Singular value decomposition: Difference between revisions

Revision as of 22:11, 1 June 2025

Revision as of 22:12, 1 June 2025

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<math>\mathbf{A}_k = \sum_{j=1}^k \sigma_j\mathbf{u}_j \mathbf{v}_j^T </math>

gives an image with the best 2-norm error out of all rank k approximations. Thus, the task becomes finding an approximation that balances retaining perceptual fidelity with the number of vectors required to reconstruct the image. Storing <math>A_k\mathbf{A}_k</math> requires only <math>k(n + m + 1)</math> numbers compared to <math>nm</math>. This same idea extends to color images by applying this operation to each channel or stacking the channels into one matrix.

Since the singular values of most natural images decay quickly, most of their variance is often captured by a small <math>k</math>. For a 1528 × 1225 greyscale image, we can achieve a relative error of <math>.7%</math> with as little as <math>k = 100</math>.<ref>{{Cite book |author1=Holmes |first=Mark |title=Introduction to Scientific Computing and Data Analysis, 2nd Ed |publisher=Springer |year=2023 |isbn=978-3-031-22429-4}}</ref> In practice, however, computing the SVD can be too computationally expensive and the resulting compression is typically less storage efficient than a specialized algorithm such as [[JPEG]].