Two-dimensional singular-value decomposition

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).
SVD.
Let matrix formula_1 contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix formula_2 and Gram matrix formula_3
and compute their eigenvectors formula_6 and formula_7. Since formula_8 and formula_9 we have
If we retain only formula_11 principal eigenvectors in formula_12, this gives low-rank approximation of formula_13.
2DSVD.
Here we deal with a set of 2D matrices formula_14. Suppose they are centered formula_15. We construct row–row and column–column covariance matrices
in exactly the same manner as in SVD, and compute their eigenvectors formula_18 and formula_19. We approximate formula_20 as
in identical fashion as in SVD. This gives a near optimal low-rank approximation of formula_14 with the objective function
Error bounds similar to Eckard–Young theorem also exist.
2DSVD is mostly used in image compression and representation.