Compression and decompression based on discrete weighted transform

Applied Mathematics and Computation 335 (2018) 133–145

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Compression and decompression based on discrete weighted transform S. Jahedi∗, F. Javadi, M.J. Mehdipour Department of Mathematics, Shiraz University of Technology, P.O. Box 71555-313, Shiraz, Iran

a r t i c l e

i n f o

Keywords: Continuous function Uniform norm Compression and decompression Fuzzy partition Fuzzy transform

a b s t r a c t The purpose of this paper is to introduce and extend the concept of the weighted transform and its inverse to functions of two variables. We show that they can be applied to image compression and decompression. The quality of the reconstructed image by inverse weighted transform is also compared with some known methods such as the fuzzy transform method and the standard JPEG method. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In many problems, we are dealing with functions which have complex computation. So, it is reasonable to approximate functions by some methods which convert them into an n-dimensional vector (For more details see [1–3]). Fuzzy transform has useful applications in different fields such as, solution of ordinary and partial differential equations, image compression and decompression [5–7] and the references therein. Some authors have used fuzzy transform of functions of two variables to show that the quality of the reconstructed image is better than that of some others [1,4]. Regarding to Jahedi et al. [2], let ϕ be a continuous function from an arbitrary bounded subset E of R into (0, 1] and B = {B1 , . . . , Bn } be a ϕ -partition of E. In addition, suppose that μ is a positive finite measure such that ∫E Bk (x) dμ(x) > 0, for all k = 1, . . . , n. The linear map

FB : L1 (E, μ ) → Rn FB ( f ) = (F1 , . . . , Fn ) is called the F-transform based on a ϕ -partition and is denoted by Fϕ -transform of f. Here,



Fk =

( x )Bk ( x )d μ ( x ) , k = 1, . . . , n. E Bk ( x )d μ ( x )

E f

To reconstruct the original function f, if FB ( f ) = (F1 , . . . , Fn ) is the Fϕ -transform of f, then the linear operator TB : L1 (E, μ ) → C (E ) defined by

TB ( f ) =

n  k=1

∗

Fk

Bk

ϕ

,

Corresponding author. E-mail addresses: [email protected] (S. Jahedi), [email protected] (F. Javadi), [email protected] (M.J. Mehdipour).

https://doi.org/10.1016/j.amc.2018.04.048 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

134

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

is called the inverse Fϕ -transform of f with respect to B, where C(E) is the space of real-valued of continuous functions on E. It is proved that the original continuous function can be approximated by the inverse Fϕ -transform of f with respect to B with an arbitrary precision. By introducing fuzzy transform of functions with two variables, some authors have realized that it can be applied in image compression so that the quality of reconstructed image will be better than the image compression based on fuzzy relation equation (ICF) method. It is also compared with the reconstructed image by using the standard JPEG method. In order to improve the quality of the reconstructed image by fuzzy transform, some authors proposed several algorithms such as FTR [8], LFTR [9], DFTR [10] and DSFTR [11]. In this paper, we introduce the concept of discrete weighted transform of two variables function, which is a little modification of weighted transform of one variable functions [3,12]. In Section 2, we prove that the original function can be approximated by the discrete inverse weighted transform with desired precision. Section 3 is devoted to the applications of discrete weighted transform in gray image compression and decompression. By applying the discrete weighted transform in the proposed algorithms, i.e. FTR, LFTR, DFTR, DSFTR, we show that the discrete weighted transform of two variables function is a powerful tool for gray image compression. In fact the quality of reconstructed image by inverse weighted transform is so convenient like some known methods such as the fuzzy transform and the standard JPEG methods. 2. Basic concepts and main results Suppose that ϕ and ψ are continuous functions defined on E = [a, b] and F = [c, e] into (0, d], d ≥ 1, respectively. Definition 1. Let x0 = x1 < · · · < xn = xn+1 be nodes from E such that x1 = a, xn = b and n ≥ 2. The collection B = {B1 , . . . , Bn } is called a ϕ -partition of E if each Bk is a continuous map from E into [0, d], where d ≥ 1, for all k = 1, . . . , n and satisfies in the following statements: (i) Bk (xk ) = ϕ (xk ), for all k = 1, . . . , n. (ii) Bk (x ) = 0 whenever x ∈ (xk−1 , xk+1 ), for all k = 1, . . . , n. n (iii) k=1 Bk (x ) = ϕ (x ) for all x ∈ E. A ϕ -partition of E can be constructed by defining Bk , k = 1, . . . , n, as follows:



B1 ( x ) =

ϕ (x )



x−x2 x1 −x2



0

⎧ ⎪ ⎨ϕ (x )− Bk−1 (x ) x−x Bk (x ) = ϕ (x ) x −xk+1 k k+1 ⎪ ⎩

x ∈ [x1 , x2 ] otherwise; x ∈ [xk−1 , xk ] x ∈ [xk , xk+1 ] otherwise;

0

for k = 2, . . . , n − 1 and

Bn ( x ) =

ϕ (x ) − Bn−1 (x ) 0

x ∈ [xn−1 , xn ] otherwise.

In the sequel, suppose that the original function f is known only at some nodes (si , tj ) ∈ E × F, i = 1, . . . , N and j = 1, . . . , M. Additionally, suppose that S = {s1 , . . . , sN } and T = {t1 , . . . , tM } are sufficiently dense with respect to the chosen partitions, i.e.:

(∀k )(∃i )Bk (si ) > 0, (∀l )(∃ j )Cl (t j ) > 0 where k = 1, . . . , n and l = 1, . . . , m. Definition 2. Let f be a function given at nodes (si , tj ) ∈ E × F, i = 1, . . . , N, j = 1, . . . , M. Let B = {B1 , . . . , Bn } be a ϕ -partition of E and C = {C1 , . . . , Cm } be a ψ -partition of F, where n < N, m < M. Suppose that the sets S = {s1 , . . . , sN } and T = {t1 , . . . , tM } are sufficiently dense with respect to the chosen partitions. Then the n × m-matrix of real numbers defined as

⎛

F11 ⎜F21 Fnm [ f ] = ⎜ . ⎝ .. Fn1

F12 F22 .. . Fn2

⎞

. . . F1m . . . F2m ⎟  k=1,...,n = Fkl .. ⎟ .. l=1,...,m ⎠ . . . . . Fnm n×m

is called the discrete weighted transform of f with respect to B = {B1 , . . . , Bn } and C = {C1 , . . . , Cm }, where

M N

Fkl =

i=1 f (si , t j ) Bk (si ) Cl (t j ) . M N j=1 i=1 Bk (si ) Cl (t j )

j=1

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

135

Definition 3. Let ϕ and ψ be continuous functions defined on E and F into (0, d], d ≥ 1, respectively. Let B = {B1 , . . . , Bn } be a ϕ -partition of E and C = {C1 , . . . , Cm } be a ψ -partition of F. Suppose that f is a function given at nodes (si , tj ) ∈ E × F,

 k=1,...,n

i = 1, . . . , N, j = 1, . . . , M and Fnm [ f ] = Fkl function Fnm [ f ] defined by

Fnm [ f ] =

n  m 

Fkl

k=1 l=1

Bk Cl

ϕψ

l=1,...,m

is the discrete weighted transform of f with respect to B and C. Then the

,

where

Fnm [ f ](si , t j ) =

n  m 

Fkl

k=1 l=1

Bk (si ) Cl (t j ) , ϕ (si ) ψ (t j )

is called the discrete inverse weighted transform of f with respect to B = {B1 , . . . , Bn } and C = {C1 , . . . , Cm }. Let us recall that for f ∈ C(E × F), f u = sup(x,y )∈E×F | f (x, y )|. Now, we show that the original function f ∈ C(E × F) which is known at given nodes (si , tj ) ∈ E × F, i = 1, . . . , N and j = 1, . . . , M, can be approximated by the discrete inverse weighted transform with an arbitrary precision. Theorem 1. Let ϕ and ψ be continuous functions defined on E and F into (0, d], d ≥ 1, respectively. Suppose that f is a function given at nodes (si , tj ) ∈ S × F ⊆ E × F, i = 1, . . . , N, j = 1, . . . , M. Then for every ε > 0, there exist n, m ∈ N and a ϕ -partition B = {B1 , . . . , Bn } of E and a ψ -partition C = {C1 , . . . , Cm } of F so that S and T are sufficiently dense with respect to B and C, respectively. Then for all (s, t ) ∈ {s1 , . . . , sN } × {t1 , . . . , tM } we have (i) | f (s, t ) − Fkl | < ε for all k = 1, . . . , n and l = 1, . . . , m. (ii) |Fkl − F(k+1 ) (l+1 ) | < ε for all k = 1, . . . , n − 1 and l = 1, . . . , m − 1. (iii) f − Fnm [ f ] u < ε . Proof. Suppose that ε > 0 is given and s1 <  < sN , t1 < · · · < tM . The required ϕ -partition and ψ -partition will be obtained by choosing nodes of the partitions {x1 , . . . , xn } and {y1 , . . . , ym }, respectively, where n ≤ N and m ≤ M. Now select the nodes {(xk , yl ) |k = 1, . . . , n; l = 1, . . . , m} from S × T by the following algorithm and construct the required partition from these nodes. Algorithm: 1. k := 0 and l := 0; 2. k := k + 1, l := l + 1. Put xk := min S and yl = min T , S × T := S × T(xk , yl ). If S × T = ∅, then stop; 3. Consider set

S × T  = {(x, y ) ∈ S × T |xk < x, yl < yand| f (x, y ) − f (xk , yl )| ≤ ε /2}. If S × T  = ∅, then put k := k + 1, l := l + 1, (xk , yl ) := max(S × T  ) and S × T := S × T \ S × T  . If S × T = ∅, then go to step 2, otherwise, put n := k, m := l and stop. By giving nodes {(xk , yl ) | k = 1, . . . , n; l = 1, . . . , m}, the ϕ -partition B = {B1 , . . . , Bn } and the ψ -partition C = {C1 , . . . , Cm } can be defined. Clearly, the sets S and T are sufficiently dense with respect to the constructed partitions. We will prove (i) for some (arbitrarily chosen) (s, t) ∈ S × T. Since

| f (s, t ) − f (si , t j )| ≤ | f (xk , yl ) − f (s, t )| + | f (xk , yl ) − f (si , t j )| <

ε 2

+

ε 2

= ε,

then we have

M  N    i=1 f (si , t j )Bk (si ) Cl (t j )  | f (s, t ) − Fkl | =  f (s, t ) − j=1  M N j=1 i=1 Bk (si ) Cl (t j )   M N   j=1 i=1 f (s, t ) − f (si , t j ) Bk (si ) Cl (t j ) ≤ M N j=1 i=1 Bk (si ) Cl (t j )       t ∈(y ,y ) s ∈(x ,x ) f (s, t ) − f (si , t j ) Bk (si ) Cl (t j ) =

< ε.

j

l−1

l+1 

i

k−1

t j ∈(yl−1 ,yl+1 )

k+1 

si ∈(xk−1 ,xk+1 )

Bk (si ) Cl (t j )

Similarly, | f (s, t ) − F(k+1 )(l+1 ) | < ε for k = 1, . . . , n − 1, l = 1, . . . , m − 1 and (s, t ) ∈ [xk , xk+1 ] × [yl , yl+1 ]. This completes the proof of (i). Part (ii), clearly follows from (i). To prove part (iii), note that

136

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

Fig. 1. Reconstructed images “Cameraman” by using WTR algorithm. (a ) Original picture. (b) ρ = 0.035. (c) ρ = 0.062. (d) ρ = 0.111. (e) ρ = 0.250. (f) ρ = 0.4 4 4.

| f (s, t ) − Fnm [ f ](s, t )| ≤

n  m 

| f (s, t ) − Fkl |

k=1 l=1

Bk (s ) Cl (t ) ϕ (s ) ψ (t )

= ε.

So the proof of (iii) will be completed.

3. An application of the discrete weighted transform in image compression and decompression Finding the size of small areas of an image is usually crucial in image compression. In the other words, the authors convert image f of size N × M into new image f of size N × M , where N < N and M < M. Recently, many authors had proposed several algorithms for solving the problem such as FTR, LFTR, DFTR, DSFTR and have used fuzzy transform method in order to find the appropriate size of these areas so that the quality of the reconstructed image will be improved.  M Remember that the compression rate of the image is ρ = NNM . In this section, by applying the aforementioned algorithms, we show that weighted transform is also a powerful tool for image compression and decompression. Let P be a gray image of size N × M pixels. Define f (i, j ) := P (i, j )/(L − 1 ) for all i = 1, . . . , N and j = 1, . . . , M, where L is the length of the gray scale (for example, L = 256). We suggest to compress image f by using the discrete weighted

 k=1,...,n

transform Fnm [ f ] = Fkl

l=1,...,m

, where

M N Fkl =

i=1 f (ti , s j )Bk (ti )Cl (s j ) , M N j=1 i=1 Bk (ti )Cl (s j )

j=1

(1)

for k = 1, . . . , n and l = 1, . . . , m. For simplicity, put ti = i and s j = j in (1) we, where [a, b] = [1, N] and [c, e] = [1, M]. The compressed image Fnm can be decompressed by the discrete inverse weighted transform defined by

Fnm [ f ](i, j ) =

n  m  k=1 l=1

Fkl

Bk (i ) Cl ( j ) . ϕ (i ) ψ ( j )

(2)

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

137

Table 1 List of compression rates.

ρ (B)

M (B)

N (B)

m (B)

n (B)

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

16 8 8 8 3

16 8 8 8 3

3 2 3 4 2

3 2 3 4 2

Fig. 2. Reconstructed images “Lena” by using WTR algorithm. (a) Original picture. (b) ρ = 0.035. (c) ρ = 0.062. (d) ρ = 0.111. (e) ρ = 0.250. (f) ρ = 0.444.

Note that, Theorem 1, just guarantees the existence of integers n and m, for which | f (i, j ) − Fnm [ f ](i, j )| < ε and it does not give a practical method to achieve such integers n and m for an arbitrary ε . So we consider different compression rates for every image. List of known compression rates is reported in Table 1. Let us recall that by using the Peak Signal to Noise Ratio (shortly, PSNR), the quality of the reconstructed image can be evaluated by

P SNR = 20 log10

 255  RMSE

or P SNR = 20 log10

 max f  RMSE

,

where RMSE (Root Mean Square Error) is given by

RMSE =

    2  N M  i=1 j=1 f (i, j ) − FFnm [ f ](i, j ) N×M

.

(3)

Note that in formula (3), FFnm [ f ] is the reconstructed image obtained from the recomposition of the blocks. The proposed algorithms FTR, LFTR, DFTR and DSFTR, enable us to introduce four algorithms for image compression and decompression by weighted transform. In the sequel, we use weighted transform instead of fuzzy transform in mentioned four algorithms and call them with WTR, LWTR, DWTR, DSWTR, respectively. To demonstrate the efficiency of results, we have considered 100 images extracted from Image Database of the University of Southern California (http://sipi.usc.edu/database/). We present our results only for well known gray images “Cameraman”

138

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

Fig. 3. Reconstructed images “Cameraman” by using LWTR algorithm. (a) Original picture. (b) ρ = 0.031. (c) ρ = 0.056. (d) ρ = 0.125. (e) ρ = 0.281. (f) ρ = 0.500.

and “Lena” and the weighted functions ϕ and ψ are defined by

ϕ (x ) = ψ (x ) = 1 + exp + exp



−

 2x0.0001 − 2x0.0002 − 6x0.0003 − 24x0.0004 − 120x0.0005  − 12 + x + 2x0.0001 − 2x0.0002 − 6x − 12

+x+

3 x2 + x6 2 0.0 0 03

x2 2

+

x4 x5 + 120 24 0.0 0 04

− 120x0.0005

− 24x

+

x3 6

+

x4 24

+

x5 120



.

3.1. Image compression and decompression algorithm by dividing image into image-blocks Di Martino et al.[8] had proposed compression algorithm by dividing image into image-blocks (FTR algorithm). Now, by using weighted transform and applying FTR algorithm, we have another algorithm known as WTR algorithm. Figs. 1 and 2 demonstrate the reconstructed images by using WTR algorithms for the compression rates of Table 1. 3.2. Image compression and decompression algorithm by dividing image into block-layers In order to improve the quality of reconstructed image, Perfilieva and De Baets [13] introduced a new algorithm. They partitioned an interval [0,1], as range of the normalized pixel values of image f of size N × M, into L closed subintervals with the same width 1/L. Afterwards, Di Martino and Sessa [9] proposed a slight modification on this method and have been named as LFTR algorithm. Now by using weighted transform and applying LFTR algorithm, we have another algorithm known as LWTR algorithm. Each block can be reconstructed by using the following formula:

FFn(B )m(B ) [ fB ](i, j ) =

L  =1

F∗F, [ f ](i, j ), n ( B )m ( B ) B

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

139

Fig. 4. Reconstructed images “Lena” by using LWTR algorithm. (a) Original picture. (b) ρ = 0.031. (c) ρ = 0.056. (d) ρ = 0.125. (e) ρ = 0.281. (f) ρ = 0.500. Table 2 Various compression rates (L = 2 ). W-transform

LW-transform

Compression rates

Rows and columns

Coded rows and columns

Rows and columns

Coded rows and columns

ρ

Lρ

Lρ tot

4 12 6 8 16

3 6 2 2 3

4 8 8 12 16

2 3 2 2 2

0.562 0.250 0.111 0.062 0.035

0.250 0.140 0.062 0.028 0.015

0.500 0.281 0.125 0.056 0.031

where

F∗F, [f ] n ( B )m ( B ) B

(i, j ) =

FF, [ f ](i, j ), n ( B )m ( B ) B

if M (i, j ) = ; 

FF, [ f ](i, j ) − InfB , n ( B )m ( B ) B

otherwise.

(4)

(For more details see [9]). Note that in formula (4), FF, [ f ](i, j ) is the discrete inverse weighted transform of each block-layer. The compression n ( B )m ( B ) B rate of each image is obtained from the sum of the compression rates of all image-layers (for more details, refer to [9]). Table 2 is contained some known compression rates. Reconstructed images by LWTR algorithm for compression rates of Table 2, is shown in Figs. 3 and 4. 3.3. Image compression and decompression algorithm by restoring of image histogram Hurtik and Perfilieva [10] introduced an algorithm (DFTR algorithm) which has been improved the quality of reconstructed image by restoring the image histogram. Regarding to DFTR, the reconstructed images by DWTR algorithm for compression rates of Table 1 is shown in Fig. 5 and 6.

140

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

Fig. 5. Reconstructed images “Cameraman” by using DWTR algorithm. (a) Original picture. (b) ρ = 0.035. (c) ρ = 0.062. (d) ρ = 0.111. (e) ρ = 0.250. (f) ρ = 0.444.

3.4. Image compression and decompression algorithm by similarity measure and joining of similar blocks Finally, an algorithm is proposed such that the quality of the reconstructed image by using fuzzy transform is better than one of the most used algorithm for image compression – the JPEG method [11]. In this algorithm, quality of reconstructed image is improved by joining of similar blocks (DSFTR algorithm). Now, by using weighted transform and applying DSFTR algorithm, we have DSWTR algorithm. Let us recall that the similarity measure of two image functions f, g is evaluated as:

S( f, g) = exp



−

|F f in − G f in | +

n

k=1

|Fkl − Gkl | 

nm + 1

,

(5)

where Fkl and Gkl are computed hierarchically by components of discrete weighted transform of images f and g, respectively. =1,...,n The lowest (first) level is computed by F (1 ) [ f ] = F[ f ] = (Fk,l )kl=1 and for a highest level , we use recursive formula ,...,m





(−1 ) ) F ( ) [ f ] = F[F (−1) ] = F11 , . . . , Fn((−1 . −1 ) m (−1 )

The reconstructed images by DSWTR algorithm for compression rates of Table 1, is shown in Figs. 7 and 8.

4. An example Tables 3 and 4 report the PSNR of reconstructed images by four methods WTR, LWTR, DWTR and DSWTR for the source images “Cameraman” and “Lena”. Tables 5 and 6 report the PSNR of reconstructed images by five methods FTR, LFTR, DFTR, DSFTR and JPEG for the source images “Cameraman” and “Lena”. The comparison of PSNR of the reconstructed images by the discrete weighted transform, the fuzzy transform and the standard JPEG method are shown in Figs. 9 and 10. Following example shows that the quality of reconstructed image by the discrete inverse weighted transform depends on the choice of the weighted functions ϕ and ψ .

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

141

Fig. 6. Reconstructed images “Lena” by using DWTR algorithm. (a) Original picture. (b) ρ = 0.035. (c) ρ = 0.062. (d) ρ = 0.111. (e) ρ = 0.250. (f) ρ = 0.444. Table 3 PSNR in different rates for the image “Cameraman”. Discrete inverse weighted transform based on ϕ -partition and ψ -partition

ρ (B)

WTR

DWTR

DSWTR

ρ (B)

ρWTR

ρ tot

LWTR

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

20.6667 21.6315 23.5602 25.2855 28.7621

25.2335 28.2675 30.4180 37.4418 43.2144

26.8268 30.3485 34.4348 41.3289 48.4627

0.035 0.062 0.111 0.250 0.562

0.015 0.028 0.062 0.014 0.250

0.031 0.056 0.125 0.281 0.500

23.9351 24.9186 25.8758 26.9786 29.6298

Table 4 PSNR in different rates for the image “Lena”. Discrete inverse weighted transform based on ϕ -partition and ψ -partition

ρ (B)

WTR

DWTR

DSWTR

ρ (B)

ρWTR

ρ tot

LWTR

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

23.1194 24.3508 26.9105 29.2252 31.9273

25.5634 27.2617 29.6876 32.9875 38.9727

28.2685 29.2749 34.0793 36.8998 42.2741

0.035 0.062 0.111 0.250 0.562

0.015 0.028 0.062 0.014 0.250

0.031 0.056 0.125 0.281 0.500

23.6285 24.4655 26.0859 27.3353 31.3906

Example 1. Consider the functions ϕ 1 and ψ 1 defined by

ϕ1 ( x ) = 1 +

1 , 1 + x2

ψ1 ( y ) = 1 +

1 . 1 + y2

The PSNR of the reconstructed images by four algorithms WTR, LWTR, DWTR, DSWTR and using the discrete inverse weighted transform based on ϕ 1 -partition B1 , and ψ 1 -partition C1 , are shown in Tables 7 and 8. The comparison of

142

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

Fig. 7. Reconstructed images “Cameraman” by using DSWTR algorithm. (a) Original picture. (b) ρ = 0.035. (c) ρ = 0.062. (d) ρ = 0.111. (e) ρ = 0.250. (f) ρ = 0.444.

Table 5 PSNR in different rates for the image “Cameraman”. Compression and decompression by inverse and fuzzy transform

ρ (B)

FTR

DFTR

DSFTR

JPEG

ρ (B)

ρ tot

LFTR

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

20.5610 21.4724 23.4738 25.1991 28.5837

25.2099 28.1786 30.3388 37.1980 42.9238

26.7820 30.1645 34.2725 41.1984 48. 1849

25.5207 28.4293 33.4379 38.8007 45.5878

0.035 0.062 0.111 0.250 0.562

0.031 0.056 0.125 0.281 0.500

23.9213 24.9011 25.8472 26.9677 29.6063

Table 6 PSNR in different rates for the image “Lena”. Compression and decompression by inverse and fuzzy transform

ρ (B)

FTR

DFTR

DSFTR

JPEG

ρ (B)

ρ tot

LFTR

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

22.5526 23.6493 26.3127 28.6263 31.0768

25.1053 27.2014 29.6427 32.6210 38.5320

28.2013 29.0535 33.9729 36.8116 42.2456

29.8727 32.4369 35.7345 37.5461 38.4881

0.035 0.062 0.111 0.250 0.562

0.031 0.056 0.125 0.281 0.500

23.5535 24.4125 26.0332 27.3229 31.3674

Tables 7 and 8 with Tables 3 and 4, respectively, shows that the improvement of the quality of reconstructed image by the discrete inverse weighted transform depends on the choice of the weighted functions.

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

143

Fig. 8. Reconstructed images “Lena” by using DSWTR algorithm. (a) Original picture. (b) ρ = 0.035. (c) ρ = 0.062. (d) ρ = 0.111. (e) ρ = 0.250. (f) ρ = 0.444.

Fig. 9. Trend of the PSNR for image “Cameraman”.

144

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

Fig. 10. Trend of the PSNR for image “Lena”. Table 7 PSNR in different rates for the image “Cameraman”. Discrete inverse weighted transform based on ϕ 1 -partition and ψ 1 -partition

ρ (B)

WTR

DWTR

DSWTR

ρ (B)

ρWTR

ρ tot

LWTR

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

20.6510 21.5780 23.5056 25.2793 28.6512

25.2235 28.2220 30.4103 37.3531 43.0202

26.8126 30.2361 34.3430 41.2436 48.2771

0.035 0.062 0.111 0.250 0.562

0.015 0.028 0.062 0.014 0.250

0.031 0.056 0.125 0.281 0.500

23.9311 24.9180 25.8689 26.9541 29.5807

Table 8 PSNR in different rates for the image “Lena”. Discrete inverse weighted transform based on ϕ 1 -partition and ψ 1 -partition

ρ (B)

WTR

DWTR

DSWTR

ρ (B)

ρWTR

ρ tot

LWTR

0.035156 0.062500 0.140625 0.250 0 0 0 0.444444

23.0792 24.2143 26.8529 29.1599 31.7392

25.5491 27.2347 29.6676 32.6443 38.8090

28.2403 29.1280 34.0209 36.8488 42.2649

0.035 0.062 0.111 0.250 0.562

0.015 0.028 0.062 0.014 0.250

0.031 0.056 0.125 0.281 0.500

23.5720 24.4392 26.0605 27.3407 31.3884

5. Conclusion This paper investigated the application of discrete weighted transform which is a modification of discrete fuzzy transform in image compression and decompression. We prove that a continuous function on a rectangle which is known at given nodes can be approximated by discrete weighted transform with an arbitrary precision. The application part of this paper demonstrated that using discrete weighted transform in W TR, LW TR, DW TR and DSW TR methods gives us appropriate algorithms which improve the PSNR results in image compression and decompression. An example has shown that the quality of the reconstructed image depends on the choice of functions ϕ and ψ . Ethical approval S. Jahedi declares that she has no conflict of interest. F. Javadi declares that she has no conflict of interest. M.J. Mehdipour declares that he has no conflict of interest. This article does not contain any studies with human participants or animal performed by any of the authors.

S. Jahedi et al. / Applied Mathematics and Computation 335 (2018) 133–145

145

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

I. Perfilieva, Fuzzy transforms: theory and applications, Fuzzy Sets Syst. 157 (8) (2006) 993–1023. S. Jahedi, M.J. Mehdipour, R. Rafizadeh, Approximation of integrable function based on ϕ -transform, Soft Comput. 18 (10) (2013) 2015–2022. S. Jahedi, F. Javadi, M.J. Mehdipour, Weighted transform and approximation of some functions on unbounded sets, Soft Comput. 21 (2017) 3579–3585. M. Stepnicka, R. Valasek, Fuzzy transform for functions with two variabes, in: J. Ramik, V. Novak (Eds.), Methods for Decision Support in Environment with Uncertainty – Applications in Economics, Business and Engineering, IRAFM, University of Ostrava, Czech Republic, 2003, pp. 96–102. F.D. Martino, V. Loia, S. Sessa, Fuzzy transforms for compression and decompression of color videos, Inf. Sci. 180 (2010) 3914–3931. F.D. Martino, V. Loia, S. Sessa, A segmentation method for images compressed by fuzzy transforms, Fuzzy Sets Syst. 161 (1) (2010) 56–74. F.D. Martino, V. Loia, S. Sessa, Fuzzy transforms method in prediction data analysis, Fuzzy Sets Syst. 180 (1) (2011) 146–163. F.D. Martino, V. Loia, I. Perfilieva, S. Sessa, An image coding/decoding method based on direct and inverse fuzzy transforms, Int. J. Approx. Reason. 48 (2008) 110–131. F.D. Martino, S. Sessa, Layers image compression and reconstruction by fuzzy transform, in: Proceedings of the Computational Intelligence in Image Processing, 2013, pp. 107–130. (chapter 6). P. Hurtik, I. Perfilieva, Image compression methodology based on fuzzy transform, in: Proceedings of Seventh International Joint Conference on Complex, Intelligent and Software Intensive Systems, Ostrava, Czech Republic, 2012, pp. 5–7. September. P. Hurtik, I. Perfilieva, Image compression methodology based on fuzzy transform using block similarity, in: Proceedings of Eighth Conference of the European Society for Fuzzy Logic and Technology, Milano, Italy, 2013, pp. 11–13. September. S. Jahedi, F. Javadi, Approximation of discrete data by discrete weighted transform, J. Math. Ext. 8 (1) (2014) 87–96. I. Perfilieva, B. De Baets, Fuzzy transforms of monotone functions with application to image compression, Inf. Sci. 180 (2010) 3304–3315.