Table 12 Definition of mathematical symbols with context/usage in the proposed study.
From: A hybrid filtering and deep learning approach for early Alzheimer’s disease identification
Symbol | Definition | Context/Usage |
|---|---|---|
I | Input image affected by noise | Image preprocessing (denoising using NLM filtering) |
u | Pixel values in the noisy image | Used in Equation (1) as input |
v | Pixel values after denoising | Output of NLM filtering |
x | Position of a pixel in the image | Filtering equations |
y | Position of a neighboring pixel | Weighting in NLM filtering |
\(\Omega\) | Neighborhood around pixel x | Defines averaging area |
w(x, y) | Weight between pixels x and y | Determines contribution in denoising |
C(x) | Normalization factor for weights | Ensures weights sum to 1 |
P(x) | Patch centered at x | For similarity computation in NLM filtering |
h | Filtering strength parameter | Controls decay in Gaussian kernel (Equation 2) |
Z(x) | Normalization constant in weight calculation | For Gaussian kernel normalization |
f(x, y) | Pixel intensity at (x, y) | Used in sharpening (Laplacian) filters |
\(\nabla ^2 f\) | Laplacian of image f | Second-order derivative for sharpening |
g(x, y) | Output pixel after sharpening | Final result of Laplacian-based sharpening |
\(W_1, \ldots , W_9\) | Pixel weights in sharpening filter mask | Part of filter kernel |
TP, TN, FP, FN | True Positives, True Negatives, False Positives, False Negatives | Performance metrics |
Accuracy | Overall correctness of prediction | Performance evaluation |
Precision | Proportion of true positive predictions | Performance evaluation |
Sensitivity / Recall | Ability to detect positive instances | Performance evaluation |
F1 Score | Harmonic mean of precision and recall | Performance evaluation |
Specificity | Ability to detect negative instances | Performance evaluation |
Matthews Correlation Coefficient (MCC) | Measure of the quality of binary and multiple classifications | Performance evaluation |
\(L(Y, \hat{Y})\) | Loss function | Sparse categorical cross-entropy |
Y | True label | Used in loss calculation |
\(\hat{Y}\) | Predicted label | Used in loss calculation |
\(\alpha ^k_{ij}\) | Grad-CAM++ weight at location (i, j) in feature map k | Used in explanation of model’s prediction |
\(y^c\) | Score for class c | Class output score used in Grad-CAM++ |
\(A^k_{ij}\) | Activation at pixel (i, j) in feature map k | Intermediate CNN output |
\(L^c_{\text {Grad-CAM++}}\) | Grad-CAM++ localization map for class c | Visualization of important regions in MRI |
ReLU | Rectified Linear Unit activation | Used in CNNs and Grad-CAM++ |
