Lecture
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.
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Digital image processing
Terms: Digital image processing