🧠 Objective

This lecture explores the application of eigenvalues and eigenvectors in image processing using Principal Component Analysis (PCA). We will:

• Understand PCA and its reliance on eigen decomposition.
• Load and process an image.
• Use PCA to compress and reconstruct the image.
• Visualize the effect of PCA using eigenvectors.

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📦 Prerequisites

Install the following Python packages if you haven’t already:

pip install numpy opencv-python matplotlib

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🖼️ Step 1: Load and Convert Image to Grayscale

import cv2
import numpy as np
import matplotlib.pyplot as plt

# Load the image in grayscale mode
img = cv2.imread('img.jpg', cv2.IMREAD_GRAYSCALE)
if img is None:
    raise FileNotFoundError("Image not found. Please check the filename and path.")

• cv2.imread() reads the image file.
• cv2.IMREAD_GRAYSCALE loads it in grayscale (single-channel image).
• We raise an error if the file is not found to prevent further issues.

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🔄 Step 2: Normalize and Reshape the Image

img = img / 255.0  # Normalize pixel values between 0 and 1
original_shape = img.shape
print(f"Original image shape: {original_shape}")

• Normalization makes the data suitable for numerical computations.
• The shape is stored to understand the structure (rows × columns).

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🎯 Step 3: Center the Data (Zero Mean)

mean = np.mean(img, axis=0)  # Mean of each column
centered_img = img - mean    # Centering

• PCA requires the data to be centered (zero mean).
• We subtract the mean of each column (each pixel column).

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📐 Step 4: Compute Covariance Matrix

cov_matrix = np.cov(centered_img, rowvar=False)

• Covariance matrix shows how features vary together.
• rowvar=False: treats rows as samples and columns as features.

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🧮 Step 5: Eigen Decomposition

eigenvalues, eigenvectors = np.linalg.eigh(cov_matrix)

• np.linalg.eigh() is used for symmetric matrices (like covariance matrices).
• It returns:
  • eigenvalues: magnitude of variance in each principal direction.
  • eigenvectors: directions of maximum variance.

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🔢 Step 6: Sort Eigenvalues and Eigenvectors

idx = np.argsort(eigenvalues)[::-1]  # Descending order

# Reorder eigenvalues and eigenvectors
eigenvalues = eigenvalues[idx]
eigenvectors = eigenvectors[:, idx]

• PCA uses components with the highest eigenvalues.
• We sort in descending order to retain most significant components first.

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✂️ Step 7: Project Data to Lower Dimensions

num_components = 50  # Number of principal components
projection = np.dot(centered_img, eigenvectors[:, :num_components])

• Project the centered image onto top num_components eigenvectors.
• This step compresses the image data.

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🔁 Step 8: Reconstruct the Image

reconstructed_img = np.dot(projection, eigenvectors[:, :num_components].T) + mean

• We reverse the projection and add the mean back to approximate the original image.
• The quality depends on how many components we retained.

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📊 Step 9: Display Original and Reconstructed Images

plt.figure(figsize=(10, 5))

plt.subplot(1, 2, 1)
plt.title("Original Image")
plt.imshow(img, cmap='gray')

plt.subplot(1, 2, 2)
plt.title(f"Reconstructed Image ({num_components} components)")
plt.imshow(reconstructed_img, cmap='gray')

plt.tight_layout()
plt.show()

• matplotlib is used to plot both images.
• You can visually compare original vs compressed output.

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📌 Key Concepts Recap

Concept	Description
Eigenvalue	Measure of variance in the data in the direction of its corresponding eigenvector.
Eigenvector	A principal axis in the data space — direction of maximum variance.
Covariance Matrix	Square matrix showing covariance (interdependence) between features.
PCA	Reduces the dimensions of data using eigen decomposition while preserving the most variance.

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🔍 Visualization of Compression

By changing the number of components used (e.g., 10, 20, 100), observe how the quality of reconstruction improves with more components.

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🧪 Exercises

1. Try different values of num_components and plot the error.
2. Apply PCA to a color image by treating each RGB channel separately.
3. Use this technique on image datasets like MNIST or CIFAR-10 for dimensionality reduction.
4. Plot cumulative explained variance using eigenvalues.