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5.4. STATISTICAL DESCRIPTION OF DISCRETE IMAGES

Lecture



Statistical methods for the description of continuous images given in Ch. 1, can be directly applied to the description of discrete images. In this section, expressions for moments of discrete images are obtained. The joint probability density models are given in the next section.

The average value of the matrix describing a discrete image is a matrix

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.1)

If this matrix is ​​transformed by columns into a vector, then the average value of this vector is

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.2)

Correlation of two image elements with coordinates   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES and   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES defined as

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.3)

The covariance of two elements of the image is

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.4)

Finally, the dispersion of the image element is

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.5)

If the image matrix is ​​converted to a vector   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES , the correlation matrix of this vector can be expressed through the correlations of the elements of the matrix   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES :

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES , (5.4.6a)

or

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.6b)

Expression

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES (5.4.7)

is a correlation matrix   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES th and   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES th matrix columns   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES and measures   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . Consequently,   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES can be represented as a block matrix

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES (5.4.8)

Covariance matrix of a vector   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES can be obtained on the basis of its correlation matrix and the vector of average values ​​using the ratio

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.9)

Dispersion matrix   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES array of numbers   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES By definition, it is a matrix whose elements are equal to the variances of the corresponding elements of the array. Matrix elements   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES You can directly select from the matrix blocks   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES :

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . (5.4.10)

If a discrete image is represented by an array, stationary in a broad sense, its correlation function can be written as

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES , (5.4.11)

Where   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES and   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . Accordingly, the blocks of the covariance matrix (5.4.9) will be connected by the relations

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES ,   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES , (5.4.12а)

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES ,   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES , (5.4.12b)

Where   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . Thus, for a stationary in a broad sense array

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES (5.4.13)

The matrix (5.4.13) is block Toeplitz [11]. Finally, if the image correlation function can be written as a product of the correlation functions of rows and columns, then the covariance matrix of the vector   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES representing the image can be written as a direct product of covariance matrices for rows and columns:

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES (5.4.14)

Where   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES - covariance matrix of matrix columns   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES measuring   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES , but   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES - covariance matrix of matrix rows   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES with sizes   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES .

Consider the case when the covariance matrix of rows of the matrix   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES has the following form:

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES (5.4.15)

Where   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES - dispersion of image elements. This covariance matrix is ​​an analogue of the continuous autocovariance function of the form   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES - describes the Markov process. In fig. 5.4.1 shows the values ​​of the correlation coefficients of the elements of a typical image line obtained by Davisson [12]. Experimental points are well approximated by the covariance function of a Markov process with a parameter   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . Similarly, the values ​​of the correlation coefficients in the direction perpendicular to the rows agree well with the Markov covariance function with   5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES . If the covariance function can be represented as (5.4.14), then the diagonal correlation coefficients should be equal to the product of the corresponding correlation coefficients along the lines of the image and in the direction perpendicular to them. In this example, it turned out that such an approximation is sufficiently accurate in the range from zero to five discretization steps.

By analogy with the continuous energy spectrum (1.8.11), one can determine the discrete spectral density of a discrete stationary two-dimensional random field representing an image as the result of a two-dimensional discrete Fourier transform of the autocorrelation function of this field. Then, by virtue of equality (5.4.11), we will have

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES (5.4.16)

  5.4.  STATISTICAL DESCRIPTION OF DISCRETE IMAGES

Fig. 5.4.1. An example of correlation dependencies between adjacent image elements.

In fig. 5.4.2 shows the energy spectra of Markov processes.


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Digital image processing

Terms: Digital image processing