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10.7. TILTED TRANSFORMATION

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   10.7.  TILTED TRANSFORMATION the oblique transform coincides with the Hadamard transform of the second order. In this way,

  10.7.  TILTED TRANSFORMATION . (10.7.1)

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

  10.7.  TILTED TRANSFORMATION , (10.7.2а)

or

  10.7.  TILTED TRANSFORMATION , (10.7.2b)

Where   10.7.  TILTED TRANSFORMATION and   10.7.  TILTED TRANSFORMATION - the actual coefficients that should be chosen so that the matrix   10.7.  TILTED TRANSFORMATION 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   10.7.  TILTED TRANSFORMATION . From the condition of orthogonality   10.7.  TILTED TRANSFORMATION follows that   10.7.  TILTED TRANSFORMATION . Thus, the quadratic oblique transformation matrix is

  10.7.  TILTED TRANSFORMATION . (10.7.3)

It is easy to check that the matrix   10.7.  TILTED TRANSFORMATION 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   10.7.  TILTED TRANSFORMATION has the appearance

  10.7.  TILTED TRANSFORMATION . (10.7.4)

As in the construction of the matrix   10.7.  TILTED TRANSFORMATION , coefficients   10.7.  TILTED TRANSFORMATION and   10.7.  TILTED TRANSFORMATION 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   10.7.  TILTED TRANSFORMATION th and   10.7.  TILTED TRANSFORMATION th order:

  10.7.  TILTED TRANSFORMATION , (10.7.5)

Where   10.7.  TILTED TRANSFORMATION - unit matrix   10.7.  TILTED TRANSFORMATION th order. Permanent   10.7.  TILTED TRANSFORMATION and   10.7.  TILTED TRANSFORMATION can be found from recurrence relations [26]

  10.7.  TILTED TRANSFORMATION , (10.7.6a)

  10.7.  TILTED TRANSFORMATION , (10.7.6b)

  10.7.  TILTED TRANSFORMATION (10.7.6b)

or by the formulas

  10.7.  TILTED TRANSFORMATION , (10.7.7a)

  10.7.  TILTED TRANSFORMATION . (10.7.7b)

In fig. 10.7.1 shows the graphs of the basic functions of the oblique transformation for   10.7.  TILTED TRANSFORMATION . An example of the spectrum obtained by this transformation is shown in Fig. 10.7.2.

  10.7.  TILTED TRANSFORMATION

Fig. 10.7.1. Basic functions of the oblique transform   10.7.  TILTED TRANSFORMATION .

  10.7.  TILTED TRANSFORMATION

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