and remove it from ℂ if it has zero significance value with respect to any one of the selected features.
6. From the remaining features of ℂ, select feature 𝒜_j that maximizes the following condition: (14) As a result of that, 𝕊 ← 𝕊∪{𝒜_j} and ℂ ← ℂ\𝒜_j.

The MRMS criterion based feature selection algorithm reported above is unsupervised in nature since it does not require any a priori knowledge about the segmented regions of brain MR image. The algorithm runs until the desired number of features m is selected. In practice, we find that the following definition works well: (15) where d represents the total number of features or subbands corresponding to the given image.

Rough-Fuzzy Clustering for Segmentation

In the proposed method, the robust rough-fuzzy c-means (rRFCM) [40] algorithm is used for segmentation of brain MR images. However, other clustering algorithms such as hard c-means [54], fuzzy c-means [35], and rough-fuzzy c-means (RFCM) [39] can also be used for this purpose. The rRFCM adds the concepts of fuzzy memberships, both probabilistic and possibilistic, of fuzzy sets, and lower and upper approximations of rough sets into c-means algorithm. While the integration of both probabilistic and possibilistic memberships of fuzzy sets enables efficient handling of overlapping clusters in noisy environment, the rough sets deal with uncertainty, vagueness, and incompleteness in cluster definition.

Let X = {x₁, ⋯, x_j, ⋯, x_n} be the set of n objects and V = {v₁, ⋯, v_i, ⋯, v_c} be the set of c centroids, where x_j ∈ ℜ^m and v_i ∈ ℜ^m. In the rRFCM, each of the clusters β_i is represented by a cluster center v_i, a possibilistic lower approximation $\underset{\_}{A}({\beta }_i)$ and a probabilistic boundary region $B({\beta }_i)=\{\overline{A}({\beta }_i)\backslash \underset{¯}{A}({\beta }_i)\}$, where $\overline{A}({\beta }_i)$ denotes the upper approximation of cluster β_i. According to the definitions of lower approximation and boundary of rough sets [55], if an object $x_j\in \underset{¯}{A}({\beta }_i)$, then $x_j\notin \underset{¯}{A}({\beta }_k),\forall k\ne i$, and x_j ∉ B(β_i), ∀i. That is, the object x_j is contained in β_i definitely. Hence, the memberships of the objects in lower approximation of a cluster should be independent of other centroids and clusters. Also, the objects in lower approximation should have different influence on the corresponding centroid and cluster. From the standpoint of compatibility with the cluster prototype, the membership of an object in the lower approximation of a cluster should be determined solely by how far it is from the prototype of the cluster, and should not be coupled with its location with respect to other clusters. As the possibilistic membership ν_ij given by (20) depends only on the distance of object x_j from cluster β_i, it allows optimal membership solutions to lie in the entire unit hypercube rather than restricting them to the hyperplane given by (21).

On the other hand, if x_j ∈ B(β_i), then the object x_j possibly belongs to cluster β_i and potentially belongs to other clusters. Hence, the objects in boundary regions should have different influence on the centroids and clusters, and their memberships should depend on the positions of all cluster centroids. So, in the rRFCM, the membership values of objects in lower approximation are identical to (20) of possibilistic c-means [37], while those in boundary region are the same as (19) of fuzzy c-means [35]. Hence, the rRFCM algorithm partitions X into c clusters by minimizing following objective function: (16) (17) (18)

The parameters ω and (1−ω) correspond to the relative importance of lower and boundary regions, while $\acute{m_1}\in [1,\infty )$ and $\acute{m_2}\in [1,\infty )$ are the probabilistic and possibilistic fuzzifiers, respectively. The probabilistic μ_ij and possibilistic ν_ij membership functions are given by (19) (20) (21) (22) where the scale parameter η_i is given by (23) which represents the zone of influence or size of the cluster β_i. Typically, K is chosen to be 1. The centroid is calculated based on the weighting average of the possibilistic lower approximation and probabilistic boundary. The centroid calculation for the rRFCM is as follows: (24)

The rRFCM algorithm starts by randomly choosing c objects as the centroids of the c clusters. The possibilistic memberships ν_ij of all the objects are calculated using (20). The scale parameters η_i for c clusters are obtained using (23). Let ν_i = (ν_i1, ⋯, ν_ij, ⋯, ν_in) be the fuzzy cluster β_i associated with the centroid v_i. After computing ν_ij for c clusters and n objects, the values of ν_ij for each object x_j are sorted and the difference of two highest memberships of x_j is compared with a threshold value δ₁. Let ν_ij and ν_kj be the highest and second highest memberships of x_j. If (ν_ij−ν_kj) > δ₁, then $x_j\in \underset{¯}{A}({\beta }_i)$. In addition, by properties of rough sets, $x_j\in \overline{A}({\beta }_i)$. Otherwise, x_j ∈ B(β_i) and x_j ∈ B(β_k) if ν_ij > δ₂. Furthermore, x_j is not part of any lower bound. After assigning each object in lower approximations or boundary regions of different clusters based on both δ₁ and δ₂, the memberships μ_ij for the objects lying in boundary regions are computed for c clusters using (19). The new centroids of different clusters are calculated as per (24). The above procedure is repeated until no more new assignments can be made.

The performance of the rRFCM depends on the values of two thresholds δ₁ and δ₂, which determine the cluster labels of all the objects. In other word, the rRFCM partitions the data set into two classes, namely, lower approximation and boundary, based on the values of δ₁ and δ₂. The thresholds δ₁ and δ₂ control the size of granules of rough-fuzzy clustering. In practice, the following definitions work well: (25) where n is the total number of objects, ν_ij and ν_kj are the highest and second highest memberships of object x_j. That is, the value of δ₁ represents the average difference of two highest possibilistic memberships of all the objects in the data set. A good clustering procedure should make the value of δ₁ as high as possible. On the other hand, the objects with (ν_ij−ν_kj) ≤ δ₁ are used to calculate the threshold δ₂: (26) where $\acute{n}$ is the number of objects that do not belong to lower approximations of any cluster and ν_ij is the highest membership of object x_j. That is, the value of δ₂ represents the average of highest memberships of $\acute{n}$ objects in the data set.

Optimum Value of Wavelet Decomposition Level

The wavelet transform, at each level, generates four subbands, namely, approximation, horizontal, vertical, and diagonal. The approximation part is iteratively decomposed as the decomposition level is increased. Hence, if an input image is decomposed upto lth level, total (3l+1) number of subbands are generated. To find out the optimum value of decomposition level l, extensive experiments are carried out by varying l = 1 to 4 on several brain MR images.

Fig 8 reports the heat maps for comparative performance analysis of different decomposition levels with respect to Jaccard index, sensitivity, and specificity on both BrainWeb and IBSR data sets. From the results reported in Fig 8, it can be seen that the segmentation quality increases with the increase of decomposition level upto 2 irrespective of the segmentation metrics and data sets used. However, the performance of the proposed method detoriates for l ≥ 3. The proposed brain MR image segmentation method for l = 2 performs better than that of other levels in 24, 19, and 20 cases, out of 25 cases each, with respect to Jaccard index, sensitivity, and specificity, respectively. Out of total 75 cases, the proposed segmentation method with l = 2 provides better results in 63 cases, while that with l = 1 and l = 3 attains in only 10 and 2 cases, respectively. Hence, each image is decomposed upto second level without degrading the segmentation quality.

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Fig 8. Heat maps for comparative performance analysis of different decomposition levels of wavelet analysis (from left to right: Jaccard index, sensitivity, and specificity).

https://doi.org/10.1371/journal.pone.0123677.g008

Importance of Wavelet Based Feature Extraction Method

The proposed brain MR image segmentation method uses wavelet for feature extraction. The features are extracted using Daubechies 6-tap wavelet decomposing upto second level, which yields seven subbands as features. However, the proposed unsupervised feature selection algorithm reduces this dimension to three. To establish the importance of wavelet based analysis over gray level, that is, the performance of the proposed method over the method ℳ₁, extensive experimentation is done on several brain MR images. The rRFCM algorithm is used as the clustering algorithm for both methods.

Figs 9, 10, and 11 present the heat maps for comparative performance analysis of wavelet based feature extraction and gray level with respect to three quantitative indices, namely, Jaccard index, sensitivity, and specificity. From the results reported in Figs 9, 10, and 11, it can be seen that the performance of the proposed method is better than that of the ℳ₁ in most of the cases, irrespective of the input images and quantitative indices used. Out of total 75 cases, the ℳ₁ performs better than the proposed method in only 13 cases. The second and third columns of Figs 4 and 5 compare qualitatively the performance of wavelet based analysis and gray level, that is, the proposed and ℳ₁ methods. All the results reported in second and third columns of Figs 4 and 5, and first and eleventh columns of heat maps presented in Figs 9, 10, and 11 confirm that the features derived by wavelet transform produce segmented images more promising than do the conventional gray level segmentation. The wavelet analysis provides a multiscale representation that resembles a hierarchical framework for interpreting the image information. At different scales, the details of an image generally characterize different physical structures of the image. In image processing, wavelet decomposition provides information from each scale to be analyzed separately. Hence, the feature extraction scheme based on multiscale analysis has several potential advantages over traditional gray level segmentation.

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Fig 9. Heat maps obtained by different methods with respect to Jaccard index.

https://doi.org/10.1371/journal.pone.0123677.g009

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Fig 10. Heat maps obtained by different methods with respect to sensitivity.

https://doi.org/10.1371/journal.pone.0123677.g010

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Fig 11. Heat maps obtained by different methods with respect to specificity.

https://doi.org/10.1371/journal.pone.0123677.g011

Effectiveness of Skull Stripping Algorithm

In the proposed segmentation method, the skull stripping algorithm is used to separate the background from major tissues of brain. To establish the effectiveness of the proposed skull stripping algorithm, the performance of the proposed method is compared with that of the methods ℳ₂ and ℳ₃. While second, third, and eleventh columns of heat maps presented in Figs 9, 10, and 11 compare the performance of the ℳ₂, ℳ₃, and the proposed method for three major tissue classes, namely, cerebrospinal fluid (CSF), gray matter, and white matter, Fig 12 presents that of only background region. From the results reported in Figs 9, 10, and 11, it can be seen that the proposed method with proposed skull stripping algorithm performs significantly better than the methods ℳ₂ and ℳ₃ for segmenting the CSF, gray matter, and white matter. The performance of the proposed method is better than that of ℳ₂ in 25, 16, and 22 cases with respect to Jaccard index, sensitivity, and specificity, respectively, while the proposed skull stripping algorithm attains higher performance compared to the BET [41] of ℳ₃ in 16, 21, and 13 cases, respectively.

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Fig 12. Heat maps for comparative performance analysis of the proposed method (skull stripping), the method ℳ₂ (without skull stripping), and the method ℳ₃ (masking using BET) for background separation (from left to right: Jaccard index, sensitivity, and specificity).

https://doi.org/10.1371/journal.pone.0123677.g012

The heat maps of Fig 12 compare the performance of the proposed method, ℳ₂, and ℳ₃ for segmentation of background region. From the results reported in Fig 12, it can be seen that the proposed method always attains better results than that of the ℳ₂ with respect to Jaccard index and sensitivity. However, the specificity value of the proposed method is lesser as compared to that of the ℳ₂ in most of the cases. Since the proposed skull stripping method adds some boundary portion of the CSF from brain image into its background, the false positive count is increased for the background. So, out of total 25 cases, the proposed method attains higher specificity values in only five cases compared to that of the ℳ₂. Hence, the proposed method performs better than the ℳ₂ in 55 cases out of total 75 cases, irrespective of the quantitative indices used. On the other hand, the proposed method provides better results than that of the ℳ₃ in most of the cases irrespective of the quantitative indices and data sets used. In other words, the performance of the proposed skull stripping algorithm is better than that of the BET [41] in 18, 15, and 17 cases with respect to Jaccard index, sensitivity, and specificity, respectively. Considering all the results reported in Figs 9, 10, 11, and 12, out of total 150 cases, the proposed method attains best performance in 79 cases, irrespective of the quantitative indices and images used, while both ℳ₂ and ℳ₃ achieve it in only 26 and 45 cases, respectively. In Figs 4 and 5, the qualitative performance analysis of the proposed method, ℳ₂, and ℳ₃ is reported. All the results reported in second, fourth, and fifth columns of Figs 4 and 5, and Figs 9, 10, 11, and 12 establish the effectiveness of the proposed skull stripping algorithm over existing BET [41] in the proposed brain MR image segmentation method.

Importance of Unsupervised Feature Selection

The proposed brain MR image segmentation method uses unsupervised feature selection algorithm based on the MRMS criterion to reduce the dimensionality of the data sets. Using the feature selection algorithm, the feature dimension can be reduced from d to m using (15). In order to establish the importance of unsupervised feature selection algorithm, extensive experimentation is carried out and the corresponding results are reported in Figs 9, 10, and 11. The performance of the proposed feature selection method is also compared with that of the feature selection algorithm proposed by Huang and Aviyente [50]. The eleventh, fourth, and fifth columns of heat maps of Figs 9, 10, and 11 compare the performance of the proposed method, ℳ₄, and ℳ₅, respectively.

From the results reported in fourth, fifth, and eleventh columns of heat maps of Figs 9, 10, and 11, it is seen that the proposed method achieves better results than ℳ₄ in 25, 25, and 24 cases with respect to Jaccard index, sensitivity, and specificity, respectively, while the performance of the proposed method is better than that of ℳ₅ in 20, 15, and 18 cases, respectively. Out of total 75 cases, the proposed method attains better results in 52 cases, while the methods ℳ₄ and ℳ₅ achieve it only in 1 and 22 cases, respectively. The second, sixth, and seventh columns of Figs 4 and 5 depict the comparative performance analysis of three methods qualitatively. All the results reported in Figs 4, 5, 9, 10, and 11 confirm that the unsupervised feature selection algorithm is effective in reducing the dimension of the feature space without losing segmentation quality. The better performance of the proposed unsupervised method over existing feature selection algorithm of Huang and Aviyente [50] is achieved due to the fact that the proposed method selects features based on their individual relevance as well as significance, whereas the method of Huang and Aviyente [50] considers only redundancy or similarity among them without considering their relevance values. In effect, the existing method selects a set of nonrelevant features, which degrades the quality of segmented images.

Importance of Robust Rough-Fuzzy C-Means

Further, the performance of the proposed method is extensively compared with that of the methods ℳ₆, ℳ₇, and ℳ₈. The only difference among these methods is the clustering algorithm used. While the proposed method uses robust rough-fuzzy c-means (rRFCM) [40] algorithm, other methods, namely, ℳ₆, ℳ₇, and ℳ₈, use hard c-means (HCM) [54], fuzzy c-means (FCM) [35], and rough-fuzzy c-means (RFCM) [39], respectively.

The sixth, seventh, and eleventh columns of heat maps reported in Figs 9, 10, and 11 compare the performance of the proposed method with that of the methods ℳ₆, ℳ₇, and ℳ₈ with respect to three quantitative indices on several brain MR images. The second, third, fourth, and fifth columns of Figs 6 and 7 compare the performance of different methods qualitatively. All the results reported in Figs 9, 10, and 11 establish the fact that the proposed method attains better performance in 18, 10, and 17 cases compared to other methods with respect to Jaccard index, sensitivity, and specificity, respectively, while the method ℳ₈ achieves it in 7, 3, and 2 cases, respectively. On the other hand, the method ℳ₇ provides better performance in 12 and 3 cases with respect to sensitivity and specificity, respectively, while the method ℳ₆ attains highest specificity in only 3 cases. Out of total 75 cases, the proposed method achieves better performance in 45 cases, irrespective of the brain MR images and quantitative indices used. From the segmented images reported in second, third, fourth, and fifth columns of Figs 6 and 7, it can also be seen that there is a significant improvement in the segmentation results obtained using the proposed method as compared to other methods. In this regard, it should be noted that some of the existing clustering algorithms such as possibilistic c-means [37] fail to produce multiple segments of the input image as they generate coincident clusters even when they are initialized with final prototypes of the hard c-means.

The best performance of the proposed method is achieved due to the fact that the probabilistic membership function of the rRFCM handles efficiently overlapping partitions, while the possibilistic membership function of lower approximation of a cluster helps to discover arbitrary shaped cluster. Moreover, the concept of possibilistic lower approximation and fuzzy boundary of the rRFCM algorithm deals with uncertainty, vagueness, and incompleteness in class definition. In effect, good segmented regions are obtained using the proposed brain MR image segmentation algorithm.

Comparative Performance Analysis

Finally, the performance of the proposed method is compared with that of both FSL [56] and SPM [57]. Results are reported in Figs 9, 10, and 11 with respect to Jaccard index, sensitivity, and specificity on both BrainWeb and IBSR data sets, while the qualitative performance analysis is reported in second, sixth, and seventh columns of Figs 6 and 7. The segmented outputs generated by the proposed method, FSL, and SPM, establish the fact that the proposed method generates more promising outputs than that obtained using the existing FSL and SPM. From the results reported in ninth, tenth, and eleventh columns of heat maps reported in Figs 9, 10, and 11, it can be seen that the proposed method provides better results than the FSL in 13, 6, and 11 cases, out of 13 cases each, for BrainWeb data set and in 10, 4, and 6 cases, out of 12 cases each, for IBSR data set with respect to Jaccard index, sensitivity, and specificity, respectively. In brief, the proposed method attains better performance than the FSL in 23, 10, and 17 cases, respectively, irrespective of the data sets used. On the other hand, the proposed method performs better than the SPM in 16 cases with respect to Jaccard index, while the performance of the proposed method detoriates compared to the SPM with respect to both sensitivity and specificity. The SPM achieves better results than the proposed method in 9, 24, and 19 cases, out of 25 cases each, with respect to Jaccard index, sensitivity, and specificity, respectively.

The better performance of the proposed method with respect to Jaccard index and lower sensitivity values obtained using the proposed method indicate that the proposed method attains lower ratio of false positive to true positive, which leads to lower false discovery rate (FDR), compared to both FSL and SPM in most of the cases, irrespective of the images used. The FDR is a multiple-hypothesis testing error measure indicating the expected proportion of false positives among the set of significant results. The FDR is particularly useful in the analysis of high-throughput data such as MRI. Out of total 25 cases, the proposed method attains lower FDR values in 17 and 23 cases than SPM and FSL, respectively. Moreover, the performance of the proposed method is compared with that of both FSL and SPM with respect to likelihood ratio positive (LR+), which is defined as the ratio of sensitivity and (1—specificity). Out of total 25 cases, the proposed method achieves higher LR+ values in 16 cases than the FSL, while SPM attains higher LR+ values in 19 cases than the proposed method.

The comparative performance analysis is also reported in terms of p-value computed through the sign test. The proposed method attains p-value of 9.72E-006, which is statistically significant considering 0.05 as the level of significance, for both Jaccard index and FDR with respect to the FSL. Also, it achieves lower p-value of 5.39E-02 for specificity and FDR, and that of 1.15E-01 for LR+ and Jaccard index, with respect to FSL and SPM, respectively. On the other hand, the SPM provides significant p-values of 2.98E-08 for sensitivity and 2.04E-03 for both specificity and LR+ with respect to the proposed method, while the FSL attains lower p-value of 1.15E-01 for sensitivity.