The electrocardiogram is the most effective examination for the diagnosis of cardiac pathologies, which affect the human heart [9] . This observation examination of the activity of the cardiac muscle tissue, given its limited function(its duration does not exceed a few seconds, carried out only during the subject’s rest, etc.), has evolved into a much more effective technique which is the Holter test. The Holter represents the recording of the electrocardiogram signal for 24 hours or more and therefore contains the maximum amount of information on the cardiac activity of the different patients.

A normal ECG recording represents a normal heart cycle; it is characterized by a spectrum of frequencies distributed below 40 - 50 Hz and by waves and time intervals which repeat in a quasi-periodic manner as presented in Figure 2.

This figure shows that a normal ECG has four maximums. It is this fundamental property that we will exploit for the development of our new model.

Cantor is the main creator of the set theory which he introduced in the early 1880s [11] . It was while working on problems of convergence of trigonometric series, in the 1870s, that he came to define a notion of derivation of sets of real numbers. His work has had several consequences and applications, particularly in algebra and signal processing.

By applying the consequences of set theory to the ECG signal vector, we deduce that there exist within the set P elements of this vector, subsets that can be organized in windows of dimension M times N in accordance to their value (intensity of future pixels); So if we consider P consisting of four windows F_i, each window F_i grouping elements close to the four specificities of the ECG signal that are the P, T, U waves and the QRS complex, then:

C a r d { P } = ∑ i = 1 4 c a r d { F i } (1)

The windows will therefore consist of close values, which has the effect of increasing the inter-pixel correlation, with the possibility of performing filtering on complex parts of the signal (P, T, U, QRS waves), this is illustrated in Figure 3.

The noise is considered in this case as an integral part of the image and compressed in the high frequencies during the decompression operation; the effect of the noise is therefore seen lessened, justified by the triangular inequality. Indeed, the data of the vector ECG put in 2D is coded on 8 bits, that is to say values of positive pixels ranging between 0 and 255 (0 and 1 in certain cases), the positive variant of the inequality theorem triangular (F₁ and F₂ positive), defined by Cauchy allows us to write that:

‖ F 1 ‖ 2 + ‖ F 2 ‖ 2 ≤ ‖ F 1 + F 2 ‖ 2 (2)

By performing an algebraic extension of this equation, we obtain:

‖ F 1 ‖ 2 + ‖ F 2 ‖ 2 < ‖ P ‖ 2 (3)

Because F 1 ∪ F 2 = P with F i ≠ ∅ (4)

If we note E { X } , the statistical error committed during the reconstruction (decompression) by the method implemented on a set or subset X, We deduce that:

c a r d E { F 1 } + c a r d E { F 2 } < c a r d E { P } (5)

This inequality shows that the probability of introducing errors or parasites in a process is reduced if one proceeds by processing piece by piece, rather than detrimental global processing; Hence the need for compression by windows or sub-set, with a quantizing effect on noise (reconstruction error) [12] .

Discrete Wavelet Transform (DWT) is a multi-resolution/multi-frequency representation [13] [14] . It makes it possible to efficiently analyze signals where phenomena of very different scales combine. The stages of the transform follow a filtering hierarchy. We then obtain a decomposition of the image into sub-bands with different filters (low pass h, and high pass g), consequently the need for a bi-dimensional DWT. This requires the use of a separable two-dimensional DWT (rows + columns). The input image is broken down each time into four sub-images (approximate image CA, horizontal detail DH, vertical detail DV and diagonal detail DD) with different low-pass and high-pass filters. The reconstruction is done using quadrature mirror filters, represented by their impulse responses (h and g) [15] [16] . This principle is illustrated in Figure 4.

SPIHT coding (set partitioning in hierarchical trees), as presented in Figure 5 is an efficient coding and is currently widely used in the specialised literature. The principle is to use a comparison threshold to say a coefficient that is an approximation or a detail. A tree branch hierarchy (in the following, we will write to simplify that i = i, j) is thus set up from a zero coefficient or parent denote X(i). A set noted O(i) represents the first two sons from X(i), all descendants including O(i) belong to the set D(i) and the set L(i) is defined by L ( i ) = D ( i ) − O ( i ) [17] [18] .

Thus, a coefficient resulting from the decorrelation will be considered as an approximation or detail depending on whether it is lower or higher than the comparison threshold given by:

k = log 2 ( max i | x i | ) ,     0 ≤ i ≤ K (6)

k represents the number of decor-related samples.

The coding will be carried out gradually in the following manner:

s k ( x i ) = { 1 ,       if   | x i | ≥ 2 k 0       if   not (7)

To implement the new approach, we chose to split our ECG signal into four blocks or subsets, corresponding to the four critical parts of an ECG signal (P, T, U, QRS). Thus, the block or subset F₁ will comprise as many P waves as the actual examination sequence will have.

Also, considering I the initial ECG vector, then I can be written as an empirical and non-matrix sum of four blocks or subsets:

I = I 1 + I 2 + I 3 + I 4 (8)

The algorithm first evaluates the maximums of the basic sequence noted, M₁, M₂, M₃ and M₄, corresponding to the different QRS, T, P, U waves. The data for the entire total sequence will therefore be classified into four blocks according to their proximity to the maximums M_i. To ensure the harmonization of the dimensions of the blocks, the following operation is carried out:

I = ( I 1 + R 1 ) + ( I 2 + R 2 ) + ( I 3 + R 3 ) + ( I 4 + R 4 ) (9)

with R_n number of zeros to be completed in block I_n to have matrices of the same dimensions. This choice depends on the matrix block having the largest number of data. Ultimately,

I = ∑ k = 1 4 I k + R k ( 0 ) (10)

If we set I k = ∑ i ∑ j X i j (11),

Then, we will get,

I = ∑ i ∑ j a i j + ∑ i ∑ j b i j + ∑ i ∑ j c i j + ∑ i ∑ j d i j (12)

with I i j = { a i j     si   I i j ≤ M 4 b i j       si   M 4 < I i j ≤ M 3 c i j       si   M 3 < I i j ≤ M 2 d i j     si   M 2 < I i j ≤ M 1 (13)

i and j being the lines and columns respectively ∑ ​ ∑ ​ representing an empirical sum in the sense of subsets and not arithmetic. Figure 6 presents the grouping into subsets of four blocks based on QRS, T, P and U. Figure 7 presents the new proposed algorithm and Figure 8 shows the 2D compression scheme using the model in subsets

Considered the most reliable way to measure real quality, subjective evaluation directly involves the human observer. The latter is called upon to judge the quality of the images or signals presented to it according to an evaluation grid with several levels of appreciation. At the end of these tests, a subjective score called MOS (Mean Opinion Score) is obtained. We have chosen to implement the simple stimulus protocol in this trial as presented in Figure 9.

The single stimulus method, called “Single Stimulus Continuous Quality Scale (SSCQS)” is used to judge the quality of one stimulus at a time. The images are presented one by one with a lag time between two presentations. This time allows the observer to note the quality of the image or signal. Scales with 5, 6, 7 or 100 levels of appreciation can be used. Table 1 presents an example of a scale, this method is very widely used to validate a quantitative or objective metric [20] , and it is what was used in the context of this article.

Compression fidelity was assessed by the percent roots meansquare difference which is proportional to the quality. Its definition and use come from the field of audio/video signals, it is defined as follows:

PRD = ∑ i = 1 N ( y i − y ′ i ) 2 ∑ i = 1 N y i ∗ 100 (15)

where, y i is the original sample and y ′ i the reconstructed sample. Where N is the maximum signal length. A value of PRD tending towards 0 corresponds to a perfectly reconstructed signal, and it decreases as a function of the distortion. The PRD, therefore, links the mean square error to the maximum energy of the signal.