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10.5. TRANSFORMATION ADAMARA

Lecture



The Hadamard transform [16, 17] is based on the Hadamard square matrix [18], the elements of which are equal to plus or minus one, and the rows and columns form orthogonal vectors. Normalized Hadamard Matrix   10.5.  TRANSFORMATION ADAMARA th order satisfies the relation

  10.5.  TRANSFORMATION ADAMARA . (10.5.1)

Among the Hadamard orthonormal matrices, the second-order matrix is ​​the smallest

  10.5.  TRANSFORMATION ADAMARA . (10.5.2)

It is known that if the Hadamard matrix is ​​of order   10.5.  TRANSFORMATION ADAMARA (Where   10.5.  TRANSFORMATION ADAMARA ) exists then   10.5.  TRANSFORMATION ADAMARA divided by 4 without remainder [19]. It has not yet been possible to determine whether Hadamard matrices exist for arbitrary   10.5.  TRANSFORMATION ADAMARA satisfying this condition, but for almost all admissible   10.5.  TRANSFORMATION ADAMARA reaching 200, found the rules for constructing the corresponding matrices. It is most easy to construct such matrices with   10.5.  TRANSFORMATION ADAMARA where   10.5.  TRANSFORMATION ADAMARA - whole. If a   10.5.  TRANSFORMATION ADAMARA - Hadamard matrix   10.5.  TRANSFORMATION ADAMARA th order, then the matrix

  10.5.  TRANSFORMATION ADAMARA (10.5.3)

is also a Hadamard matrix, but order   10.5.  TRANSFORMATION ADAMARA . In fig. 10.5.1 shows the Hadamard matrices of the fourth and eighth order, constructed using the relation (10.5.3).

  10.5.  TRANSFORMATION ADAMARA

Fig. 10.5.1. Fourth and eighth unordered Hadamard matrices.

Harmut [20] proposed a frequency interpretation of Hadamard matrices having a block structure (10.5.3). The number of sign changes along each row of the Hadamard matrix, divided by two, is called the sequence row. You can build a Hadamard matrix of order   10.5.  TRANSFORMATION ADAMARA in which the number of character changes in the lines takes values ​​from 0 to   10.5.  TRANSFORMATION ADAMARA . Unitary matrices with such characteristics are called matrices with the sequential property.

The rows of the Hadamard matrix, described by the relation (10.5.3), can be considered as a sequence of samples of rectangular periodic oscillations (signals) whose period is a multiple of   10.5.  TRANSFORMATION ADAMARA . Such continuous functions, called Walsh functions [21], are associated with impulse Rademacher functions [22]. Consequently, the Hadamard matrix describes the transformation associated with the expansion of functions in a family of rectangular basis functions, and not in sines and cosines characteristic of the Fourier transform.

For symmetric Hadamard matrices of order   10.5.  TRANSFORMATION ADAMARA the two-dimensional Hadamard transform can be represented as a series

  10.5.  TRANSFORMATION ADAMARA , (10.5.4)

Where

  10.5.  TRANSFORMATION ADAMARA . (10.5.5)

Variables   10.5.  TRANSFORMATION ADAMARA and   10.5.  TRANSFORMATION ADAMARA equal to the numbers in the binary representation of numbers   10.5.  TRANSFORMATION ADAMARA and   10.5.  TRANSFORMATION ADAMARA respectively. So for example, if   10.5.  TRANSFORMATION ADAMARA then   10.5.  TRANSFORMATION ADAMARA and   10.5.  TRANSFORMATION ADAMARA . If the Hadamard matrix is ​​ordered, that is, its rows are rearranged in ascending order of the sequence, then there is another form of the Hadamard transform. In this case

  10.5.  TRANSFORMATION ADAMARA , (10.5.6)

Where

  10.5.  TRANSFORMATION ADAMARA , (10.5.7)

and

  10.5.  TRANSFORMATION ADAMARA (10.5.8)

Graphs of basis functions of the Hadamard transform with an ordered matrix for   10.5.  TRANSFORMATION ADAMARA are presented on fig. 10.5.2. Basic images formed with the help of the matrix product of basic Hadamard transform vectors of size   10.5.  TRANSFORMATION ADAMARA shown in Fig. 10.5.3. In fig. 10.5.4 gives an example of the Hadamard transform (for an ordered Hadamard matrix).

  10.5.  TRANSFORMATION ADAMARA

Fig. 10.5.2. Hadamard transform basis functions with   10.5.  TRANSFORMATION ADAMARA .

  10.5.  TRANSFORMATION ADAMARA

Fig. 10.5.3. Hadamard transform baseline images   10.5.  TRANSFORMATION ADAMARA .

Black color corresponds to the value +1, white - value -1.

  10.5.  TRANSFORMATION ADAMARA

Fig. 10.5.4. Hadamard transform image "Portrait".

a - the original image; b - Hadamard spectrum in a logarithmic scale along the amplitude axis; c - spectrum with limited highest harmonics

created: 2016-09-09
updated: 2024-11-14
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Digital image processing

Terms: Digital image processing