RU2764377C1 - METHOD FOR INTEGER EXPANSION OF NEARLY EVERYWHERE MEASURABLE RANDOM FINITE FUNCTIONS AND COMPRESSION OF IMAGES IN SPACES Lp(0,1], p GREATER THAN 0 AND LESS THAN 1, AND S(0,1] BY SYSTEMS OF COMPRESSIONS AND SHIFTS OF A SINGLE FUNCTION BY FOURIER SERIES - Google Patents

Abstract

FIELD: computing technology.

SUBSTANCE: invention relates to computing technology and can be used in television and radio transmitting, television and radio receiving apparatuses, measuring equipment, phase-metering systems, in the field of information technology. The method for integer expansion of nearly everywhere measurable random finite functions and compression of images in spaces

and

by systems of compressions and shifts of a single function by Fourier series consists in a recording of a digital signal

being supplied in the form of a function from the output of the object under study as an element of space

to the input of a personal computer. The sequence of values of said function is recorded in the electronic computing unit; the elements of spaces

are approximated by systems of compressions and shifts of a single function in Fourier series with integer coefficients. A function

is introduced as a generating function for the encoding system in the form of a table or formula. The elements of the system whereby the signal will be encoded are calculated. A cycle for calculating Fourier-type coefficients is formed; the signal is encoded; the signal is restored by decoding.

EFFECT: reduction in errors during image compression.

1 cl, 3 dwg

Description

The invention relates to computer technology and can be used in television and radio transmitting, television and radio receivers, measuring equipment, phase-metric systems, when compiling computer programs that require expansion (including integer expansion) of rapidly growing non-summable functions (nuclear energy) in a series on functional systems, as well as in various areas of information technology.

There is a method of image vector compression (RF Patent No. (19)2 646 348, Published: 03/02/2018, Bull. No. 7), which includes the creation of a reference vector of target image characteristics based on the image vector, and the reference vector includes information about the target image characteristics from image vector; compressing the image vector with an auto-encoder to obtain a compressed image vector based on the image vector; decompressing the compressed image vector with an autoencoder to obtain a lossy image vector based on the compressed image vector; generating a lossy image target feature vector based on the lossy image vector; comparing the target image feature reference vector with the lossy image feature vector by determining a discrepancy parameter, and using the discrepancy parameter to train the autoencoder such that the loss of information in the lossy image vector associated with the target features is reduced due to the increased loss of information associated with additional image characteristics.

However, there is no information in this patent on how a target image performance reference vector is created based on the image vector. In modern technologies, this vector is usually created using the Haar system, trigonometric system and wavelet analysis [1, 2]. But with the help of the autoencoder, a lossy image vector is obtained based on the image vector. As a rule, this is obtained by removing small coefficients in the expansion in the Haar system, trigonometric system and wavelet systems, and smaller losses are achieved by removing smaller (in absolute value) coefficients in the expansion in these systems. We have a different principle of image compression. We just have an approximation (creation of a reference vector of target image characteristics) based on the image vector, moreover, with integer components, while the initial components of the approximate vector remain and the subsequent ones are added if the initial approximation does not suit us. Perhaps this will be the optimal approximation based on the predetermined number of vector components.

Also known is a method for compressing digital information using a reference electrical signal (RF patent No. 2 482 604, published: 05/20/2013, bull. No. 14) using a reference electrical signal, which uses a pre-selected reference electrical compression signal S(N) and the reference electrical signal of the recovery keys K(N), which are changed using an arithmetic logic unit (ALU) by electrical signals that correspond to information elements, the digital codes of which are compressed and, as a result, a modified reference electrical compression signal S(n) and a modified reference electrical the signal of the recovery keys K(n), with the help of which the original electrical signals are subsequently restored, which correspond to the information elements, the digital codes of which were compressed, while in the process of compression, the digital bits of the reference electrical signal reflect any changes in the reference signal and, therefore, contain the complete inform information about the electrical signals that were received for compression.

However, here, the compression of images is obtained due to the fact that when decomposing the source signal according to the above systems, many coefficients are simply equal to zero.

по системам сжатий и сдвигов одной функции рядами типа Фурье с целыми коэффициентами (патент РФ № 2 681 367, опубликовано: 06.03.2019, бюл. № 7). Технический результат изобретения заключается в том, что сжатие образов можно осуществить за счет того, что при кодировании получаются целочисленные коэффициенты и много коэффициентов равно нулю. В то время как в [2] рассматриваются другие системы функций, для которых целочисленные коэффициенты получить невозможно и ошибки при вычислении коэффициентов искажают исходный сигнал. При этом получается сжатие образов без отбрасывания малых коэффициентов и коэффициенты или 0 или больше или равно 1 по абсолютной величине.The author has previously obtained a method for compressing digital information by compressing one-dimensional images by approximating the elements of spaces

on systems of contractions and shifts of one function by Fourier-type series with integer coefficients (RF patent No. 2 681 367, published: 06.03.2019, bull. No. 7). The technical result of the invention lies in the fact that image compression can be carried out due to the fact that when encoding integer coefficients are obtained and many coefficients are equal to zero. While in [2] other systems of functions are considered, for which it is impossible to obtain integer coefficients and errors in calculating the coefficients distort the original signal. This results in image compression without discarding small coefficients and coefficients or 0 or greater than or equal to 1 in absolute value.

In the claimed method, a simple approximation is implemented (creating a reference vector of target image characteristics) based on the image vector, moreover, with integer components, while the initial components of the approximate vector remain and subsequent ones are added if the initial approximation does not suit us.

In the following, we will use the terms from the examples above.

The technical problem is the need to create algorithms in which, when encoding signals (including those consisting of non-summable functions), a vector with integer components is obtained, which was not the case before. Note that earlier in the mathematical literature there were no results on the expansion of non-summable functions into Fourier-type series, moreover, there are no specific algorithms for expanding non-summable functions into series in terms of functional systems.

The technical result of the present invention lies in the fact that the compression of images (including those consisting of non-summable functions) can be carried out due to the fact that when encoding integer coefficients are obtained and many coefficients are equal to zero. While in [2] other systems of functions are considered, for which it is impossible to obtain integer coefficients and errors in calculating the coefficients distort the original signal. In this case, we obtain image compression without discarding small coefficients. We have coefficients or 0 or greater than or equal to 1 in absolute value.

In the claimed method, a simple approximation is implemented (creating a reference vector of target image characteristics) based on the image vector, moreover, with integer components, while the initial components of the approximate vector remain and subsequent ones are added if the initial approximation does not suit us. New systems of functions [3-7] are used and the coefficients of the encoded signal are calculated in a different way. Thus, an optimal approximation is provided based on a predetermined number of vector components. Note that in [7] the coefficients are not integer and they are obtained differently. In addition, in the inventive method, during intermediate calculations, inaccuracy of calculations is allowed, which is corrected in subsequent calculations.

(всюду ниже коротко будем обозначать через

) и

([13] стр. 66) по системам из сжатий и сдвигов одной функции рядами типа Фурье с целыми коэффициентами реализуют следующим образом.The method is illustrated by the drawing: Fig.1 is an example of a practical implementation of the compression of a digital signal obtained from a function, where the step function is the obtained approximation of the original signal. Signal compression method by approximation of space elements

(everywhere below we will briefly denote by

) and

([13] p. 66) in systems of contractions and shifts of one function by Fourier-type series with integer coefficients is implemented as follows.

в виде функции с выхода исследуемого объекта поступает как элемент пространства

на вход персонального компьютера. Как правило, это аналитически (в виде формулы) заданная функция, вектор или таблица, где указано, на каком множестве какие значения эта функция принимает. Как правило, эти множества задаются в виде интервалов. Затем в электронно-вычислительный блок записывают последовательность значений этой функции.digital signal

in the form of a function from the output of the object under study comes as an element of space

to the input of a personal computer. As a rule, this is an analytically (in the form of a formula) given function, a vector or a table, which indicates on which set what values this function takes. As a rule, these sets are specified as intervals. Then, a sequence of values of this function is written into the electronic computing unit.

по системам из сжатий и сдвигов одной функции рядами типа Фурье с целыми коэффициентами. Для этого принимают, например, допущения: по оси ox время, а по оси oy значение сигнала в данный момент времени, точность приближения

Вводят функцию

(как образующую функцию для системы кодирования) в виде таблицы или формулы. При этом

таким образом обеспечивают большой выбор систем, по которым осуществляют обработку исходного сигнала

. Пусть

([13] стр. 66)), и N достаточно большое целое, положительное, фиксированное число. Заметим, что рассуждения ниже верны и для пространства

произвольных измеримых почти всюду конечных функций заданных на множестве

Представим функцию

в виде

, гдеThen, in the electronic computing unit, the elements of the spaces are approximated

over systems of contractions and shifts of one function by Fourier-type series with integer coefficients. To do this, take, for example, assumptions: on the ox axis, the time, and on the y axis, the value of the signal at a given time, the accuracy of the approximation

Enter function

(as a generating function for the coding system) in the form of a table or formula. Wherein

thus provide a large selection of systems by which the processing of the original signal is carried out

. Let

([13] p. 66)), and N is a sufficiently large integer, positive, fixed number. Note that the reasoning below is also true for the space

arbitrary measurable almost everywhere finite functions given on the set

Imagine the function

as

, where

Заметим, что

сходимость ряда понимается в смысле сходимости по квазинорме пространства

Очевидно, что существует последовательность целых чисел

такая, что wherein

notice, that

the convergence of a series is understood in the sense of convergence in the quasi-norm of the space

Obviously there is a sequence of integers

such that

– элементы системы (где

и

при

а

подбирается специальным образом) по которой будет кодироваться сигнал, где

Then calculate

– elements of the system (where

and

at

a

selected in a special way) according to which the signal will be encoded, where

которые вычисляют исходя из

как коэффициенты Фурье от

с умножением на

где

– характеристическая функция интервала

и умножением на 2^k, а затем берут целую часть полученного числа, если это число больше или равно нуля, и целую часть плюс один, если это число меньше нуля.Then a loop is formed to calculate the Fourier-type coefficients

which are calculated from

as the Fourier coefficients of

multiplied by

where

is the characteristic function of the interval

and multiplying by 2 ^k , and then take the integer part of the resulting number if this number is greater than or equal to zero, and the integer part plus one if this number is less than zero.

notice, that

зависит от индекса

Для

индекс

Далее, для каждого последующего

индекс

остается без изменений, если

в противном случае индекс

увеличивается на 1 и, соответственно, к функции

добавляется следующий блок функций

При этом

при

и фиксированном

.In this case, the index

depends on index

For

index

Further, for each subsequent

index

remains unchanged if

otherwise index

increases by 1 and, accordingly, to the function

the following block of functions is added

Wherein

at

and fixed

.

Если точность приближения достигнута, то вычисления прекращают, в противном случае формируют новый цикл с

Выводят для запоминания коэффициенты

где

– целые числа, то есть кодируют сигнал. Так как коэффициенты целые и многие равны 0 (нулевые коэффициенты игнорируем), получают сжатие образа (сигнала). Затем восстанавливают (то есть декодируют) путем составления суммы

где

мы получили, а

заранее известны, так как мы заранее установили, что кодирование и раскодирование происходит с участием системы

Таким образом, получаем удовлетворяющее поставленным условиям приближениеCheck the accuracy of the approximation

If the approximation accuracy is reached, then the calculations are stopped, otherwise a new cycle is formed with

Display coefficients for memorization

where

- integers, that is, they encode the signal. Since the coefficients are integers and many are equal to 0 (zero coefficients are ignored), the image (signal) is compressed. Then recover (i.e. decode) by adding up the sum

where

we got and

are known in advance, since we have established in advance that encoding and decoding takes place with the participation of the system

Thus, we obtain an approximation that satisfies the stated conditions

имеем, что

Note that for some

we have that

This algorithm is also valid for

Example 1

To confirm the practical implementation, consider an example of compression of the output signal of a harmonic frequency doubler in the form of a hyperbola.

Очевидно, что это несуммируемая функция. В работе [8] показано каким точно пространствам принадлежит данная функция

Мы рассмотрим эту функцию в пространстве

Пусть

иConsider the function

Obviously, this is a non-summable function. In [8], it is shown exactly which spaces the given function belongs to

We will consider this function in the space

Let

and

Let

Let's construct the sum

с указанием номеров по

и индексов

нулевые коэффициенты мы игнорируем. В данном случае

В качестве

мы взялиTable 1 gives the values of the coefficients

with numbers for

and indices

we ignore zero coefficients. In this case

As

We took

и

оставалось равным 0 до

включительно, начиная с

мы добавили

и вычисления продолжались до

включительно до заданной точности.where

and

remained at 0 until

inclusive, starting from

we added

and the calculations continued until

up to the specified accuracy.

Table 1.

l i j

l i j

l i j

l i j

one 0 0 2 25 5 eight -9 49 6 sixteen -eight 73 6 40 -one 2 one one 3 26 5 9 7 50 6 17 eight 74 6 41 2 3 one 2 -2 27 5 10 -5 51 6 eighteen -6 75 6 43 2 4 2 one 5 28 5 eleven 5 52 6 nineteen 5 76 6 44 -one 5 2 2 -3 29 5 12 -4 53 6 twenty -6 77 6 45 one 6 2 3 2 thirty 5 thirteen 3 54 6 21 4 78 6 46 -one 7 3 one 2 31 5 14 -3 55 6 22 -5 79 6 47 one eight 3 3 3 32 5 15 2 56 6 23 5 80 6 49 2 9 3 4 -4 33 5 sixteen -3 57 6 24 -3 81 6 52 -one 10 3 5 2 34 5 17 one 58 6 25 4 82 6 53 one eleven 3 6 -one 35 5 eighteen -2 59 6 26 -3 83 6 56 -one 12 3 7 one 36 5 nineteen 3 60 6 27 3 84 6 57 2 thirteen 4 3 4 37 5 21 one 61 6 28 -2 85 6 58 one 14 4 4 -4 38 5 22 -one 62 6 29 3 86 6 59 one 15 4 5 6 39 5 23 one 63 6 thirty -2 87 6 61 one sixteen 4 6 -4 40 5 24 -one 64 6 31 3 88 6 63 one 17 4 7 3 41 5 25 one 65 6 32 -one eighteen 4 eight -2 42 5 27 2 66 6 33 3 nineteen 4 9 2 43 5 28 one 67 6 34 -one twenty 4 10 -2 44 5 29 one 68 6 35 2 21 4 eleven 2 45 5 31 one 69 6 36 -2 22 4 thirteen 2 46 6 thirteen 10 70 6 37 one 23 4 15 one 47 6 14 -eight 71 6 38 -2 24 5 7 10 48 6 15 10 72 6 39 2

The approximation error satisfies the given approximation accuracy in the space metric

In the issues of image compression [1, 2], interest arose in systems of the type

- произвольная суммируемая функция, определенная на R.where

is an arbitrary summable function defined on R.

рассмотрено впервые.Systems of contractions and shifts of one function are considered, in particular, in [1-7]. But the expansion with integer coefficients in systems (1.1) has not been considered anywhere by other authors. And the integer expansion in spaces

considered for the first time.

Thus, in the claimed invention, a simple approximation (creation of a reference vector of target image characteristics) is carried out based on the image vector, moreover, with integer components, while the initial components of the approximate vector remain and the subsequent ones are added if the initial approximation does not suit us. New systems of functions [3-7] are used and the coefficients of the encoded signal are calculated differently. This method is an optimal approximation based on a predetermined number of vector components. In addition, with us, during intermediate calculations, a possible inaccuracy of calculations is allowed, which is corrected in subsequent calculations. Note that both here and in [1, 2, 7] systems of functions obtained from contractions and shifts of one function are used. What forms the basis of modern technologies in this area. The systems of functions considered by us are not orthonormal.

Systems of contractions and shifts of one function are considered, in particular, in [1-7]. But the expansion with integer coefficients in systems (1.1) has not been considered anywhere, except for the author's works [9-12].

а на фиг. 2 дан эскиз графика функции

In FIG. 1 given a sketch of the graph of the function

and in fig. 2 given a sketch of the graph of the function

Example 2

Очевидно, что это несуммируемая функция и эта функция не принадлежит ни одному из пространств

Мы рассмотрим эту функцию в пространстве

[13]. Это метрическое полное, сепарабельное пространство с квазинормой

Пусть

иTo confirm the practical implementation, consider an example of compression of the output signal of a harmonic frequency doubler in the form of an exponentially growing function. Consider the function

Obviously, this is a non-summable function and this function does not belong to any of the spaces

We will consider this function in the space

[thirteen]. This is a metric complete, separable space with the quasi-norm

Let

and

Применяя алгоритм примера 1, получим суммуLet

Applying the algorithm of example 1, we get the sum

При этомIn FIG. 3 shows a sketch of the graph of the function

Wherein

Literature

[1] Jia R.Q., and Micchelli C. Using the refinement equation for the construction of pre-wavelets 2: Powers of two, in “Curves and Surfaces (P.J. Laurent, A. LeMehaute, and L.L.Schumaker, Eds.). Academic Press. new york. 1991. P. 209-246.

[2] Daubechies I. Ten lectures on wavelets. SIAM. Philadelphia. 1992.

// Journal of Approximation Theory. 1998. V. 94. P. 42-53.[3] Filippov VI On the completeness and other properties of some function system in

// Journal of Approximation Theory. 1998. V. 94. P. 42-53.

// Изв. РАН, сер. Матем. 2012. Т. 76. N 6. С.193-206.[4] Filippov V.I. Representation systems obtained from contractions and shifts of a single function in multidimensional spaces

// Izv. RAS, ser. Mat. 2012. V. 76. N 6. P. 193-206.

// Известия Вузов. Математика, 2018, 62:1, 87-92.[5] Filippov V.I. On generalizations of the Haar system and other systems of functions in spaces

// Izvestiya Universities. Mathematics, 2018, 62:1, 87-92.

by series of translates and dilates of one function// Journal of Approximation Theory. 1995. V.82. №1. P. 15-29.[6] Fillipov VI, and Oswald P. Representation in

by series of translates and dilates of one function// Journal of Approximation Theory. 1995. V.82. No. 1. P. 15-29.

[7] Kudryavtsev A. Yu. On the rate of convergence of orthorecursive expansionsovernon-orthogonal wavelets//Izvestiya: Mathematics, 2012, 76(4), 688-701.

[8] Filippov V.I. Multimodular spaces and their properties // Izvestiya Vuzov. Mathematics. 2017. 61:12. pp. 57-65.

[9] Filippov V.I. A method for compressing one-dimensional images by approximating the elements of Lp spaces by systems of compressions and shifts of one function by Fourier-type series with integer coefficients// (RF patent No. 2 681 367, published: 06.03.2019 Bull. No. 7). http://www1.fips.ru/wps/PA_FipsPub/res/Doc/IZPM/RUNWC1/000/000/002/681/367/%D0%98%D0%97-02681367-00001/document.pdf

[10] Filippov V.I. Fourier-type series with integer coefficients in systems of contractions and shifts of one function in the spaces Lp, p≥1// Izv. universities. Mat., 2019, No. 6, pp. 58–64.

[11] Filippov V.I. Integer expansion in systems of contractions and shifts of one function // Izv. RAN. Ser. Mat., 84:4 (2020), pp. 187–197.

[12] Filippov V.I. Series with Integer Coefficients by Systems of Contractions and Shifts of One Function// Lobachevskii Journal of Mathematics, 2020, Vol. 41, no. 11, pp. 2143–2148.

[13] Kantorovich L.V., Akilov G.P. Functional analysis. Ed. "The science". Moscow. 1972.

Claims (7)

A method for integer expansion of arbitrary measurable almost everywhere finite functions and compression of images in the spaces L _p (0,1], 0<p<1, and S(0,1] in systems of contractions and shifts of one function by Fourier-type series, including writing a digital signal ƒ in the form of a function from the output of the object under study comes as an element of the space L _p (S[0,1]) to the input of a personal computer; then, a sequence of values of this function is recorded in the electronic computing unit; after which, the elements are approximated in the electronic computing unit spaces L _p (S[0,1]) by systems of contractions and shifts of one function by Fourier-type series with integer coefficients, under assumptions along the x-axis - time, and along the y-axis - the value of the signal at a given moment of time, the approximation accuracy ε>0; then the function ψ is introduced as a generating function for the coding system in the form of a table or formula, while ψ∈L ₂ [0,1],

ψ(t)=0, tΔT, T=[0,1], thus provide a large selection of systems, which carry out the processing of the original signal ƒ; under the conditions ƒ∈L _p [0,l], 0<р<1, (ƒ(t)∈S[0,1],

and N is a sufficiently large integer, positive, fixed number, represent the function ƒ in the form

, where

, k=0, 1, 2, …, while

and

defined as follows:

wherein

k=0, 1, 2, …; then calculate

- elements of the system by which the signal will be encoded, where i≥0, k=0, 1, ..., 2ⁱ-one; then a loop is formed to calculate the Fourier-type coefficients

which are calculated from g_k+1,r, as the Fourier coefficients of g_k+1,r With multiplication by

, where

- characteristic function of the interval

and multiplying by 2k, and then taking the integer part of the resulting number if this number is greater than or equal to zero, and the integer part plus one if this number is less than zero; provided that

where index r depends on index k, for k=0 index r=0, then for each subsequent k+1, k=1, 2, …, index r remains unchanged if

wherein

at

and fixed r, check the accuracy of the approximation

while the coefficients

integers and many are 0; further, if the accuracy of the approximation is reached, then the calculations are stopped, otherwise a new cycle is formed with g _k,r , the coefficients are deduced for memorization

where Z are integers, that is, they encode the signal; then recover the signal by decoding by summing up

where

we have obtained, a ψ _i,j (t) are known in advance, since we have established in advance that encoding and decoding occur with the participation of the system [ψ _i,j (t)}, thus obtaining an approximation that satisfies the conditions set

, ||ƒ-R _k || _p <ε.

Images