10.9. SINGULAR CONVERSION

The singular transformation [34] is a two-dimensional unitary transformation based on the singular decomposition of matrices (see Chapter 5). A direct singular transformation is by definition equal to

, (10.9.1)

and the inverse transform

. (10.9.2)

String Conversion Matrix provides a diagonalization operation

, (10.9.3)

Where - diagonal matrix, elements which are eigenvalues ​​of the matrix . Similarly

. (10.9.4)

Substituting expression (10.9.2) into equalities (10.9.3) and (10.9.4), we get

, (10.9.5)

where elements of the diagonal matrix are numbers called the singular values ​​of the matrix and equal to the square root of the corresponding eigenvalues .

The image matrix can be written in a very compact form using the matrix product of vectors obtained by singular decomposition. According to equality (10.1.14b),

, (10.9.6)

Where and are vectors consisting of elements columns of matrices and .

Using a singular transformation, the image matrix containing items can be fully described values ​​representing coefficients . However, it should be noted that the specific values ​​of the elements of the transformation matrices in rows and columns in this case depend on the image elements.

In fig. 10.9.1 shows an example of a singular transformation of the image. Shown here are works and as well as the corresponding transformation matrices in rows and by columns . The singular values ​​of the image in question are shown in Fig. 10.9.2. In fig. 10.9.3 shows several matrix products .

Fig. 10.9.1. Singular transformation of the image "Portrait". All arrays from elements are derived from size arrays element by means of bilinear interpolation.

and - the initial image, a matrix ; b - the result of a singular transformation matrix ; in - matrix ; g - matrix ; d - the matrix consisting of the modules of the elements of the matrix ; е - the matrix consisting of the modules of the elements of the matrix .

Fig. 10.9.2. Singular values ​​of the image "Portrait".

Fig. 10.9.3. Basic images for the image "Portrait".

Figures a, b, c, d, d, e correspond , , , , , .