Editing Singular value decomposition (section)

=== Image compression ===
[[File:Svd compression.jpg|thumb|Singular-value decomposition (SVD) image compression of a 1996 Chevrolet Corvette photograph. The original RGB image (upper-left) is compared with rank 1, 10, and 100 reconstructions.|292x292px]]One practical consequence of the low-rank approximation given by SVD is that a greyscale image represented as an <math>m \times n</math> matrix <math>A</math>, can be efficiently represented by keeping the first <math>k</math> singular values and corresponding vectors. The truncated decomposition

<math>A_k = \mathbf{U}_k \mathbf{\Sigma}_k \mathbf{V}^T_k</math>

gives an image which minimizes the [[Frobenius norm|Frobenius error]] compared to the original image. Thus, the task becomes finding a close approximation <math>A_k</math> that balances retaining perceptual fidelity with the number of vectors required to reconstruct the image. Storing <math>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]].