Alzheimer’s Disease MRI Brain Segmentation Using Pythagorean Fuzzy Sets

AD is a classic degenerative age-related disorder. Memory loss, mood swings, cognitive decline, and difficulties in speaking, writing, and walking are all symptoms observed in clinical practice. It is a major disease that threatens the health of elderly people [1]. More than 50 million individuals worldwide are currently affected and is expected to rise to 150 million by 2050. Currently, there is no therapy for AD in humans because the pathogenic mechanism has not been fully explained. One major reason is that dementia is difficult to diagnose because, by the time noticeable memory loss and neurological impairment have set in, the disease is in an irreversibly advanced stage [2]. First, there is an asymptomatic period, followed by moderate cognitive impairment (MCI), which eventually progresses into full-blown Alzheimer's disease. Mild cognitive impairment is frequently misdiagnosed as a natural part of aging and the treatment is delayed. Delaying the onset of AD, altering the course of the disease, and avoiding early intervention measures depend on a correct diagnosis. Hence, the development of novel and effective treatments for Alzheimer's disease depends on its early identification.

Neuronal damage in dementia is permanent, and the resulting structural alterations in the brain can be studied using multifractal frameworks [3]. Many in-depth studies on this condition have been conducted in recent years using computer-aided diagnostics. Early diagnosis of AD has shown better detection results by combining the dimensionality reduction of features method with an efficient classifier, and most studies have followed this trend. There is some indication that Ada-SVM outperforms AdaBoost and Free Surfer in extracting the hippocampus border from the brain image, as demonstrated by Morra et al. [4]. Farhana's research focused on early-stage AD and how to back up the disease's multistage categorization. MRI images were preprocessed (using watershed segmentation and median filtering), and automatic feature extraction was performed using models of convolutional neural networks (CNNs) that were pre-trained with benchmarks, including Alex Net, VGG16, VGG19, ResNet18, and ResNet50. The healthcare business generates massive volumes of data, and algorithm-based systems are essential for making sense of it and improving prediction and diagnosis [5]. In AD, most brain MRI scans are segmented using deep network models trained on CNN data. However, identifying Alzheimer's from the brain structure is complex because of the tissue's penetration; hence, the segmentation of tissues from the brain and then applying classification on the extracted tissues to enhance the prediction of Alzheimer’s.

Fuzzy sets address the problem of vagueness by providing a degree of belongingness and non-belongingness [6, 7]. Many algorithms, namely bias-corrected FCM, enhanced FCM, Kerneled FCM, spatially contained FCM, possibilistic FCM, FCM with spatial information, fuzzy local information C-means, robust kernelized FCM, automatic FCM, meta-heuristic based FCM, and fast generalized FCM based on the fuzzy sets for brain image segmentation were available in the literature [8]. These algorithms effectively handle the vagueness of the images. Rough sets have been suggested to deal with the uncertainty in tissue borders [9, 10]. The algorithms yielded enhanced outcomes in comparison to fuzzy sets and demonstrated the ability to tackle the noise found in the regions [11].

A powerful tool, namely, soft sets (SS) with a mathematical foundation, was proposed by Molodstov to address uncertainty. The parameterization tool in soft sets aids decision-making and has been applied in the medical field. Maji et al. [12] defined and investigated many foundational concepts of soft set theory. They also generalized fuzzy soft sets (FSSs) from crisp soft sets. Soft sets have been applied to brain image segmentation using soft fuzzy c-means [13]. However, the proposed model was ineffective at reducing noise in the images. Intuitionistic fuzzy sets (IFS) with three-degree memberships were proposed as a solution to noise and uncertainty. These degrees include deterministic, non-deterministic, and hesitancy, which were used for clustering [14]. To incorporate local spatial information in an intuitive fuzzy manner, Atanassov [15] suggested that an enhanced intuitionistic fuzzy set is a novel approach. Dubey et al. [16] proposed a rough set based on IFS for segmenting brain MRI images. Verma et al. [17] suggested an enhanced intuitionistic fuzzy c-means (IIFCM) to consider the local spatial information in an intuitively fuzzy manner. To address the limitations of the IFS approach, Yager [18] created a Pythagorean fuzzy set (PFS). PFS developed upon IFS. The PFS add-on expands IFS's use and adaptability of the IFS. With PFS, it can be seen not just how much consensus exists among regions but also how vague that consensus is. Segmenting brain images suggests using an intuitive fuzzy c-means (IFCM) method. Verma suggested enhanced intuitionistic fuzzy c-means (IIFCM) as a novel approach. The authors used IFCM to manage ambiguity. However, the findings indicate that the technique is extremely susceptible to ambiance and makes little use of local spatial data. Improved intuitionistic fuzzy c-means were created to address the insensitivity noise and parameter flexibility selection during tuning. Compared to other existing approaches, the IIFCM approach significantly improves performance. To take this into account, local spatial information is intuitively fuzzy.

Peng et al. [19] investigated the properties of PF aggregation operators, including rigidity, inertia, and predictability. Furthermore, the researchers suggested a PF ranking system based on a scale of excellence and incompetence for addressing MAGDM issues, such as ambiguity and an abundance of attributes. Careful analysis of the PFSMs also led to the recommendation of a novel algorithm for decision-making (DM) based on a ‘choice matrix’ (CHMX) and a weighted CHMX. Operations including complement, union, intersection, addition, multiplication, necessity, and possibility are delineated in PFS.

Group decision-making, a crucial tool for facilitating medical diagnosis, has been proposed for application to PFSMs. Due to its computational simplicity, PFSMs will be integrated into hospital information systems to help doctors make well-informed decisions based on symptom values. Guleria and Bajaj [20], Ejegwa and Onasanya [21], Hashim et al. [22], and Srivastava [23] proposed various methods based on soft sets. The standardized Rough Intuitive sets were applied to MRI brain images using the Fuzzy C-Means method. Some recent advancements in SSs and FSSs, as well as their applications in decision-making, are discussed by Ali et al. [24], Feng et al. [25], and Kirisci [26].

The Pythagorean fuzzy c-means method was recently proposed for image segmentation [27, 28]. It used min-max parameters to fuzzify and de-fuzzify the image data and used distance measures for clustering. However, the dependency with min-max is not suitable for all types of image data; therefore, a generalized Pythagorean fuzzy c-mean is proposed to extract the tissues of brain AD images. The method uses histogram peaks as initial centroids that help determine the initial degrees of PFS. Using these initial degrees, constrained fuzzy c-means are applied with the Pythagorean objective function, which reduces clustering errors and provides higher data inclusion than intuitionistic fuzzy sets. Kirisci and Simsek [29] proposed advancements in techniques for addressing group decision-making challenges.

The important contributions of this study are as follows:

•   The use of PFS was explored in this study for extracting tissues from Alzheimer’s disease images.

•   An initial centroid is required for each fuzzy-based clustering technique. In this study, centroids are calculated based on the peaks and valleys of the input image histogram.

•   PFS aids in identifying the starting cluster regions, reducing the number of segmentation iterations.

•   The rapid convergence of PFS is achieved as the starting centroids are calculated based on the image data itself, and the segmentation accuracy is improved with PFS.

The paper is divided into several sections, starting with the background in Section II, the proposed methodology on PFS in Section III, Experimental procedures and outcomes in Section IV, and a summary and future directions in Section V.

This chapter focuses mostly on the process of segmenting brain MRI images utilizing PFS. Clustering always depends on the membership function and no single membership function suits all datasets. Hence, to avoid the dependency with the membership we used an approach to calculate initial regions with the image data itself using the thresholds computations [9] and initial centroids using histogram. By identifying the key areas independently of the membership functions used as data for the Pythagorean fuzzy c-means, the proposed method offers a fresh perspective. Further, the approach used the histogram peaks and valleys to determine the initial centroids for the clusters which assists in avoiding the cluster mistakes that occur with random selection. With the proposed method the clustering mistakes are reduced and the dependency on the membership functions is eliminated. Also, the Pythagorean fuzzy c-means has the advantage of addressing ambiguity and uncertainty in the data compared to IFS and offers broader data representation. The detailed process of the proposed method is explained in Figure 3.

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Figure 3. The flow of the proposed method of work

With the histogram peaks calculation, the peaks of brain images are computed and these are used as initial centroids for clustering mentioned in section 3.1. Also, three initial regions are determined by calculating the thresholds obtained from the brain image as presented in section 3.2 continued by the updated constrained partition fuzzy Pythagorean fuzzy c-means for performing the clustering. The primary difference between IFS and PFS lies in the degree of uncertainty measurement given in Eq. (3). The intuitionistic fuzzy c-means clustering is modified to apply the uncertainty equation of PFS.

3.1 Histogram-based initial centroid selection

The clustering algorithms are significantly sensitive to the initial cluster. To identify the regions based on distance, the initial centroids are calculated from the histogram peaks. The calculation of histogram peaks is given in previous study [25]. The sample Figure 4 displays the brain images and corresponding prominent peaks and valleys labelled. The author proposed to use the peaks of each histogram as the initial centroids as the peak represents the most repeated value and the bin around it represents the proximity of values near the cluster. Choosing the peaks as initial centroids will give stable regions initially that avoid the problem of local minima (usually occurs when wrong centroids are used).

4.png

Figure 4. (a, c) ADNI sample Image (b, d) image valleys and peaks

3.2 Determination of Pythagorean fuzzy regions

Since Fuzzy membership functions are crucial for making clusters, they have a large effect on segmentation efficiency. The initial cluster regions are determined by pixel thresholds calculated with the distances. Let the image of size m×n, is represented as E={P(x)/P(x) indicates pixel value of the i^th image element}.

First, the distance between each pixel P(x) to unique intensity values of the image L_kis calculated using Eq. (4) given below:

${{d}_{i}}\left( k \right)=\frac{\sqrt{\mathop{\sum }_{k\epsilon N~{{H}_{i}}}{}^{\left( {{x}_{i}}-{{L}_{k}} \right)}/{}_{\left| N~{{H}_{i}} \right|}}}{{{L}_{max}}-{{L}_{min}}}$                (4)

where, N H_i is the area surrounding the pixel x_i and | N H_i | is the number of distinct pixels in the image. L_max reaches the maximum intensity and L_min is the least intensity value. A distance vector can be calculated using this formula d_i(k)={d_i(L₁), d_i(L₂) ..., d_i(L_k)} each x_i^th pixel. To estimate two thresholds, we employ the maximum and minimum distances, respectively denoted by d_imax and d_imin.

$t1=\frac{1}{M*N}\underset{i=1}{\overset{M*N}{\mathop \sum }}\,{{d}_{imin}}$                (5)

$t2=\frac{1}{M*N}\underset{i=1}{\overset{M*N}{\mathop \sum }}\,{{d}_{imax}}$                (6)

The t1 values show how far apart on average the pixels are from the center of each cluster, whereas the t2 values show how far apart on average the pixels are from the furthest edges of the clusters. These thresholds serve to estimate three membership functions of the PFS. The belonging region B(P_x), non-belonging NB(P_x), and in-deterministic I(P_x).

If a pixel’s distance to a cluster obtained in the histogram, denoted by the symbol d_i(C_j), is smaller than the threshold value t1, then the pixel most likely belongs to the cluster and hence, is located in the determined region B(C_j). Pixels in the NB(C_j) hesitancy region have distances between t1 and t2, indicating that they might either be part of the cluster or not. A pixel is considered to be outside of the cluster and outside of the deterministic region I(C_j) if its distance from the cluster center is larger than the threshold value t2. As a result, the pixels can be roughly divided into three regions using these thresholds.

$x_i=\left\{\begin{array}{c}B\left(C_j\right) ; i f d_i\left(C_j\right) \leq t 1 \\ N\left(C_j\right) ; \text { if } t 1 \leq d_i\left(C_j\right) \leq t 2 \\ I\left(C_j\right) ; \text { otherwise }\end{array}\right.$                (7)

The computation of membership value ${{\alpha }_{P}}\left( x \right)$ of the PFS is calculated using the pixels in the B(C_j) using Eq. (8).

$\alpha_P(x)=\frac{1}{\sum_{j=1}^n\left(\frac{\left\|B_p(x)-C_j\right\|}{\sum_{c=1}^{n, c!=j}\left\|B_p(x)-C_c\right\|}\right)^{2 / m-1}}$                (8)

where, C_j represents the center of the j^th cluster. Here, N B_P (x), and π_P (x) are derived from the triangular membership function, and C_c is the arithmetic mean of B_P (x) and C_c, representing the values of the centroid of areas that do not cluster into j^th. Eqs. (8)-(10) may be used to calculate π_P (x) and γ_P (x) from π_P (x) and NB_P (x), respectively.

$\beta_P(x)=\frac{1}{\sum_{j=1}^n\left(\frac{\left\|H\left(e_i\right)-C_j\right\|}{\sum_{c=1}^{n, c!=j}\left\|H\left(e_i\right)-C_c\right\|}\right)^{2 / m-1}}$                (9)

${{\pi }_{p}}\left( x \right)=\sqrt{1-{{\left( {{\alpha }_{p}}\left( x \right) \right)}^{2}}~+~{{\left( {{\beta }_{p}}\left( x \right) \right)}^{2}}}$                (10)

The following equations are used to revise the PFS partition matrix μ_{P F S} P (x) and the cluster centroid C(μ_P (x)).

μ_{P F S} P(x)=α²_P(x)+β²_P(x)+π²_P(x)                (11)

With the PFS partitioning matrix, the uncertainty of the pixels is modelled and this allows the pixels to more accurately participate in clustering.

$C\left( \mu \left( p\left( x \right) \right) \right)=\frac{\mathop{\sum }_{p\left( x \right)\epsilon E}{{\left( \mu \left( p\left( x \right) \right) \right)}^{m}}\left( p\left( x \right) \right)}{\mathop{\sum }_{p\left( x \right)\epsilon E}{{\left( \mu \left( p\left( x \right) \right) \right)}^{m}}}$                (12)

Use Eq. (12) to locate the missing centroids for π(p(x)) and γ(p(x)). The user-specified fuzzification constant is denoted here by m. A sharp and binary membership function is obtained if m is near 1; otherwise, it becomes fuzzy and blurry as m increases [22]. With, large datasets that can be successfully segmented 1.5<m<3 plus 2 is typically thought of [23, 24]. Like many other clustering techniques, Pythagorean fuzzy c-means relies on iterative refinement. Repeatedly applying Eqs. (4)-(6) to identify the three regions and Eq. (8) to obtain the new centroids, the process concludes once the clusters reach stability. Stability in a cluster is determined by a Hamming distance between the first set of clusters and a similarity coefficient ${{\mu }_{PFS~}}P{{\left( x \right)}^{l~}}$ and$~{{\mu }_{PFS}}P{{\left( x \right)}^{l+1}}$. The formula for the hammering distance is:

$S C=\sum_{i=1}^{M * N}\left|\mu_{P F S} P(x)^l-\mu_{P F S} P(x)^{l+1}\right|$                (13)

3.3 PFCM algorithm for tissue segmentation

In this section, the steps of the PFCM clustering are proposed. The following Algorithm 1 describes the steps for the segmentation of brain tissues.

Algorithm1

Computer-aided detection of Alzheimer’s with the use of brain images has tremendously assisted clinicians. Due to numerous factors like artifacts in brain MRI, varied pixel intensity, and images with poor contrast Alzheimer’s detection increased complexity. Here, we introduce a novel PFCM algorithm for Segmenting Brain Images. The histogram peaks of the image are used as initial regions and centroids. This avoids the local minima and the initial regions are determined without depending on the membership function making the algorithm generalized. Cluster stability is improved, and runtime is shortened, thanks to the regions’ contributions. The extracted regions serve as inputs in the membership function calculations of the Pythagorean fuzzy soft sets clustering handling uncertainty in the brain images. Further, the extracted tissues can be used as input to deep learning models that can classify the Alzheimer's stages more accurately.