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entropy

Article

Reliable Recurrence Algorithm for High-Order

Krawtchouk Polynomials

Khaled A. AL-Utaibi 1,*,† , Sadiq H. Abdulhussain 2,† , Basheera M. Mahmmod 2,† ,

Marwah Abdulrazzaq Naser 3,† , Muntadher Alsabah 4,† and Sadiq M. Sait 5,†

Citation: AL-Utaibi, K.A.;

Abdulhussain, S.H.; Mahmmod, B.M.;

Naser, M.A.; Alsabah, M.; Sait, S.M.

Reliable Recurrence Algorithm for

High-Order Krawtchouk

Polynomials. Entropy 2021,23, 1162.

https://doi.org/10.3390/e23091162

Academic Editor: Boris Ryabko

Received: 3 August 2021

Accepted: 1 September 2021

Published: 3 September 2021

Publisher’s Note: MDPI stays neutral

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Attribution (CC BY) license (https://

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1Department of Computer Engineering, University of Ha’il, Ha’il 682507, Saudi Arabia; alutaibi@uoh.edu.sa

2Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq;

sadiqhabeeb@coeng.uobaghdad.edu.iq (S.H.A.); basheera.m@coeng.uobaghdad.edu.iq (B.M.M.)

3Department of Architectural Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq;

marwahabdalkhafaji@gmail.com

4Department of Electronic and Electrical Engineering, University of Shefﬁeld, Shefﬁeld S1 4ET, UK;

mqalsabah@gmail.com

5Department of Computer Engineering, Interdisciplinary Research Center for Intelligent Secure Systems,

Research Institute, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia;

sadiq@kfupm.edu.sa

*Correspondence: alutaibi@uoh.edu.sa

† These authors contributed equally to this work.

Abstract:

Krawtchouk polynomials (KPs) and their moments are promising techniques for applica-

tions of information theory, coding theory, and signal processing. This is due to the special capabilities

of KPs in feature extraction and classiﬁcation processes. The main challenge in existing KPs recur-

rence algorithms is that of numerical errors, which occur during the computation of the coefﬁcients

in large polynomial sizes, particularly when the KP parameter (

p

) values deviate away from 0.5 to 0

and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefﬁcients

of KPs in high orders. In particular, this paper discusses the development of a new algorithm and

presents a new mathematical model for computing the initial value of the KP parameter. In addition,

a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal

recurrence algorithm was derived from the existing

n

direction and

x

direction recurrence algorithms.

The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP

coefﬁcients. First, the KP coefﬁcients were computed for one partition after dividing the KP plane into

four. To compute the KP coefﬁcients in the other partitions, the symmetry relations were exploited.

The performance evaluation of the proposed recurrence algorithm was determined through different

comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polyno-

mial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable

and computes lesser coefﬁcients when compared to the existing algorithms across wide ranges of

parameter values of

p

and polynomial sizes

N

. The results also show that the improvement ratio of

the computed coefﬁcients ranges from 18.64% to 81.55% in comparison to the existing algorithms.

Besides this, the proposed algorithm can generate polynomials of an order

∼

8.5 times larger than

those generated using state-of-the-art algorithms.

Keywords:

discrete Krawtchouk polynomials; Krawtchouk moments; propagation error; energy

compaction; computation cost

1. Introduction

Digital image processing plays an essential role in several aspects of our daily lives.

Image signals are subject to several processes such as transmission [

1

], enhancement [

2

],

transformation [

3

], hiding [

4

], and compression [

5

,

6

]. Similarly to image processing, speech

signals processing is also essential [

7

], and involves several stages such as transfer [

8

], acqui-

sition [

9

], and coding [

10

]. Pattern recognition, which is considered an automated process,

Entropy 2021,23, 1162. https://doi.org/10.3390/e23091162 https://www.mdpi.com/journal/entropy

Entropy 2021,23, 1162 2 of 24

is widely used in various applications such as computer vision [

11

], statistical data analysis

[

12

], information retrieval [

13

], shot boundary detection [

14

], and bio-informatics [

15

].

However, the accuracy of extracting the signiﬁcant features in these essential signal pro-

cessing approaches is crucial [

16

]. Feature extraction, in particular, is used to reduce the

dimensionality of the signals to a ﬁnite size [

17

,

18

]. Speciﬁcally, a ﬁnite number of features

can be used to represent the signals. These ﬁnite features can be considered the most

signiﬁcant ones and need to be extracted using efﬁcient methods. As such, to achieve

the best signal representation, a fast and robust feature extraction mechanism becomes

necessary. To this end, such features’ extraction mechanism needs to meet the desired

accuracy concerns by extracting the most signiﬁcant features efﬁciently with low processing

times. Furthermore, the energy compaction and localization of the signals can also be

considered as essential factors in signal compression [

19

]. This is attributed to the fact that

using fewer effective coefﬁcients results in a more accurate representation of the signals.

Hence, orthogonal polynomials are an effective tool that can be applied to meet these

desired requirements and features characterization.

Continuous and discrete orthogonal polynomials are commonly used in many signal

processing applications and feature characteristics. Continuous orthogonal polynomials

are used in speech and image applications, for example, in pattern recognition, robot vision,

face recognition, object classiﬁcation, hiding information, data compression, template

matching, and in edge detection for image data compression [

20

–

23

]. The performance of

orthogonal polynomials is evaluated according to their ability to extract distinct features

from signals in a fast and efﬁcient way. This special ability of feature extraction can be

quantiﬁed using properties such as a) energy compaction; b) signal representation without

redundancy; c) numerical stability; and d) localization [24–26].

Discrete orthogonal polynomials are widely used to extract features from images [

27

].

There are different types of discrete polynomials. Examples of these include discrete

Tchebichef polynomials [

28

], Chebyshev polynomials [

29

], discrete Charlier polynomials

[

26

], discrete Krawtchouk polynomials (DKPs), and discrete Meixner polynomials [

30

].

Among these polynomials, DKPs are widely exploited in image processing. This is due

to their salient characteristics, which can be used to extract local features from images.

Speciﬁcally, by exploiting the localization property of the DKPs, images can be efﬁciently

represented by using a ﬁnite number of features [

31

]. The localization property is carried

out by controlling the parameter value (

p

) of the DKPs. Typically, discrete orthogonal mo-

ments are generated using DKPs. Discrete orthogonal moments are extensively exploited in

image and signal processing [

11

,

32

–

34

], coding theory [

35

], and information theory [

36

,

37

].

However, reconstructing a signal using moments and maintaining the orthogonality has

to date been considered a challenging task. In addition to this, the discretization error is

another challenge that appears when reconstructing the signal, especially when moments

are used in the implementation. This discretization error, however, increases with the

moments’ order. For example, when the order of the moments increases, the discretization

error increases accordingly. As such, the accuracy of the moments’ computation is reduced,

resulting in an inaccurate representation of images [38–40].

Several studies have been performed on the discrete Krawtchouk polynomials and

methods developed to efﬁciently compute their coefﬁcients, for example see [

25

,

31

,

41

–

44

].

These research works utilize a three-term recurrence algorithm [

25

]. In addition, the hy-

pergeometric series and gamma functions are widely applied in image processing [

25

].

However, the aforementioned research works use functions that require a long time to

execute and process the signals. Furthermore, these functions become numerically unstable

when the order of the moments increases. Instead, a three-terms recurrence algorithm

can be applied to come up with the aforementioned time and accuracy issues. To this

end, Yap et al. [

41

] presented a recurrence algorithm in the

n

direction to calculate the

Krawtchouk polynomial coefﬁcients (KPCs). Due to the propagation error, this recurrence

algorithm becomes unstable—especially when the polynomial size increases. In general,

such a propagation error increases through the computation of polynomial coefﬁcients.

Entropy 2021,23, 1162 3 of 24

This is attributed to the fact that pitfalls may happen even when small errors in ﬂoating

numbers occur. As such, there is an essential need to reduce the number of recurrences,

especially when the polynomial size is increased. Furthermore, such a reduction could also

lead to a reduction in the propagation error, thereby leading to a more stable computation

of polynomial coefﬁcients, as desired. The work in [

31

] proposes a modiﬁed recurrence

algorithm in the

n

direction (RAN) by partitioning the KP array into two partitions. There-

fore, only 50% of the coefﬁcients need to be computed. However, the partitioning of the

KP array generates a larger polynomial size, which is undesirable. On a similar basis, the

work in [

42

] proposes a recurrence algorithm in the

x

direction (RAX) by partitioning the

KP plane into two partitions. Speciﬁcally, the

x

direction of the recurrence algorithm is

used to compute the KPCs. The results show that the RAX algorithm outperforms the

RAN algorithm. It is worth noting that the RAN and RAX algorithms use a symmetric

property to compute the polynomial coefﬁcients of the second portion. A novel bi-recursive

relation algorithm in the n direction (BRRN) was proposed in [

43

]. In this method, the KP

array is divided into four partitions. However, the KPC coefﬁcients are computed for two

partitions only, i.e., 50% of the coefﬁcients are computed. Then, a symmetric property is

used to compute the KPCs for the remaining partitions. The results indicate that the BRRN

algorithm provides higher gain than the RAX algorithm for limited values of parameter

p

,

i.e., the polynomial size. Abdulhussain et al. [

44

] developed an algorithm and presented

new properties of orthogonal polynomials such that the KP plane is divided into four por-

tions and only the KPCs for one portion are computed. For this, the size of the generated

polynomials is increased, but it is still limited, especially for parameter

p

less than 0.25 and

greater than 0.75. This is because the initial values or sets become zero as the polynomial

size increases. Recently, the work in [

25

] proposed a recurrence relation algorithm that has

the ability to compute KPCs with very large sizes. However, the proposed algorithm is

limited to the parameter value of p=0.5.

The existing algorithms suffer from the following limitations: (1) no initial value is

provided; (2) the propagation error is high; and (3) the implementation of these algorithms

is limited to a speciﬁc value of the parameter

p

. They also suffer from numerical instabilities,

especially when the polynomials orders and sizes become high. Therefore, an advanced

and reliable recurrence algorithm for high-order polynomials and large sizes is required.

Therefore, a new recurrence algorithm is presented in this paper, which handles the

numerical instabilities issue of using high orders of polynomials and large sizes. The

proposed algorithm is able to compute the KPCs for all values of the parameter

p

. In

addition to this, this paper presents the development of a new mathematical model for

computing the initial value of

p

. In particular, the initial value is accurately computed for all

values of parameter

p

. Furthermore, a new relation to compute the values of the initial sets

is derived. To this end, a diagonal recurrence relation is introduced. The proposed diagonal

recurrence algorithm is derived from the existing

n

and

x

directions of the recurrence

algorithm. The diagonal and the existing recurrence algorithm are exploited to compute

the KP coefﬁcients. The KP coefﬁcients are then used for one partition after dividing the KP

plane into four partitions. To compute the KP coefﬁcients in other partitions, a symmetric

property relation is utilized.

Organization of the paper

: This paper is organized as follows. Section 2presents

the mathematical formulations of the orthogonal polynomials and moments. In Section

3, the methodology of the proposed recurrence algorithm is provided. This methodology

involves providing a discussion about the initial value selection of parameter

p

. In addition,

this section explains how the Krawtchouk polynomial’s coefﬁcients can be computed. In

order to characterize the performance of the proposed approaches, Section 4provides the

numerical results. Finally, conclusions are discussed in Section 5.

Notation: In this paper, the operator transpose is denoted by

(·)T

and

(a

b)

denotes the

binomial coefﬁcients.

Entropy 2021,23, 1162 4 of 24

2. Preliminaries

This section presents the Krawtchouk polynomials and their recurrence relation. To

this end, the

n

-th order of the Krawtchouk polynomials based on the hypergeometric series

is given as

ˆ

Kp

n(x) = 2F1−n−x

−N+1;1

p. (1)

The weighted function

ω(x

,

p)

and the norm function

ρ(n

,

p)

are used to generate the

weighted and normalized KP coefﬁcients as given in [

31

]. To this end, the weighted and

normalized KP coefﬁcients can be written as in (2) and (3), respectively:

ω(x,p) = N−1

xpx(1−p)N−x−1(2)

ρ(n,p) = (−1)n1−p

pnn!

(−N+1)n(3)

The Pochhammer symbol

(·)c

, which is known as an ascending or rising factorial function,

can be written as [45]

(a)c=Γ(a+c)

Γ(a)=a(a+1)(a+2)· · · (a+c−1), (4)

where

Γ(

.

)

denotes the Gamma function. To this end, using the weight and norm functions,

the weighted and normalized Krawtchouk polynomials of the

n

-th order for a signal of

size Nare given as [31]

Kp

n(x)=sω(n,x)

ρ(n,x)ˆ

Kp

n(x), (5)

Kp

n(x)=sN−1

nN−1

x p

1−pn+x

2F1−n−x

−N+1;1

p, (6)

n,x=0, 1, . . . , N−1; p∈(0, 1),

where 2F1describes the hypergeometric series and can be written as

2F1−n−x

−N+1;1

p=

∞

∑

k=0

(−n)k(−x)k

(−N+1)k,k!1

pk

. (7)

3. Proposed Recurrence Algorithm

This section describes the methodology of the proposed recurrence algorithm.

3.1. Computing the Initial Value

The problem with traditional approaches for computing the initial value in the

n=

0

and

x=

0 directions is the numerical instability. For example, the traditional methods

provide zero values of the initial

Kp

0(0)

—which is unstable. The initial value can be

computed as

Kp

0(0)=q(1−p)N−1. (8)

The expression in

(8)

makes the initial value (

Kp

0(0)

) decrease to zero for different

values of parameter

p

, especially for large polynomial sizes

N

. Figure 1shows the values

of

Kp

0(0)

for different values of parameter

p

and size

N

. The results show that the value of

Entropy 2021,23, 1162 5 of 24

Kp

0(0)

starts to fall to zero when

p

becomes larger than 0.1. Speciﬁcally, as the values of

parameter

p

increase, the value of

Kp

0(0)

falls to zero. For example, for

p=

0.15, the value

of

Kp

0(0)

becomes zero for

N>

5000 while for

p=

0.4, the value of

Kp

0(0)

becomes zero

earlier for

N>

2000. This makes it impossible to compute the rest of the KP coefﬁcients’

values. Therefore, there is an essential need to ﬁnd an efﬁcient method for computing the

initial value of

p

, which prevents the initial value (

Kp

0(0)

) from dropping to zero. To this

end, this paper identiﬁes the suitable non-zero values in the KP plane that need to be used

as an initial value. As such, there is an essential need to plot the values of coefﬁcient

Kp

0(x)

for different values of parameter

p

. Figure 2shows the plots of the values of coefﬁcient

Kp

0(x)

for different values of parameter

p

. Clearly, the results in Figure 2show that the

values of

Kp

0(x)

start with a small number and gradually reach the peak. Then, the values

drop to a very small number. In addition, we observe that using non-small values as an

initial set of parameter pseems useful to compute other values of the KP coefﬁcients.

Figure 1.

The computation of the initial values (

Kp

0(0)

) as a function of different polynomial sizes

using Equation (8).

Figure 2. Plots of Kp

0(x)for a wide range of parameter pand N=500.

Figure 2demonstrates that the peak values can be located at

x=N p

. In this paper,

the value of

x=N p

is denoted by

x0

. To this end, a general formula for computing

Kp

0(x0)

can be written as

Entropy 2021,23, 1162 6 of 24

Kp

0(x0)=Kp

0(Np)×sω(N p;p)

ρ(0; p),

Kp

0(x0)=1×s(N−1

Np )pNp(1−p)−Np+N−1

1,

Kp

0(x0)=sN−1

Np pN p (1−p)−N p+N−1. (9)

Computing

Kp

0(x0)

using expression

(9)

may also lead to unstable numerical values

of coefﬁcients with errors, especially for high-order polynomials. This is because the

binomial coefﬁcients function tends to be very large and close to inﬁnity. To demonstrate

this behavior, Figure 3is provided.

Figure 3.

The computation of the initial values (

Kp

0(x0)

) as a function of different polynomial sizes

using Equation (9).

Figure 3shows that the initial values (

Kp

0(x0)

) are still inaccurate where these values

record either NaN or Inf values. This is due to the nature of the polynomial coefﬁcients

that are obtained by the expression in

(9)

. Thus, the initial values (

Kp

0(x0)

) seem difﬁcult to

be computed for large polynomial sizes (

N

). To overcome this issue, this paper proposes

an efﬁcient and suitable approach that makes the value of

Kp

0(x0)

commutable. It is

worth noting that the values obtained from the polynomial coefﬁcients’ formula, especially

the Gamma function, should be reduced when the coefﬁcients become large since their

argument value is increased. This can be achieved using

arg =exp(ln(arg)) = eln (arg)

and

the initial values can be computed as

Kp

0(x0) = e0.5×z, (10)

where zis given as

z=ln Γ(N)−log Γ(x0+1)−log Γ(N−x0)−x0ln1−p

p+ (N−1)ln(1−p).

Entropy 2021,23, 1162 7 of 24

A proof of expression (10) is presented in Appendix A.

Figure 4shows a plot of the proposed initial values of (

Kp

0(x0)

) in the developed

expression in

(10)

for various values of parameter

p

as a function of polynomials size

N

.

The results show that the proposed initial values are more computable for wide ranges of

parameter

p

and large polynomial sizes

N

, as desired. Hence, this signiﬁes the feasibility

of the proposed formula for practical implementations compared with state-of-the-art

equations.

Figure 4.

The computation of the initial sets’ values (

Kp

0(x0)

) as a function of different polynomial

sizes using Equation (10).

3.2. The Fundamental Computation of the Initial Values

Typically, for any orthogonal polynomial, the computation of coefﬁcients requires

the evaluation of a signiﬁcant number of fundamental initials. Thus, based on the ﬁrst

initial value

Kp

0(x0)

, computed using the proposed formula, the KP coefﬁcients are ob-

tained

Kp

0(x1)

,

Kp

1(x0)

, and

Kp

1(x1)

(see Figure 5). Therefore, this section shows how the

aforementioned coefﬁcients values are computed.

First,

Kp

0(x1)

is computed using the proposed derived formula, which provides the

two terms relation between the

Kp

0(x0)

and

Kp

0(x1)

. To this end, this relation/ratio between

the coefﬁcients can be formulated as

Kp

0(x1)

Kp

0(x0)=v

u

u

t

(N−1

Np+1)pN p+1(1−p)−N p+N−2

(N−1

Np )pNp (1−p)−Np+N−1,

=v

u

u

u

t

(N−1)!

(Np+1)!(N−N p−2)!

(N−1)!

(Np)!(N−N p−1)!

·1−p

p,

=sN−Np −1

Np +1·p

1−p, (11)

where x1=x0+1=Np +1. Thus, the expression in (11) can be further simpliﬁed to:

Entropy 2021,23, 1162 8 of 24

Kp

0(x1) = sN−Np −1

Np +1·p

1−pKp

0(x0). (12)

Figure 5.

The fundamental computation of initial values according to the

x

and

n

directions in the

KP plane.

Then,

Kp

1(x0)

and

Kp

1(x1)

can be computed using a two-term recurrence relation

with

Kp

0(x0)

and

Kp

0(x0)

, respectively. To derive the recurrence relation of the proposed

approach, the following formulas are used:

Kp

0(x)=qωp

K(x,N), (13)

Kp

1(x)=qωp

K(x,N),p(N−1)−x

p(N−1)p(1−p). (14)

From (13) and (14), Kp

1(x)can be simpliﬁed to:

Kp

1(x)=p(N−1)−x

p(N−1)p(1−p)Kp

0(x). (15)

Using the expression in (15), Kp

1(x0)and Kp

1(x1)can be further simpliﬁed to

Kp

1(x0)=p(N−1)−Np

p(N−1)p(1−p)Kp

0(x0)

=p

p(N−1)p(1−p)Kp

0(x0)(16)

Kp

1(x1)=p(N−1)−(Np +1)

p(N−1)p(1−p)Kp

0(x1)

=p+1

p(N−1)p(1−p)Kp

0(x1). (17)

Entropy 2021,23, 1162 9 of 24

3.3. The Computation of the Initial Sets

In this section, the computation of the initial sets is discussed. These initial sets are

shown in Figure 6. Figure 6shows parts of the KP coefﬁcients, which are covered in this

section. The initial sets are deﬁned in the ranges of x=x0,x1and n=2, 3, . . . , x.

Figure 6. A diagram shows the location of initial sets in the KP plane.

To compute the values of the initial set, the recurrence in the

n

direction is used. To

this end, the formulation of recurrence is given as

Kp

n(x)=α1n,xKp

n−1(x)−α2n,xKp

n−2(x), (18)

α1n,x=(N−2n+1)p+n−x−1

pp(1−p)n(N−n),

α2n,x=s(n−1)(N−n+1)

n(N−n),

x=x0,x1and n=2, 3, . . . , x.

After computing the initial sets, the values in the ranges

x=x0

,

x1

and

n=

0, 1,

. . .

,

x

are used as the initials to compute the rest of the KP coefﬁcients values.

3.4. Computation of the Coefﬁcients Values for KP

In this section, the rest of the coefﬁcient values are computed. These coefﬁcients are

shown in Figure 7. As depicted in Figure 7, there are two main parts, which are located at

the left (Part 1) and the right (Part 2) sides of the initial sets. In addition, the coefﬁcients are

located at the right side of the initial sets and can be divided into three sub-parts. These

parts are Part 2-1, Part 2-2, and Part 2-3. The detailed description of the computation of

each part is presented in the following subsections.

Entropy 2021,23, 1162 10 of 24

Figure 7.

A diagram shows the parts’ locations in the KP coefﬁcients’ plane in the

x

and

n

directions

based on the proposed algorithm.

3.4.1. Computation of the Coefﬁcients Located at Part 1

In this section, the values of KP coefﬁcients in Part 1, shown in Figure 7, are computed

using a backward

x

recurrence relation. The backward recurrence relation is obtained from

the traditional recurrence relation in the xdirection as

Kp

n(x−1) = β1n,xKp

n(x)−β2n,xKp

n(x+1), (19)

β1n,x=−(N−2x−1)p−n+x

pp(1−p)x(N−x),

β2n,x=s(N−x−1)(x+1)

x(N−x),

n=0, 1, . . . , x0and x=x0,x0−1, . . . , n.

The values of KP coefﬁcients become unstable as the index of xgoes towards n. This

is because the values of the coefﬁcients tend to be less than 10

−7

. To overcome this issue,

the condition of a threshold value is used to stop the recurrence for each value of index

n

.

The proposed condition is given by

|Kn(x)|<10−5and |Kn(x+1)|<10−7. (20)

3.4.2. Computation of the Coefﬁcients Located at Part 2-1

In this section, the values of KP coefﬁcients in Part 2-1, given in Figure 7, were

computed using a forward xrecurrence relation as given in (21):

Kp

n(x+1) = γ1n,xKp

n(x)−γ2n,xKp

n(x−1)(21)

γ1n,x=(N−2x−1)p−n+x

pp(1−p)(x+1)(N−x−1)

γ2n,x=−s(N−x)x

(x+1)(N−x−1)

n=0, 1, . . . , x0and x=x0,x0+1, . . . , N−n−1

The aforementioned recurrence relation, which is used to compute the values in Part

2-1, is subject to the following condition:

|Kn(x)|<10−5and |Kn(x+1)|<10−7. (22)

Entropy 2021,23, 1162 11 of 24

3.4.3. Computation of the Coefﬁcients Located at Part 2-2

This section presents two new recurrence relations’ approaches to compute the KP

coefﬁcient values diagonally. This diagonal calculation is given in Part 2-2 Figure 7. The

values in the diagonal of Figure 7are then used to compute the coefﬁcients’ values in Part

2-3 in Figure 7. This is because the recurrence relation in the

n

direction cannot be used to

compute the coefﬁcients’ values. Consequently, some values in Part 2 become zero, which

results from the condition used to prevent the occurrence of unstable values.

This paper derives the recurrence relations provided in Figure 8. From Figure 8a, it can

be seen that the elements computed for

x0

and

x1

can be used to compute the coefﬁcients

along the main diagonal

n=x

and

n=x−

1. Furthermore, to compute the coefﬁcients’

values of KP

Kp

n(x+

1

)

, the coefﬁcient value

Kp

n+1(x)

is computed using the

n

direction

recurrence algorithm. The similarity across the main diagonal (

n=x

) is exploited for

simplicity where

Kp

n(x+

1

) = Kp

n+1(x)

. To this end, the KP coefﬁcients along

n=x−

1

are computed as

Kp

n(x+1) = δ1n,x,Kp

n(x)−δ2n,x,Kp

n−1(x), (23)

δ1n,x=(N−2x−1)p−n+x

pp(1−p)x(N−x),

δ2n,x=−s(N−x)x

(x+1)(N−x−1),

x=x0+1, x0+2, N

2−1 and n=x.

Figure 8.

A diagram shows the coefﬁcients’ locations that are used to compute the values in Part 2-2.

To compute the values at the main diagonal where

n=x

, a new recurrence relation

approach is developed. This is achieved by combining both

n

and

x

directions recurrences.

Suppose that the values at

(n,x+1), and (n−1, x+1)

are known (see circulated values

I

and

K

in Figure 8d). Then, the value at

n+

1,

x+

1 (see circulated values

L

in Figure 8d)

can be computed using the ndirection recurrence relation as

Kp

n+1(x+1) = α1n+1,x+1,Kp

n(x+1)−α2n+1,x+1,Kp

n−1(x+1). (24)

The value at

(n−

1,

x+

1

)

can be computed using the

x

direction recurrence relation as

Kp

n−1(x+1) = γ1n−1,x,Kp

n−1(x)−γ2n−1,x,Kp

n−1(x−1). (25)

Entropy 2021,23, 1162 12 of 24

Substituting Equation

(25)

in

(24)

yields the following general expression of the recur-

rence relation:

Kp

n+1(x+1) = α1n+1,x+1Kp

n(x+1)−α2n+1,x+1γ1n−1,xKp

n−1(x)−γ2n−1,xKp

n−1(x−1)

=α1n+1,x+1Kp

n(x+1)−α2n+1,x+1γ1n−1,xKp

n−1(x) + α2n+1,x+1γ2n−1,xKp

n−1(x−1)

=η1n,x,Kp

n(x+1)−η2n,xKp

n−1(x) + η3n,xKp

n−1(x−1)(26)

η1n,x=α1n+1,x+1=(N−2n−1)p+n−x−1

pp(1−p)(n+1)(N−n−1)

η2n,x=α2n+1,x+1γ1n−1,x=sn(N−n)((N−2x−1)p+x−n+1)2

p(1−p)(n+1)(N−n−1)(x+1)(N−x−1)

η3n,x=α2n+1,x+1γ2n−1,x=sn(N−n)x(N−x)

(n+1)(N−n−1)(x+1)(N−x−1)

x=x1,x1+1, · · · ,N/2 −1; and n=x

This recurrence relation is termed as the four-term recurrence relation in the

n− −x

direction. This new development approach is used to compute the KP coefﬁcients in the

range

x=x1+

1,

x1+

2,

. . .

,

N/

2

+

1 and

n=x−

1, and

x=x1+

1,

x1+

2,

. . .

,

N/

2 and

n=xas shown in Figure 9:

Figure 9. A diagram shows a location of the coefﬁcients in Part 2-2.

3.4.4. Computation of the Coefﬁcients Located at Part 2-3

This section presents the computation of the KP coefﬁcients located at Part 2-3 in

Figure 7. These values are computed using

(21)

in the ranges

n=x1

,

x1+

1,

N/

2

−

2 and

n+2≤x≤N−n+1. However, the following condition should be met:

|Kn(x)|<10−5and |Kn(x+1)|<10−7. (27)

Entropy 2021,23, 1162 13 of 24

3.5. Computation of the Rest of the KP Coefﬁcients

This subsection provides the computation of the rest of the KP coefﬁcients. To this

end, the rest of the coefﬁcients can be computed using a similarity relation of the KP. The

coefﬁcients in the ranges

x=

0, 1,

. . .

,

N/

2

−

1 and

n=x+

1,

x+

2,

. . .

,

N−x−

1 are

given as Kp

n(x)=Kp

x(n). (28)

The coefﬁcients in the ranges

x=

0, 1,

. . .

,

N−

1 and

n=N−x

,

N−x+

1,

. . .

,

N−

1

are computed using the following expression:

Kp

n(x)= (−1)N−n−x−1Kp

N−n(N−x). (29)

In addition, to calculate the KP coefﬁcients for

p>

0.5, ﬁrstly the value of

p

is set to

1

−p

. Then, the KP coefﬁcients are computed using the proposed methodology. Finally,

the following formula is applied for all coefﬁcients [44]:

Kp

n(x)= (−1)nKp

n(N−x−1). (30)

3.6. Summary of the Proposed Algorithm

In this subsection, a summary of the proposed algorithm is presented. To this end, a

ﬂow chart of the proposed recurrence is shown in Figure 10. In addition to this, a pseudo-

code is presented (see Algorithm 1) for more clariﬁcation. In addition, 3D plots of the KP

coefﬁcients are given in this subsection.

Figure 10. Flowchart of the proposed algorithm.

Entropy 2021,23, 1162 14 of 24

Algorithm 1 Computation of Krawtchouk polynomials using the proposed algorithm.

Input: N,p

Nrepresents the size of the Krawtchouk polynomial,

prepresents the parameter of the Krawtchouk polynomials.

Output: Kp

n(x)

1: Flag=False

2: if p>0.5 then

3: Flag=True; p←p−1

4: end if

5: x0←N p,x1←x0+1

6: Compute Kp

0(x0)using (10)

7: Compute Kp

0(x1)using (12)

8: Compute Kp

1(x0)and Kp

1(x1)using (16) and (17)

9: .Compute initial set

10: for x=x0:x1do

11: for n=2 : xdo

12: Compute Kp

n(x)using (18)

13: end for

14: end for

15: .Compute coefﬁcient values in Part 1

16: for n=0 : x0do

17: for x=x0:−1 : ndo .inner loop

18: Compute Kp

n(x)using (19)

19: if |Kn(x)|<10−5and |Kn(x+1)|<10−7then

20: Exit inner loop

21: end if

22: end for

23: end for

24: .Compute coefﬁcient values in Part 2-1

25: for n=0 : x0do

26: for x=x0:N−n−1do .inner loop

27: Compute Kp

n(x)using (21)

28: if |Kn(x+1)|<10−7and |Kn(x)|<10−5then

29: Exit inner loop

30: end if

31: end for

32: end for

33: .Compute coefﬁcient values in Part 2-2

34: for x=x0:N/2 −1do

35: n←x

36: Compute Kp

n(x)using (23)

37: end for

38: for x=x1:N/2 −1do

39: n←x

40: Compute Kp

n(x)using (26)

41: end for

42: .Compute coefﬁcient values in Part 2-3

43: for n=x1:N/2 −2do

44: for x=n+2 : N−n−1do .inner loop

45: Compute Kp

n(x)using (21)

46: if |Kn(x)|<10−5and |Kn(x+1)|<10−7then

47: Exit inner loop

48: end if

49: end for

50: end for

51: Compute the rest of the coefﬁcients using the similarity relations (28) and (29)

52: if Flag=True then

53: Apply (30)

54: end if

Figures 11 and 12 show a 3D plot of the KP coefﬁcients, which are generated using

the proposed recurrence algorithm with

N=

2000 and different values of the

p

parameter

ranging between <0.5 and >0.5, respectively.

Entropy 2021,23, 1162 15 of 24

Figure 11. 3D plot of the KP coefﬁcients computed for N=2000 and p<0.5.

Figure 12. 3D plot of the KP coefﬁcients computed for N=2000 and p>0.5.

4. Numerical Results and Analyses

This section presents the results obtained using the proposed recurrence algorithm.

In addition, a comprehensive comparison is conducted with the existing recurrence algo-

rithms. The comparison is carried out in terms of the energy compaction, reconstruction

error, and computation cost.

Entropy 2021,23, 1162 16 of 24

4.1. Energy Compaction Analysis

The order of moments

n

impacts the process of signal reconstruction, energy com-

paction, and information retrieval. The order of the KP moments is given by

n=

0, 1, 2,

. . .

,

N−

1. The energy compaction is utilized to check the impact of using KP

to transform a large fraction of the signal energy into relatively few coefﬁcients of moments.

To ﬁnd the impact of using the KP parameter (

p

) on the energy compaction property, the

procedure given by [

46

] is employed. The stationary Markov sequence with length

N

and

zero mean is analyzed. A matrix Lwith covariance coefﬁcients (ρ) is deﬁned as [27]:

L=

1ρ· · · ρN−1

ρ1· · · .

.

.

.

.

........

.

.

ρN−1· · · ρ1

(31)

The matrix

L

is then transformed to the Krawtchouk domain. As such, the coefﬁcients

in the main diagonal of the transformed matrix (

S

) are computed. The matrix

S

represents

the variance σ2

l, which can be computed as

S=RLRT, (32)

where

R

denotes the KP matrix and

(·)T

refers to the matrix transpose operation. In

addition, the normalized restriction error (Jm) can be computed using:

Jm=∑N−1

q=mσ2

q

∑N−1

q=0σ2

q

, (33)

m=0, 1, . . . , N−1, (34)

where

σ2

q

represents the order of

σ2

l

sorted in descending order. In the experiment, the

normalized restriction error is performed by considering different covariance coefﬁcients

(ρ) and different values of the parameters MNP.

Figure 13 shows the normalized restriction error for different values of parameters (

p

)

with the covariance coefﬁcient

ρ=

0.93. It can be observed from Figure 13 that when the

value of

p

is equal to 1

−p

, the normalized restriction error becomes equal. For example,

the normalized restriction error for

p=

0.05 is equal to

p=

1

−

0.05

=

0.95. In addition,

the energy compaction is inﬂuenced by the KP parameter

p

. For instance, as the parameter

p

increases from 0.05 to 0.45, the performance of the KP in terms of energy compaction

is changed. Furthermore, the energy compaction at parameter

p=

0.45 shows better

performance for parameter

p=

0.05 because the normalized restriction error (

Jm

) reaches

zero values. However, small values of parameter

p

shows better performance in terms of

feature extraction, as proven in [

44

]. Thus, it can be concluded that KP provides further

performance improvement as the parameter

p

reaches 0.5. Furthermore, a more accurate

result can be achieved when the parameter pis deviates from 0.5 [44].

Figure 13. Energy compaction for different values of the parameter p.

Entropy 2021,23, 1162 17 of 24

4.2. Analysis of Reconstruction Error

In this section, the proposed recurrence algorithm is evaluated by carrying out recon-

struction error analysis (REA). This REA was conducted for the proposed and the existing

works. The REA was performed using an image formed from 16 well-known images as

shown in Figure 14. In addition, the comparison was performed on an image with a large

size, i.e., (4096

×

4096). Different values of parameter

p

were considered in the analysis.

These values are p=0.1, 0.2, 0.3, and 0.4.

Figure 14. Evaluation of an image used for REA.

First, the WNKP (

R

) is generated using the proposed and existing algorithms. Then,

the KMs (

ψ

) of the image are computed. Then, the image is reconstructed from the

computed moments using a limited number of moments. Finally, the normalized mean

square error (NMSE) is calculated between the input image and the reconstructed version

of the image. Hence, the NMSE is given as [44]

NMSE(I,IRec) = ∑x,y(I(x,y)−IRec (x,y))2

∑x,y(I(x,y))2, (35)

where parameters

I

and

IRec

denote the original image and the reconstructed image, respectively.

The NMSE and the reconstructed image for

p=

0.1

and p=

0.2 are shown in

Figure 15

and Figure 16, respectively. The ﬁrst row depicts the reconstructed images by utilizing the

FRK [

44

], while the second row represents the reconstructed images using the proposed

algorithm. The FRK algorithm is unable to reconstruct the image of the low-order moments.

In addition, it is unable to reconstruct the image with high orders. However, the proposed

algorithm is capable of fully reconstructing the image of different order values. In addition,

the NMSE is minimized in the proposed algorithm using a moment order of 680, which

is

∼

16%. Figure 15 shows that the proposed algorithm achieves an NMSE of 0.72 while

the FRK algorithm records a value of 0.84. This implies that the proposed algorithm

outperforms the FRK algorithm. Moreover, the NMSE reaches zero when the proposed

algorithm is used, while it records 0.64 when the FRK is used. The limitation with the FRK

algorithm is due to the initial set that was computed, which makes the KPCs values tend

towards zero, and thus, the NMSE is increased. Figure 17 provides a plot of experiments

using

p=

0.3. The results show that the proposed algorithm provides better NMSE than

the FRK starting from a moment order of 680. In addition, the performance improvement

Entropy 2021,23, 1162 18 of 24

increases when the moment order increases until it reaches the full order of (4096). At the

full order, the NMSE using the proposed algorithm reaches 0, while it reaches 0.18 using

the FRK algorithm.

Figure 15.

The NMSE performance comparing the proposed algorithm and the RAK algorithm [

44

]

with p=0.1.

Figure 16.

The NMSE performance comparing the proposed algorithm and the RAK algorithm [

44

]

with p=0.2.

Entropy 2021,23, 1162 19 of 24

Figure 17.

The NMSE performance comparing the proposed algorithm and the RAK algorithm [

44

]

with p=0.3.

Figure 18 shows a new performance evaluation using

p=

0.4. The results show that

the proposed algorithm has the ability to accurately generate the KP coefﬁcients while

for the FRK, the KP coefﬁcients remain inaccurate. This is attributed to the zero initial

value obtained in the FRK algorithm. It is also worth noting that the proposed algorithm is

able to generate KP coefﬁcients for large polynomial sizes and at a high polynomial order,

which was experimentally found to be greater than 8192.

Figure 18.

The NMSE performance comparing the proposed algorithm and the RAK algorithm [

44

]

with p=0.4.

Entropy 2021,23, 1162 20 of 24

Finally, this paper provides a comparison of a maximum polynomial size between the

proposed and existing algorithms. The maximum size is obtained for different values of

the parameter

p

. For each recurrence algorithm, the polynomial (

R

) is computed ﬁrst with

a size and order of N×N. Then, the following criterion is applied:

Err =sum(diagonal(R×RT)−I(N)) <Th, (36)

where

Th

is the threshold value over which 10

−5

is chosen.

I(N)

is the identity matrix with

a size of

N×N

. It is worth noting that the maximum value of

N

is set to 20,480. Table 1

lists the obtained maximum size for each algorithm for different values of parameter

p

.

This table demonstrates that the proposed algorithm is capable of generating an orthogonal

polynomial with a large size and for different values of parameter

p

. The proposed

algorithm outperforms the existing recurrence algorithms for all values of the parameters

p

considered. For example, at

p=

0.05, the proposed algorithm generates a size of 82

×

larger than the RAN algorithm, 243

×

larger than the RAX algorithm, and 16

×

larger than

the FRK algorithm. Thus, the proposed recurrence algorithm can be used to process large

signals sizes quickly and accurately.

Table 1. Maximum size of KP that is generated using the proposed and existing algorithms.

pAlgorithm pAlgorithm

RAN RAX FRK Proposed RAN RAX FRK Proposed

0.05 248 84 1236 20,480 0.55 926 932 2428 20,480

0.10 324 132 2250 20,480 0.60 808 812 2880 20,480

0.15 392 196 2252 20,480 0.65 706 708 3368 20,480

0.20 462 276 2980 20,480 0.70 618 618 3058 20,480

0.25 538 436 3400 20,480 0.75 538 490 3400 20,480

0.30 618 676 3058 20,480 0.80 462 318 2980 20,480

0.35 710 1234 3368 20,480 0.85 390 202 2252 20,480

0.40 814 1428 2880 20,480 0.90 322 140 2250 20,480

0.45 936 1220 2428 20,480 0.95 240 88 1236 20,480

4.3. Computation of the Cost Analysis

The computation cost is considered an important factor to evaluate the performance

of the proposed recurrence algorithm [

44

]. Thus, the computation cost of the proposed

algorithm is compared with the existing works. These algorithms the are recurrence

algorithm in the

x

direction (RAX), the recurrence algorithm in the

n

direction (RAN),

and FRK. The computation cost is performed using the number of computed coefﬁcients.

Table 2shows the ratio of computed coefﬁcients (RCCs) using the proposed algorithm for

different polynomials’ sizes. Full moments’ orders are considered. From Table 2, it can

be observed that the RCC for small values of

p

and 1

−p

achieves a low percentage. This

percentage increases as the value of

p

increases towards 0.5. The average RCC for

p=

0.05

and

p=

0.95 is 9.22% while for

p=

0.5, the average RCC is 20.35%. This is because the

effective coefﬁcient is shaping a circle for

p=

0.5, and they are shaping a rotated ellipse as

the parameter

p

deviates towards 0 or 1, which allows the number of effective coefﬁcients

to be reduced. Consequently, the percentage of the computed coefﬁcients is reduced.

Entropy 2021,23, 1162 21 of 24

Table 2. The ratio comparison of the computed coefﬁcients of the proposed recurrence algorithm.

Krawtchouk Parameter (p)

0.05,

0.95

0.1,

0.9

0.15,

0.85

0.2,

0.8

0.25,

0.75

0.3,

0.7

0.35,

0.65

0.4,

0.6

0.45,

0.55 0.5

Polynomial size (N)

1024 10.03 13.31 15.56 17.24 18.52 19.51 20.24 20.75 21.05 21.16

2048 9.48 12.74 14.99 16.68 17.97 18.96 19.69 20.20 20.50 20.60

3072 9.25 12.51 14.76 16.45 17.74 18.73 19.47 19.97 20.27 20.38

4096 9.13 12.38 14.63 16.32 17.61 18.60 19.34 19.85 20.14 20.24

5120 9.05 12.30 14.54 16.23 17.53 18.52 19.25 19.76 20.06 20.16

6144 8.99 12.24 14.48 16.17 17.47 18.46 19.19 19.70 20.00 20.10

7168 8.95 12.19 14.44 16.13 17.42 18.41 19.15 19.65 19.95 20.05

8192 8.91 12.15 14.40 16.09 17.38 18.38 19.11 19.62 19.92 20.02

Average 9.22 12.48 14.73 16.41 17.71 18.70 19.43 19.94 20.24 20.34

Table 3demonstrates the performance improvement ratio between the proposed and

the existing algorithms, namely RAN, RAX, and FRK. The RAN and RAX algorithms

compute 50% of the KPCs for all values of parameter

p

because the

nx

plane is divided

into two portions. On the other hand, the FRK algorithm computes 25% of the KPCs as it

divides the

nx

plane into four partitions. The improvement ratio between the proposed

and existing algorithms is obtained as

Improvement Ratio =1−RCC of the proposed algorithm

RCC of the existing algorithm . (37)

According to Table 3, the proposed polynomial shows an improvement ratio of 18.64%

at

p=

0.5 when compared to the FRK algorithm and 59.32% when compared to the RAN

and the RAX algorithm. The improvement ratio increases as the value of the parameter

p

deviates towards 0 and 1. For example, the results show that at parameter

p=

0.25, the

proposed method achieves an improvement ratio of 29.18% compared to the FRK algorithm,

and 64.59% compared to other algorithms. In addition, a maximum improvement ratio of

63.11% is achieved by the proposed algorithm when it compares with the FRK algorithm

while it is 81.55% when it compares with the RAN and RAX algorithms.

Table 3.

Comparing the improvement ratio of the computed coefﬁcients of the proposed algorithm

and the existing algorithms.

Krawtchouk Polynomial Parameter (p)

0.05,

0.95

0.1,

0.9

0.15,

0.85

0.2,

0.8

0.25,

0.75

0.3,

0.7

0.35,

0.65

0.4,

0.6

0.45,

0.55 0.5

Proposed 9.22 12.48 14.73 16.41 17.71 18.70 19.43 19.94 20.24 20.34

FRK 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00

RAN and RAX 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00

Improvement

over

FRK

63.11 50.09 41.10 34.35 29.18 25.22 22.28 20.25 19.05 18.64

Improvement

over

RAN and RAX

81.55 75.05 70.55 67.18 64.59 62.61 61.14 60.12 59.52 59.32

5. Conclusions

This paper described that state-of-the-art algorithms suffer from high error and that

the implementation of these algorithms is limited to a speciﬁc value of parameter

p

. In

Entropy 2021,23, 1162 22 of 24

addition to this, the state-of-the-art algorithms do not provide any computation of the initial

value. Furthermore, to date, no algorithm is proposed for computing the KP coefﬁcients

with high-order polynomial and large polynomial sizes. To address these limitations,

a new recurrence relation was proposed in this paper. To this end, a new initial value

was presented and derived. In addition to this, a new diagonal recurrence relation was

introduced. The proposed algorithm divided the KP plane into four triangles and only

the coefﬁcients in the upper triangle are computed. The coefﬁcients in the upper triangle

were divided into fundamental initial sets, initial sets, Part-1 and Part-2, respectively. The

n

-recurrence,

x

-recurrence, backwards

x

-recurrence, and diagonal recurrence relations

were used to compute the values of the coefﬁcients in the upper triangle. In addition, the

identities were used to compute the values of the coefﬁcients in the other triangles. The

proposed algorithm was evaluated and compared with the existing recurrence algorithms.

The comparison was carried out based on the reconstruction error, energy compaction, and

computation cost. The experimental results showed that the proposed algorithm achieves

a remarkable improvement over the existing algorithms in terms of the maximum size

generated and the number of coefﬁcients computed. Although the proposed algorithm

outperforms state-of-the-art algorithms, the computational complexity is still high and can

be further reduced. This can be achieved by implementing the proposed algorithm in a

multi-processing environment (parallelizing) rather than in sequential form, as considered

in this paper. Our future work is also directed towards implementing the proposed

algorithm for KP with other orthogonal polynomials. This is expected to result in a new

OP that has the potential of using orthogonal polynomials as well as the properties of

KP coefﬁcients.

Appendix A. Proof of Equation (10)

This section presents a proof through Equation (10):

Kp

0(x0) =sN−1

Np pN p (1−p)−N p+N−1(A1)

=N−1

Np pN p (1−p)−N p+N−11/2

(A2)

=(N−1)!

(x0)!(N−x0−1)!×px0(1−p)N−1

(1−p)x01/2

(A3)

= Γ(N)

Γ(x0+1)Γ(N−x0)×1−p

p−x0

(1−p)N−1!1/2

(A4)

(A5)

By taking the exponential and natural log (elog), then (A5) can be simpliﬁed to:

Kp

0(x0) =elogΓ(N)

Γ(x0+1)Γ(N−x0)×1−p

p−x0(1−p)N−11/2

(A6)

=e

1

2logΓ(N)

Γ(x0+1)Γ(N−x0)×1−p

p−x0(1−p)N−1(A7)

=e

1

2log Γ(N)

log Γ(x0+1)log Γ(N−x0)×1−p

p−x0(1−p)N−1(A8)

=e1

2log Γ(N)−log Γ(x0+1)−log Γ(N−x0)−x0log(1−p

p)+(N−1)log(1−p). (A9)

Author Contributions:

Conceptualization, S.H.A., K.A.A. and B.M.M.; methodology, S.H.A. and

B.M.M.; software, S.H.A., K.A.A. and M.A.N.; validation, M.A., S.M.S. and M.A.N.; formal analysis,

M.A.; investigation, M.A. and K.A.A.; writing—original draft preparation, S.H.A., M.A., M.A.N. and

B.M.M.; writing—review and editing, K.A.A., S.M.S. and M.A.; visualization, S.H.A. and B.M.M.;

Entropy 2021,23, 1162 23 of 24

supervision, S.H.A. and S.M.S.; project administration, S.H.A. All authors have read and agreed to

the published version of the manuscript.

Funding: This research received no external funding.

Data Availability Statement: All data are available within the manuscript.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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