An Exploration: Alzheimer’s Disease Classification Based on Spectral Matching of Shape Features

In recent days, Alzheimer’s disease (AD) is the most well-known degenerative disease, and it proceeds slowly. The age-specific mortality rate has grown, as has a global interest in dementia-related studies. AD is among the most well-known diseases among the elderly, and it causes dementia symptoms such as sensory impairments (such as reasoning, memorizing, organizing, and judgment) [1]. It is observed that the AD occurrence rate is 3% for people aged over 60 years and 58% for people aged over 80 years globally. As people live longer, the percentage of people with Alzheimer's disease rises considerably. It is noted that approximately 30 million people were affected by AD in 2006. This figure is expected to reach 0.1 billion by 2050 [2]. It causes the hippocampal to shrink while interconnected cavities expand. The stage of the disease determines these severity levels of interruption. Brain scans (MRI images) revealed considerable expansion in the brain cortex, as well as expansion of the ventricles, throughout the later AD cycle [3, 4].

Mild Cognitive Impairment (MCI) is a term used to describe early-stage Alzheimer's disease patients. Not all MCI patients will develop AD. MCI is a stage between being healthy and having AD. During this stage, a person's mental abilities slowly change in ways that only they and their families can see. Furthermore, hippocampus segmentation and identification are not only subjective to the operator's perspective but also time-consuming. On the other hand, due to neuronal death, ventricular enlargement is a prominent feature of AD [5]. Ventricles are located in the brain's center. These are loaded with cerebrospinal fluid (CSF), which regulates brain metabolism, and are bordered by grey and white matter (WM). Dementia illnesses frequently impair the covering of GM and WM structures. As a result, hemisphere shrinkage rates on cognitive tests show a stronger link than medial temporal lobe atrophy rates [6], and there is a big difference between normal people and people with moderate cognitive impairment or AD. A recent investigation into identifying region-specific biomarkers for Alzheimer's disease and amnestic MCI identification shows that the heterogeneity of MCI types may make categorization even more difficult [7].

Recent research on surface representation has demonstrated that spectral shape description outperforms Euclidean surface modeling [8-10]. Associations are described as graphs in the spectral-based shape-matching technique, and an Eigen decomposition on such graphs allows us to match comparable features. After matching surfaces are created, vertex coordinates serve as shape descriptors. A variational autoencoder achieves non-linear low-dimensional shape embedding (VAE). The Variational Autoencoder (VAE) [11, 12] is a prominent generative model that allows us to formulate picture creation tasks in probabilistic graphical models containing latent constructs. The potential to regulate the distribution of the latent vector, which is an independent unit Gaussian random variable, is the key attribute of VAE. Numerous strategies for improving VAE performance have been suggested. For example, Gordon et al. [13], Hua and Chen [14] suggested conditioning variational autoencoders on either class labels or a range of visual qualities. Their studies show that they can generate realistic images with various looks. Furthermore, two models can be trained concurrently in the GAN framework: a generator network used to map a noise variable to data space and a discriminator network meant to discriminate between samples from actual training data and generated samples provided by the generator. Simultaneously, an MLP model was trained to mimic a non-linear class imbalance. Deep convolutional techniques are a popular unsupervised machine learning method. Many customs may benefit from a trained generative classic. This simple method is for understanding the dataset core and using random inputs to produce realistic dataset visuals. These are available in compressed form for the model's training purposes.

Another alternative is to develop reused local features obtained from unlabeled data, which can be used for classification purposes. In this study, we provide a strategy for training the variational autoencoder (VAE) to increase the system's effectiveness. However, the particular interest of this work is improving the quality of the resulting images to make them look natural and less blurred. Rather than the unsatisfactory per-pixel loss functions, we use objective functions explicitly on deep feature consistency principles. With learning convolutional operations, deep feature consistency can assist in capturing essential perceptual properties like spatial correlation. Our specific contributions are as follows:

·Using a shared latent embedding space, our model smoothly links with two modes: variational autoencoder (VAE) and generative adversarial network (GAN). They are also tested for performance gains.

·We confirm that trained latent models get theoretical and valuable data from MRI images, which can then be used to change the properties of medical images.

·Finally, we provide an efficient feature extraction method for AD disease classification studies that outperform traditional techniques.

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Figure 1. Simplified model for Alzheimer’s Disease classification

3.1 Extraction of shape features

Considering a baseline surface mesh Sr and a population of n surfaces{S_i}_i=1..n, spectral matching for every surface mesh as S_i and S_r is accomplished as. Firstly, the initial map generation for the two surfaces is computed [20]. Then, according to Lombaert et al. [21], this initial map is utilized to create a smooth map for two surface meshes in the second stage. Vertices and nearby spots within every surface mesh are treated as vertices and edges of a graph in this case. Later, this group is then used to generate a graph Laplacian of each surface mesh. Finally, the Laplacian operator L_i, in general, on each surface is demarcated as follows:

$L_i=G_i^{-1}\left(D_i-W_i\right)$         (1)

here, W_i is the weight matrix calculated from the measured distance of nodes that are connected. Here, D_irefers to a diagonal matrix whose members are determined. Finally, G_i denotes the weighting matrix according to Shakeri et al. [22]. The spectral components are provided via the eigen decomposition of the Laplacian matrix L_i. The final smooth map between two surfaces, S_i and S_r, is created in the following phase, given an initial map c. The association graph is then given a Laplacian matrix, which generates an eigenvectors group to obtain two surface meshes.

In this VAE, the encoder E(x) job is to turn the image data into the same amount of latent vector data. Decoder D(z) task is to generate an output image. Finally, the pretrained VGGNetΦ(x) task extracts the relevant features needed for classification. In this current study, we use discriminator Dis(x) as a classifier for generating real and generated images. To increase classification accuracy, we applied VAE, which requires KL divergence loss $\mathcal{L}_{k l} \leftarrow D_{K L}(q(Z \mid X) \| p(Z))$ [23]. The other is a feature reconstruction loss. Explicitly, for a pre-trained network, Φ using input and output images and measuring the reconstruction loss of the lth hidden layer is expressed as $\mathcal{L}_{r e c}=\mathcal{L}^1+\mathcal{L}^2+\cdots+\mathcal{L}^l$, where denotes the feature reconstruction loss.

Rather than providing information to the discriminator, using the VGGNet as the discriminator's input is the better idea. The goal is to allow for more steady training while utilizing less information as feasible and employing a GAN model with pre-trained VGGNet architecture.

3.2 Neural network architecture

Both the autoencoder and discriminator networks, as illustrated in Figure 2, are deep residual CNNs based on Radford et al. [24]. By instead employing deterministic spatial functions like max-pooling, we build four convolutional layers in the encoder network with a 4×4 kernel and a 2×2 stride to accomplish spatial downsampling. Following each convolutional layer comes a batch normalizing and, for a gradient allowing, using LeakyReLU successively. After each convolutional layer, a residual block is added, and all of them comprise two 3×3 kernels of identical layers. Finally, the encoder includes two fully-connected layers (FCL) that will be utilized to compute the KL divergence loss and sample latent variable z. We employ 4 CLs with 3×3 kernels and 1×1 stride for the decoder. We also recommend that replicated padding be used instead of traditional zero-padding, such that the input map fill-out with the input edge replica. Every convolutional layer except the last one is covered by a residual layer, just like the encoder.

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(a) Encoder

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(b) Decoder

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(c) Residual block

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(d) Discriminator network

Figure 2. Simplified block diagram of encoder, decoder and discriminator network

We employ the nearest neighbor approach on a scale of 2 for upsampling rather than the partial stride convolution operation used by prior publications [25]. In addition, batch normalization is used to assist in stabilizing the entire training, and LeakyReLU is used as the activation function. DCGAN's architectural advancements inspire the discriminator's design. To accomplish spatial downsampling, we use convolutional layers with a 4×4 kernel and a 2×2 stride and append at the end of CL with a residual block. Later, discarding the sigmoid at last and stride CL of size 4×4 is employed for output incorporating a gradient of -0.01 and discriminator of 0.01.

3.3 Feature reconstruction loss

This is expressed as the mean squared error between feature representations in a DCNN as Φ. As in [26], we use VGGNet [27] as the loss network in our experiment. The primary idea of feature reconstructive loss is to search for evenness in the learnt feature space between real and generated images. Consider that Φ_l(x) represents the l^th hidden layer. Φ_l(x) also signifies a block array of [C_l×W_l×H_l] dimensions. C_lindicates the filter count, W_l means the width, and H_l denotes the height of the l^th layer feature map. The squared Euclidean distance is a simple way to express the feature reconstruction loss (L^l) between two images (x and $\ddot{x}$).

$\mathcal{L}_l=\frac{1}{2 C_l W_l H_l} \sum_{c=1}^{c_l} \sum_{w=1}^{W_l} \sum_{h=1}^{H_l}\left(\Phi_l(x)_{c, w, h}-\Phi_l(\ddot{x})_{c, w, h}\right)^2$            (2)

Instead of merely using features from a single layer, we use visual information from many levels and integrate the VGGNet'soutputs. For example, the expression for the last reconstruction loss (RL) is calculated as follows:

$\mathcal{L}_{r e c}=\sum_{l=1}^L \frac{100}{C_l^2} \mathcal{L}_l$        (3)

where, $\mathcal{L}_l$ and C_l are the loss of features and filter count, and L is the total number of CLs of the model. In addition, to regulate the spread of the latent variable z, we use the KL divergence loss $\mathcal{L}_{kl}$ to regularize the encoder network. For training VAE accurately, we have to reduce KL divergence loss and reconstruction loss at every layer.

$\mathcal{L}_{\text {vae }}=\alpha \mathcal{L}_{k l}+\beta \mathcal{L}_{\text {rec }}$          (4)

where, α and β denote weighting values, respectively. It should be noted that VGGNet is solely utilized for feature extraction throughout the training process. In our current work, the MRI latent representation is the latent variable of the autoencoder.

3.4 Adversarial loss

In this, we added a VAE context to the GAN to generate the required outputs distributed throughout real images. WGAN is the basis of our adversarial training. To increase training stability even further, rather than directly feeding original images, we extracted features from the first layer of pretrained VGGNet and provided them to the discriminator. With the help of deep networks, to achieve a reconstructed image that is the same as the original. Later, the sigmoid is discarded, and 1 and -1 should be used as labels for real and generated images. We discovered that GAN can fail and that VAE is likely to take precedence when using too small labels, such as 1 and -1, during testing. Finally, as per Equation 5, our model may be trained end-to-end by adding three losses, and Algorithm 1 is used for the training approach.

$\mathcal{L}_{\text {vae }}=\alpha \mathcal{L}_{k l}+\beta \mathcal{L}_{\text {rec }}+\mathcal{L}_{G A N}$             (5)

where, $\mathcal{L}_{k l}$ denotes KL divergence loss, $\mathcal{L}_{r e c}$ denotes reconstruction loss and $\mathcal{L}_{G A N}$ denotes adversarial loss.

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Algorithm 1: Pseudo code for VAE-WGAN training model

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Input: feature parameter; Φ, pretrained model

W_Encoder; W_Decoder; W_{Discriminator} $\leftarrow$ Initialize parameters

repeat

X $\leftarrow$ mini-batch dataset images

$Z \leftarrow$ Encoder $(X)$

$\mathcal{L}_{k l} \leftarrow D_{K L}(q(Z \mid X) \| p(Z))$

$\widehat{\mathrm{X}} \leftarrow \operatorname{Decoder}(Z)$

$\mathcal{L}_{r e c} \leftarrow\|\Phi(\mathrm{X})-\Phi(\widehat{\mathrm{X}})\|^2$

$\mathcal{L}_{\text {GAN }} \leftarrow$ Discriminator $(X)-$ Discriminator $(\widehat{\mathrm{X}})$

$W_{\text {Encoder }} \stackrel{+}{\leftarrow}-\nabla_{W_{\text {Encoder }}}\left(\mathcal{L}_{k l}+\mathcal{L}_{\text {rec }}-\mathcal{L}_{\text {GAN }}\right)$

$W_{\text {Decoder }} \stackrel{+}{\leftarrow}-\nabla_{W_{\text {Decoder }}}\left(\mathcal{L}_{\text {rec }}-\mathcal{L}_{\text {GAN }}\right)$

$W_{\text {Discriminator }} \stackrel{+}{\leftarrow}-\nabla_{W_{\text {Disiminance }}}\,\,\,\,\,\,\, \mathcal{L}_{G A N}$

$W_{\text {Discriminator }} \leftarrow$ clip $\left(W_{\text {Discriminator }},-c, c\right)$

until convergence of parameters

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