5.2. SINGULAR DECOMPOSITION OF THE MATRIX

It is known that any matrix size having rank can be represented as a weighted sum of matrices of unit size . Such a representation is called singular decomposition [6–8]. The following sections will discuss the use of this method for image processing.

When singular decomposition using a unitary matrix size and unitary matrix size such that

, (5.2.1)

where is the matrix

(5.2.2)

measures and its diagonal elements are called singular values ​​of the matrix . Since the matrices and unitary then and . therefore

. (5.2.3)

Columns of the unitary matrix are eigenvectors symmetric matrix i.e.

(5.2.4)

Where - nonzero eigenvalues ​​of the matrix . Similar to the matrix row are eigenvectors symmetric matrix i.e.

(5.2.5)

Where - the corresponding nonzero eigenvalues ​​of the matrix . It is easy to verify that equality (5.2.3) is consistent with (5.2.4) and (5.2.5).

Matrix decomposition given by relation (5.2.3) can be represented as a series

. (5.2.6)

Matrix products of eigenvectors form a set of matrices of unit rank, each of which is multiplied by a weighting factor, which is the corresponding singular value of the matrix . The consistency of decomposition (5.2.6) with the above relations can be shown by substituting it into equality (5.2.1). The result is

. (5.2.7)

Note that the product gives a column vector element of which is equal to one, and all the others are zeros. The row vector resulting from the calculation of the product , has a similar look. Therefore, in the right-hand side of (5.2.7), a diagonal matrix is ​​formed, whose elements are equal to the singular values ​​of the matrix .

The matrix expansion (5.2.3) and the equivalent representation in the form of a series (5.2.6) can be found for any matrix. Therefore, such a decomposition can be directly applied to the processing of discrete images presented in the form of matrices. In addition, these formulas can be used to decompose matrices of linear image transformations. The use of the singular value decomposition method for correcting and encoding images is discussed in subsequent chapters of the book.