New Orthogonal Multiwavelet Filters Derived by Matrix Spectral Factorization

New Orthogonal Multiwavelet Filters Derived By Matrix Spectral Factorization Vasil Kolev∗ , Todor Cooklev and Fritz Keinert Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Bl. 2, Acad. G. Bonchev St., 1113 Sofia, Bulgaria Wireless Technology Center, Purdue University, Fort Wayne, IN 46805, USA Department of Mathematics, Iowa State University, Ames, IA 50011, USA

ABSTRACT

Keywords: Supercompact multiwavelet spectral factorization classical and exact Bauer’s method signal denoising image compression subband-based edge detection

The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer’s method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

arXiv:2608.11518v1 [cs.CV] 12 Aug 2026

ARTICLE INFO

1. Introduction

Multiwavelets, as extensions of scalar wavelets, can possess important features such as orthogonality, compact support, symmetry and high-order vanishing moments. Their structure makes them useful in mathematical theory and many applications. Supercompact multiwavelets have support [0, 1] and retain the spatial compactness and orthogonality of Haar wavelets [1]. They are simple to implement in applications on a finite interval and/or for analysis of signals that are piecewise continuous, since no special treatment of external or internal boundaries is necessary. This compactness becomes more important where strong gradients occur, such as near shock waves and in boundary regions. Numerical evidence also illustrates that these supercompact multiwavelets could be utilized without modification even near boundaries [2]. Supercompact multiwavelets are orthogonal vectors of 𝑟 scaling functions 𝝓(𝑡) = [𝜙0 , 𝜙1 , … , 𝜙𝑟−1 ]∗ and 𝑟 wavelet functions 𝝍(𝑡) = [𝜓0 , 𝜓1 , … , 𝜓𝑟−1 ]∗ , which belong to 𝐿2 ([0, 1])𝑟 [3] and satisfy matrix recursion equations with only two coefficients √ [ ] 𝝓(𝑡) = 2 𝐻0 𝝓(2𝑡) + 𝐻1 𝝓(2𝑡 − 1) , √ [ ] 𝝍(𝑡) = 2 𝐺0 𝝓(2𝑡) + 𝐺1 𝝓(2𝑡 − 1) . The multiscaling symbols 𝐻(𝑧) = 𝐻0 + 𝐻1 𝑧−1 and 𝐺(𝑧) = 𝐺0 + 𝐺1 𝑧−1 are linear matrix polynomials. The orthogonality conditions ⟨𝜙𝑖 , 𝜙𝑗 ⟩ = ⟨𝜓𝑖 , 𝜓𝑗 ⟩ = 𝛿𝑖𝑗 and ⟨𝜙𝑖 , 𝜓𝑗 ⟩ = 0 for 𝑖, 𝑗 = 0 ∶ 𝑟 − 1, can be expressed in terms of the coefficients as 𝐻0 𝐻0∗ + 𝐻1 𝐻1∗ = 𝐺0 𝐺0∗ + 𝐺1 𝐺1∗ = 𝐼,

𝐻0 𝐺0∗ + 𝐻1 𝐺1∗ = 0.

The Hermitian transpose of a matrix polynomial 𝐴(𝑧) = product filter 𝑃 (𝑧) is defined as the autocovariance function

∑

∑ ∗ −𝑗 𝑗 ∗ 𝑗 𝐴𝑗 𝑧 is given by 𝐴(𝑧) = 𝑗 𝐴𝑗 𝑧 . The matrix (1)

𝑃 (𝑧) = 𝐻(𝑧)𝐻(𝑧)∗ = 𝑃1 𝑧−1 + 𝑃0 + 𝑃−1 𝑧, Keinert)

vasil.kolev@iict.bas.bg, kolev_acad@abv.bg (V. Kolev); cooklevt@pfw.edu (T. Cooklev); keinert@iastate.edu (F.

ORCID (s): 0000-0002-4619-3579 (T. Cooklev); 0000-0003-3116-967X (F. Keinert)

Kolev et al.: Preprint submitted to Elsevier

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Multiscaling Function

HCL(z) Matrix Spectral Factorization

Matrix Product Filter

New Multiscaling Functions

PCL(z) = H(z)H(z)T

PCL(z)

H (1)(z) QR

G (1)(z)

H (2)(z) QR

G (2)(z)

Figure 1: The main steps in the construction of new supercompact multiscaling and multiwavelet functions

which is a positive semidefinite polynomial matrix. The matrix product coefficients are 𝑃0 = 𝐻0 𝐻0∗ + 𝐻1 𝐻1∗ ,

𝑃1 = 𝐻0 𝐻1∗ ,

𝑃−1 = 𝑃1∗ = 𝐻1 𝐻0∗ .

(2)

It is Hermitian, and satisfies the half band condition 𝑃 (𝑧) + 𝑃 (−𝑧) = 2𝐼 [4]. The stability and causality conditions translate to requiring that 𝑃 (𝑧) has all its poles and zeros inside the unit circle. Matrix Spectral Factorization (MSF) refers to finding a matrix spectral factor 𝐻(𝑧), given 𝑃 (𝑧). The existence of such a factor is guaranteed under a condition on 𝑃 (𝑧) due to Paley and Wiener, requiring that log(det(𝑃 (𝑧))) have a finite integral over an interval of length 2𝜋. MSF is a key step for the construction of minimum phase spectral factors from matrix product filters. Unlike the spectral factorization of scalar Hermitian polynomial functions, where an explicit formula exists, there is no explicit expression for matrix spectral factorization. However, for low-degree para-Hermitian polynomial matrix polynomials a fast algorithm has been constructed in [5, 6]. Our main goal in this paper is to present a new approach for the construction of new supercompact orthogonal multiwavelets by matrix spectral factorization of a suitable product filter. The main steps are shown in Fig. 1. The new multiwavelets are obtained from the product filter of the Chui-Lian (CL) multiwavelet filter [7]. This immediately leads to three main problems: 1. Does the Fast Bauer’s method for matrix spectral factorization converge in this case? 2. Is possible to obtain an orthogonal multiscaling filter with exact coefficients from existing multifilters? 3. Do the newly derived multiwavelets have better coding gain than other well-known orthogonal or biorthogonal multifilters? We have the following main contributions: • Obtain the matrix product filter of the CL orthogonal multiwavelet filter; • Apply Fast Bauer’s method for spectral factorization to find two new supercompact multiscaling functions with exact recursion coefficients; • Find the complementary new supercompact multiwavelet functions; • Compare the performance of new multiwavelet filters in subband-based edge detection, 1D signal denoising and grayscale and color image denoising and compression with with the Integer Haar, Alpert, GHM, SA4 and CL multiwavelet filter banks. Kolev et al.: Preprint submitted to Elsevier

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2. Theory of Matrix Spectral Factorization Spectral factorization is an important mathematical tool in signal processing, control theory, and other areas. The fundamental theorem of the spectral factorization is the Fejér-Riesz theorem for positive definite functions. The terminology comes from prediction theory, where the nonnegative function plays the role of a spectral density for a multidimensional stationary stochastic process. Fejér [8] first showed the importance of the class of trigonometric polynomials that admit only positive real values; as a theorem it was proved by Riesz [9]. The definition of a trigonometric univariate polynomial is 𝜈(𝑧) =

𝑁 ∑

𝜈𝑘 𝑧𝑘 .

𝑘=−𝑁

When the function 𝜈(𝑧) is real for all 𝑧 ∈ 𝕋 , where 𝕋 is the unit circle in the complex plane, then the coefficients satisfy 𝜈 𝑘 = 𝜈−𝑘 for all 𝑘. If 𝜈(𝑧) ≥ 0 for all 𝑧 ∈ 𝕋 , the spectral factorization of the function is 𝜈(𝑧) = 𝑝(𝑧)𝑝(𝑧)∗ , ∑ 𝑘 where 𝑝(𝑧) = 𝑁 𝑘=0 𝑝𝑘 𝑧 is called a scalar spectral factor. Matrix spectral factorization (MSF) refers to finding a matrix spectral factor 𝐻(𝑧), given 𝑃 (𝑧), so that 𝑃 (𝑧) = 𝐻(𝑧)𝐻(𝑧)∗ . Such a factorization exists if 𝑃 (𝑧) is positive definite on 𝕋 , absolutely integrable and satisfies the Paley-Wiener condition [10, 11, 12] | 𝜋 | 𝑖𝜔 | | |∫ ln det 𝑃 (𝑒 ) 𝑑𝜔| < ∞. | | −𝜋 There is a unique minimal phase solution 𝐻𝑀 (𝑧). All other solutions are of the form 𝐻(𝑧) = 𝐻𝑀 (𝑧)𝑈 , where 𝑈 is a unitary matrix multiplier. Usually, 𝑃 (𝑧) is a half-band filter with 𝑃0 = 𝐼, 𝑃2𝑘 = 0 for 𝑘 ≠ 0.

2.1. Classical Bauer’s method of Matrix Spectral Factorization Let the multiscaling function 𝐻(𝑧) and its complex conjugate 𝐻 ∗ (𝑧) be 𝐻(𝑧) = 𝐻0 + 𝐻1 𝑧−1 + ⋯ + 𝐻𝑚 𝑧−𝑚 , 𝐻 ∗ (𝑧) = 𝐻0∗ + 𝐻1∗ 𝑧 + ⋯ + 𝐻𝑚∗ 𝑧𝑚 ,

(3)

where 𝐻𝑘 are real matrices of size 𝑟 × 𝑟 and 𝑧 ∈ ℂ. Let 𝕋 = {𝑧 ∶ |𝑧| = 1} be the unit circle in the complex plane. For 𝑧 ∈ 𝕋 we have 𝑧−1 = 𝑧. Then the matrix product 𝑃 (𝑧) = 𝐻(𝑧)𝐻 ∗ (𝑧) = 𝑃𝑚∗ 𝑧−𝑚 + ⋯ 𝑃1∗ 𝑧−1 + 𝑃0 + 𝑃1 𝑧 + ⋯ 𝑃𝑚 𝑧𝑚

(4)

is called the matrix product filter, where each matrix coefficient is determined by ∗ 𝑃𝑘 = 𝐻𝑚−𝑘 𝐻𝑚∗ + 𝐻𝑚−𝑘−1 𝐻𝑚−1 + ⋯ + 𝐻0 𝐻𝑘∗ .

(5)

Supercompact multifilters represent the simplest case 𝑚 = 1, with the coefficients 𝑃0 and 𝑃1 shown in (2). Bauer’s matrix spectral factorization finds the Cholesky factor 𝐿𝑛 of the (𝑛 + 1) × (𝑛 + 1) banded block Toeplitz matrix ⎛ 𝑃0 ⎜𝑃 ∗ 𝑇𝑛 = ⎜ 1 0 ⎜ ⎝0

𝑃1 𝑃0 ⋱ 0

0 𝑃1 ⋱ 0

0 0 ⋱ 𝑃1∗

0⎞ 0⎟ = 𝐿𝑛 𝐿∗𝑛 , 0⎟ ⎟ 𝑃0 ⎠

Kolev et al.: Preprint submitted to Elsevier

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where ⎛𝐻 (0) ⎜ 0(1) 𝐻 𝐿𝑛 = ⎜ 1 ⎜ 0 ⎜ ⎝ 0

0 𝐻0(1) ⋱ 0

0 0 ⋱ 𝐻1(𝑛)

0 ⎞ ⎟ 0 ⎟ 0 ⎟ ⎟ 𝐻0(𝑛) ⎠

is a lower bidiagonal matrix. The last row of 𝐿𝑛 converges to the true matrix coefficients 𝐻0(𝑛) → 𝐻0 , 𝐻1(𝑛) → 𝐻1 as 𝑛 → ∞. The classical Bauer’s method is very effective for nonsingular matrix product filters P(z). Convergence can be very fast, so n does not need to be large to accurately estimate the spectral factors. However, since in our applications the matrix product filters are required to possess multiple zeros on the unit circle, the method converges very slowly and requires a very large Toeplitz matrix. In one example, it required a matrix of size around 106 × 106 for achieving a precision of 10−13 . In addition, for larger multiplicity r⩾ 3 high precision often cannot be achieved. In one example in [6] with r=5, m=1 and a decuple zero singularity, the maximum achievable accuracy in the classical Bauer’s method was around 10−3 .

2.2. Fast Bauer’s method of Matrix Spectral Factorization Obviously the Cholesky decomposition (6) can be realized row by row. If we assume that 𝐻0(𝑛) , 𝐻1(𝑛) have already

been found, with the initial value 𝐻0(0) obtained as the Cholesky factor of 𝑃0 , then it follows that [ ]−∗ 𝐻1(𝑛+1) = 𝑃1∗ 𝐻0(𝑛)

(7)

[ ]∗ from 𝑃1∗ = 𝐻1(𝑛+1) 𝐻0(𝑛) , and [ ]∗ [ ]∗ 𝐻0(𝑛+1) 𝐻0(𝑛+1) = 𝑃0 − 𝐻1(𝑛+1) 𝐻1(𝑛+1)

(8)

[ ]∗ [ ]∗ from 𝑃0 = 𝐻0(𝑛+1) 𝐻0(𝑛+1) + 𝐻1(𝑛+1) 𝐻1(𝑛+1) . Then (7) substituted into (8) leads to [ ]−∗ [ ]−1 [ ]∗ 𝐻0(𝑛) 𝑃1 . 𝐻0(𝑛+1) 𝐻0(𝑛+1) = 𝑃0 − 𝑃1∗ 𝐻0(𝑛)

(9)

[ ]∗ If we define 𝑋 (𝑛+1) = 𝐻0(𝑛+1) 𝐻0(𝑛+1) , then [ ]−1 𝑋 (𝑛+1) = 𝑃0 − 𝑃1∗ 𝑋0(𝑛) 𝑃1 .

(10)

The limit 𝑛 → ∞ leads to the solution 𝑋 (𝑛+1) → 𝑋 of the nonlinear matrix equation (NME) (11)

𝑋 = 𝑃0 − 𝑃1∗ 𝑋 −1 𝑃1 .

Our focus in this paper is the supercompact case m=1, but we note that the general case can be reduced to this by replacing the original coefficients with block matrices. For m=3, for example, we set ⎡𝑃 ∗ ⎢ 3 𝑃̂−1 = 𝑃̂1∗ = ⎢ 0 ⎢0 ⎣

𝑃2∗ 𝑃3∗ 0

𝑃1∗ ⎤ ⎥ 𝑃2∗ ⎥ , 𝑃3∗ ⎥⎦

⎡ 𝑃0 ̂ 𝑃0 = ⎢𝑃1∗ ⎢ ∗ ⎣𝑃2

𝑃1 𝑃0 𝑃1∗

𝑃2 𝑃1 𝑃0

⎤ ⎥, ⎥ ⎦

⎡𝑃3 𝑃̂1 = ⎢𝑃2 ⎢ ⎣𝑃1

0 𝑃3 𝑃2

0⎤ 0 ⎥. ⎥ 𝑃3 ⎦

The resulting coefficients 𝐻̂ 0 , 𝐻̂ 1 have the form ⎡𝐻3 𝐻̂ 1 = ⎢ 0 ⎢ ⎣0

𝐻2 𝐻3 0

𝐻1 ⎤ 𝐻2 ⎥ , ⎥ 𝐻3 ⎦

⎡𝐻0 𝐻̂ 0 = ⎢𝐻1 ⎢ ⎣𝐻2

Kolev et al.: Preprint submitted to Elsevier

0 𝐻0 𝐻1

0⎤ 0 ⎥. ⎥ 𝐻0 ⎦ Page 4 of 18

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Once 𝐻̂ 0 , 𝐻̂ 1 have been computed, we can extract the original 𝐻0 , 𝐻1 , 𝐻2 , 𝐻3 as submatrices. Setting up the symmetrical nonsingular matrix 𝑋 from the above Fast Bauer’s method with symbolic entries 𝑥𝑖𝑗 converts (11) into a system of nonlinear equations. For small matrix sizes this can be solved with exact values by a computer algebra system (CAS). For larger r, (11) could be solved numerically by fixed point iteration, which is identical to the original Bauer’s method, but Newton’s method converges much faster and gives more accurate results. See [6] for more details.

3. Design of the supercompact multiwavelet filter 3.1. The matrix product filter of the CL multiscaling function The matrix coefficients of the orthogonal CL scaling functions [7] ] [( ( ) √ ) ( √ ) 0 2 + √7 3 1 −1 2 − √7 0 −2 1 + 𝑧 + 𝑧 𝐻CL (𝑧) = 1 3 4 0 2− 7 2+ 7 0 lead to the matrix product filter [( √ ) ( 4√ 1 +√ 7 −1 8 1 𝑃CL (𝑧) = 𝑧 + 0 8 −(1 + 7) − 7

) ( 0 4√ + 8 1+ 7

√ ) ] −(1 + √ 7) 𝑧 − 7

with det(𝑃CL (𝑧)) = (4 −

√ (1 + 𝑧)4 7) , 32𝑧2

which satisfies the regularity criterion that det(𝑃 (𝑧)) has a quadruple zero at 𝑧 = −1.

3.2. New supercompact multiscaling functions from the Exact Bauer’s method In order to obtain multiscaling functions with exact coeficients, we solve eq.(11) by an appropriate CAS, for example the Matlab Symbolic Toolbox. The solution is a symmetric matrix 1 𝑋= 8

(

√4 7+1

) √ 7+1 . 4

The standard Cholesky factor is √ ( 4 2 √ 𝐻0(1) = 8 7+1

√

)

0

√ 8−2 7

The square root has two possible values √ √ √ 8 − 2 7 = ±( 7 − 1) which leads to two solutions

𝐻0(1) =

√ ( 2 √4 8 7+1

√0 7−1

𝐻0(2) =

√ ( 2 √4 8 7+1

√0 −( 7 − 1)

and

)

Kolev et al.: Preprint submitted to Elsevier

)

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Figure 2: (a) CL multiscaling function 𝝓𝐶𝐿 (red) and multiwavelet function 𝝍 𝐶𝐿 (blue); (b) New supercompact multiscaling function 𝝓(1) (red) and multiwavelet function 𝝍 (1) (blue); (c) New supercompact multiscaling function 𝝓(2) (red) and multiwavelet function 𝝍 (2) (blue);

[ ]−∗ The second coefficients 𝐻1(𝑖) is found from (7) as 𝐻1(𝑖) = 𝑃1∗ 𝐻0(𝑖) . We call the resulting supercompact multiscaling function New1 and New2. The resulting coeficients, as well as the coeficients of the corresponding multiwavelet functions and the determinants of the symbols are tabulated in Tab 1. It is important to note that the determinants of both multiscaling functions include a factor of (1 + 𝑧−1 )∕2, which according to [4] leads to regular scaling functions and a desirable supercompact multiscaling function.

3.3. New supercompact multiwavelet functions In order to obtain the wavelet matrix coefficients 𝐺0 and 𝐺1 we can employ the QR decomposition as in [5], or in the present case, the orthogonal vectors method. This multiwavelet possesses the all the desirable properties of the Alpert multiwavelet. The recursion coefficients of the new orthogonal multiscaling and multiwavelet functions obtained by QR decomposition are shown in Table 1; the functions are shown in Fig. 2.

4. Performance Analysis 4.1. Numerical errors The classical Bauer’s method is applied to the spectral factorization of the matrix product filter. The numerical errors obtained for the matrix product filter 𝑃 (𝑧) and the matrix spectral factor 𝐻(𝑧) are defined as | | (𝑛) 𝜖𝐻 = |𝐻 − 𝐻 (𝑛) | | | | (𝑛) (𝑛) | 𝜖𝑃 = |𝑃 − 𝑃 | . | | Kolev et al.: Preprint submitted to Elsevier

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New Orthogonal Multiwavelet Filters ... Table 1 The matrix coefficients of the two new multiscaling and multiwavelet functions New1 and New2. Multiscaling Functions 𝐻 (1) (𝑧) 𝐻 (2) (𝑧) ) ) √ ( √ ( 4 0 4 0√ 2 2 (1) (2) √ √ √ 𝐻0 = 8 𝐻0 = 8 7+1 7−1 7+1 1− 7 ) ) √ ( √ ( 4 0 4 0√ 2 2 (2) (1) √ √ √ 𝐻1 = 8 𝐻1 = 8 −( 7 + 1) −( 7 + 1) 1 − 7 7−1 det(𝐻 (1) (𝑧))√= det(𝐻0(1) + 𝐻1(1) 𝑧−1 ) =

7−1 √ (1 + 𝑧−1 )2 2

det(𝐻 (2) (𝑧)) =√ det(𝐻0(2) + 𝐻1(2) 𝑧−1 ) = 1−√ 7 (1 + 𝑧−1 )2 2

Multiwavelet Functions √

(

𝐺(1) (𝑧)

)

√0 √−4 −( 7 − 1) 7+1 ) √ ( 0 4 2 (1) √ √ 𝐺1 = 8 7−1 7+1

𝐺0(1) =

2 8

det(𝐺(1) (𝑧))√= det(𝐺0(1) + 𝐺1(1) 𝑧−1 ) =

7−1 √ (1 − 𝑧−1 )2 2

2 8

(

𝐺(2) (𝑧)

) √0 √−4 −( 7 − 1) −( 7 + 1) ) √ ( 0 4 2 (2) √ √ 𝐺1 = 8 7 − 1 −( 7 + 1)

𝐺0(2) =

√

det(𝐺(2) (𝑧))√= det(𝐺0(2) + 𝐺1(2) 𝑧−1 ) =

7−1 √ (1 − 𝑧−1 )2 2

The values are shown in Fig. 3, in a log-log plot. The convergence of the two errors 𝜖𝐻 and 𝜖𝑃 is well modeled by the (𝑛) linear fit 𝑝𝜖𝐻 (𝑥) = −1.0101𝑛 and 𝑝𝜖𝑃 (𝑥) = −2.0454𝑛 respectively, where the logarithm of the error log(𝜖𝐻 ) has slope

(𝑛) (−1) (a red line), while log(𝜖𝐻 ) has a slope of (−2) (cyan line). In [6], after applying the fast Bauer’s method using a general discrete algebraic Riccati equation (GDARE) in (𝑛) Matlab R2018a, the best achieved precision for the matrix spectral factor 𝜖𝐻 = 6.16×10−5 and the matrix product filter

(𝑛) 𝜖𝑃(𝑛) = 6.66×10−14 , while with Matlab R2010a,b the best precisions are 𝜖𝐻 = 6.7077×10−5 and 𝜖𝑃(𝑛) = 3.3307×10−16 , respectively. It is important to note that the classical Bauer’s method for spectral factorization obtained errors for 𝑛 = 10, 000 of (𝑛) 𝜖𝐻 = 8.396 × 10−5 and 𝜖𝑃(𝑛) = 2.055 × 10−9 shown in Fig. 3. They can easily be modeled by regression functions. The (𝑛) (𝑛) empirical results show minimal possible errors 𝜖𝐻 = 9.6145 × 10−17 and 𝜖𝐻 = 6.979 × 10−5 that can be achieved for 𝑛 = 38.639.

4.2. Statistical Characteristic Analysis of the Test Images 4.2.1. Histograms The histogram of an image is a one-dimensional statistical characteristic of the given image and represents the frequency of intensity in separate groups of pixels in the image. The groups usually contain the same number of adjacent intensity levels. The most representative (precise) are histograms on all possible single levels. As an application, the histogram values are in some (engineering) range of intensities of useful (measured) quantities, of course appropriately quantized. Visually, the histogram allows for different interpretations by pixels and groups of pixels, for example how the brightness (luminance) correlates with the number of pixels in it, as well as some easily apparent characteristics such as contrast, dynamic range and saturation effects, which make problems easily visible and facilitate the analysis of the image (and the problem it reflects). The color of images is represented by color models. The most popular of them is RGB, but we can also mention many others, YUV, HSV, etc., emphasizing different characteristics of color. While a halftone image is a scalar function over the discrete area of pixels, color images are usually vector functions. RGB is a 3-dimensional function, one vector for each of the 3 channels (all halftones).

Kolev et al.: Preprint submitted to Elsevier

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Figure 3: Influence of the size 𝑛 of the Toeplitz matrix on the precision of the matrix spectral factor (in log-log format): (𝑛) The numerical error of matrix spectral factor 𝜖𝐻 (red) and of the matrix product filter 𝜖𝑃(𝑛) (cyan)

For better decomposition at multiple levels of detail and better encoding as well, in the future we could apply a suitable 3D transform based on 3D convolution and extraction of color channels by a inflated 3D convNet (I3D) model [26] instead of extraction of RGB channels. The abscissa 𝑂𝑥 shows the pixel intensities in the range from 0 to 255 for each color channel (R, G, B). The value 0 corresponds to the "black" level, and 255 to the "white" level. If the image is halftone, successive levels of "gray" are placed between them. If the channel is color, the "gray" level represents the corresponding base colors (R, G, B) in the given color image. The ordinate 𝑂𝑦 shows the frequency of the corresponding pixel intensity levels. Color histograms are triplets of halftone histograms, one for each color (red (R), green (G), and blue (B) channels. The size of the intervals in the figures is not specified and the value of each pixel is displayed separately. Some characteristic tests for seven color and seven grayscale images with different sizes and textures are illustrated in Fig. 4(a)–(f) and Fig. 5(a)–(g). • Bart Run has an extremely low dynamic range with only 14 intensity values; • Cameraman has 2 isolated spikes, as can be see from the histogram in Fig. 4(b). The big peak on the far left side of the graph means that the image is made up of a lot of shadows or black tones. Therefore the image has low brightness. The peaks in the middle of the histogram indicate lots of midtones which mean muted color grays; • Kiel has distributed intensity values with 2 isolated peaks. The histogram in Fig. 4(c) shows peaks that are cut off on the left or right side, which means an underexposed image; Kolev et al.: Preprint submitted to Elsevier

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• House is underexposed (intensity values are clustered) and has high contrast for all three colors. The blue channel, see Fig. 4(d), has many pixels in the high intensity region, while the red and green spectra remain mostly low. This means that the image has shades of blue, but not much red and green. • Cara has saturated and underexposed pixel values (intensity values are clustered) and has also an isolated peak for all three colors. This is shown in Fig. 4(e), where the histogram peaks in the middle of each color indicate a large presence of dark pixels for each color. Each color has peaks scattered around the middle, indicating a midtone, which means a muted color. • Voit has properly exposed normal contrast (intensity values are scattered). • House has low contrast histogram by a slight increase in contrast. The big peak on the right side shown in Fig. 5(a). • Couple has low contrast histogram gaps caused by a slight increase in contrast shown in Fig. 5(b). • Brick wall has a normal contrast and many pixels in the low intensity region which means that the image has shades or dark color tones shown in Fig. 5(c). • Texture 1 has low high and overexposed contrast for R, G and B channels respectively. The blue channel has many pixels with in the lowest intensity region, while the red and green channels remain mostly low shown in Fig. 5(d). • Subwater has normal contrast for channels, blur the red and blue channels and isolated peak shown in Fig. 5(e). • Texture 4 Texture 4 has extremely low dynamic range for R, G, B channels with 2 isolated peaks shown in Fig. 5(f). • Texture 5 histogram gaps caused by a slight increase in contrast shown in Fig. 5(g).

4.2.2. Adjacent Pixel Correlations Neighboring pixel correlation refers to the relationship between two adjacent pixels in horizontal and vertical coordinates, respectively. The more the pixel values remain concentrated near the linear fit, the stronger the correlation between neighboring pixels in the image; conversely, the more pixels are scattered throughout the plot, the weaker the correlation between neighboring pixels in the image. The correlation coefficient 𝑅2 is a primary measure of how well a regression model fits the data. This correlation coefficient is simply the square of Pearson’s correlation coefficient 𝑅. To check the existence of a high correlation among neighboring pixels in the test images, a pixel correlation analysis test has performed. The correlation analysis of adjacent pixels of this encryption scheme is shown in Fig. 6(a)-(f). High correlation among pixels in the original image is represented by the linear fit it converges to. The scatterplot of Fig. 6 leads to the values of 𝑅2 for the regression models shown in Table 2. The correlation coefficient 𝑅2 , which is the square of the Pearson correlation coefficient, is a basic measure of how well a regression model fits the data; when 𝑅2 = 1, the regression line predicts a perfect fit of the pixel values. The values of the regression lines for all test images are shown in Table 2. Fig. 6 shows the scatter plot of all regression models from Table 2. The images Bart Run and Cara have quite a bit of scatter, but the scatter plot of Fig. 6(a) and (e) (red, green, and blue channels) shows a clear linear trend with correlation 88% − 98%. The images Kiel and Voit have a moderate positive correlation in the interval 62% − 68%. It’s not a very strong relationship, as some values show no correlation. Obviously, for these two images there is quite a lot of scatter, but the trend is reasonably linear, despite the fact that the big number of data points makes it difficult to do a full correlation assessment. Fig. 5(b) and (f) (red, green, and blue channels) shows that there is no correlation in some subset pixels and a strong correlation in the whole pixels data set. The images Bart Run and Cara have quite a large scatter, but the scatter plot in Fig. 6(a) and (e) (RGB channels) shows a clear linear trend with a correlation of 88% − 98%. The two images Kiel and Voit have a moderate positive correlation in the interval 62% − 68%. This is not a very strong relationship, as some values show no correlation. It is clear that there is quite a bit of scatter for these two images, but the trend is relatively linear, despite the large number Kolev et al.: Preprint submitted to Elsevier

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Figure 4: Test images and their histograms; (a)Bart Run(128 × 128); (b)Cameraman(256 × 256); (c)Kiel(512 × 512);(d)House(256 × 256); (e)Cara(256 × 256);(f)Voit(256 × 256).

of data points, which makes it difficult to fully assess the correlation. Fig. 6(b) and (f) (RGB channels) show that there is no correlation in some subsets of pixels and a strong correlation in the entire pixel dataset. The grayscale images House,Cameraman, Texture 5 and color image (for R,G,B channels)Subwater have a strong correlation shown of Fig. 7. The grayscale image House and color images (for R,G,B channels) Texture 1, Texture 4 there is no correlation of pixels and a strong correlation in the entire pixel dataset shown of Fig. 7.

4.3. General multifilter characteristics We compare the performance of the proposed new multifilters with some well-known orthogonal and biorthogonal filter banks.The results are given in Table 3, with the criteria being coding gain (CG), Sobolev smoothness (S) [13], symmetry/anti-symmetry (S/A) and length. The CG is the ratio of arithmetic and geometric means of channel variances and is a good indication of the performance in signal processing. To estimate the CG, the variance is computed using a first order Markov model AR(1) with intersample autocorrelation coefficient 𝜌 = 0.95 [20]. As shown in Table 3, the supercompact multiwavelets have smaller CGs than SA4 and Geronimo-Hardin-Massopust (GHM), but better CGs, by at least a quarter dB, than even CL and most other orthogonal and biorthogonal multifilters, including designs with three or more matrix coefficients. Kolev et al.: Preprint submitted to Elsevier

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Figure 5: Test images and their histograms; (a)House(256 × 256); (b)Couple(256 × 256); (c)Brick Wall(1024 × 1024); (d) Texture 5 (256 × 256); (e) Texture 1(512 × 512); (f) Subwater(256 × 128); (g)Texture 4(1024 × 1024).

4.4. Applications 4.4.1. Subband-based edge detection Edge detection is one of the fundamental operations in image processing and computer vision. It is used to identify the edges in an image, for image segmentation and data extraction, and for easy processing as well. Since the choice of edge detection operator has to be responsive to gradual changes, the Canny method is used, which provides good edge detection. It uses two different threshold types to detect strong and weak edges [21]. The location and orientation of edges in an image can be determined from the encoded subbands of the original image [22]. Note that in many edge detection algorithms, the detail subbands which can be used in multiwavelet subband-based edge detection are using vector quantization (VQ) via 𝑘-means-based deep network architecture [23] to encode the multiwavelet coefficients. The main advantage of VQ is the construction of the code book consisting of code vectors. The goal of quantization is to obtain sufficiently small multiwavelet coefficients with a quantized distribution so that they can be encoded at a low bit rate. This motivated us to consider a simple example of a multiwavelet subbandbased edge detection. The Canny method is applied to the present multiwavelet filters for one decomposition level of a simple image, where we obtain sixteen subbands. The results from the subband-based edge detection are shown in Fig. 8. The contour lines of the SA1 filter show that this is an excellent detector, better than the other six transformations.

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Figure 6: Pixel correlations and linear fit lines for the gray and color (RGB channels) images; (a) Bart Run (128x128) (b) Cameraman (256x256) (c) Kiel (512x512) (d) House (256×256); (e) Cara (256×256); (f) Voit (256×256).

4.4.2. Image Compression ̃ 𝑦) images of size 𝑁 2 is The measurement for comparative analysis between the original 𝐈(𝑥, 𝑦) and the noise 𝐈(𝑥, the mean square error (MSE), 1 ∑ ̃ 2, 𝑀𝑆𝐸 = (𝐈 − 𝐈) 𝑁 2 (𝑥,𝑦) which in decibels on a logarithmic scale is the Peak Signal-to-Noise Ratio (PSNR) 2552 ) 𝑀𝑆𝐸 for 𝑥, 𝑦 = 1 ∶ 𝑁. PSNR of the compression ratio (CR) is defined as the ratio between the original and compressed image size. Results are shown in Table 5. The structural information of natural images is based on the human visual system (HVS). Quality assessment of compressed images is obtained through the structural similarity index measures (SSIM) [24] and its multiscale 𝑃 𝑆𝑁𝑅 = 10 log10 (

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New Orthogonal Multiwavelet Filters ... Table 2 The coefficients of the linear regression lines for the gray and color images The coefficients of the linear regression equation 𝑦 = 𝑝1 𝑥 + 𝑝2 𝑝1 𝑝2

Images

Bart Run Cameraman Kiel House Couple Brick wall Texture 5 House

Cara

Voit

Texture 1

Subwater

Texture 4

R G B R G B R G B R G B R G B R G B

0.9912 0.9887 0.8271 0.9586 0.8927 0.9053 0.9727 0.9326 0.9408 0.9668 0.9763 0.9797 0.9861 0.7972 0.8058 0.7870 0.5361 0.4606 0.6607 0.9761 0.9762 0.9689 0.9927 0.9947 0.9923

Coefficient of determination 𝑅2

1.3412 1.3424 23.5374 5.7160 12.9530 9.5085 2.9083 9.8814 7.8723 4.7158 2.2668 2.1628 1.8061 24.2278 21.2176 22.8056 35.3482 34.0091 17.9257 2.5480 3.4688 3.6210 0.7649 0.5201 0.7514

0.9824 0.9774 0.6841 0.9189 0.7970 0.8195 0.9461 0.8697 0.8852 0.9347 0.9531 0.9598 0.9723 0.6356 0.6493 0.6194 0.2874 0.2121 0.4365 0.9527 0.9530 0.9387 0.9854 0.9894 0.9846

Table 3 Comparison of CGs, S, S/A, for autocorrelation matrix 𝑅𝑥 of AR(1) and 𝜌 = 0.95 Multifilter Biorthogonal Hermitian cubic spline (bih34n) [14] Biorthogonal (dual to Hermitian cubic spline) (bih32s) [15] Integer Haar [16] (supercompact) CL [7] Alpert [17] (supercompact) New2 (supercompact) New1 (supercompact) SA4 [18] GHM [19]

CG

S

S/A

length

1.51

2.5

yes

3/5

1.01

2.5

yes

3/5

1.83 2.06 2.13 2.24 2.38 3.73 4.41

0.5 1.06 0.5 1.28 1.28 0.99 1.5

yes yes yes yes yes yes yes

2 3 2 2 2 4 4

extension, the multiscale structural similarity index (MS-SSIM) [25], where the objective quality evaluation at each image scale contributes to the overal scorel. The SSIM metric combines local image structure, luminance, and contrast into a single local quality score. In this metric, structures are patterns of pixel intensities, especially among neighboring pixels, after normalizing for luminance and contrast. Because the human visual system is good at perceiving structure, the SSIM quality metric agrees more closely with the subjective quality score. The SSIM perceived quality by quantifying the SSIM between an image and Kolev et al.: Preprint submitted to Elsevier

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a reference image [24]: ̃ = 𝑚𝐾 (𝐈, 𝐈)𝑣 ̃ 𝐾 (𝐈, 𝐈) ̃ = 𝑆𝑆𝐼𝑀(𝐈, 𝐈)

(2𝜇𝑥 𝜇𝑦 + 𝐶1 )(2𝜎𝑥𝑦 + 𝐶2 ) (𝜇𝑥2 + 𝜇𝑦2 + 𝐶1 )(𝜎𝑥2 + 𝜎𝑦2 + 𝐶2 )

,

̃ is the luminance distortion, 𝑣𝐾 (𝐈, 𝐈) ̃ are the contrast components, and 𝜇𝑥 , 𝜇𝑦 , 𝜎𝑥 , 𝜎𝑦 and 𝜎𝑥𝑦 are the where 𝑚𝐾 (𝐈, 𝐈) ̃ 𝑦) with 𝐶1 = 2.55, 𝐶2 = 7.65. local means, standard deviations, and cross-covariance for images 𝐈(𝑥, 𝑦) and 𝐈(𝑥, MS-SSIM [25] expands on the SSIM index by combining luminance information with the variance and crosscorrelation components at scale 𝐾. The multiple scales account for variability in the perception of image details caused by factors such as viewing distance from the image, distance from the scene to the sensor, and resolution of the image acquisition sensor. The MS-SSIM index for scale 𝐾 is given by [25] ̃ 𝛼𝐾 𝑀𝑆 − 𝑆𝑆𝐼𝑀 = 𝑚𝐾 (𝐈, 𝐈)

𝐾 ∏

̃ 𝛽𝑘 𝑟𝑘 (𝐈, 𝐈) ̃ 𝛾𝑘 , 𝑣𝑘 (𝐈, 𝐈)

𝑘=1

where ̃ = 𝑟𝐾 (𝐈, 𝐈)

𝜎𝑥𝑦 + 𝐶3 𝜎𝑥 𝜎𝑦 + 𝐶 3

,

and 𝐶3 = 𝐶2 ∕2 is the structure component at scale 𝐾 (where 𝐾 = 1 is the full-resolution image). For 𝐾 = 5, the exponents are 𝛽1 = 𝛾1 = 0.0448, 𝛽2 = 𝛾2 = 0.2856, 𝛽3 = 𝛾3 = 0.3001, 𝛽4 = 𝛾4 = 0.2363, and 𝛼5 = 𝛽5 = 𝛾5 = 0.1333. The best quality values for structural information are obtained for SSIM = MS-SSIM = 1. Image compression of the gray level images Bart Run, Cameraman and Kiel, with sizes 1282 , 2562 , and 5122 pixels respectively, is applied using balanced GHM, CL, SA4, Alpert, and the two new supercompact multifilters. The balancing for GHM and CL is according to [17]; for SA4, Alpert, and the new supercompact multifilters it is of the Haar type. Results are shown in Tables 4–9. All minimal MSE and maximal PSNR, SSIM and MS-SSIM values are marked in bold. Table 4 shows results for the gray level image Bart Run. When CR = 4, all multiwavelet filter banks from decomposition levels 2 through 5 achieve nearly constant SSIM and MS-SSIM. The lowest MSE and highest values for PSNR, SSIM and MS-SSIM are obtained by the Integer Haar multiwavelet. The supercompact multiwavelets compared with other multiwavelets (GHM, SA4 and CL) leads to better PSNR, SSIM and MS-SSIM. When CR = 8, nearly constant SSIM and MS-SSIM are achieved for all multiwavelet filter banks from the third to the fifth level decomposition. The lowest MSE and highest values for PSNR are achieved by using the Alpert multiwavelet. We need to note that better MSE and PSNR do not lead to better SSIM and MS-SSIM. The highest SSIM is by using CL multiwavelet while and highest MS-SSIM values are achieved for 3 and more level decompositions from the New1 supercompact multiwavelet. The next two CR = 16 and 32 show that at the first compression level all the measured values for the Alpert, New1 and New2 multiwavelets are equal. This comes from the similarity in structure of the pairs of scaling and wavelet functions. Moreover, Alpert and New1 multiwavelet filters achieve best PSNRs, SSIMs and MS-SSIMs. For CR =16, best MSEs and PSNRs values are for the New1 multiwavelet at 3 and more decomposition level, while for CR = 32 it is the Alpert multiwavelet. With respect to SSIMs and MS-SSIMs for CR = 16, the Alpert multiwavelet achieved the best values for 3 and more decomposition levels, while for CR= 32 the best values are from 1 to 3 decomposition levels with the New1 multiwavelet filter. Table 5 shows results for the gray level image Cameraman. For CR = 4 GHM has better MS-SSIM, but for CR = 8, 16 and 32 the New1 supercompact multifilter filter for 3 and more decomposition levels achieved better values than all the rest. It is important to note that all supercompact multiwavelets have nearly constant values for PSNR, SSIM and MS-SSIM for 4 and more decomposition levels. Table 6 shows results for the gray level image Kiel. For 4 and more decomposition levels and CR = 4, 8 and 16 all multiwavelets achieved nearly constant MSEs and PSNRs. Better results are obtained for the two supercompact Alpert and New1 multiwavelet filters. For the three considered CRs, the New1 multiwavelet has better values for MS-SSIM, MSE and PSNR. Kolev et al.: Preprint submitted to Elsevier

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Table 7 shows results for the color image House. The PSNR values of all multiwavelet filters for CR = 4 and 8 for 4 and more decomposition levels and for CR = 16 and 32 for 5 and 6 decomposition levels are nearly constant, For all CRs, the best PSNR, SSIM and MS-SSIM with 3 and more decomposition levels were achieved again by the New1 multiwavelet filter. Moreover, with CR = 4, its MS-SSIM values equal those the of the SA4 multiwavelet filter, which is longer with more vanishing moments. Table 8 shows results for the color image Cara. The best measured values for CR = 4 of all decomposition levels are achieved by the New1 multiwavelet. The SSIM and MS-SSIM values of GHM, SA4 and Alpert multiwavelets are equal for 2 and more levels. We need to note that for CR = 4 and 8 at 3 and more decomposition levels all multifilters have nearly constant SSIM and MS-SSIM values. For CR = 8, best measured values are achieved again by the New1 multiwavelet, except for 1 and 2 decomposition levels, and the MS-SSIM values are equal with GHM and Alpert for 3 and more levels. For CR = 16, for 1 and 2 decomposition levels the SA4 multifilter has the best SSIM values, for the rest the best measured values are achieved by the New1 multiwavelet. For CR = 32, the New1 multiwavelet achieves all best measured values, except MSE and PSNR at 1, 2 and 3 decomposition levels. The MS-SSIM values of the supercompact multiwavelet filters are nearly constant for more than 4 decomposition levels. Table 9 shows results for the color image Voit. For CR = 4 and 8, all multiwavelet filters at 4 and more decomposition levels have nearly constants measured values. With a minimal difference from the SA4 multiwavelet, all best measured values are achieved again by the New1 multiwavelet filter. For CR = 4 and 8, the New1 multiwavelet filter has the best MSE, PSNR and SSIM for 3 and more decomposition levels. The best MS-SSIM, is achieved by the SA4 multiwavelet, except at the 2nd level. For CR = 32, from 1 to 3 decomposition level the best SSIM is with the SA4 multiwavelet filter, while the best MS-SSIM values are for CL, New1 and SA4 multiwavelets, respectively. For 4 and more decomposition level, the New1 multiwavelet has the best SSIM and MS–SSIM values, while its PSNR values are equal with Alpert multiwavelet. Table 10 shows results for the color image Texture 1. For CR = 2 for decomposition levels bigger than 1, the New1 multifilter has max PSNR, SSIM and MS-SSIM values. For CR = 4, the SA4 multifilter has max PSNR and SSIM values for all decomposition levels, which are slightly higher (0.03dB and 0.020 respectively) than New1, while New1 has max MS-SSIM slightly higher (0.20dB, 0.020dB and 0.03dB, respectively) than New2. For CR = 8, SA4 has max PSNR and SSIM values for all decomposition level, which are slightly higher (0.03dB, 0.04dB and 0.012dB, respectively) than New1. Table 11 shows results for the color image Texture 4. Since the image allows superb values at higher compression levels, the beginning value is CR = 64. For CR=64, the max PSNR value at the first decomposition level is achieved by the SA4 multifilter, slightly higher (by 0.04dB) than New1. At the second decomposition level, New1 is better. For decompositin levels 3 to 7, Integer Haar has the max PSNR levels, significantly higher (by 0.64dB – 1.36dB) than New1. For SSIM and MS-SSIM, the maximal values at the first 3 decomposition levels are for New1 and New2, and at levels 4 to 7 the Integer Haar multifilter has negligibly higher levels (by 0.003dB). For CR = 128, max PSNR at the first decomposition level is for SA4, higher by 0.02dB than New1. The New1 multifilter has superior values for PSNR, SSIM and MS-SSIM at levels 2 to 7 (6.5dB better than CL and SA4, 12dB better than GHM and 6.5dB better than New2). For CR = 256, SA4 has the maximal PSNR values at the first and second decomposition levels, better by 0.03dB and 0.05dB respectively) than New1. New1 has max SSIM and MS-SSIM values. From levels 3 to 7 all maximal values are for New1, bigger by around 5.0dB than SA4 and CL, and around 8.0dB better than GHM. For CR = 512, SA4 achieves the maximal PSNR values at the first and second decomposition levels, negligibly higher (by 0.03dB and 0.04dB, respectively) than New1. At the second level, New1 have max SSIM and MS-SSIM values. From levels 3 to 7 New1 has all maximal values, bigger by around 5.0dB than SA4 and CL, and around 6.0dB better than GHM. The multifilter New1 applied to the image Texture 4, which has a strongly combed histogram, achieves better results for all metrics, with minimal differences for all compression levels. Table 12 shows results for the grayscale image Brick wall. For CR=16, at the first decomposition level SA4 has max PSNR values, while at levels 3 to 7 CL is best, negligibly higher (by 0.01dB) than New1. The max SSIM is for the GHM multifilter, while the SA4 multifilter has max MS-SSIM, negligible higher (by 0.004dB) than New1. For CR=32 and 64, SA4 has the max PSNR, SSIM and MS-SSIM values at decomposition levels 1 and 2, whuile for levels 3 to 7 the SA4 multifilter New1 has all max values.

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Table 13 shows results for the grayscale image Texture 5. For CR = 4, 8 and 16, the New1 multifilter has max PSNR, SSIM and MS-SSIM values at all decomposition levels. For CR=32, the SA4 multifilter has maximal values for the first and second decomposition levels, while New1 has max values for the third to seventh decomposition levels. Table 14 shows results for the color image Subwater. For CR= 8 all maximal values are obtained for SA4, slightly higher (by 0.17dB and 0.004dB, respectively) than New1; the values of MS-SSIM are equal. For CR=16, max PSNR, SSIM ans MS-SSIM values are obtained by New1, except at the second level, where the values are negligible lower (by 0.06dB, 0.0007dB and 0.0001dB, respectively) than SA4. Note that SA4 and New1 multifilters have constant MS-SSIM from the third to the seventh decomposition level. For CR=32, all maximal values are obtained for New1, significantly higher than the others (by around 1.00dB for CL, around 0.70dB) for SA4 and even by 3.00dB for GHM).

4.4.3. Signal denoising Multiwavelet denoising is a non-linear process, which depends heavily on the choice of a thresholding parameter. For a better comparison of the new supercompact and orthogonal multifilters New1 and New2 with orthogonal GHM, SA4, CL, Alpert and Integer Haar multifilters, noise reduction in 1D and 2D is considered.

1D denoising We apply denoising with seven orthogonal multifilters (GHM, SA4, CL, Alpert, Integer Haar, New1

and New2) to three 1D three test signals s(t) (Piece-Polynomial and Cusp with a length of 1024 values and HeaviSine with 4096), with two different additive white Gaussian noise (AWGN) values (𝜎=10, 15). The original, noisy signals are shown in Fig. 9(a)-(f). The denoised signals, with a zoom of some parts are shown in Fig. 10(a)-(f). The Piece-Polynomial noisy signal is more complex; it consists of exponential and others polynomial segments with edges and jumps. The zoom images of the simulation plots give us enough information to recognize the quality of approximation for all multifilters. For all three test signals and both noise values, Fig. 10(c) and (d) show that the New2 multifilter has the best plots for cruder numerical approximations. The scaling part of the HeaviSine signal shows better approximations of sudden changes by both the New2 and New1 multifilters. The zoom plots of Piece-Polynomial and Cusp signals for 𝜎=10, 15 show better approximation using the New1 and GHM multifilters, as shown in Fig. 10(a),(b),(c) and (d). Using the Alpert multifilter for Piece-Polynomial and Cusp signals leads to spurious peak values. To complete the analysis, we use the maximal absolute error (MAE): 𝜖 = ||𝑠(𝑡) − 𝑠(𝑡)|| ̂ ∞.

(13)

The MAE values are tabulated in Table 15. Since the Piece-Polynomial noisy signal is more complex, its MAEs are bigger than for other signals. We see that the minimal MAE values for 𝜎=10 are obtained by both the New1 and New2 multifilters, for 𝜎=15 only by New1. The noise suppression results obtained for the Cusp signal shows again minimal MAE values for both New1 and New2 multifilters for 𝜎=10, 15. The results for the HeaviSine signal show minimal MAE value using for 𝜎=10, 15. A comparison of the results shows that the proposed New1 and New2 multifilters can provide more effective noise suppression and better approximation than the other five multifilters.

2D denoising Image denoising is the process of removing noise from a noisy image while preserving the structure of the original image. The best way to test the effect of noise on a standard grayscale image is to add AWGN. We apply thresholding methods to suppress noise from grayscale images of size 256 × 256 and 512 × 512 pixels shown in Fig.4 and Fig.5 and the multiwavelet decomposition for seven orthogonal multiwavelets GHM, SA4, CL, Alpert, Integer Haar, New1 and New2. The AWGN has variances 𝜎=10, 20. In order to suppress the noise coefficients in all test images, soft or hard vector thresholding is applied at five decomposition levels. Best results are shown in boldface. Since a single one criterion is not enough to compare denoising quality, four well-defined criteria are introduced: MSE, PSNR, SSIM, and MS-SSIM. Table 16 gives denoising results for the soft and hard thresholds of a noisy Cameraman image with 𝜎=10. The SA4 multifilter has max PSNR, SSIM and MS-SSIM values for all measures only for a soft threshold at the first decomposition level. The New1 multifilter achieves highest PSNR, SSIM and MS-SSIM values from the second to the fifth decomposition level for both thresholding methods, while the New2 multifilter achieves only for the hard thresholding method. Table 17 gives denoising results with the soft and hard thresholds and 𝜎=10 for the Couple image. The results shows that for all decomposition levels the highest PSNR and SSIM values are achieved by the SA4 multifilter with Kolev et al.: Preprint submitted to Elsevier

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soft thresholding, while New1 achieves the highest MS-SSIM values. With hard thresholding, the New1 multifilter achieves the highest PSNR, SSIM and MS-SSIM values for all decomposition levels. Table 18 gives denoising results with soft and hard thresholding and 𝜎=10 for the Kiel image. The results show that, except for the first level of decomposition, the multifilter New1 has highest PSNR, SSIM and MS-SSIM values for both thresholding methods. Table 19 gives denoising results with soft and hard thresholding and 𝜎=20 for the House image. The New1 multifilter achieves the highest PSNR, SSIM and MS-SSIM values for soft thresholding from the second to fifth decomposition levels, as well as for hard thresholding from the third to the fifth level . Table 20 gives denoising results with soft and hard thresholding and 𝜎=20 forthe Kiel image. The results show that for soft thresholding the highest PSNR, SSIM, and MS-SSIM values are achieved by New1 from the second to the fifth decomposition levels, and for hard thresholding from the third to the fifth level. From the results, we can see that in the case of small noise variance 𝜎=10 the New1 multifilter has the best denoising. Finally, MS-SSIM, which generalizes SSIM over all decomposition levels, gives us information that the New1 multiwavelet filter has better HVPs for image quality and structure.

5. Conclusions Two new orthogonal bases for supercompact multiscaling functions with exact recursion coefficients are obtained from the CL matrix product filter by the fast Bauer’s method of matrix spectral factorization. The complementary supercompact multiwavelet functions are also found by QR decomposition. The two new multifilters, even with very short support, show relatively good CG performance, and better Sobolev smoothness than other well-known orthogonal and biorthogonal multiwavelets. Considering the highest MSE, PSNR, SSIM, and MS-SSIM(in bold) for CR at all decomposition levels for image compression and denoising, the new supercompact multiwavelet New1 (in same case also New2) achieves better performance than the other multiwavelet filters. The measure MS-SSIM tells us that by applying New1 multifilters, better HVPs for image quality and structure can be achieved in some cases. From comparison with the well-known orthogonal Alpert, GHM, SA4, CL and Integer Haar multiwavelets, we can conclude that the multifilters New1 and New2 can provide more effective noise suppression of some one-dimensional signals, can be also useful in image compression and subband-based edge detection applications as well. In the future, these new supercompact bases could be used for solving other different mathematical and engineering applications.

6. Conflict of interest The author declares that there is no conflict of interest.

7. Data availability statement Data available on request from the authors.

8. Acknowledgments The authors would like to thank the editor, two anonymous reviewers for their careful reading, their constructive and valuable comments and very helpful suggestions which helped us to improve the manuscript.

References [1] R. M. Beam, R. F. Warming, Multiresolution analysis and supercompact multiwavelets, SIAM Journal on Scientific Computing 22 (4) (2000) 1238–1268. [2] D. Lee, R. M. Beam, R. F. Warming, Supercompact multiwavelets for flow field simulation, Computers & Fluids 30 (7) (2001) 783–805. [3] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996. [4] T. Cooklev, A. Nishihara, M. Kato, M. Sablatash, Two-channel multifilter banks and multiwavelets, in: 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings, Vol. 5, 1996, pp. 2769–2772. [5] V. Kolev, T. Cooklev, F. Keinert, Matrix spectral factorization for SA4 multiwavelet, Multidimensional Systems and Signal Processing 29 (4) (2018) 1613–1641.

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New Orthogonal Multiwavelet Filters ... [6] V. Kolev, T. Cooklev, F. Keinert, Bauer’s spectral factorization method for low order multiwavelet filter design, Journal of Computational and Applied Mathematics 441, article 115713 (2024). [7] C. K. Chui, J.-A. Lian, A study of orthonormal multi-wavelets, Appl. Numer. Math. 20 (3) (1996) 273–298. [8] L. Fejér, Über trigonometrische Polynome, J. Reine Angew. Math. (Crelles J.) 146 (1916) 53–82. [9] F. Riesz, Über ein Problem des Herrn Carathéodory, J. Reine Angew. Math. (Crelles J.) 146 (1916) 83–87. [10] R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Colloquium Publications AMS, New York, 1933. [11] N. Wiener, On the factorization of matrices, Commentarii Mathematici Helvetici 29 (1) (1955) 97–111. [12] A. H. Sayed, T. Kailath, A survey of spectral factorization methods, Numerical Linear Algebra With Applications 8 (6–7) (2001) 467–496. [13] Q. Jiang, On the regularity of matrix refinable functions, SIAM J. Math. Anal. 29 (5) (1998) 1157–1176. [14] V. Strela, A note on construction of biorthogonal multi-scaling functions, in: Wavelets, Multiwavelets, and their Applications (San Diego, CA, 1997), Vol. 216 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1998, pp. 149–157. [15] A. Z. Averbuch, V. A. Zheludev, T. Cohen, Multiwavelet frames in signal space originated from Hermite splines, IEEE Trans. Signal Process. 55 (3) (2007) 797–808. [16] K. Rajakumar, T. Arivoli, Lossy image compression using multiwavelet transform for wireless transmission, Wireless Personal Communications 87 (2) (2016) 315–333. [17] V. Strela, A. Walden, Signal and image denoising via wavelet thresholding: Orthogonal and biorthogonal, scalar and multiple wavelet transforms, in: W. J. Fitzgerald, R. L. Smith, A. T. Walden, P. C. Young (Eds.), Nonlinear and Nonstationary Signal Processing, Cambridge University Press, 2001, pp. 395–441. [18] L. Shen, H. H. Tan, J. Y. Tham, Symmetric-antisymmetric orthonormal multiwavelets and related scalar wavelets, Appl. Comput. Harmon. Anal. 8 (3) (2000) 258–279. [19] J. S. Geronimo, D. P. Hardin, P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (3) (1994) 373–401. [20] P. P. Vaidyanathan, Theory of optimal orthonormal subband coders, IEEE Transactions on Signal Processing 46 (6) (1998) 1528–1543. [21] J. Canny, A computational approach to edge detection, IEEE Transactions on pattern analysis and machine intelligence (6) (2009) 679–698. [22] N. Mohsenian, N. M. Nasrabadi, Edge-based subband vq techniques for images and video, IEEE transactions on circuits and systems for video technology 4 (1) (2002) 53–67. [23] S.-K. Im, K.-H. Chan, Vector quantization using k-means clustering neural network, Electronics Letters 59 (7) (2023) e12758. [24] R. D. M., H. S. S, Understanding and simplifying the structural similarity metric, in: 15th IEEE International Conference on Image Processing, 2008, pp. 1188–1191. [25] Z. Wang, E. P. Simoncelli, A. C. Bovik, Multiscale structural similarity for image quality assessment, in: The Thirty-Seventh Asilomar Conference on Signals, Systems & Computers, Vol. 2, 2003, pp. 1398—-1402. [26] Huang X., Chan K.H., Wu W., Sheng H. and Ke W., Fusion of multi-modal features to enhance dense video caption, Sensors, vol. 23, no.12, p.5565, 2023.

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Figure 7: Pixel correlations and linear fit lines for the gray and color (RGB channels) images; (a)House (256 × 256); (b)Couple(256×256); (c)Brick Wall(1024×1024); (d)Texture 5(256×256); (e)Texture 1(512×512); (f)Subwater(256× 128); (g)Texture 4(1024 × 1024).

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Figure 8: Subband-based edge detection for a one decomposition level of following orthogonal transforms by using the Canny method: (a) Original image; (b) GHM multiwavelet; (c) SA4 multiwavelet; (d) CL multiwavelet; (e) Alpert multiwavelet; (f) New1 multiwavelet; (g) New2 multiwavelet; (h) Integer Haar multiwavelet.

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Table 4 Results for Bart Run. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

Bart Run

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 86.167 28.78 0.9031 0.9509 1.0416 47.95 0.9935 0.9995 0.5866 50.45 0.9967 0.9997 0.6630 49.92 0.9966 0.9997 0.6096 50.28 0.9967 0.9997 1.1408 47.56 0.9948 0.9995 0.6234 50.18 0.9972 0.9997

Decomposition level 2 3 4 3.0603 2.4312 2.3896 43.27 44.27 44.35 0.9830 0.9857 0.9859 0.9984 0.9986 0.9986 1.0153 1.0305 1.0559 48.07 48.00 47.89 0.9932 0.993 0.9929 0.9994 0.9994 0.9994 0.5192 0.4995 0.5065 50.98 51.15 51.08 0.9969 0.9969 0.9969 0.9997 0.9997 0.9997 0.5090 0.4955 0.4993 51.06 51.18 51.15 0.9970 0.9971 0.9971 0.9997 0.9997 0.9997 0.4987 0.4923 0.4983 51.15 51.21 51.16 0.9972 0.9972 0.9971 0.9997 0.9997 0.9997 0.9912 0.9570 0.9782 48.17 48.32 48.23 0.9949 0.9951 0.9950 0.9994 0.9994 0.9994 0.4246 0.3891 0.3919 51.85 52.23 52.20 0.9977 0.9978 0.9978 0.9998 0.9998 0.9998

5 2.3896 44.35 0.9859 0.9986 1.0548 47.9 0.9929 0.9994 0.5111 51.05 0.9969 0.9997 0.5040 51.11 0.997 0.9997 0.5021 51.12 0.9971 0.9997 0.9921 48.17 0.9949 0.9994 0.3984 52.13 0.9978 0.9998

5385.1 10.82 0.2538 0.6421 9.9828 38.14 0.9507 0.9953 9.8477 38.20 0.9531 0.9936 10.4090 37.96 0.9458 0.9955 9.9606 38.15 0.9494 0.9958 14.023 36.66 0.9244 0.9918 12.76 37.07 0.9411 0.9932

15.463 36.24 0.9241 0.9912 6.9655 39.70 0.9633 0.9957 6.6463 39.91 0.9668 0.9956 6.3780 40.08 0.9658 0.9962 6.4132 40.06 0.9646 0.9960 11.193 37.64 0.9381 0.9919 8.2607 38.96 0.9593 0.9950

11.071 37.69 0.9428 0.9936 6.9234 39.73 0.9632 0.9954 6.0143 40.34 0.9695 0.9956 5.7534 40.53 0.9680 0.9960 6.0197 40.34 0.9681 0.9960 10.767 37.81 0.9444 0.9931 7.6042 39.32 0.9620 0.9952

11.505 37.52 0.9404 0.9934 6.815 39.80 0.9634 0.9955 5.931 40.40 0.9699 0.9957 5.7271 40.55 0.9683 0.9961 5.9374 40.39 0.9684 0.9962 10.575 37.89 0.9460 0.9932 7.4608 39.40 0.9628 0.9953

11.127 37.67 0.9425 0.9936 6.9071 39.74 0.9634 0.9954 5.973 40.37 0.9696 0.9956 5.7040 40.57 0.9682 0.9960 5.9642 40.38 0.9682 0.9962 10.776 37.81 0.944 0.9931 7.5038 39.38 0.9625 0.9952

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Table 4(continued) Results for Bart Run. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

Bart Run

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 11837 7.399 0.0142 0.4809 189.23 25.36 0.766 0.9008 240.87 24.31 0.7795 0.8490 64.561 30.03 0.7730 0.9622 64.561 30.03 0.7730 0.9622 64.561 30.03 0.7730 0.9622 317.99 23.11 0.7559 0.8150

Decomposition level 2 3 4 189.7 27.636 25.712 25.35 33.72 34.03 0.7820 0.8719 0.8798 0.8622 0.9792 0.9806 19.975 18.572 18.818 35.13 35.44 35.39 0.9033 0.9099 0.9089 0.9850 0.9869 0.9863 22.234 18.827 18.763 34.66 35.38 35.40 0.8946 0.9076 0.9076 0.9829 0.9852 0.9848 20.539 17.289 17.243 35.00 35.75 35.76 0.8943 0.9111 0.9113 0.9852 0.9881 0.9879 20.152 17.055 17.054 35.09 35.81 35.81 0.8969 0.9107 0.9105 0.9856 0.9878 0.9874 29.797 27.838 28.114 33.39 33.68 33.64 0.8478 0.8597 0.8586 0.9743 0.9777 0.9770 26.172 23.221 23.302 33.95 34.47 34.46 0.8760 0.8878 0.8872 0.9791 0.9816 0.9811

5 25.596 34.05 0.8802 0.9806 18.889 35.37 0.9086 0.9862 18.888 35.37 0.9072 0.9846 17.264 35.76 0.9113 0.9878 17.251 35.76 0.9096 0.9872 28.25 33.62 0.8581 0.9768 23.494 34.42 0.8862 0.9809

17368 5.733 0.00814 0.4401 5415.3 10.79 0.3788 0.4941 5269.8 10.91 0.378 0.5236 5018 11.13 0.3735 0.6502 5018 11.13 0.3735 0.6502 5018 11.13 0.3735 0.6502 5524.4 10.71 0.3684 0.4943

4704.3 11.41 0.2671 0.5499 47.809 31.34 0.8026 0.9528 56.575 30.60 0.7439 0.9347 44.849 31.61 0.7993 0.9596 44.08 31.69 0.8029 0.9609 57.571 30.53 0.7503 0.9389 58.614 30.45 0.7495 0.9393

50.794 31.07 0.7976 0.9542 38.058 32.33 0.842 0.9663 37.993 32.33 0.8338 0.9652 32.648 32.99 0.8462 0.9718 32.927 32.96 0.8462 0.9713 51.497 31.01 0.7693 0.9495 47.93 31.32 0.8003 0.9579

73.563 29.46 0.7643 0.9338 37.967 32.34 0.8412 0.9681 38.712 32.25 0.8311 0.9650 32.806 32.97 0.8463 0.9711 32.803 32.97 0.8478 0.9728 51.015 31.05 0.7703 0.9510 48.135 31.31 0.801 0.9586

51.504 31.01 0.7942 0.9531 37.859 32.35 0.8424 0.9664 37.714 32.37 0.8342 0.9654 32.498 33.01 0.8463 0.9722 32.601 33.00 0.8469 0.9717 51.224 31.04 0.7692 0.9500 47.572 31.36 0.8023 0.9585

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Table 5 Results for Cameramsan. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

cameraman

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 65.773 29.95 0.846 0.9778 6.4772 40.02 0.9646 0.9946 6.3258 40.12 0.9637 0.9941 6.1808 40.22 0.9637 0.9952 6.1103 40.27 0.964 0.9953 8.0861 39.05 0.9584 0.9944 7.045 39.65 0.9616 0.9935

2 9.814 38.21 0.9596 0.9945 5.6798 40.59 0.9656 0.9945 5.2186 40.96 0.9655 0.9939 5.1158 41.04 0.9662 0.9941 5.0703 41.08 0.9664 0.9942 7.0528 39.65 0.9585 0.9920 5.9673 40.37 0.9628 0.9933

4488 11.61 0.1478 0.6785 39.525 32.16 0.9050 0.9826 41.284 31.97 0.8995 0.9793 40.38 32.07 0.8986 0.9859 39.706 32.14 0.9004 0.9863 52.833 30.90 0.8819 0.9822 48.803 31.25 0.8869 0.9742

54.003 30.81 0.8883 0.98 27.101 33.8 0.9197 0.9838 26.052 33.97 0.9192 0.9819 25.849 34.01 0.9193 0.9841 25.503 34.06 0.9204 0.9844 37.008 32.45 0.8995 0.9793 32.369 33.03 0.9078 0.9794

Decomposition level 3 4 5 7.9468 7.7663 7.7578 39.13 39.23 39.23 0.963 0.9633 0.9633 0.9947 0.9947 0.9947 5.7046 5.7165 5.7208 40.57 40.56 40.56 0.9659 0.9659 0.9659 0.9945 0.9945 0.9945 5.1314 5.129 5.1226 41.03 41.03 41.04 0.9663 0.9664 0.9665 0.9942 0.9941 0.9941 4.9445 4.9299 4.9375 41.19 41.20 41.20 0.9672 0.9672 0.9672 0.9946 0.9945 0.9945 4.9072 4.8982 4.9033 41.22 41.23 41.23 0.9675 0.9675 0.9674 0.9947 0.9946 0.9946 7.0942 7.1256 7.1481 39.62 39.6 39.59 0.9585 0.9584 0.9583 0.9923 0.9923 0.9923 5.8743 5.8631 5.8739 40.44 40.45 40.44 0.9633 0.9634 0.9633 0.9935 0.9936 0.9935 37.34 32.41 0.9086 0.9835 26.908 33.83 0.9188 0.9820 25.502 34.06 0.9203 0.9809 24.791 34.19 0.9215 0.9831 24.488 34.24 0.9226 0.9836 36.229 32.54 0.8990 0.9743 31.122 33.20 0.9080 0.9783

35.658 32.61 0.9111 0.9833 26.987 33.82 0.919 0.9818 25.39 34.08 0.9209 0.9813 24.635 34.22 0.9222 0.9839 24.351 34.27 0.9231 0.9841 36.612 32.49 0.8976 0.9753 31.10 33.20 0.9077 0.9787

35.56 32.62 0.9113 0.9833 27.012 33.82 0.919 0.9818 25.433 34.08 0.9208 0.9815 24.693 34.21 0.9221 0.9838 24.413 34.25 0.9228 0.9841 36.775 32.48 0.8971 0.975 31.151 33.20 0.9075 0.9787

6 7.7579 39.23 0.9634 0.9947 5.7204 40.56 0.9659 0.9945 5.1235 41.04 0.9665 0.9941 4.9356 41.20 0.9672 0.9945 4.9044 41.22 0.9674 0.9946 7.1492 39.59 0.9583 0.9923 5.874 40.44 0.9633 0.9935 35.538 32.62 0.9113 0.9833 27.013 33.82 0.9189 0.9818 25.427 34.08 0.9208 0.9815 24.698 34.20 0.922 0.9838 24.429 34.25 0.9228 0.9841 36.796 32.47 0.8971 0.9751 31.15 33.20 0.9074 0.9787

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Table 5(continued) Results for Cameraman. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

cameraman

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 9398.3 8.4 0.05697 0.5289 180.76 25.56 0.6773 0.9446 203.43 25.05 0.6668 0.9383 202.99 25.06 0.665 0.9601 199.72 25.13 0.6665 0.9605 234.83 24.42 0.6541 0.9566 229.89 24.52 0.655 0.9284

Decomposition level 2 3 4 5 206.16 97.873 89.405 88.794 24.99 28.22 28.62 28.65 0.6939 0.8304 0.837 0.8379 0.9337 0.9586 0.959 0.959 75.635 71.015 71.511 71.684 29.34 29.62 29.59 29.58 0.8468 0.8472 0.8458 0.8455 0.9618 0.9562 0.9564 0.956 75.691 70.133 70.075 70.233 29.34 29.67 29.68 29.67 0.8445 0.8466 0.8486 0.8481 0.9594 0.9550 0.956 0.9555 75.024 67.809 67.454 67.673 29.38 29.82 29.84 29.83 0.8436 0.8472 0.8492 0.8488 0.9641 0.9582 0.962 0.962 73.971 66.886 66.615 66.842 29.44 29.88 29.90 29.88 0.8448 0.8497 0.8516 0.8515 0.9651 0.9594 0.9631 0.9628 105.86 99.378 100.62 101.19 27.88 28.16 28.1 28.08 0.8131 0.8135 0.81 0.8088 0.954 0.9460 0.9444 0.9441 97.676 87.721 87.61 87.707 28.23 28.7 28.71 28.7 0.8190 0.8212 0.8193 0.8191 0.9495 0.9467 0.9479 0.9479

6 88.729 28.65 0.8379 0.959 71.656 29.58 0.8456 0.9559 70.217 29.67 0.8482 0.9556 67.704 29.82 0.8488 0.9622 66.858 29.88 0.8516 0.9630 101.3 28.07 0.8087 0.9442 87.699 28.7 0.8191 0.9478

12589 7.131 0.02968 0.4926 4094.8 12.01 0.4219 0.6840 4190.9 11.91 0.4191 0.6926 4250.4 11.85 0.4214 0.7224 4250.4 11.85 0.4214 0.7224 4250.4 11.85 0.4214 0.7224 4295.2 11.80 0.4115 0.6747

3469.7 12.73 0.1307 0.5191 180.79 25.56 0.7478 0.9209 188.21 25.38 0.7432 0.9178 186.87 25.42 0.7467 0.931 183.77 25.49 0.7503 0.9319 247.83 24.19 0.7057 0.9141 238.78 24.35 0.7034 0.8994

175.25 25.69 0.7616 0.9201 147.59 26.44 0.7635 0.9133 149.68 26.38 0.7670 0.9152 143.71 26.56 0.7734 0.9269 141.41 26.63 0.7752 0.9263 211.87 24.87 0.727 0.8954 185.24 25.45 0.733 0.9046

214.42 24.82 0.7325 0.9122 145.72 26.50 0.7673 0.9179 150.33 26.36 0.7676 0.9184 144.55 26.53 0.7739 0.9274 142.56 26.59 0.7756 0.9276 209 24.93 0.7383 0.9092 188.02 25.39 0.7408 0.9052

179.14 25.60 0.7593 0.92 147.17 26.45 0.7637 0.9132 149.38 26.39 0.7672 0.9156 143.11 26.57 0.7753 0.9283 140.93 26.64 0.7768 0.9285 208.59 24.94 0.7307 0.8941 185.13 25.46 0.7336 0.9036

175.67 25.68 0.7615 0.9203 147.62 26.44 0.7636 0.9131 149.54 26.38 0.767 0.9154 143.5 26.56 0.7735 0.9268 141.35 26.63 0.7755 0.9264 211.29 24.88 0.7273 0.8962 185.47 25.45 0.7331 0.9044

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Table 6 Results for Kiel. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

Kiel

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

Decomposition level 3 4 5 28.619 27.844 27.761 33.56 33.68 33.70 0.9853 0.9855 0.9855 0.9917 0.9917 0.9916 20.825 20.849 20.854 34.94 34.94 34.94 0.983 0.983 0.983 0.9909 0.9907 0.9908 19.694 19.679 19.70 35.19 35.19 35.19 0.9824 0.9824 0.9823 0.9908 0.9907 0.9906 19.713 19.634 19.631 35.18 35.20 35.20 0.9841 0.9841 0.9842 0.9914 0.9914 0.9914 19.605 19.547 19.542 35.21 35.22 35.22 0.9842 0.9842 0.9843 0.9914 0.9915 0.9915 26.337 26.378 26.423 33.93 33.92 33.91 0.9783 0.9781 0.978 0.9886 0.9884 0.9883 22.629 22.644 22.664 34.58 34.58 34.58 0.9817 0.9817 0.9817 0.9899 0.9898 0.9899

1 235.84 24.40 0.9432 0.9688 24.975 34.16 0.9846 0.9913 25.139 34.13 0.9805 0.9903 25.241 34.11 0.9844 0.9926 25.028 34.15 0.9847 0.9927 29.828 33.38 0.9803 0.9911 27.443 33.75 0.9824 0.99

2 34.756 32.72 0.9844 0.9911 21.198 34.87 0.9835 0.9913 20.194 35.08 0.9826 0.9911 20.372 35.04 0.9842 0.9918 20.25 35.07 0.9844 0.9919 26.691 33.87 0.9788 0.9893 23.167 34.48 0.9818 0.9902

5841.7 10.47 0.2796 0.6034 118.98 27.38 0.9411 0.9677 123.00 27.23 0.9271 0.9645 123.4 27.22 0.9408 0.9748 122.38 27.25 0.9420 0.9751 137.67 26.74 0.9344 0.9724 132.02 26.92 0.9351 0.964

156.69 26.18 0.9397 0.9669 82.231 28.98 0.9476 0.9732 80.573 29.07 0.9441 0.9719 81..216 29.03 0.9482 0.9747 80.805 29.06 0.9490 0.9751 102.07 28.04 0.9322 0.9672 90.962 28.54 0.9408 0.9693

109.85 27.72 0.9482 0.9722 78.496 29.18 0.9475 0.9722 76.112 29.32 0.9440 0.9710 76.271 29.31 0.9486 0.9734 75.999 29.32 0.9492 0.9738 98.629 28.19 0.931 0.9652 86.343 28.77 0.9416 0.9687

104.61 27.93 0.9491 0.9724 78.299 29.19 0.9469 0.971 75.721 29.34 0.9435 0.9703 75.632 29.34 0.9484 0.9731 75.447 29.35 0.9491 0.9736 98.492 28.20 0.9307 0.9645 85.976 28.79 0.941 0.968

104.08 27.96 0.9492 0.9723 78.345 29.19 0.9468 0.9709 75.791 29.33 0.9434 0.9701 75.664 29.34 0.9484 0.9732 75.487 29.35 0.9489 0.9735 98.688 28.19 0.9305 0.9643 86.114 28.78 0.9408 0.9679

6 27.758 33.70 0.9855 0.9916 20.859 34.94 0.983 0.9908 19.698 35.19 0.9823 0.9907 19.619 35.20 0.9842 0.9914 19.539 35.22 0.9843 0.9915 26.422 33.91 0.978 0.9883 22.668 34.58 0.9817 0.9899 104.06 27.96 0.9492 0.9723 78.355 29.19 0.9467 0.9708 75.785 29.33 0.9434 0.9702 75.654 29.34 0.9483 0.9732 75.451 29.35 0.9488 0.9734 98.733 28.19 0.9303 0.9643 86.134 28.78 0.9408 0.9679

Page 25 of 18

New Orthogonal Multiwavelet Filters ...

Table 6(continued) Results for Kiel. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1 Kiel New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 11418 7.555 0.1151 0.4868 536.64 20.83 0.8016 0.8826 548.81 20.74 0.7703 0.8838 560.92 20.64 0.7958 0.9193 558.71 20.66 0.7982 0.9201 571.14 20.56 0.7904 0.917 578.87 20.5 0.7879 0.8696

2 549 20.74 0.8160 0.8957 202.31 25.07 0.8867 0.9405 201.50 25.09 0.8840 0.9395 201.99 25.08 0.8878 0.9451 201.49 25.09 0.8875 0.9451 242.55 24.28 0.8645 0.9334 223.72 24.63 0.8767 0.9341

Decomposition level 3 4 5 259.13 238.48 236.06 24.00 24.36 24.40 0.8871 0.8902 0.8908 0.9384 0.9393 0.9392 182.95 181.18 181.18 25.51 25.55 25.55 0.8895 0.8881 0.8875 0.9393 0.9365 0.9358 180.29 178.04 177.97 25.57 25.63 25.63 0.8857 0.8855 0.8850 0.9381 0.9371 0.9365 180.07 176.93 176.94 25.58 25.65 25.65 0.8930 0.8932 0.8929 0.9446 0.9434 0.9431 180.21 177.25 177.28 25.57 25.64 25.64 0.8931 0.8934 0.8927 0.9445 0.9436 0.9428 225.26 223.77 224.12 24.60 24.63 24.63 0.8655 0.8651 0.8636 0.9298 0.9285 0.927 201.15 198.84 199.08 25.10 25.15 25.14 0.8810 0.8794 0.8787 0.9342 0.9322 0.9315

6 235.88 24.40 0.8909 0.9391 181.26 25.55 0.8876 0.9358 177.99 25.63 0.8849 0.9365 176.88 25.65 0.8929 0.9433 177.19 25.65 0.8928 0.9429 224.32 24.62 0.8634 0.9268 199.13 25.14 0.8785 0.9316

Page 26 of 18

New Orthogonal Multiwavelet Filters ...

Table 7 Results for House. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

House

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

Decomposition level 3 4 5 5.9222 5.8497 5.8459 40.46 40.51 40.51 0.9593 0.9598 0.9599 0.9952 0.9952 0.9952 4.3717 4.3892 4.3903 41.80 41.78 41.78 0.9689 0.9688 0.9688 0.9956 0.9956 0.9957 4.2743 4.2803 4.2847 41.89 41.88 41.88 0.9691 0.9691 0.9690 0.9951 0.9951 0.9951 4.291 4.2871 4.2909 41.85 41.86 41.85 0.9689 0.9690 0.9689 0.9956 0.9956 0.9956 4.229 4.2339 4.2352 41.92 41.92 41.92 0.9695 0.9694 0.9694 0.9957 0.9957 0.9957 5.8072 5.8293 5.8379 40.51 40.49 40.48 0.9584 0.9582 0.9582 0.9939 0.9939 0.9939 5.3174 5.3297 5.3319 40.90 40.89 40.89 0.9619 0.9618 0.9617 0.9942 0.9942 0.9942

1 62.454 30.43 0.8683 0.9746 4.9972 41.22 0.9643 0.9953 5.0617 41.16 0.9636 0.9946 5.1499 41.07 0.9630 0.9957 5.0737 41.14 0.9636 0.9958 6.4048 40.09 0.9543 0.9945 6.0433 40.35 0.957 0.9939

2 6.664 39.95 0.9547 0.9949 4.3791 41.79 0.9688 0.9957 4.3374 41.82 0.9686 0.9952 4.3863 41.76 0.9682 0.9956 4.3199 41.83 0.9687 0.9957 5.848 40.48 0.9580 0.9940 5.3726 40.86 0.9615 0.9941

5510.5 10.93 0.2712 0.657 16.127 36.10 0.8964 0.9843 16.409 36.02 0.8921 0.9826 16.549 35.97 0.8920 0.9866 16.37 36.02 0.8931 0.9868 20.271 35.08 0.8770 0.9843 19.548 35.24 0.8790 0.9812

21.225 34.90 0.8740 0.9824 12.194 37.33 0.9200 0.9874 12.007 37.39 0.9188 0.9865 12.091 37.34 0.9188 0.9876 11.96 37.39 0.9197 0.9879 16..374 36.00 0.8955 0.9828 14.883 36.42 0.9033 0.9837

16.415 36.01 0.8956 0.9853 12.057 37.38 0.9213 0.9872 11.671 37.51 0.9213 0.9866 11.612 37.52 0.9223 0.9878 11.525 37.56 0.9229 0.9880 16.168 36.06 0.8970 0.9826 14.521 36.53 0.9058 0.9838

15.993 36.12 0.8981 0.9856 12.15 37.35 0.9207 0.9871 11.686 37.51 0.9212 0.9865 11.615 37.52 0.9222 0.9878 11.534 37.55 0.9228 0.9879 16.27 36.03 0.8965 0.9826 14.609 36.51 0.9051 0.9837

15.972 36.13 0.8982 0.9856 12.167 37.34 0.9206 0.9872 11.709 37.50 0.9211 0.9865 11.63 37.51 0.9221 0.9878 11.537 37.55 0.9227 0.9879 16.315 36.02 0.8963 0.9825 14.609 36.50 0.9051 0.9837

6 5.8459 40.51 0.9599 0.9952 4.3899 41.78 0.9688 0.9957 4.2846 41.88 0.9690 0.9951 4.2912 41.85 0.9689 0.9956 4.2352 41.92 0.9694 0.9957 5.8389 40.48 0.9581 0.9939 5.3317 40.89 0.9617 0.9942 15.972 36.13 0.8982 0.9856 12.169 37.34 0.9206 0.9872 11.707 37.50 0.9211 0.9865 11.634 37.51 0.9220 0.9878 11.541 37.55 0.9227 0.9879 16.323 36.02 0.8962 0.9825 14.613 36.50 0.9051 0.9837

Page 27 of 18

New Orthogonal Multiwavelet Filters ...

Table 7(continued) Results for House. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

House

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 11054 7.738 0.0391 0.5900 198.44 25.37 0.7350 0.9111 190.88 25.51 0.7426 0.9088 196.03 25.38 0.7221 0.9476 197.17 25.37 0.7222 0.9477 201.1 25.29 0.7208 0.9482 236.19 24.64 0.7222 0.8881

Decomposition level 2 3 4 5 170.5 38.224 35.033 34.845 26.07 32.38 32.74 32.77 0.7398 0.8271 0.8326 0.833 0.9162 0.9691 0.9697 0.9696 26.90 25.683 26.03 26.116 33.88 34.08 34.02 34.01 0.8481 0.8541 0.8529 0.8527 0.9727 0.9728 0.9721 0.9722 26.63 24.528 24.533 24.663 33.91 34.27 34.27 34.24 0.8448 0.8527 0.8529 0.8524 0.9712 0.9718 0.9715 0.9715 26.219 23.883 23.806 23.877 33.97 34.37 34.39 34.37 0.8478 0.8573 0.8575 0.8571 0.9735 0.9742 0.9741 0.9741 25.827 23.695 23.707 23.763 34.03 34.41 34.41 34.40 0.8502 0.8591 0.8590 0.8588 0.9741 0.9747 0.9747 0.9748 37.967 35.901 36.254 36.519 32.36 32.60 32.56 32.53 0.8214 0.8253 0.8244 0.8236 0.9656 0.9636 0.9634 0.9633 33.76 31.283 31.52 31.558 32.87 33.20 33.16 33.16 0.8282 0.8336 0.8326 0.8323 0.9664 0.967 0.9666 0.9666

6 34.84 32.77 0.833 0.9696 26.126 34.01 0.8525 0.9722 24.662 34.24 0.8524 0.9715 23.887 34.37 0.8571 0.9741 23.775 34.39 0.8587 0.9748 36.56 32.53 0.8234 0.9633 31.56 33.16 0.8322 0.9666

15440 6.261 0.02841 0.5604 5465.5 10.97 0.3528 0.6279 5586.6 10.88 0.3526 0.6347 5710.4 10.79 0.3419 0.6331 5710.4 10.79 0.3419 0.6331 5710.4 10.79 0.3419 0.6331 5738.7 10.76 0.3423 0.6235

4641.1 11.60 0.2525 0.5453 64.412 30.15 0.7860 0.9507 73.51 29.58 0.7792 0.9485 66.371 30.00 0.7828 0.9543 64.631 30.12 0.7853 0.9551 103.05 28.12 0.7521 0.9425 90.77 28.65 0.7594 0.9421

78.700 29.35 0.7779 0.9434 54.317 30.88 0.7935 0.9482 53.516 30.93 0.7949 0.9500 48.02 31.37 0.8017 0.9550 47.434 31.43 0.8032 9551 82.71 29.04 0.7599 0.9355 68.398 29.85 0.7698 0.9427

102.12 28.26 0.7605 0.9369 53.237 30.96 0.7956 0.9518 53.266 30.94 0.7958 0.9510 48.518 31.33 0.8017 0.9561 47.651 31.41 0.8037 0.9564 80.823 29.14 0.7643 0.9397 68.696 29.83 0.7731 0.9453

80.223 29.27 0.7766 0.9433 53.817 30.92 0.7940 0.9485 52.708 30.99 0.7961 0.9509 47.634 31.41 0.8025 0.9551 47.051 31.46 0.8041 0.9555 81.065 29.13 0.7627 0.9372 68.174 29.86 0.7706 0.9431

78.789 29.34 0.7779 0.9433 54.285 30.88 0.7935 0.9481 53.512 30.93 0.7948 0.9500 47.921 31.38 0.802 0.9550 47.361 31.44 0.8033 0.9553 82.416 29.06 0.7605 0.9358 68.329 29.85 0.7700 0.9426

Page 28 of 18

New Orthogonal Multiwavelet Filters ...

Table 8 Results for Cara. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

Cara

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 14.669 36.47 0.9471 0.9927 1.2479 47.19 0.9847 0.9979 1.2791 47.08 0.9843 0.9976 1.2582 47.16 0.9847 0.9980 1.2397 47.22 0.9849 0.9980 1.5073 46.37 0.982 0.9975 1.4994 46.39 0.9821 0.9972

Decomposition level 2 3 4 5 1.5443 1.3911 1.3727 1.372 46.27 46.72 46.78 46.78 0.9829 0.9841 0.9843 0.9843 0.998 0.9981 0.9981 0.9981 1.1258 1.1206 1.1208 1.1209 47.64 47.66 47.66 47.66 0.9860 0.9861 0.9860 0.9860 0.9981 0.9981 0.9981 0.9981 1.1236 1.1074 1.1065 1.1052 47.65 47.71 47.71 47.72 0.9858 0.9860 0.9860 0.9860 0.9978 0.9978 0.9978 0.9978 1.1111 1.0896 1.0852 1.0847 47.70 47.78 47.80 47.80 0.9861 0.9864 0.9864 0.9864 0.9980 0.9980 0.9980 0.9980 1.0961 1.0759 1.072 1.0717 47.76 47.84 47.85 47.85 0.9863 0.9866 0.9866 0.9866 0.9981 0.9981 0.9981 0.9981 1.3779 1.3716 1.3728 1.3728 46.76 46.79 46.78 46.78 0.9832 0.9832 0.9832 0.9832 0.9973 0.9973 0.9973 0.9973 1.332 1.3149 1.3131 1.3131 46.91 46.96 46.97 46.97 0.9836 0.9837 0.9837 0.9837 0.9974 0.9974 0.9974 0.9974

6 1.3719 46.78 0.9843 0.9981 1.121 47.65 0.9860 0.9981 1.1052 47.72 0.9860 0.9978 1.0846 47.80 0.9864 0.9980 1.0716 47.85 0.9866 0.9981 1.3733 46.78 0.9832 0.9973 1.3133 46.97 0.9837 0.9974

4659.8 11.56 0.05861 0.5297 4.7746 41.37 0.9638 0.9946 5.0704 41.11 0.9619 0.9940 4.8966 41.26 0.9628 0.9951 4.7765 41.37 0.9634 0.9952 6.6138 39.96 0.9538 0.9935 6.7467 39.87 0.9542 0.9925

5.9153 40.44 0.9611 0.9945 3.2036 43.10 0.9708 0.9953 3.2022 43.10 0.9702 0.9949 3.1377 43.20 0.9710 0.9950 3.0599 43.30 0.9715 0.9951 4.7232 41.42 0.9616 0.9934 4.3785 41.75 0.9637 0.9938

3.8907 42.26 0.9689 0.9952 3.1579 43.16 0.9712 0.9951 3.0538 43.31 0.9711 0.9949 2.9518 43.46 0.9719 0.9952 2.8819 43.56 0.9724 0.9952 4.624 41.52 0.9618 0.9930 4.1973 41.93 0.9644 0.9940

4.0861 42.05 0.968 0.9952 3.1461 43.18 0.9713 0.9951 3.0691 43.29 0.9711 0.9948 2.9816 43.42 0.9718 0.9951 2.9094 43.52 0.9723 0.9952 4.5804 41.56 0.9623 0.9930 4.1929 41.94 0.9647 0.9938

3.9051 42.24 0.9688 0.9952 3.1508 43.17 0.9713 0.9951 3.0491 43.32 0.9712 0.9949 2.9554 43.46 0.9719 0.9952 2.8828 43.56 0.9724 0.9952 4.6141 41.53 0.9619 0.9930 4.1946 41.94 0.9644 0.9940

3.8908 42.26 0.9689 0.9952 3.1576 43.17 0.9712 0.9951 3.0534 43.31 0.9711 0.9949 2.9505 43.46 0.9719 0.9952 2.8794 43.57 0.9725 0.9952 4.619 41.52 0.9618 0.9930 4.1969 41.93 0.9644 0.9940

Page 29 of 18

New Orthogonal Multiwavelet Filters ...

Table 8(continued) Results for Cara. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

Cara

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 8248.7 9.065 0.03618 0.503 44.06 31.71 0.8648 0.9668 47.142 31.42 0.8598 0.9691 43.916 31.73 0.8621 0.9751 43.581 31.77 0.8625 0.9753 48.679 31.30 0.8571 0.9739 54.35 30.81 0.8485 0.9602

Decomposition level 2 3 4 5 38.541 11.345 10.24 10.138 32.29 37.62 38.07 38.11 0.8883 0.9421 0.9453 0.9457 0.9728 0.9894 0.9899 0.9899 9.0035 8.2593 8.3198 8.3657 38.62 39.00 38.97 38.95 0.9475 0.9496 0.9495 0.9492 0.9902 0.9895 0.9895 0.9895 9.0648 8.0847 8.0233 8.0603 38.59 39.09 39.12 39.10 0.9462 0.9494 0.9496 0.9495 0.9896 0.9894 0.9896 0.9897 8.9286 7.7402 7.6756 7.6529 38.66 39.28 39.32 39.33 0.9463 0.9503 0.9504 0.9505 0.9905 0.9904 0.9909 0.9909 8.7188 7.5929 7.5075 7.4977 38.76 39.36 39.41 39.42 0.9474 0.9512 0.9514 0.9514 0.9907 0.9906 0.9910 0.9910 14.159 13.083 13.242 13.257 36.67 37.01 36.96 36.96 0.9265 0.9291 0.9283 0.9282 0.9852 0.9849 0.9846 0.9846 13.133 11.83 11.86 11.861 36.99 37.44 37.43 37.43 0.9311 0.9344 0.9338 0.9336 0.9862 0.9864 0.9864 0.9865

6 10.135 38.12 0.9457 0.9899 8.3696 38.94 0.9492 0.9895 8.0547 39.10 0.9495 0.9897 7.6628 39.33 0.9504 0.9909 7.5047 39.41 0.9514 0.9910 13.293 36.94 0.9280 0.9846 11.881 37.43 0.9336 0.9865

10501 8.051 0.01585 0.521 4664.8 11.57 0.3833 0.5896 4721.1 11.52 0.3841 0.5860 4761 11.49 0.3867 0.6056 4761.00 11.49 0.3867 0.6056 4761 11.49 0.3867 0.6056 4770.8 11.48 0.3792 0.5835

3488 12.82 0.06779 0.3933 22.996 34.56 0.9111 0.9780 24.749 34.25 0.9051 0.9765 23.861 34.41 0.905 0.9789 23.364 34.50 0.9070 0.9795 35.318 32.73 0.8801 0.9710 33.309 32.97 0.8852 0.9703

21.981 34.76 0.9165 0.9796 18.425 35.53 0.9210 0.9784 18.254 35.57 0.9198 0.9798 17.015 35.88 0.9208 0.9826 16.845 35.92 0.9226 0.9831 29.859 33.45 0.886 0.9694 26.146 34.02 0.8946 0.9731

27.742 33.75 0.9044 0.9764 18.123 35.60 0.9221 0.9799 18.57 35.49 0.9191 0.9800 17.324 35.80 0.9204 0.9823 17.139 35.84 0.9218 0.9826 29.309 33.53 0.8888 0.9719 26.202 34.01 0.8964 0.9740

22.497 34.66 0.9153 0.9796 18.228 35.58 0.9216 0.9783 18.076 35.61 0.9203 0.9798 17.001 35.88 0.921 0.9822 16.803 35.93 0.9228 0.9827 29.573 33.50 0.8871 0.9696 26.049 34.03 0.8953 0.9729

22.009 34.76 0.9165 0.9796 18.414 35.54 0.9210 0.9783 18.272 35.56 0.9197 0.9798 16.96 35.89 0.9211 0.9826 16.787 35.93 0.9228 0.9831 29.675 33.48 0.8865 0.9694 26.042 34.03 0.8950 0.9730

Page 30 of 18

New Orthogonal Multiwavelet Filters ...

Table 9 Results for Voit. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

Voit

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 192.89 25.29 0.7299 0.9547 27.374 33.77 0.9055 0.9874 27.639 33.73 0.9032 0.9854 27.554 33.74 0.9036 0.9888 27.205 33.80 0.9047 0.9891 30.78 33.26 0.896 0.9874 29.85 33.39 0.8971 0.9847

2 34.812 32.73 0.8896 0.9867 24.717 34.22 0.9136 0.9881 24.311 34.29 0.9134 0.9865 24.036 34.33 0.9136 0.9881 23.891 34.36 0.9144 0.9883 28.92 33.54 0.9014 0.9853 26.893 33.85 0.9053 0.9856

3941.9 12.20 0.2293 0.6774 82.81 28.97 0.7800 0.9581 85.127 28.86 0.7739 0.9543 85.654 28.83 0.7712 0.9665 84.412 28.89 0.774 0.9672 93.636 28.44 0.757 0.9626 90.456 28.59 0.7632 0.9514

105.44 27.93 0.7473 0.9559 64.558 30.06 0.8197 0.9668 64.228 30.08 0.8167 0.9630 63.808 30.10 0.8172 0.9665 63.328 30.14 0.8191 0.9672 75.327 29.39 0.7939 0.9588 69.434 29.74 0.804 0.9604

Decomposition level 3 4 5 31.389 31.084 31.061 33.18 33.22 33.22 0.898 0.8988 0.8989 0.9876 0.9877 0.9877 24.37 24.369 24.377 34.28 34.28 34.27 0.9147 0.9148 0.9148 0.9882 0.9882 0.9882 23.967 23.917 23.932 34.35 34.36 34.35 0.9147 0.9149 0.9148 0.9865 0.9866 0.9866 23.544 23.432 23.424 34.42 34.44 34.45 0.915 0.9155 0.9155 0.9879 0.9879 0.9878 23.427 23.347 23.338 34.45 34.46 34.46 0.9159 0.9162 0.9163 0.9882 0.9881 0.9881 29.171 29.252 29.260 33.50 33.49 33.49 0.9011 0.9009 0.9009 0.9845 0.9845 0.9846 26.702 26.65 26.652 33.88 33.89 33.89 0.906 0.9061 0.9062 0.9856 0.9856 0.9855 82.277 29.01 0.7846 0.9632 62.664 30.18 0.8242 0.9669 62.134 30.22 0.8216 0.9636 61.202 30.29 0.8228 0.9661 60.746 30.32 0.8247 0.9666 74.947 29.41 0.7959 0.957 67.948 29.83 0.8067 0.9603

80.29 29.11 0.789 0.9637 62.671 30.18 0.8242 0.9669 61.891 30.24 0.8223 0.9636 60.714 30.32 0.8239 0.966 60.379 30.35 0.8256 0.9667 75.224 29.40 0.7953 0.9567 67.661 29.85 0.8072 0.9605

80.138 29.12 0.7894 0.9638 62.667 30.18 0.8242 0.9669 61.934 30.24 0.8222 0.9636 60.674 30.32 0.824 0.966 60.354 30.35 0.8257 0.9667 75.275 29.39 0.7952 0.9567 67.633 29.85 0.8073 0.9605

6 31.056 33.22 0.8989 0.9877 24.378 34.27 0.9148 0.9882 23.931 34.35 0.9148 0.9866 23.429 34.44 0.9154 0.9879 23.342 34.46 0.9163 0.9881 29.267 33.49 0.9009 0.9846 26.655 33.89 0.9061 0.9855 80.112 29.12 0.7894 0.9638 62.688 30.18 0.8241 0.9669 61.933 30.24 0.8222 0.9636 60.696 30.32 0.8239 0.966 60.379 30.35 0.8256 0.9667 75.314 29.39 0.7951 0.9567 67.637 29.85 0.8073 0.9605

Page 31 of 18

New Orthogonal Multiwavelet Filters ...

Table 9(continued) Results for Voit. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

Voit

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 7554.6 9.367 0.0826 0.6229 395.03 22.17 0.5099 0.8262 405.90 22.06 0.5011 0.8423 415.65 21.97 0.4804 0.8864 414.61 21.98 0.4808 0.8861 417.16 21.95 0.4808 0.8871 422.81 21.88 0.4923 0.8197

2 375.82 22.39 0.5001 0.8494 129.48 27.04 0.6940 0.9278 130.05 27.02 0.6914 0.9208 128.04 27.08 0.6921 0.9269 127.11 27.12 0.6939 0.9280 149.21 26.42 0.6636 0.9144 139.29 26.72 0.6754 0.9178

9930.8 8.178 0.0484 0.5877 3805.5 12.35 0.2487 0.6103 3939.5 12.20 0.24 0.6241 4044.6 12.08 0.2242 0.6053 4044.6 12.08 0.2242 0.6053 4044.6 12.08 0.2242 0.6053 4029.1 12.10 0.2329 0.6029

3519.9 12.69 0.1632 0.5275 236.61 24.42 0.5628 0.8725 244.59 24.27 0.5619 0.8595 236.81 24.41 0.5604 0.8754 236.46 24.42 0.5611 0.8760 273.36 23.79 0.5342 0.8639 264.94 23.93 0.5406 0.8535

Decomposition level 3 4 5 164.29 154.67 153.91 26.00 26.27 26.29 0.6575 0.6707 0.6718 0.9211 0.9228 0.9232 121.36 121.23 121.23 27.32 27.32 27.32 0.7097 0.7098 0.7099 0.9293 0.9295 0.9295 120.54 119.55 119.59 27.35 27.38 27.38 0.7073 0.7092 0.7095 0.9236 0.9235 0.9233 118.22 116.63 116.52 27.43 27.49 27.49 0.7084 0.7112 0.7115 0.9293 0.9300 0.9299 117.48 116.14 116.05 27.46 27.51 27.51 0.7108 0.7132 0.7134 0.9305 0.9308 0.9307 143.56 144.00 144.33 26.59 26.58 26.57 0.6741 0.6736 0.673 0.9121 0.9112 0.9106 131.03 130.01 129.68 26.98 27.02 27.03 0.6887 0.6903 0.6907 0.9189 0.9201 0.9199 283.21 23.64 0.5327 0.8622 201.48 25.12 0.6022 0.8811 201.47 25.11 0.6009 0.8706 197.71 25.20 0.5999 0.8799 197.54 25.20 0.6008 0.8805 238.9 24.38 0.5636 0.8544 222.82 24.68 0.5734 0.8621

248.98 24.20 0.5633 0.8701 199.9 25.15 0.6038 0.8805 197.79 25.20 0.6052 0.8716 192.21 25.32 0.6061 0.8817 192.33 25.32 0.6067 0.8820 238.69 24.39 0.5634 0.8527 218.29 24.77 0.5753 0.8637

246.14 24.25 0.5664 0.871 199.67 25.16 0.6043 0.8804 197.64 25.20 0.6054 0.8713 191.70 25.33 0.6069 0.8817 191.93 25.33 0.6075 0.8819 239.17 24.38 0.563 0.8520 216.56 24.8 0.5771 0.8631

6 153.84 26.29 0.6719 0.9232 121.27 27.32 0.7098 0.9295 119.62 27.38 0.7094 0.9233 116.58 27.49 0.7113 0.9299 116.12 27.51 0.7133 0.9306 144.43 26.57 0.6729 0.9107 129.72 27.03 0.6908 0.9199 245.9 24.25 0.5667 0.8711 199.86 25.15 0.6042 0.8803 197.75 25.20 0.6053 0.8713 191.90 25.33 0.6067 0.8816 192.16 25.32 0.6073 0.8818 239.67 24.37 0.5628 0.8522 216.82 24.8 0.5768 0.863

Page 32 of 18

New Orthogonal Multiwavelet Filters ...

Table 10 Results for Texture 1. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

2:1

Alpert

New1

New2

Integer Haar

GHM

Texture 1

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 79.713 29.31 0.9875 0.9919 46.476 31.63 0.9854 0.9922 46.341 31.65 0.9788 0.9905 213.6 24.97 0.9768 0.9829 46.024 31.67 0.9853 0.9938 46.177 31.66 0.9854 0.9922 45.865 31.69 0.9844 0.9936

2 53.973 30.99 0.9898 0.9939 43.135 31.95 0.9847 0.9924 43.049 31.97 0.98 0.9909 274.26 23.88 0.8952 0.9545 42.953 31.97 0.9848 0.9928 44.069 31.85 0.9843 0.9920 45.222 31.74 0.9825 0.9918

432.58 21.94 0.9051 0.9442 204.65 25.17 0.9301 0.9626 206.16 25.14 0.9049 0.9568 326.72 23.13 0.9435 0.9658 205.68 25.15 0.9299 0.9705 206.37 25.14 0.9296 0.9622 207.13 25.12 0.9245 0.9690

230.44 24.66 0.9466 0.9702 184.38 25.62 0.9288 0.9648 185.71 25.59 0.9114 0.9586 366.72 22.62 0.8529 0.9328 185.71 25.59 0.9272 0.9657 190.66 25.47 0.9243 0.9617 189.59 25.50 0.9201 0.9625

Decomposition level 3 4 5 51.129 50.784 50.761 31.22 31.25 31.25 0.9900 0.9900 0.9900 0.9941 0.9941 0.9941 42.608 42.554 42.556 32.00 32.01 32.01 0.9848 0.9848 0.9848 0.9924 0.9924 0.9924 42.432 42.344 42.334 32.03 32.04 32.04 0.98 0.98 0.98 0.9908 0.9908 0.9908 312.27 361.87 437.23 23.31 22.64 21.79 0.8301 0.7752 0.7414 0.9156 0.8599 0.7942 42.32 42.242 42.223 32.03 32.04 32.04 0.9848 0.9847 0.9847 0.9926 0.9926 0.9926 44.158 44.243 44.24 31.84 31.83 31.83 0.9844 0.9845 0.9845 0.9920 0.9920 0.9920 45.112 45.123 45.112 31.75 31.75 31.75 0.9825 0.9825 0.9825 0.9916 0.9916 0.9916 212.9 25 0.9488 0.9717 181.18 25.7 0.9296 0.9650 182.11 25.68 0.9120 0.9586 401.02 22.22 0.7824 0.8902 182.16 25.67 0.9277 0.9649 190.61 25.47 0.9246 0.9612 187.42 25.55 0.9203 0.9614

210.92 25.04 0.9493 0.9718 180.82 25.71 0.9296 0.9649 181.55 25.69 0.9122 0.9586 450.07 21.70 0.725 0.833 181.61 25.69 0.9276 0.9648 191.07 25.46 0.9247 0.9613 187.32 25.55 0.9205 0.9614

210.76 25.05 0.9493 0.9718 180.81 25.71 0.9296 0.9649 181.53 25.69 0.9122 0.9586 525.38 21.00 0.6897 0.7671 181.53 25.69 0.9277 0.9648 191.08 25.46 0.9248 0.9613 187.29 25.55 0.9205 0.9614

6 50.754 31.25 0.9900 0.9941 42.554 32.01 0.9848 0.9924 42.33 32.04 0.98 0.9908 485.49 21.31 0.7329 0.7681 42.227 32.04 0.9847 0.9926 44.238 31.83 0.9845 0.9920 45.108 31.75 0.9825 0.9916 210.73 25.05 0.9493 0.9718 180.78 25.71 0.9296 0.9649 181.52 25.69 0.9122 0.9586 573.66 20.60 0.6807 0.7410 181.53 25.69 0.9277 0.9648 191.07 25.46 0.9248 0.9613 187.27 25.55 0.9205 0.9614

Page 33 of 18

New Orthogonal Multiwavelet Filters ...

Table 11 Results for Texture 4. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

64:1

Alpert

New1

New2

Integer Haar

GHM

Texture 4

SA4

CL

128:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 1.10E+04 7.54 0.2919 0.7652 5995.0 10.36 0.4384 0.8289 6035 10.33 0.4443 0.8326 6039 10.32 0.4362 0.8236 6039 10.32 0.4362 0.8236 6064 10.31 0.4421 0.8311 6039 10.32 0.4362 0.8236

2 4985.0 11.16 0.3080 0.6724 28.03 33.72 0.9844 0.9828 37.53 32.48 0.9838 0.9789 19.85 35.16 0.9875 0.9975 7.116 39.72 0.9882 0.9983 40.49 32.14 0.9830 0.9790 8.795 38.80 0.9880 0.9981

Decomposition level 3 4 5 58.32 15.81 12.58 30.52 36.19 37.20 0.9815 0.9976 0.9957 0.9716 0.9921 0.9926 4.139 3.3107 3.5373 42.01 42.99 42.70 0.9983 0.9963 0.9945 0.9962 0.9962 0.9954 1.913 1.191 1.193 45.40 47.50 47.49 0.9995 0.9995 0.9986 0.9989 0.9991 0.9989 37.37 86.73 119.8 32.41 28.75 27.35 0.9937 0.9758 0.9497 0.9936 0.9813 0.9672 0.5232 0.4045 0.4921 50.95 52.07 51.22 0.9998 0.9998 0.9998 0.9996 0.9996 0.9995 1.642 0.9366 0.9678 46.09 48.52 48.39 0.9996 0.9997 0.9996 0.9992 0.9995 0.9995 0.4513 0.3109 0.2893 51.59 53.21 53.54 0.9998 0.9999 0.9999 0.9996 0.9998 0.9998

6 12.00 37.41 0.9953 0.9928 3.77 42.42 0.9945 0.9951 1.222 47.39 0.9985 0.9988 141 26.64 0.9135 0.9625 0.4427 51.68 0.9998 0.9996 1.135 47.70 0.9995 0.9994 0.3237 53.05 0.9999 0.9997

7 12.00 37.41 0.9952 0.9928 3.745 42.45 0.9944 0.9951 1.251 47.29 0.9985 0.9988 179.2 25.60 0.8859 0.9658 0.4368 51.74 0.9998 0.9996 1.108 47.80 0.9995 0.9994 0.3197 53.10 0.9999 0.9997

1.30E+04 6.99 0.2381 0.7586 9496 8.36 0.3120 0.7853 9540 8.34 0.3168 0.7950 9551 8.34 0.3114 0.7847 9551 8.34 0.3114 0.7847 9575 8.33 0.3130 0.7922 9551 8.34 0.3114 0.7847

8189 9.00 0.2386 0.6887 1607 16.18 0.6811 0.8825 1710 15.88 0.6488 0.8553 1554 16.32 0.6895 0.9100 1553 16.32 0.6895 0.9102 1707 15.91 0.6827 0.8808 1555 16.31 0.6893 0.9096

1352 16.90 0.5278 0.7073 29.78 33.41 0.9748 0.9739 28.20 33.66 0.9839 0.9820 41.02 32.01 0.9930 0.9925 5.55 40.69 0.9987 0.9979 37.31 32.45 0.9795 0.9774 6.976 39.70 0.9984 0.9975

42.65 31.85 0.9831 0.9742 22.44 34.64 0.9749 0.9740 15.62 36.22 0.9865 0.9856 143.1 26.58 0.9125 0.9614 3.475 42.73 0.9979 0.9974 23.18 34.54 0.9843 0.9813 3.863 42.28 0.9969 0.9971

42.63 31.85 0.9826 0.9741 22.18 34.69 0.9741 0.9742 15.91 36.14 0.9864 0.9854 181.2 25.55 0.8850 0.9648 3.353 42.89 0.9979 0.9974 22.46 34.68 0.9826 0.9816 3.732 42.43 0.9969 0.9971

57.81 30.53 0.9768 0.9683 20.07 35.13 0.9777 0.9779 16.94 35.87 0.9934 0.9871 88.57 28.66 0.9751 0.9805 2.956 43.43 0.9987 0.9981 20.02 35.18 0.9903 0.9854 3.422 42.80 0.9983 0.9979

44.90 31.63 0.9841 0.9734 20.75 34.99 0.9757 0.9757 15.61 36.23 0.9899 0.9860 121.6 27.29 0.9490 0.9664 2.869 43.56 0.9985 0.9980 20.08 35.17 0.9869 0.9840 3.308 42.95 0.9982 0.9978

Page 34 of 18

New Orthogonal Multiwavelet Filters ...

Table 11(continued) Results for Texture 4. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

256:1

Alpert

New1

New2

Integer Haar

GHM

Texture 4

SA4

CL

512:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 1.41E+04 6.66 0.1936 0.6961 1.18E+04 7.42 0.2406 0.7621 1.18E+04 7.41 0.2422 0.7600 1.19E+04 7.39 0.2370 0.7431 1.19E+04 7.39 0.2370 0.7431 1.19E+04 7.39 0.2414 0.7546 1.19E+04 7.39 0.2370 0.7431

2 1.10E+04 7.71 0.2070 0.7198 6094 10.29 0.4089 0.7943 6175 10.23 0.4048 0.7877 6164 10.24 0.4230 0.8101 6164 10.24 0.4230 0.8101 6223 10.19 0.4131 0.7978 6164 10.24 0.4230 0.8101

Decomposition level 3 4 5 4983 188.4 94.64 11.16 25.41 28.38 0.2806 0.8938 0.9424 0.5966 0.8823 0.9365 129.1 62.03 64.08 27.06 30.22 30.08 0.9504 0.9417 0.9329 0.9373 0.9405 0.9323 155.1 61.75 52.84 26.26 30.24 30.92 0.9314 0.9692 0.9685 0.9183 0.9614 0.9617 78.59 98.25 130.5 29.18 28.21 26.98 0.9756 0.9721 0.9458 0.9858 0.9776 0.9632 53.05 15.89 15.05 30.90 36.12 36.36 0.9788 0.9929 0.9902 0.9883 0.9926 0.9909 208.1 81.83 79.52 24.98 29.02 29.14 0.9390 0.9520 0.9408 0.9253 0.9537 0.9466 67.16 19.00 17.91 29.87 35.35 35.60 0.9767 0.9916 0.9892 0.9863 0.9919 0.9905

6 86.37 28.78 0.9505 0.9428 72.09 29.57 0.9268 0.9253 52.23 30.97 0.9637 0.9616 153.9 26.26 0.9085 0.9568 18.74 35.41 0.9839 0.9864 90.38 28.59 0.9184 0.9329 20.98 34.92 0.9808 0.9855

7 86.37 28.78 0.9468 0.9425 70.10 29.69 0.9241 0.9266 53.44 30.87 0.9617 0.9606 191.1 25.32 0.8807 0.9604 17.72 35.65 0.9839 0.9870 87.29 28.74 0.9202 0.9362 19.78 35.17 0.9810 0.9862

1.48E+04 6.44 0.1744 0.6838 1.36E+04 6.81 0.2092 0.7466 1.36E+04 6.79 0.2094 0.7419 1.37E+04 6.78 0.2026 0.7207 1.37E+04 6.78 0.2026 0.7207 1.37E+04 6.78 0.2056 0.7302 1.37E+04 6.78 0.2026 0.7207

1.27E+04 7.11 0.1956 0.7316 9585 8.32 0.2908 0.7514 9684 8.28 0.2820 0.7520 9680 8.28 0.2956 0.7580 9680 8.28 0.2956 0.7580 9728 8.26 0.2953 0.7685 9680 8.28 0.2956 0.7580

8213 8.99 0.2181 0.6184 1751 15.78 0.6348 0.8420 2083 15.00 0.5581 0.7533 1747 15.79 0.6476 0.8690 1745 15.80 0.6485 0.8704 1905 15.41 0.6384 0.8447 1750 15.78 0.6470 0.8678

159.7 26.11 0.8906 0.8875 155.9 26.21 0.8515 0.8666 105.2 27.92 0.9382 0.9373 179.8 25.59 0.8982 0.9460 56.05 30.65 0.9436 0.9588 199.3 25.15 0.8371 0.8780 62.12 30.21 0.9386 0.9589

158.3 26.15 0.8844 0.8874 149.6 26.39 0.8526 0.8703 108.2 27.80 0.9333 0.9352 214.2 24.83 0.8714 0.9511 51.34 31.03 0.9448 0.9614 193.2 25.28 0.8451 0.8819 56.73 30.60 0.9414 0.9619

1631 16.06 0.4446 0.6104 130.7 26.98 0.9128 0.9097 144.5 26.55 0.9317 0.9184 124.6 27.18 0.9558 0.9721 49.30 31.21 0.9734 0.9841 193.4 25.28 0.9318 0.9242 60.02 30.35 0.9713 0.9827

199.0 25.15 0.8571 0.8629 134.2 26.87 0.8702 0.8847 108 27.81 0.9426 0.9365 152.5 26.3 0.9376 0.9559 44.09 31.69 0.9696 0.9755 178.1 25.64 0.8936 0.9051 53.95 30.82 0.9662 0.9730

Page 35 of 18

New Orthogonal Multiwavelet Filters ...

Table 12 Results for Brick wall. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

Brick wall

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 5617 10.64 0.5177 0.7018 288.7 23.53 0.9648 0.8836 319.4 23.09 0.9231 0.8869 298.8 23.38 0.9594 0.9162 296.1 23.42 0.9593 0.9165 332 22.92 0.9533 0.8750 307.2 23.26 0.9581 0.9138

2 278.9 23.68 0.9870 0.9006 111.8 27.65 0.9854 0.9463 109.1 27.75 0.9856 0.9432 167.6 25.89 0.9782 0.9340 109.1 27.75 0.9848 0.9467 112.2 27.63 0.9835 0.9431 112.4 27.62 0.9831 0.9422

Decomposition level 3 4 5 141.7 135 133.8 26.62 26.83 26.87 0.9860 0.9859 0.9859 0.9383 0.9414 0.9420 106.7 105.8 105.5 27.85 27.89 27.90 0.9850 0.9852 0.9851 0.9493 0.9499 0.9501 103.4 102.5 102.2 27.99 28.02 28.04 0.9843 0.9842 0.9841 0.9463 0.9467 0.9469 252.1 366.5 459.3 24.12 22.49 21.51 0.9285 0.8381 0.8100 0.8850 0.8149 0.7861 103.5 102.7 102.5 27.98 28.01 28.02 0.9844 0.9846 0.9845 0.9491 0.9496 0.9497 107.1 106.8 106.8 27.83 27.84 27.85 0.9829 0.9830 0.9830 0.9463 0.9468 0.9469 106.2 105.3 105.2 27.87 27.91 27.91 0.9830 0.9832 0.9832 0.9449 0.9454 0.9456

6 133.6 26.87 0.9859 0.9421 105.5 27.90 0.9851 0.9501 102.2 28.04 0.9841 0.9469 495.9 21.18 0.8058 0.7817 102.5 28.03 0.9845 0.9497 106.8 27.85 0.9830 0.9469 105.3 27.91 0.9832 0.9454

7 133.6 26.87 0.9859 0.9421 105.4 27.90 0.9851 0.9501 102.2 28.04 0.9841 0.9469 511.3 21.04 0.8046 0.7801 102.5 28.03 0.9845 0.9497 106.8 27.85 0.9830 0.9469 105.2 27.91 0.9832 0.9456

7665 9.29 0.3534 0.7400 3205 13.07 0.5909 0.6891 3327 12.91 0.6012 0.6929 3344 12.89 0.5870 0.6540 3344 12.89 0.587 0.654 3451 12.75 0.5871 0.6808 3344 12.89 0.5871 0.6539

2719 13.79 0.5730 0.6433 171.8 25.78 0.9726 0.9117 175.2 25.7 0.9751 0.9109 217.7 24.75 0.9664 0.9086 173 25.75 0.9722 0.9166 191.2 25.32 0.968 0.9093 192.1 25.3 0.9686 0.9115

232.3 24.47 0.9697 0.8972 149.9 26.37 0.9735 0.9214 145.4 26.5 0.9733 0.9191 282.6 23.62 0.9193 0.8656 145.3 26.51 0.9735 0.9215 155 26.23 0.9699 0.9163 152.4 26.3 0.9707 0.9156

194.5 25.24 0.9716 0.9096 145.7 26.50 0.9742 0.9239 141.5 26.62 0.9735 0.9210 523.5 20.94 0.7953 0.7629 141.6 26.62 0.9738 0.9231 151.7 26.32 0.9703 0.9177 147.8 26.43 0.9710 0.9171

194.4 25.24 0.9717 0.9097 145.7 26.50 0.9742 0.9239 141.5 26.62 0.9735 0.9210 538.9 20.82 0.7941 0.7612 141.5 26.62 0.9738 0.9232 151.6 26.32 0.9703 0.9177 147.8 26.43 0.9710 0.9171

200 25.12 0.9713 0.9075 146.6 26.47 0.9742 0.9234 142.5 26.59 0.9733 0.9204 394.9 22.17 0.8290 0.7967 142.5 26.59 0.9738 0.9228 152.1 26.31 0.9703 0.9175 148.5 26.41 0.9711 0.9170

195.4 25.22 0.9716 0.9092 145.8 26.49 0.9742 0.9238 141.6 26.62 0.9734 0.9209 487 21.26 0.7997 0.7673 141.7 26.62 0.9738 0.9231 151.7 26.32 0.9702 0.9177 147.9 26.43 0.9709 0.9171

Page 36 of 18

New Orthogonal Multiwavelet Filters ...

Table 13(continued) Results for Brick wall. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

64:1

Alpert

New1 Brick wall New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 9198 8.494 0.1961 0.6883 5513.0 10.72 0.4275 0.6928 5624 10.63 0.4359 0.7105 5644 10.62 0.4280 0.6696 5644 10.62 0.4280 0.6696 5779 10.51 0.4263 0.6910 5644 10.62 0.4280 0.6695

2 5278 10.91 0.4234 0.6511 624.0 20.18 0.7833 0.7199 649.3 20.01 0.7845 0.7042 650.1 20.00 0.7709 0.7310 641.2 20.06 0.7674 0.7299 692 19.73 0.7541 0.6892 647 20.02 0.7617 0.7253

Decomposition level 3 4 5 576.6 308.7 288.3 20.52 23.24 23.53 0.8255 0.9446 0.9481 0.7417 0.8630 0.8697 216.5 201.3 198 24.78 25.09 25.16 0.9552 0.9591 0.9595 0.8904 0.8968 0.8981 218.5 203 199.2 24.74 25.06 25.14 0.9614 0.9608 0.9609 0.8912 0.8948 0.8958 337.1 439.2 528.9 22.85 21.70 20.90 0.9047 0.8160 0.7858 0.8469 0.7807 0.7510 215.7 201.1 197.6 24.79 25.10 25.17 0.9596 0.9608 0.9608 0.8987 0.9012 0.9017 279 242.1 235.6 23.67 24.29 24.41 0.9380 0.9473 0.9487 0.8691 0.8841 0.8867 244.6 222.2 218.3 24.25 24.66 24.74 0.9530 0.9557 0.9557 0.8910 0.8943 0.8946

6 284.4 23.59 0.9493 0.8715 197.5 25.18 0.9597 0.8984 198.6 25.15 0.9611 0.8960 564.9 20.61 0.7813 0.7465 197.0 25.19 0.9609 0.9019 234.9 24.42 0.9487 0.8871 217.9 24.75 0.9558 0.8946

7 283.8 23.60 0.9495 0.8718 197.4 25.18 0.9597 0.8984 198.5 25.15 0.9611 0.8961 580.2 20.50 0.7801 0.7449 196.9 25.19 0.9609 0.9019 234.7 24.43 0.9488 0.8871 217.8 24.75 0.9558 0.8947

Page 37 of 18

New Orthogonal Multiwavelet Filters ...

Table 13 Results for Texture 5. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

4:1

Alpert

New1

New2

Integer Haar

GHM

Texture 5

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

Decomposition level 3 4 5 16.25 16.22 16.21 36.02 36.03 36.03 0.9616 0.9617 0.9617 0.9968 0.9968 0.9968 7.82 7.82 7.81 39.20 39.20 39.20 0.9793 0.9793 0.9793 0.9981 0.9981 0.9981 12.89 12.89 12.89 37.03 37.03 37.03 0.9669 0.9669 0.9669 0.9968 0.9968 0.9968 213.98 575.5 1043 24.83 20.53 17.95 0.8160 0.7121 0.6869 0.9495 0.8567 0.7826 3.65 3.65 3.65 42.51 42.51 42.51 0.9900 0.9900 0.9900 0.9991 0.9991 0.9991 8.91 8.92 8.92 38.63 38.63 38.63 0.9755 0.9755 0.9755 0.9977 0.9977 0.9977 7.91 7.89 7.89 39.15 39.16 39.16 0.9780 0.9780 0.9780 0.9977 0.9977 0.9977

1 67.96 29.81 0.9055 0.9922 11.63 37.48 0.9693 0.9972 21.75 34.76 0.9462 0.9949 29.08 33.50 0.9633 0.9965 5.71 40.57 0.9835 0.9985 8.28 38.95 0.9766 0.9978 7.81 39.20 0.9777 0.9976

2 16.95 35.84 0.9605 0.9967 7.95 39.13 0.9788 0.9981 13.21 36.92 0.9661 0.9967 63.98 30.07 0.9298 0.9905 3.73 42.41 0.9898 0.9991 8.67 38.75 0.9758 0.9977 7.9 39.15 0.9780 0.9977

1569.8 16.17 0.4348 0.8711 54.59 30.76 0.8931 0.9860 95.17 28.35 0.8366 0.9816 61.65 30.23 0.9070 0.9882 40.33 32.07 0.9136 0.9889 44.37 31.66 0.9074 0.9864 49.65 31.17 0.9002 0.9856

59.01 30.42 0.9100 0.99100 29.02 33.50 0.9410 0.9936 46.01 31.50 0.9143 0.9900 80.89 29.05 0.9002 0.9865 20.62 34.99 0.9537 0.9950 44.23 31.67 0.9107 0.9867 47.34 31.38 0.9003 0.9862

51.79 30.99 0.9141 0.9913 27.94 33.67 0.9435 0.9939 43.40 31.76 0.9183 0.9905 229.31 24.53 0.7934 0.9448 19.31 35.27 0.9555 0.9952 47.84 31.33 0.9051 0.9857 46.84 31.42 0.9012 0.9868

51.48 31.01 0.9146 0.9914 27.90 33.67 0.9435 0.9939 43.32 31.76 0.9182 0.9905 590.77 20.42 0.6947 0.8515 19.27 35.28 0.9556 0.9952 47.99 31.32 0.9051 0.9857 46.88 31.42 0.9011 0.9868

51.46 31.02 0.9146 0.9914 27.87 33.68 0.9435 0.9939 43.32 31.76 0.9182 0.9905 1058.3 17.88 0.6709 0.7771 19.27 35.28 0.9556 0.9952 47.968 31.32 0.9052 0.9857 46.90 31.42 0.9010 0.9868

6 16.21 36.03 0.9617 0.9968 7.81 39.20 0.9793 0.9981 12.89 37.03 0.9669 0.9968 1412.5 16.63 0.6741 0.7496 3.65 42.51 0.9900 0.9991 8.92 38.63 0.9755 0.9977 7.89 39.16 0.9780 0.9977 51.462 31.02 0.9146 0.9914 27.86 33.68 0.9435 0.9939 43.32 31.76 0.9182 0.9905 1427.7 16.58 0.6588 0.7441 19.27 35.28 0.9556 0.9952 47.968 31.32 0.9052 0.9857 46.889 31.42 0.9011 0.9868

Page 38 of 18

New Orthogonal Multiwavelet Filters ...

Table 13(continued) Results for Texture 5. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

GHM

Texture 5

SA4

CL

32:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 5720 10.56 0.1313 0.6833 221.13 24.68 0.7388 0.9544 341.3 22.80 0.6459 0.9508 208.5 24.94 0.7655 0.9610 195.64 25.22 0.7651 0.9609 215.9 24.79 0.7518 0.9521 214.67 24.81 0.7568 0.9577

2 170.5 25.81 0.8197 0.9729 80.21 29.09 0.8777 0.9819 116 27.49 0.8391 0.9751 129.4 27.01 0.8446 0.9752 72.66 29.52 0.8808 0.9818 149.6 26.38 0.7897 0.9556 149.7 26.38 0.7772 0.9588

9408 8.4 0.0538 0.5223 1699 15.83 0.4132 0.8201 1932 15.27 0.3508 0.8272 1732 15.74 0.4198 0.8061 1732 15.74 0.4198 0.8061 1823 15.52 0.4163 0.8126 1732 15.74 0.4196 0.8061

1443 16.54 0.4411 0.8358 205.3 25.01 0.7512 0.9469 264.5 23.91 0.7098 0.9350 260.1 23.98 0.7234 0.9380 215.8 24.79 0.7344 0.9387 371.5 22.43 0.6328 0.8987 347.1 22.73 0.6340 0.9101

Decomposition level 3 4 5 118.2 116.5 116.4 27.41 27.47 27.47 0.8578 0.8594 0.8596 0.9810 0.9812 0.9813 71.77 71.32 71.22 29.57 29.60 29.61 0.8902 0.8908 0.8910 0.9845 0.9845 0.9845 102.3 101.7 101.6 28.03 28.06 28.06 0.8558 0.8560 0.8563 0.9786 0.9787 0.9787 267.8 629.1 1097 23.85 20.14 17.73 0.7545 0.6628 0.6411 0.9347 0.8403 0.7653 61.17 60.82 60.85 30.27 30.29 30.29 0.8967 0.8971 0.8972 0.9849 0.9850 0.9850 166.1 168.9 168.7 25.93 25.85 25.86 0.7724 0.7691 0.7692 0.9499 0.9493 0.9494 148.24 148.5 148.8 26.42 26.41 26.41 0.7764 0.7760 0.7756 0.9612 0.9613 0.9613 217.3 24.76 0.7761 0.9554 154.4 26.24 0.8132 0.9634 207 24.97 0.7720 0.9545 341.5 22.8 0.7021 0.9161 150.5 26.36 0.8105 0.9616 398.3 22.13 0.6186 0.8844 327.9 22.97 0.6382 0.9176

206.2 24.99 0.7836 0.9579 152.7 26.29 0.8157 0.9644 203.7 25.04 0.7757 0.9556 700.4 19.68 0.6192 0.8205 148.9 26.4 0.8126 0.9624 411.7 21.98 0.6038 0.8824 329.9 22.95 0.6331 0.9171

205.4 25.01 0.7843 0.9582 152.3 26.30 0.8161 0.9645 203.5 25.05 0.7761 0.9557 1168 17.46 0.5989 0.7446 148.7 26.41 0.8127 0.9623 412.2 21.98 0.6031 0.8822 331.3 22.93 0.6319 0.9169

6 116.4 27.47 0.8596 0.9813 71.2 29.61 0.8910 0.9845 101.6 28.06 0.8564 0.9787 1466 16.47 0.6299 0.7323 60.84 30.29 0.8971 0.9850 168.7 25.86 0.7691 0.9495 148.6 26.41 0.7757 0.9613 205.4 25.01 0.7843 0.9582 152.1 26.31 0.8162 0.9646 203.4 25.05 0.7760 0.9557 1538 16.26 0.5881 0.7112 148.7 26.41 0.8127 0.9623 412.1 21.98 0.6031 0.8823 331 22.93 0.6323 0.9170

Page 39 of 18

New Orthogonal Multiwavelet Filters ...

Table 14 Results for Subwater. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

CR

Multiwavelet GHM

SA4

CL

8:1

Alpert

New1

New2

Integer Haar

GHM

Subwater

SA4

CL

16:1

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 4140 6.91 0.0569 0.3204 0.8359 31.45 0.6581 0.6651 1.004 30.91 0.6566 0.6652 1.443 29.86 0.6554 0.6651 0.9442 31.10 0.6569 0.6650 1.114 30.61 0.6557 0.6646 1.385 29.98 0.6532 0.6644

Decomposition level 2 3 4 0.9046 0.703 0.6931 31.22 31.95 32.00 0.6589 0.6601 0.6601 0.6656 0.6657 0.6657 0.4812 0.4567 0.4561 33.05 33.19 33.20 0.6617 0.6619 0.6619 0.6659 0.6659 0.6659 0.5448 0.5036 0.5002 32.68 32.91 32.93 0.6611 0.6614 0.6615 0.6657 0.6658 0.6658 2.786 7.515 15.13 27.95 25.11 23.11 0.6518 0.6399 0.6309 0.6641 0.6561 0.6420 0.5117 0.4716 0.4686 32.87 33.11 33.13 0.6613 0.6616 0.6617 0.6659 0.6659 0.6659 0.8457 0.844 0.8431 31.41 31.41 31.42 0.6585 0.6585 0.6585 0.6651 0.6651 0.6651 0.9685 0.9087 0.9023 31.01 31.20 31.22 0.6572 0.6577 0.6577 0.6650 0.6651 0.6651

5 0.6917 32.00 0.6602 0.6657 0.4569 33.19 0.6619 0.6659 0.499 32.94 0.6615 0.6658 39.26 20.45 0.6255 0.6150 0.4687 33.13 0.6617 0.6659 0.8438 31.41 0.6585 0.6651 0.9031 31.22 0.6577 0.6651

7027 5.35 0.0103 0.3318 43.59 20.07 0.6121 0.6073 37.81 20.50 0.6109 0.6218 13.6 23.38 0.6079 0.6555 13.6 23.38 0.6079 0.6555 47.96 19.80 0.6038 0.6047 13.6 23.38 0.60790 0.6555

29.38 21.28 0.6278 0.6234 1.647 29.48 0.6532 0.6636 1.914 29.05 0.6513 0.6631 3.72 27.11 0.6451 0.6625 1.681 29.42 0.6525 0.6635 3.24 27.52 0.6406 0.6599 2.953 27.78 0.6430 0.6608

2.053 28.84 0.6521 0.6635 1.36 30.03 0.6557 0.6642 1.474 29.80 0.6546 0.6639 39.82 20.40 0.6214 0.6137 1.282 30.20 0.6555 0.6643 3.29 27.47 0.6403 0.6598 2.519 28.24 0.6462 0.6622

2.214 28.63 0.6512 0.6634 1.372 30.01 0.6556 0.6642 1.523 29.70 0.6543 0.6639 8.119 24.88 0.6355 0.6549 1.322 30.11 0.6552 0.6643 3.304 27.46 0.6405 0.6597 2.553 28.20 0.6460 0.6622

2.063 28.83 0.6520 0.6635 1.355 30.04 0.6557 0.6642 1.482 29.78 0.6545 0.6639 15.7 23.00 0.6268 0.6407 1.284 30.20 0.6555 0.6643 3.273 27.48 0.6405 0.6598 2.516 28.24 0.6463 0.6623

Page 40 of 18

New Orthogonal Multiwavelet Filters ...

Table 14(continued) Results for Subwater. The four lines for each multiwavelet represent MSE, PSNR, SSIM and MS-SSIM, in this order. Image

Multiwavelet

CR

GHM

SA4

CL

32:1

Alpert

Subwater New1

New2

Integer Haar

1 8976 4.63 0.0065 0.3774 3944 7.09 0.2598 0.3466 3977.4 7.08 0.2638 0.3683 3955.2 7.1 0.2626 0.3969 3955 7.10 0.2626 0.3968 4046 7.02 0.2552 0.3440 3955.2 7.10 0.2626 0.3968

Decomposition level 2 3 4 3250 9.484 6.745 7.62 24.46 25.42 0.0652 0.6322 0.6362 0.2303 0.6523 0.6553 6.575 4.264 4.018 25.48 26.72 26.89 0.6276 0.6409 0.6423 0.6531 0.6578 0.6583 7.37 4.771 4.271 25.14 26.39 26.71 0.6221 0.6382 0.6407 0.6523 0.6572 0.6585 6.846 9.572 16.943 25.35 24.4 22.77 0.6251 0.6275 0.6203 0.6543 0.6523 0.6382 5.416 3.4 3.155 26.02 27.37 27.59 0.6282 0.6430 0.6444 0.6542 0.6603 0.6606 11.35 10.11 9.914 23.88 24.22 24.27 0.6038 0.6070 0.6066 0.6463 0.6452 0.646 7.689 5.9762 5.6887 25.01 25.74 25.88 0.6164 0.6275 0.6289 0.6507 0.6557 0.6562

5 6.603 25.48 0.6366 0.6555 4.036 26.88 0.6422 0.6583 4.222 26.74 0.6409 0.6586 41.073 20.31 0.6150 0.6112 3.137 27.60 0.6446 0.6607 10.04 24.23 0.6059 0.6458 5.6662 25.89 0.6290 0.6563

Multiwavelet GHM SA4 CL Alpert Integer Haar New1 New2 GHM SA4 CL Alpert Integer Haar New1 New2 GHM SA4 CL Alpert Integer Haar New1 New2

HeaviSine

Cusp

Signal Piece-Polynomial

Table 15 The 𝜖 values for denoising of 1D Piece-Polynomial,Cusp and HeaviSine signals by using orthogonal multiwavelet filters.

Kolev et al.: Preprint submitted to Elsevier

𝜖 𝜎 = 10

𝜎 = 15

28.60 28.36 31.47 69.66 29.52 26.32 27.34 0.0900 0.0974 0.0905 0.0495 0.0562 0.0440 0.0398 0.7560 0.6690 0.7089 1.1430 0.5114 0.5496 0.4250

41.67 41.54 46.49 74.91 36.00 31.80 36.89 0.0887 0.1086 0.0919 0.0534 0.0687 0.0506 0.0464 0.8731 0.8544 0.8032 1.2355 0.7404 0.7658 0.6944

Page 41 of 18

New Orthogonal Multiwavelet Filters ...

Figure 9: Original and Noisy 1D signals with added AWGN with variance 𝜎 = 10, 15; (a) Piece-Polynomial with variance 𝜎 = 10; (b) Piece-Polynomial with variance 𝜎 = 15; (c) Cusp with variance 𝜎 = 10; (d) Cusp with variance 𝜎 = 15; (e) HeaviSine with variance 𝜎 = 10; (f) HeaviSine with variance 𝜎 = 15.

Kolev et al.: Preprint submitted to Elsevier

Page 42 of 18

New Orthogonal Multiwavelet Filters ...

Figure 10: Denoising of 1D signals with added AWGN with variance 𝜎 = 10, 15 by using orthogonal multiwavelet banks and applied soft thresholding; (a) Piece-Polynomial with variance 𝜎 = 10; (b) Piece-Polynomial with variance 𝜎 = 15; (c) Cusp with variance 𝜎 = 10; (d) Cusp with variance 𝜎 = 15; (e) HeaviSine with variance 𝜎 = 10; (f) HeaviSine with variance 𝜎 = 15.

Kolev et al.: Preprint submitted to Elsevier

Page 43 of 18

New Orthogonal Multiwavelet Filters ...

Table 16 Comparative results of MSE,PSNR, SSIM and MS-SSIM of denoised image ’Cameraman’ of size 256 × 256 pixels for seven orthogonal multiwavelets through 𝐽 = 5 decomposition levels with AWGN and variance 𝜎 = 10. Image

Threshold

Multiwavelet GHM

SA4

CL

Soft

Alpert

New1

New2

Integer Haar

GHM

Cameraman

SA4

CL

hard

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 71.08 29.61 0.7803 0.8077 66.5 29.90 0.7892 0.8025 69.57 29.71 0.7851 0.8117 129.73 27.00 0.7508 0.7972 67.77 29.82 0.7855 0.8013 67.77 29.82 0.7855 0.8013 72.33 29.54 0.7784 0.8044

Decomposition Levels 2 3 4 72.02 75.95 78.16 29.56 29.33 29.2 0.8450 0.8539 0.8542 0.8125 0.8101 0.8043 70.00 74.23 76.64 29.68 29.42 29.29 0.8495 0.8566 0.8563 0.8165 0.8149 0.7991 72.49 76.70 78.86 29.53 29.28 29.16 0.8467 0.8553 0.8554 0.8100 0.8104 0.7972 212.88 318.79 431.91 24.85 23.10 21.78 0.751 0.7010 0.6738 0.7809 0.7246 0.6410 69.45 73.07 75.02 29.71 29.49 29.38 0.8458 0.8552 0.8558 0.8140 0.8277 0.8207 72.416 76.08 77.99 29.53 29.32 29.21 0.8397 0.8494 0.8508 0.8043 0.8016 0.7870 74.82 78.29 80.28 29.39 29.19 29.08 0.8351 0.8451 0.8459 0.8174 0.8190 0.8088

5 78.99 29.16 0.8542 0.8026 77.24 29.25 0.8565 0.7960 79.48 29.13 0.8558 0.7953 550.35 20.72 0.6599 0.5467 75.66 29.34 0.8559 0.8168 78.77 29.17 0.8513 0.7862 80.97 29.05 0.8451 0.8033

62.96 30.14 0.7733 0.8121 59.7 30.37 0.7806 0.8033 59.84 30.36 0.7783 0.8170 95.12 28.35 0.7715 0.8004 59.42 30.39 0.7797 0.8028 59.42 30.39 0.7797 0.8028 61.24 30.26 0.7727 0.8075

56.45 30.61 0.8337 0.8270 53.38 30.86 0.8393 0.8285 53.12 30.88 0.8390 0.8228 146.3 26.48 0.8003 0.8006 52.45 30.93 0.8388 0.8201 52.49 30.93 0.8346 0.8142 54.26 30.79 0.8292 0.8264

55.46 30.69 0.8435 0.8370 52.65 30.92 0.8464 0.8324 52.07 30.97 0.8483 0.8318 435.01 21.75 0.7255 0.6176 51.36 31.02 0.8493 0.8364 51.49 31.01 0.8457 0.8241 53.2 30.87 0.8389 0.8311

55.52 30.69 0.8430 0.8352 52.64 30.92 0.8463 0.8336 52.14 30.96 0.8480 0.8311 228.29 24.55 0.7649 0.7708 51.35 31.03 0.8491 0.8383 51.57 31.01 0.8449 0.8207 53.17 30.87 0.8385 0.8342

55.43 30.69 0.8435 0.8374 52.65 30.92 0.8464 0.8326 52.06 30.97 0.8483 0.8320 326.6 22.99 0.7392 0.7074 51.36 31.03 0.8494 0.8367 51.49 31.01 0.8457 0.8244 53.21 30.87 0.8389 0.8311

Page 44 of 18

New Orthogonal Multiwavelet Filters ...

Table 17 Comparative results of MSE,PSNR, SSIM and MS-SSIM of denoised image ’Couple’ of size 256 × 256 pixels for seven orthogonal multiwavelets through 𝐽 = 5 decomposition levels with AWGN and variance 𝜎=10. Image

Threshold

Multiwavelet GHM

SA4

CL

Soft

Alpert

New1

New2

Integer Haar

GHM

Couple

SA4

CL

hard

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 92.29 28.48 0.8162 0.9488 87.91 28.69 0.8206 0.9505 91.44 28.52 0.8169 0.9460 155.72 26.21 0.7496 0.9435 89.19 28.63 0.8164 0.9509 89.19 28.63 0.8164 0.9509 95.04 28.35 0.8051 0.9493

Decomposition Levels 2 3 4 100.35 107.19 110.23 28.12 27.83 27.71 0.8049 0.7955 0.7936 0.9370 0.9238 0.9181 98.62 105.69 108.22 28.19 27.89 27.79 0.8060 0.7967 0.7953 0.9398 0.9275 0.9241 102.99 110.09 112.76 28.00 27.71 27.61 0.7998 0.7895 0.7885 0.9339 0.9181 0.9133 239.46 320.22 394.98 24.34 23.08 22.17 0.6423 0.5611 0.5394 0.8982 0.7995 0.7082 100.26 106.99 109.52 28.12 27.84 27.74 0.8002 0.7904 0.7896 0.9425 0.9290 0.9245 106.46 113.41 116.14 27.86 27.58 27.48 0.7885 0.7820 0.7828 0.9339 0.9162 0.9116 107.04 114.19 116.87 27.84 27.55 27.45 0.7864 0.7782 0.7775 0.9364 0.9197 0.9135

5 111.19 27.67 0.7935 0.9181 109.12 27.75 0.7951 0.9236 113.7 27.57 0.7883 0.9130 479.76 21.32 0.5347 0.6526 110.37 27.70 0.7895 0.9238 117.09 27.45 0.7832 0.9113 117.73 27.42 0.7777 0.9130

77.31 29.25 0.8324 0.9542 73.9 29.44 0.8366 0.9529 74.33 29.42 0.8357 0.9550 109.94 27.72 0.7959 0.9488 73.91 29.44 0.8360 0.9533 73.91 29.44 0.8360 0.9533 76.616 29.29 0.8308 0.9541

75.38 29.36 0.8384 0.9549 72.06 29.55 0.8420 0.9547 72.47 29.53 0.8408 0.9539 155.68 26.21 0.7447 0.9268 71.85 29.57 0.8423 0.9547 73.08 29.49 0.8382 0.9521 74.57 29.4 0.8361 0.9548

75.66 29.34 0.8378 0.9538 72.12 29.55 0.8420 0.9545 72.68 29.52 0.8402 0.9520 344.7 22.76 0.6743 0.7615 71.89 29.56 0.8424 0.9548 73.34 29.48 0.8383 0.9521 74.78 29.39 0.8358 0.9540

75.63 29.34 0.8379 0.9539 72.1 29.55 0.8421 0.9546 72.65 29.52 0.8402 0.9522 210.01 24.91 0.6960 0.8716 71.88 29.56 0.8425 0.9548 73.29 29.48 0.8384 0.9522 74.75 29.39 0.8359 0.9540

75.66 29.34 0.8378 0.9538 72.12 29.55 0.8420 0.9545 72.68 29.52 0.8402 0.9520 269.04 23.83 0.6792 0.8082 71.893 29.56 0.8424 0.9548 73.34 29.48 0.8383 0.9521 74.78 29.39 0.8358 0.9540

Page 45 of 18

New Orthogonal Multiwavelet Filters ...

Table 18 Comparative results of MSE,PSNR, SSIM and MS-SSIM of denoised image ’Kiel’ of size 512 × 512 pixels for seven orthogonal multiwavelets through 𝐽 = 5 decomposition levels with AWGN and variance 𝜎=10. Image

Threshold

Multiwavelet GHM

SA4

CL

Soft

Alpert

New1

New2

Integer Haar

GHM

Kiel

SA4

CL

hard

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 105.02 27.92 0.9139 0.8155 102.14 28.04 0.9026 0.8183 104.94 27.92 0.9178 0.8120 248.68 24.17 0.9000 0.8081 103.02 28.00 0.9069 0.8184 103.02 28.00 0.9069 0.8184 107.76 27.81 0.9162 0.8164

Decomposition Levels 2 3 4 114.96 121.17 123.68 27.53 27.3 27.21 0.9247 0.9236 0.9221 0.8069 0.7988 0.7941 114.43 120.9 123.5 27.55 27.31 27.21 0.9271 0.9268 0.9250 0.8085 0.7996 0.7949 116.99 123.39 125.82 27.45 27.22 27.13 0.9200 0.9175 0.9164 0.8042 0.7944 0.7924 369.09 470.63 555.71 22.46 21.40 20.68 0.8484 0.7684 0.7140 0.7658 0.6911 0.6106 114.26 120.64 122.91 27.55 27.32 27.23 0.9263 0.9261 0.9252 0.8121 0.8085 0.8095 118.5 124.74 127.12 27.39 27.17 27.09 0.9192 0.9195 0.9192 0.8063 0.7977 0.7937 119.93 126.47 128.7 27.34 27.11 27.03 0.9227 0.9191 0.9181 0.8091 0.8058 0.8047

5 124.52 27.18 0.9220 0.7924 124.3 27.19 0.9249 0.7922 126.59 27.11 0.9162 0.7894 630.52 20.13 0.6970 0.5539 123.66 27.21 0.9252 0.8109 127.87 27.06 0.9198 0.7878 129.46 27.01 0.9181 0.8057

75.26 29.37 0.9209 0.8244 72.91 29.50 0.9032 0.8222 73.15 29.49 0.9322 0.8257 167.83 25.88 0.9063 0.8147 72.757 29.51 0.9088 0.8221 72.76 29.51 0.9088 0.8221 74.41 29.41 0.9234 0.8227

71.62 29.58 0.9426 0.8296 69.49 29.71 0.9400 0.8291 69.15 29.73 0.9416 0.8277 241.1 24.31 0.8913 0.7932 68.89 29.75 0.9408 0.8249 69.58 29.71 0.9384 0.8223 70.74 29.63 0.9445 0.8272

71.11 29.61 0.9518 0.8385 69.16 29.73 0.9487 0.8316 68.59 29.77 0.9501 0.8341 451.25 21.59 0.7918 0.6503 68.24 29.79 0.9518 0.8379 68.9 29.75 0.9497 0.8283 70.11 29.67 0.9526 0.8386

71.21 29.61 0.9504 0.8345 69.17 29.73 0.9482 0.8306 68.64 29.76 0.9490 0.8308 314.31 23.16 0.8462 0.7522 68.35 29.78 0.9503 0.8323 69.00 29.74 0.9481 0.8270 70.22 29.67 0.9512 0.8354

71.12 29.61 0.9517 0.8384 69.16 29.73 0.9487 0.8312 68.57 29.77 0.9502 0.8350 384.3 22.28 0.8069 0.6975 68.26 29.79 0.9517 0.8374 68.91 29.75 0.9496 0.8289 70.13 29.67 0.9525 0.8384

Page 46 of 18

New Orthogonal Multiwavelet Filters ...

Table 19 Comparative results of MSE,PSNR, SSIM and MS-SSIM of denoised image ’House’ of size 256 × 256 pixels for seven orthogonal multiwavelets through 𝐽 = 5 decomposition levels with AWGN and variance 𝜎=20. Image

Threshold

Multiwavelet GHM

SA4

CL

Soft

Alpert

New1

New2

Integer Haar

GHM

House

SA4

CL

hard

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 142.45 26.59 0.5531 0.6539 130.61 26.97 0.5651 0.6466 138.84 26.71 0.5577 0.6554 141.88 26.61 0.5528 0.6480 133.63 26.87 0.5597 0.6476 133.63 26.87 0.5597 0.6476 143.67 26.56 0.5475 0.6517

Decomposition Levels 2 3 4 96.18 96.29 101.75 28.30 28.30 28.06 0.7296 0.7803 0.7856 0.6569 0.6459 0.6264 87.73 90.28 95.79 28.70 28.57 28.32 0.7388 0.7842 0.7878 0.6643 0.6473 0.6351 100.21 99.7 105.1 28.12 28.14 27.91 0.7272 0.7799 0.7862 0.6531 0.6477 0.6297 132.45 200.02 283.49 26.91 25.12 23.61 0.6944 0.6912 0.6868 0.6505 0.5989 0.5288 87.54 85.925 89.234 28.71 28.79 28.63 0.7335 0.7837 0.7917 0.6644 0.6758 0.6801 100.62 101.43 105.88 28.10 28.07 27.88 0.7095 0.7612 0.7754 0.6520 0.6508 0.6391 101.04 100.53 104.21 28.09 28.11 27.95 0.7072 0.7585 0.7668 0.6625 0.6673 0.6627

5 104.51 27.94 0.7872 0.6176 98.45 28.20 0.7887 0.6320 107.66 27.81 0.7875 0.6172 384.37 22.28 0.6927 0.4256 91.48 28.52 0.7932 0.6740 108.33 27.78 0.7771 0.6247 106.35 27.86 0.7706 0.6537

166.77 25.91 0.5216 0.6563 156.48 26.19 0.5336 0.6457 161.72 26.04 0.5262 0.6589 137.42 26.75 0.5559 0.6485 157.12 26.17 0.5307 0.6463 157.12 26.17 0.5307 0.6463 163.3 26.00 0.5218 0.6509

118.57 27.39 0.6514 0.6721 109.25 27.75 0.6620 0.6739 113.67 27.57 0.6575 0.6681 101.72 28.06 0.721 0.6607 107.16 27.83 0.6669 0.6640 113.38 27.59 0.6502 0.6579 115.52 27.50 0.6471 0.6662

109.77 27.73 0.6850 0.6801 102.2 28.04 0.6884 0.6795 104.28 27.95 0.6930 0.6767 284.32 23.59 0.7405 0.5093 98.07 28.22 0.7035 0.6870 105.27 27.91 0.6918 0.6646 106.86 27.84 0.6831 0.6806

110.57 27.69 0.6826 0.6790 102.48 28.02 0.6876 0.6786 105.08 27.92 0.6907 0.6772 139.34 26.69 0.7417 0.6396 99.04 28.17 0.6996 0.6785 106.22 27.87 0.6859 0.6674 107.55 27.81 0.6795 0.6805

109.79 27.73 0.6850 0.6796 102.17 28.04 0.6884 0.6792 104.22 27.95 0.6930 0.6768 200.79 25.10 0.7385 0.5943 98.1 28.21 0.7034 0.6861 105.33 27.91 0.6915 0.6665 106.88 27.84 0.6830 0.6826

Page 47 of 18

New Orthogonal Multiwavelet Filters ...

Table 20 Comparative results of MSE,PSNR, SSIM and MS-SSIM of denoised image ’Kiel’ of size 512 × 512 pixels for seven orthogonal multiwavelets through 𝐽 = 5 decomposition levels with AWGN and variance 𝜎=20. Image

Threshold

Multiwavelet GHM

SA4

CL

Soft

Alpert

New1

New2

Integer Haar

GHM

Kiel

SA4

CL

hard

Alpert

New1

New2

Integer Haar

Kolev et al.: Preprint submitted to Elsevier

1 272.71 23.77 0.7821 0.7432 261.74 23.95 0.7550 0.7447 269.97 23.82 0.8041 0.7408 397.34 22.14 0.7625 0.7376 266.38 23.88 0.7635 0.7448 266.38 23.88 0.7635 0.7448 279.29 23.67 0.7870 0.7435

Decomposition Levels 2 3 4 268.66 281.78 289.61 23.84 23.63 23.51 0.8254 0.8385 0.8360 0.7304 0.7120 0.6992 264.28 278.39 286.46 23.91 23.68 23.56 0.8282 0.8447 0.8416 0.7331 0.7145 0.7010 271.63 285.83 293.45 23.79 23.57 23.46 0.8211 0.8283 0.8275 0.7282 0.7063 0.6973 507.77 625.36 723.45 21.07 20.17 19.54 0.7572 0.6895 0.6297 0.6973 0.6087 0.5148 266.5 281.11 288.16 23.87 23.64 23.53 0.8250 0.8398 0.8394 0.7374 0.7212 0.7160 282.67 296.43 303.67 23.62 23.41 23.31 0.8061 0.8221 0.8246 0.7269 0.7024 0.6943 281.94 297.11 304.12 23.63 23.40 23.30 0.8229 0.8262 0.8249 0.7312 0.7130 0.7051

5 292.65 23.47 0.8356 0.6950 289.35 23.52 0.8413 0.7007 296.15 23.42 0.8273 0.6955 805.9 19.07 0.6121 0.4615 290.8 23.49 0.8398 0.7216 306.29 23.27 0.8269 0.6920 306.81 23.26 0.8254 0.7106

242.43 24.28 0.7892 0.7518 232.79 24.46 0.7550 0.7479 235.51 24.41 0.8170 0.7536 303.59 23.31 0.7678 0.7428 232.7 24.46 0.7650 0.7485 232.7 24.46 0.7650 0.7485 239.53 24.34 0.7947 0.7504

215.75 24.79 0.8453 0.7577 206.91 24.97 0.8418 0.7572 207.86 24.95 0.8441 0.7549 344.75 22.76 0.8029 0.7245 205.71 25.00 0.8412 0.7533 211.29 24.88 0.8313 0.7482 213.74 24.83 0.8507 0.7550

212.67 24.85 0.8775 0.7644 204.49 25.02 0.8740 0.7597 204.5 25.02 0.8734 0.7611 548.84 20.74 0.7329 0.5912 202.26 25.07 0.8786 0.7698 207.57 24.96 0.8694 0.7551 210.63 24.90 0.8784 0.7689

212.99 24.85 0.8722 0.7605 204.37 25.03 0.8710 0.7577 204.95 25.01 0.8682 0.7555 413.51 21.97 0.7818 0.6792 202.83 25.06 0.8717 0.7575 207.95 24.95 0.8623 0.7509 210.94 24.89 0.8729 0.7594

212.68 24.85 0.8773 0.7642 204.5 25.02 0.8738 0.7588 204.51 25.02 0.8734 0.7610 482.18 21.30 0.7474 0.6277 202.36 25.07 0.8780 0.7658 207.64 24.96 0.8686 0.7556 210.72 24.89 0.8778 0.7645

Page 48 of 18

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