Singular value decomposition: Difference between revisions

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[[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>\mathbf{A}_k = \Sigma_sum_{j=1}^k \sigma_j\mathbf{u}_j \mathbf{v}_j^T =\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.