10.7. TILTED TRANSFORMATION

An orthogonal transform, called a slant transform, was proposed in [24–26]. This transformation has the following features: 1) among the basis vectors there is a vector with the same components (a constant basis vector); 2) the oblique basis vector monotonically decreases from the maximum to the minimum value by jumps of a constant value; 3) the transformation matrix has a sequential property; 4) there is a fast conversion algorithm; 5) provides a high degree of energy concentration of the image. With the length of the vector the oblique transform coincides with the Hadamard transform of the second order. In this way,

. (10.7.1)

The matrix of the oblique transformation of the fourth order is formed according to the following rule:

, (10.7.2а)

or

, (10.7.2b)

Where and - the actual coefficients that should be chosen so that the matrix was orthogonal, and the magnitude of the jumps with a change in the second oblique basis vector was constant. From the condition of constancy of the magnitude of the jump, one can find that . From the condition of orthogonality follows that . Thus, the quadratic oblique transformation matrix is

. (10.7.3)

It is easy to check that the matrix is orthonormal. In addition, it has a sequential property: the number of changes in the sign increases with the line number from 0 to 3.

Oblique transform matrix when has the appearance

. (10.7.4)

As in the construction of the matrix , coefficients and are chosen so that the oblique basis vector decreases in uniform jumps, all rows of the matrix are orthonormal vectors, and the matrix itself has a sequential property.

Summarizing the relation (10.7.4), one can obtain a recurrent formula connecting the oblique transformation matrices th and th order:

, (10.7.5)

Where - unit matrix th order. Permanent and can be found from recurrence relations [26]

, (10.7.6a)

, (10.7.6b)

(10.7.6b)

or by the formulas

, (10.7.7a)

. (10.7.7b)

In fig. 10.7.1 shows the graphs of the basic functions of the oblique transformation for . An example of the spectrum obtained by this transformation is shown in Fig. 10.7.2.

Fig. 10.7.1. Basic functions of the oblique transform .

Fig. 10.7.2. Slanting image conversion "Portrait".

a - the original image; b - the result of the transformation in a logarithmic scale along the axis of amplitudes: c - the result of the transformation with limited greatest harmonics.