Methods for effective transformation of information based on the modified Bailey-Borwein-Plaff formula

Description

The following method allows for efficient conversion of digital sequences of any dimension, which in turn is applicable in the field of digital information processing. The following are two computational methods:

• Conversion of volumetric sequence of numbers with the expression of value in a compact form,
• Inverse transform of a compact sequence into a bulk one.

The improvement of this method and its implementation in the code, together with the effective implementation of the algorithm, allows you to apply this method in practice, being a competitor to the existing methods of processing and storing information that exceeds the effectiveness of existing competitors several times.

Existing problem

Any digital information is a set of numbers, i.e. a numerical sequence that possesses the properties of entropy and each symbol of this sequence carries a transition point to other states. These points are limited and are determined by the Shannon limit, which assumes the final states for information symbols. Reaching the Shannon limit, the number of combinations of effective sequence reductions is sharply reduced, turning the flow of information into "noise".



Existing mathematical methods are based on using a dictionary to search for variants of sequence reduction: the implementation of trained neural networks that build their “dictionaries” (knowledge bases) during training, or arithmetic coding is used, where each character is the carrier of the “frequency” of appearance. Vocabulary and neural network methods of working with numerical sequences close to the Shannon limit show their extreme inefficiency.

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According to the theory of Shannon, a random independent equiprobable sequence cannot be compressed without loss. The proposed method does not conflict with the existing theory, since is a method of expressing complex, volume sequences without the use of frequencies and dictionaries.



In the case of the method described below, the digital sequence is expressed as a multidimensional representation function:







A separate element of the sequence is the "account". Therefore, χ (n1, n2) will be the “sequence count” χ in the region (n1, n2). The value of "accounts" can be real and complex.

Digital Conversion

Given that the sequences are presented in the form of multidimensional representation functions, various methods of mathematical transformations are applicable to them.



Next will be considered the work with a digital image on the example of the implementation of the model for working with images.

Matrix expression from the view model

During the transformation of the matrix of the sequence ∫ (z, v) of size Z × Z, a matrix of the transformed sequence of smaller size is formed, the elements of which are equal







where φ (zu, zv) (z, v) is the transformation itself, and zu, zv are variable spaces for the transformation.

Recovery of the original matrix from the view model

The original matrix f (z, v) can be obtained using the inverse transform







where φ ((z, v)) ^ (- 1) (zu, zv) is the inverse transform.



The conversion is complete and complete if the following conditions are met for the original matrix:







Where







There is a multi-dimensional version of the Kronecker symbol.



The sequence for work is formed from the original value, is a set of transformed values ​​formed by the “mixing formula” of the search for integer values. The basis is a minimal sequence of 200 characters in length, where this sequence is divided into 10 equal parts of 20 characters each, which is a series for convergence testing.



Each part of the sequence forms a matrix with a corresponding dimension.



Each matrix forms a group of 200-t values, which has a “common property”; it is set when the entire group is transformed. The conversion is performed according to the following algorithm:

1. The sum of the entire series of the original value is calculated:
   
   
   
   
2. Each element of the matrix is ​​multiplied by a constant a, where a = 40829.

If the numerical sequence does not converge with the primary one, then it is necessary to change the “seed” for the entire sequence in order to avoid errors; The "seed" is set for the entire series, in paragraph 1, the method for calculating the series changes to the following:







The computed “seed” restores the original sequence using the inverse transform formula, which is indicated in several paragraphs above.



If f (z, v) is a Z × Z matrix, and R (zu, zv) is its representation coefficients, then the equality is true:







The above expression is Parseval equality for representing multidimensional functions. The use of this statement allows several times to reduce the computational cost when converting an image model.



The coefficient matrix itself is calculated using a one-dimensional transform (integral case):







The resulting expression is a one-dimensional "account" of the expression.



In case of an error and violation of Parseval's equality, it is necessary to introduce corrective edits in the matrix column with an error







Articles that had a significant impact on the described studies:

• A Compendium of BBP-Type Formulas for
  
  Mathematical Constants
• Ten Problems in Experimental Mathematics