how can I compute PCA between different images, tell me its mathematics.

Computing Principal Component Analysis (PCA) between different images involves treating each image (or image patch) as a high-dimensional vector and then performing PCA to find the dominant patterns (principal components) across these vectors. Below, I’ll explain the mathematics step-by-step and provide a clear method to implement it.

Key Idea of PCA for Images

Flatten each image into a 1D vector (e.g., a 64x64 image becomes a 4096-dim vector).
Stack all images into a data matrix X (each row is an image, columns are pixel values).
Compute PCA on X to find the principal components (PCs) that explain the most variance across images.

Step-by-Step Mathematics

Let’s say you have N images, each of size H × W (height × width).

1. Flatten and Create Data Matrix

Reshape each image into a column vector of size D = H × W.
Stack all images into a N × D data matrix X: [ X = \begin{bmatrix} \text{Image}_1^{(1)} & \text{Image}_1^{(2)} & \cdots & \text{Image}_1^{(D)} \ \text{Image}_2^{(1)} & \text{Image}_2^{(2)} & \cdots & \text{Image}_2^{(D)} \ \vdots & \vdots & \ddots & \vdots \ \text{Image}_N^{(1)} & \text{Image}_N^{(2)} & \cdots & \text{Image}_N^{(D)} \end{bmatrix} ] (Each row is an image, columns are pixel intensities.)

2. Center the Data (Subtract Mean)

Compute the mean image: [ \mu = \frac{1}{N} \sum_{i=1}^N \text{Image}_i ]
Subtract the mean from each image: [ X_{\text{centered}} = X - \mu ]

3. Compute Covariance Matrix

The covariance matrix C (size D × D) measures how pixel intensities vary together: [ C = \frac{1}{N-1} X_{\text{centered}}^T X_{\text{centered}} ] (For N < D, computing C directly is inefficient—see Efficient Computation below.)

4. Eigenvalue Decomposition

Solve for the eigenvectors (V) and eigenvalues (\Lambda) of C: [ C V = V \Lambda ]
The eigenvectors (columns of V) are the principal components (PCs).
The eigenvalues (\Lambda) indicate the variance explained by each PC.

5. Project Data onto Principal Components

To reduce dimensionality, select the top k PCs (e.g., k=3).
Project the centered data onto these PCs: [ \text{Scores} = X_{\text{centered}} V_k ] (Scores is an N × k matrix representing each image in the PCA space.)

Efficient Computation (for High-Dimensional Images)

If D (number of pixels) is large (e.g., D = 4096 for 64x64 images), computing the D × D covariance matrix is expensive. Instead:

Compute the Gram matrix G = X_{\text{centered}} X_{\text{centered}}^T (size N × N).
Find its eigenvectors U and eigenvalues \Lambda: [ G U = U \Lambda ]
The PCs are then: [ V = X_{\text{centered}}^T U \Lambda^{-1/2} ] (This avoids computing the large D × D covariance matrix.)

Example in Python (using NumPy)

import numpy as np

# Example: 10 images of size 64x64 (flattened to 4096-dim vectors)
N, H, W = 10, 64, 64
X = np.random.rand(N, H * W)  # Replace with actual images

# Step 1: Center data
mean = np.mean(X, axis=0)
X_centered = X - mean

# Step 2: Compute covariance matrix (use Gram matrix trick for D >> N)
G = X_centered @ X_centered.T  # Gram matrix (N x N)
eigenvals, eigenvecs = np.linalg.eigh(G)  # Eigen decomposition

# Step 3: Get top k=3 principal components
k = 3
top_k_eigenvecs = eigenvecs[:, -k:]  # Last k eigenvectors (largest eigenvalues)
top_k_pcs = (X_centered.T @ top_k_eigenvecs) / np.sqrt(eigenvals[-k:])

# Step 4: Project data onto PCs
scores = X_centered @ top_k_pcs  # N x k matrix (PCA scores for each image)
print("PCA Scores (first 3 components):\n", scores)

Interpretation

Each row of scores gives the coordinates of an image in the PCA space.
The first PC (scores[:, 0]) captures the direction of maximum variance across images.
Plotting scores[:, 0] vs scores[:, 1] visualizes image similarities.

Key Notes

Normalization: If pixel scales vary, standardize X (subtract mean, divide by std).
Whitening: To make all PCs equally important, divide by sqrt(eigenvals).
PCA vs. SVD: In practice, PCA is often computed via SVD (more numerically stable).