Math Notes

How SVD Enables Image Compression

2026-05-08

1. What is SVD?

Singular value decomposition factorizes a matrix into three matrices: A = UΣVᵀ. The singular values in Σ describe how much information is preserved along different directions.

2. Images as Matrices

A grayscale image can be represented as a matrix where each entry corresponds to pixel intensity. Compressing the image becomes a problem of approximating this matrix with fewer numbers.

3. Low-rank Approximation

By keeping only the largest singular values and discarding the smaller ones, we can approximate the original image using a lower-rank matrix. This reduces storage while preserving the main visual structure.

4. Why Compression Works

Many images contain redundant information. The largest singular values usually capture the most important patterns, while smaller singular values often represent fine details or noise.