MAT-350 Project 2: SVD Image Compression and Rank-k Approximations

Author: Justin Paul Guida
Course: MAT-350 – Applied Linear Algebra
Date: October 16, 2025

Overview

This repository contains the MATLAB implementation and documentation for Project 2, which applies Singular Value Decomposition (SVD) to compute rank-k approximations of matrices and images.
It demonstrates how dimensionality reduction can be used for both matrix approximation and image compression, balancing storage efficiency and image quality.

Both .m and .mlx versions of the project are included:

• .m: Executable MATLAB script suitable for reproducibility
• .mlx: Interactive Live Script with explanations and formatted results
• .pdf: Exported report version for viewing without MATLAB

Objectives

1. Compute low-rank matrix approximations using SVD
2. Measure approximation quality using Root Mean Square Error (RMSE)
3. Apply rank-k SVD approximations to a grayscale image for compression
4. Analyze the relationship between compression ratio (CR), rank, and image quality
5. Compare approximations at CR = 10, 25, and 75 and identify the optimal balance

Files

File	Description
mat350-svd-compression.mlx	MATLAB Live Script with formatted text, figures, and explanations
mat350_svd_compression.m	Standard MATLAB script version, suitable for direct execution
mat350-svd-compression.pdf	PDF export of the Live Script for viewing without MATLAB
MAT 350 Project Two MATLAB Image.mat	Image dataset used for compression analysis

Methods

The project uses the Singular Value Decomposition (SVD)

[ A = U \Sigma V^T ]

and constructs approximations of the form

[ A_k = U(:,1:k)\Sigma(1:k,1:k)V(:,1:k)^T ]

for various values of k.

Implementation Steps

1. Compute SVD with MATLAB’s svd() function
2. Construct rank-1 and rank-2 matrix approximations
3. Compute RMSE between A and Aₖ
4. Apply SVD to a grayscale image for compression
5. Compare reconstructed images at different k values

Results Summary

• Low-rank approximations preserve the main structure of A while discarding smaller singular values
• Increasing rank decreases RMSE and improves image quality
• A compression ratio of CR ≈ 15 offers the best trade-off between file size and clarity

How to Run

1. Open MATLAB
2. Navigate to the project directory:

   cd path_to_folder/mat350_svd_compression
   
    3.	Run the script: mat350_svd_compression
   
   
    4.	The program will:
    •	Display RMSE calculations
    •	Show original and compressed images at multiple CR values
   

Ensure MAT 350 Project Two MATLAB Image.mat is in the same folder as the script.

Dependencies

•	MATLAB R2023a or later
•	Image Processing Toolbox (for imshow)

Educational Purpose

This repository demonstrates how linear algebra concepts, particularly SVD and orthogonal decomposition, apply to real-world problems such as data compression and dimensionality reduction. It serves as both an academic submission and a teaching reference for students studying applied linear algebra with MATLAB.