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5.2. SINGULAR DECOMPOSITION OF THE MATRIX

Lecture



It is known that any matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX size   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX having rank   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX can be represented as a weighted sum of matrices of unit size   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX . 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   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX size   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX and unitary matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX size   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX such that

  5.2.  SINGULAR DECOMPOSITION OF THE MATRIX , (5.2.1)

where is the matrix

  5.2.  SINGULAR DECOMPOSITION OF THE MATRIX (5.2.2)

measures   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX and its diagonal elements   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX are called singular values ​​of the matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX . Since the matrices   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX and   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX unitary then   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX and   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX . therefore

  5.2.  SINGULAR DECOMPOSITION OF THE MATRIX . (5.2.3)

Columns of the unitary matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX are eigenvectors   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX symmetric matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX i.e.

  5.2.  SINGULAR DECOMPOSITION OF THE MATRIX (5.2.4)

Where   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX - nonzero eigenvalues ​​of the matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX . Similar to the matrix row   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX are eigenvectors   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX symmetric matrix   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX i.e.

  5.2.  SINGULAR DECOMPOSITION OF THE MATRIX (5.2.5)

Where   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX - the corresponding nonzero eigenvalues ​​of the matrix   5.2.  SINGULAR DECOMPOSITION 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   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX given by relation (5.2.3) can be represented as a series

  5.2.  SINGULAR DECOMPOSITION OF THE MATRIX . (5.2.6)

Matrix products of eigenvectors   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX 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   5.2.  SINGULAR DECOMPOSITION 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.  SINGULAR DECOMPOSITION OF THE MATRIX . (5.2.7)

Note that the product   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX gives a column vector   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX element of which is equal to one, and all the others are zeros. The row vector resulting from the calculation of the product   5.2.  SINGULAR DECOMPOSITION OF THE MATRIX , 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   5.2.  SINGULAR DECOMPOSITION 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.


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Digital image processing

Terms: Digital image processing