SVD Image Compression and Evaluation Framework

This project provides tools for image compression using different variants of Singular Value Decomposition (SVD):

1. Truncated SVD (TSVD)
2. Compressed SVD
3. Randomized SVD (RSVD), with optional power iterations to improve accuracy.

The code supports both grayscale and colored image compression while measuring computational time, Mean Squared Error (MSE), and compression ratios for different numbers of singular values.

Table of Contents

1. Setup and Installation
2. Dependencies
3. Code Overview
4. How to Use
   • Grayscale Image Compression
   • Colored Image Compression
5. Results and Outputs
6. Evaluation Metrics

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Setup and Installation

Clone the repository and install required Python libraries:

# Clone the repository
git clone https://github.com/your-repository/svd-compression.git
cd svd-compression

# Install dependencies
pip install -r requirements.txt

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Dependencies

The following libraries are required for the project:

numpy
matplotlib
pandas
opencv-python
Pillow

Install them with:

pip install numpy matplotlib pandas opencv-python Pillow

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Code Overview

The project consists of:

1. svd.py

This module implements different variants of SVD:

• trunc_svd: Truncated SVD for grayscale images
• compressed_svd: Compressed SVD for grayscale images
• randomized_svd: Randomized SVD with power iteration for grayscale images
• trunc_svd_colored: Truncated SVD for colored images
• compressed_svd_colored: Compressed SVD for colored images
• randomized_svd_colored: Randomized SVD with power iteration for colored images

2. evaluation.py

This module provides tools to evaluate the compression results:

• measure_computational_time: Measure execution time of compression algorithms
• mse_frobenius: Compute the Mean Squared Error (MSE) for grayscale images
• mse_frobenius_colored: Compute MSE for colored images
• calculate_compression_ratio: Calculate the compression ratio
• save_compressed_image: Save the compressed image to file

3. Main Script

The main script performs compression experiments for both grayscale and colored images, evaluates performance, and visualizes results.

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How to Use

Grayscale Image Compression

The script compresses a grayscale image using Truncated SVD, Compressed SVD, and Randomized SVD (with varying power iterations). Results include:

• Computational time
• MSE error
• Compression ratio

Steps

1. Place your grayscale image in the working directory (e.g., alone.jpg).
2. Run the main script:

python main.py

3. Outputs:
   • Compressed images saved under image_results/
   • Performance metrics saved under evaluation_results/
   • Plots generated for time taken, MSE, and compression ratios

Code Snippet (Grayscale)

from PIL import Image
import matplotlib.pyplot as plt
import os
import numpy as np
import pandas as pd
import svd
evaluation

image = Image.open("alone.jpg")
gray_image = image.convert('L')
image_matrix = np.array(gray_image)/255

singular_values = [1, 5, 10, 50, 100, 200, 300, 400, 500]
ts_results = []
for k in singular_values:
    comp_time, comp_image = evaluation.measure_computational_time(svd.trunc_svd, image_matrix, k)
    mse = evaluation.mse_frobenius(image_matrix, comp_image)
    print(f"k={k}, Time={comp_time}, MSE={mse}")

Colored Image Compression

The script compresses colored images using Truncated SVD, Compressed SVD, and Randomized SVD with power iterations.

Steps

1. Place your colored image in the working directory (e.g., alone.jpg).
2. Run the main script:

python main.py

3. Outputs:
   • Compressed images saved under image_results/
   • Evaluation results stored under evaluation_results/
   • Plots for computational time, MSE, and compression ratio

Code Snippet (Colored)

image_bgr = cv2.imread('alone.jpg')
image = cv2.cvtColor(image_bgr, cv2.COLOR_BGR2RGB) / 255.0

singular_values = [1, 5, 10, 50, 100, 200]
ts_results = []
for k in singular_values:
    comp_time, comp_image = evaluation.measure_computational_time(svd.trunc_svd_colored, image, k)
    mse = evaluation.mse_frobenius_colored(image, comp_image)
    print(f"k={k}, Time={comp_time}, MSE={mse}")

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Results and Outputs

Generated Outputs

• Compressed Images: Stored in the image_results/ directory
• Evaluation Metrics: Saved as CSV or plots in evaluation_results/
• Plots: Performance comparisons:
  • Computational Time vs Singular Values
  • MSE Error vs Singular Values
  • Compression Ratio vs Singular Values

Example Plots

• Computational Time
• MSE (Mean Squared Error)
• Compression Ratio

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Evaluation Metrics

1. Computational Time: Measures how long it takes to perform the compression for different SVD variants.
2. MSE (Mean Squared Error): Measures reconstruction error between the original and compressed images.
3. Compression Ratio: Compression efficiency based on the size of the original and compressed images.

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Example Outputs

Console Log:

This is the start of the truncated svd
Running the truncated svd for 50 singular value
Time=0.123s, MSE=0.0023, Compression Ratio=0.25

This is the start of the randomized svd with q=2
Running for k=50 singular value
Time=0.089s, MSE=0.0021, Compression Ratio=0.22

Output Directory Structure:

project/
|-- image_results/
|   |-- gray_alone.jpg
|   |-- tsvd_50.jpg
|   |-- rsvd_q=2_50.jpg
|-- evaluation_results/
|   |-- time_taken_plot.png
|   |-- mse_plot.png
|   |-- compression_ratio_plot.png

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Author

Desmond Kofi Boateng

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License

This project is licensed under the MIT License. Feel free to use and modify it!