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7.7. SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES

Lecture



The speed at which information is created by some source determines the minimum amount of bandwidth required to transmit a message with a given permissible level of distortion in a received message relative to the transmitted one [45, 46]. A number of attempts were made [2, 47–50] to adapt information theory to image transfer tasks in order to determine the limits of image coding systems. This section contains the main provisions of this theory, formulated in relation to images based on the review article by Davisson [50].

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES

Fig. 7.7.1. Block diagram of the information transmission system.

In fig. 7.7.1 shows a simplified block diagram of an image transmission system. The source creates a sequence of   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES image elements, each of which is quantized into   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES levels. This sequence forms a vector   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES size   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . In the coder to each of   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES possible combinations of brightness   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES where   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES The code combination is matched.

After decoding, the combination of brightness is restored   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . Characteristics of the image transmission system can be described using conditional probability.   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES of vector   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES , at the output, provided that the vector was encoded   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . If the encoder and decoder work without errors, then the input and output vectors of the image (in the absence of channel errors) will be the same.

Conditional probability   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES describes the operation of the image transmission system in the presence of distortion. Based on this conditional probability and the distribution of a priori probabilities, we find the unconditional probability distribution of the reconstructed vectors

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.1)

Channel capacity requirements are determined by the amount of mutual information, by definition equal to

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.2)

With error-free coding, this expression is simplified:

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES , (7.7.3)

that is, the amount of mutual information is equal to the entropy of the source. If in the process of encoding distortions are made, then the recovered sequence   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES contains incomplete status information   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES and bandwidth requirements will be reduced.

Suppose the function   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES is a measure of the distortion of the reproduced image. Then for a vector from   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES elements, the average value of distortion per element will be determined by the equality

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.4)

We define for this vector the speed of creating information per one element as

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES (7.7.5)

at   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . Basically   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES and there is a minimum channel capacity necessary to transmit information created by the source, when the distortions on average should not exceed a certain maximum value   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . The speed of creating information source   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES can be found by increasing the length of the vector to infinity:

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.6)

As a rule, to find the minimum amount of mutual information, provided that the average value of the distortion should not exceed a specified limit   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES difficult both analytical and numerical methods. Received several solutions for communication channels used in practice. One of them relates to a source with a Gaussian probability distribution when estimating the distortions by the mean square measure. This solution cannot be directly applied to the problem of coding the brightness of the elements, since the brightness is a non-negative value. In addition, the rms distortion measure may not be appropriate. However, the solution obtained for the Gaussian source and the rms distortion measure allows one to indicate the limits of the coding system for any sources with given second moments. In addition, this solution is directly transferred to the transformation coding problem. Therefore, below we will consider the properties of the rate of creation of information for the case of a Gaussian source and the mean square distortion measure.

Consider a vector   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES educated   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES independent Gaussian random variables with zero means and known variance   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . The rms error is determined by the formula

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.7)

It was found [45] that the speed of creating information

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES

Thus, the rate at which information is created is equal to half the logarithm of the signal power ratio to the distortion power, if this ratio exceeds unity, and zero otherwise. If the elements of the sequence created by the Gaussian source are correlated and the covariance matrix   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES known, the speed of creating information is equal to [50]

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES , (7.7.9)

Where   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES -   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES -e eigenvalue of the matrix   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES , but   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES chosen so that

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.10)

In image processing, a special case of a two-dimensional separable Markov source is of interest, when all elements have the same dispersions equal to   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES and the correlation coefficients along the rows and columns are respectively   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES and   7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . If we assume that the degree of distortion is small, then the speed of creating information for a homogeneous case is [50]

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES , (7.7.11)

and in the two-dimensional case

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES . (7.7.12)

  7.7.  SPEED OF CREATING INFORMATION BY THE SOURCE OF IMAGES

Fig. 7.7.2. The dependence of the rate of creation of information on the magnitude of distortion in one- and two-dimensional coding of images - realizations of the Markov field.

In fig. 7.7.2 shows the graphs of the rate of creation of information on the magnitude of distortion for different values ​​of the correlation coefficients. In ch. 24 a comparison was made of the characteristics of some image coding systems with limiting characteristics.

created: 2016-09-08
updated: 2021-03-13
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Digital image processing

Terms: Digital image processing