词条 | Two-dimensional singular-value decomposition |
释义 |
Two-dimensional singular-value decomposition (2DSVD) computes the low-rank approximation of a set of matrices such as 2D images or weather maps in a manner almost identical to SVD (singular-value decomposition) which computes the low-rank approximation of a single matrix (or a set of 1D vectors). SVDLet matrix contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix and Gram matrix , and compute their eigenvectors and . Since , we have If we retain only principal eigenvectors in , this gives low-rank approximation of . 2DSVDHere we deal with a set of 2D matrices . Suppose they are centered . We construct row–row and column–column covariance matrices , in exactly the same manner as in SVD, and compute their eigenvectors and . We approximate as in identical fashion as in SVD. This gives a near optimal low-rank approximation of with the objective function Error bounds similar to Eckard–Young theorem also exist. 2DSVD is mostly used in image compression and representation. References
1 : Singular value decomposition |
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