Image Compression using Singular Value Decomposition (SVD)

Overview:
This project explores image compression using Singular Value Decomposition (SVD), a mathematical technique from linear algebra. Images are represented as matrices and approximated by keeping only the most significant singular values, which reduces data while preserving essential visual information. The project demonstrates low-rank approximations for different values of k and evaluates the trade-off between approximation quality and retained singular values. Error analysis is carried out using Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) to quantify the information loss at different compression levels.

Suitable For:

• BS Mathematics – applications of linear algebra, matrix factorization, and approximation theory

• BS Computer Science/Artificial Intelligence/Data Science– algorithmic implementation of compression using SVD

Technologies Used:

• Programming Language: Python

• Libraries & Tools: NumPy, Matplotlib, SciPy

• Techniques: Singular Value Decomposition, Low-Rank Approximation, Error Analysis

• Visualization: Original vs. compressed images, singular value plots, error vs. rank graphs

Features:

• Representing images as matrices and applying SVD factorization

• Step-by-step manual decomposition of a small matrix (toy example)

• Low-rank approximations for different k values, showing image quality changes

• Error analysis using MSE and RMSE to assess image quality

• Visual side-by-side comparison of original and compressed images

Deliverables:

• Python source code for SVD-based image compression

• Project documentation: mathematical background, methodology, and worked examples

• Presentation slides (PPT): objectives, SVD explanation, error analysis, and results with figures

• Visual assets: original vs. compressed images, error tables, and singular value plots

Support:

• Guidance for applying the method to grayscale and RGB images

• Help in interpreting error metrics and selecting the appropriate number of singular values

• Assistance in extending the project to larger datasets and real-world images

Benefits for Students:

• Gain practical experience applying SVD to real-world image compression

• Understand the mathematics of low-rank approximation in linear algebra

• Learn to measure and interpret approximation quality using MSE and RMSE

• Develop programming and data visualization skills with Python

• Build a strong capstone project connecting mathematics, computing, and image processing