SVD Image Compression

Implement image compression using Truncated Singular Value Decomposition (SVD).

SVD decomposes an image matrix $A$ (m×n) into: $A = U \Sigma V^T$

Where:

• $U$ (m×m): Left singular vectors (row patterns)
• $\Sigma$ (m×n): Diagonal matrix of singular values
• $V^T$ (n×n): Right singular vectors (column patterns)

Truncated SVD keeps only the top $k$ singular values, achieving compression: $A_k = U_k \Sigma_k V_k^T$

The compression ratio is: $\frac{m \times n}{k(m + n + 1)}$

Quality measure: The retained energy/variance is: $\text{energy} = \frac{\sum_{i=1}^{k} \sigma_i^2}{\sum_{i=1}^{r} \sigma_i^2}$

where $r$ is the rank of the original matrix.