Multi Discrete Laguerre Wavelets Transforms with The Mathematical aspects

The Linear approximation, means the order of smoothness of the void function. As for non-linear approximation, when compared to linear approximation, a significant improvement in the arrangement of approximation can be observed

The basis for wavelet transformation is the requirements for analyzing multiple solutions.This is why found a large group of wavelet families ready to do various applications, including image compression, which depends on the image content.The characteristic of a discrete wavelet transform has the efficiency of image coding due to its ability to activate data wavelet analysis is a powerful mathematical tool [3], [4], so it has been widely used in image digital processing, [5] .The approximation by wavelet was used to solve many problems in the fields of mathematics, physics, and engineering, [6][7][8][9][10]Many scholars have an interest in this topic through Debnath ,Meyer ,Morlet, [11] In this work constructed the new wavelet depended of the mother wavelet with the tow scalars c, d are parameters for translating and dilating in section 3 transform is constructed Multi Discrete Laguerre Wavelets Transform (MDLWT), with proved the orthogonal of the MDLWT in section 4 finally in section 5 proved the approximation Multi Discrete Laguerre Wavelets Transform (MDLWT), in linear and non linear approximation the theorems illustrated The Linear approximation, means the order of smoothness of the void function.As for non-linear approximation, when compared to linear approximation, a significant improvement in the arrangement of approximation can be observed.Image processing using MDLWT were done on 4 samples are displayed in table1 used high pass filter and low pass filter for analyses image and processing it during calculate important criteria that indicate the efficiency of the proposed wavelets.
Table1: The original samples with its normalized histogram

Wavelets Transformation
The wavelets are created from expansion and contraction through the two parameters a and b which are represented by the parent function from the continuous wavelets. where The basis for the above function consists of the elements

Multi Discrete Laguerre Wavelets Transform (MDLWT)
By transfer the parameters c, d to specific values, the wavelets will turn into discrete wavelets transform as, Laguerre wavelet , v is order for Laguerre polynomials and  is normalized time.

The dilation by parameter
and translation by parameter where v is the order of Laguerre polynomials In used k=2 gotten from (3) many functions but if used k=3 will be get multi numbers functions from (3), therefore, the wavelet in this work was called the multi wavelet The atoms obtained in equation ( 4) from Laguerre polynomials with weight function The function approximate with dilation and translation the weight function where ), ( From the above equations will be get two vectors 3 , , ,

Orthogonality for MDLWT
In this section the orthogonality of the wavelet constructed in section 2 will be demonstrated.

 
x L u has orthogonality with , the series of laguerre wavelets are the orthogonal with respect weight function the functions taking values wavelets belong to [0,1), then

Proof
The orthogonal of Laguerre polynomials with weight function where v is the order of Laguerre polynomials v is the order of Laguerre polynomials with weight function Finite equation( 3) and ( 4) Theorem is hold

Multi Discrete Laguerre Wavelets Transform (MDLWT) with Mathematical aspects
Definition The interval , is dyadic interval all interval in group is called dyadic sub interval

Multi Resolution Analysis (MRA) of Multi Discrete Laguerre Wavelets Transform (MDLWT)
The analysis of DMLWT is a system for determent of basis coefficients in And the MRA in wavelet space and foremost, It has the following important characteristics to clarify MRA that it has MDLWT to be ready to work in important areas of image processing such as image analysis 1 , , The scalar function () 2 L  on R→ ( − ) ,  0 = {( − )}.

Approximation by MDLWT in different Spaces
In this section it is established that the waveforms belong to the approximation area, which qualifies the smoothness of the wave in many uses in the field of image processing by finding approximate and details coefficients, which leads to extracting a suitable filter for use in analyzing the image to approximate and details coefficients then compressing the image and removing noise from it

Space
In this section some theorems proved MDLWT in   R L 2 is a square integrable functions over R belong to different approximate spaces, L P (R), Lip M (δ, P), and Besov Space    (  ())F s (R) See [11], Theorem (6.3) The approximate error of in    2 () will be .

Proof:
same above theorems, can be used properties, the Besov Space its properties  In this section used filters MDLWT in Image Processing after find it by decision Tree of Signal from MDLWT, the signal S is found its root started analysis signal from the approximation 0 AS  in level n=0 second step will get the detail in level 1 The

Conclusion
In this work, the orthogonality theory is given to give the function or wavelet constructed from Laguerre polynomials and obtained the new wavelet Multi Discrete Laguerre Wavelets Transform (MDLWT) The Linear approximation, means the order of smoothness of the void function.As for non-linear approximation, when compared to linear approximation, a significant improvement in the arrangement of approximation can be observed.
The second part of the work has been demonstrated by a number of theorems in linear and non-linear approximation in order to prove that the new wave is smooth in its use in many applications such as solutions to many numerical problems such as Variational problems, integro differential and integral equations.
Moreover to this it enhances the possibility and smoothness of its use in many areas of image processing such as pressure and de noise from the image to the end using a process new wavelet MDLWT was used after extracting the filter high pass filter and low pass filter by extracting the decision trees for the transactions where the new wavelet was used in image processing and mean square error (MSE), Peak signalto-noise ratio (PSNR), Bits per pixel ratio (BPP) and Compression ratio (CR) were calculated, and the applications were done on 4 samples which good results have been found to prove the efficiency of the new wavelet


The coefficients of wavelets are approximate coefficients denoted by v u a , and the details coefficients denoted by v u d , then (  ()) ≅ ((∑ ∑|< ,  ,linear approximation in   ()  ∈   () then

7 .
Tree decision Multi Discrete Laguerre Wavelets Transform (MDLWT) filters of MDLWT after add in MATLAB program used it in analyses image (decomposition) in compressed image with got the results of PSNR depended of MSE and PBB of the CR in level 8 table2 illustrated the effected of MDLWT in image processing in table3 shows the results of compression images in size 256 × 256 with pixels in x, y dimension