Natural Images Edge Detection using Prewitt Fractional … · 2020. 12. 18. · 3Facultad de...

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 16, Number 6 (2020), pp. 789-809©Research India Publicationshttp://www.ripublication.com/gjpam.htm

Natural Images Edge Detection using Prewitt FractionalDifferential Algorithm via Caputo and Caputo-Fabrizio

Definitions

Gustavo Asumu MBoro Nchama1, Leandro Daniel Lau Alfonso2

Augusto Pedroso Cosme3

1Universidad Nacional de Guinea Ecuatorial (UNGE), Malabo, Guinea Ecuatorial, Calle Hassan II,

2Instituto de Cibernetica Matematica y Fısica, ICIMAF, Calle 15 No. 551, entre C y D, Vedado, Habana 4, CP–10400, Cuba.

3Facultad de Ciencias Medicas, Miguel Enrıquez, Ramon Pinto y Ensenada Luyano La Habana,

Abstract

Edge detection is an image processing technique for finding the boundaries ofobjects within images. It works by detecting discontinuities in brightness. Edgedetection is used for image segmentation and data extraction in areas such as imageprocessing, computer vision, and machine vision. The first order differentiationoperators have been used in edge detection. The problem with the use of suchmethods is that they generally cause thicker edges. To avoid this undesirableeffect, second order derivative methods have been proposed. Even though thesemethods have a stronger response to fine details, they are more sensitive to noise.To solve this problem, fractional order derivatives have been introduced in the edgedetection methods. In this work, authors propose to construct Prewitt fractionalfilters in the Caputo and Caputo-Fabrizio senses to detect edges in natural images.Experimental results show that the proposed methods can suppress efficientlyundesirable effect caused by traditional techniques. In addition, the proposedmethods prove a good performance in visual quality, with higher peak signalto noise ratio.

Keywords: Caputo-Fabrizio fractional derivative; Caputo fractional derivative;Image edge detection.

790 Natural images edge detection using Prewitt fractional differential algorithm...

1. INTRODUCTION

An edge is a boundary between two regions of different gray levels. We may useedges to measure the size of objects in an image, to isolate particular objects from theirbackground and to recognize and classify objects. In image processing, many formsof edge detection algorithms have been proposed [1-13]. The first order differentiationmethods such as Roberts, Sobel and Prewitt operators have been used in edge detection.The problem with the use of first order derivative methods is that they generallycause thicker edges, resulting in the loss of image details. To avoid this undesirableeffect, second order derivative techniques have been proposed. Even though thesemethods have a stronger response to fine details, they are more sensitive to noise[14]. To avoid these inconveniences, fractional order derivatives have been introducedin the edge detection methods [15-17]. The use of such kind of operators havedemonstrated capability to preserve more low-frequency contour features in the smoothareas, maintaining high-frequency marginal features and enhancing medium-frequencytexture details [18]. This paper presents four cases, case 1: uses Sobel, Roberts, Prewitt,Canny, logarithm of Gaussian and zero-crossing edge detection methods, case 2: usesPrewitt edge detection method in the sense of Caputo fractional derivative, case 3: usesPrewitt edge detection method in the sense of Caputo-Fabrizio fractional derivative andcase 4: makes a comparison study of filters in terms of PSNR.

1.1. Basic definitions

Here we give some definitions used in our subsequent discussions:

Definition 1. A function f : [a, b]→ R is said to be absolutely continuous, denoted byf ∈ AC[a, b] on [a, b], if given ε > 0, there exists some σ > 0 such that∑

k

|f(yk)− f(xk)| < ε,

whenever {[xk, yk] : k = 1, · · · , n} is a finite collection of mutually disjointsubintervals of [a, b] with ∑

k

(yk − xk) < σ.

Definition 2. Let n ∈ N and k = 1, 2, · · · , n− 1, the space ACn[a, b] is defined by

ACn[a, b] := {f : [a, b]→ R : f (k)(t) ∈ C[a, b], f (n−1)(t) ∈ AC[a, b]}.

Definition 3. Let a, b, p ∈ R, a < b, 1 ≤ p ≤ ∞. The Sobolev space, W 1,p(a, b), isdefined by [33]

W 1,p(a, b) ={u ∈ Lp(a, b);∃g ∈ Lp(a, b) :

∫ b

a

uϕ′ = −∫ b

a

gϕ, ∀ϕ ∈ C∞0 (a, b)},

Gustavo Asumu Mboro Nchama et al. 791

where C∞0 (a, b) is the space of all infinitely differentiable functions with compactsupport in (a, b). We set

H1(a, b) = W 1,2(a, b).

Definition 4. Let a, b ∈ R, 0 < α < 1, u ∈ H1(a, b). The Caputo-Fabrizio fractionalderivative of order α is defined by

Dαaxu(x) =

M(α)

1− α

∫ x

a

e−α

1−α (x−s)u′(s)ds,

where M(α) is a function such that M(0) = M(1) = 1. In [32], Losada and Nietosuggested the following particular case

Dαaxu(x) =

1

1− α

∫ x

a

e−α

1−α (x−s)u′(s)ds (1)

=1

1− α

(u(x)− e−

α1−αxu(a)

)− α

(1− α)2

∫ x

a

e−α

1−α (x−τ)u(τ)dτ.

Taking a = 0, then the formula (1) can be approximated as

Dα0xu(x) =

1

1− α

∫ x

0

e−α

1−α (x−ξ)u′(ξ)dξ

≈ 1

1− α

N−1∑k=0

(k+1)·x

N∫k·x

N

e−α

1−α (x−ξ)u′(ξk)dξ. (2)

Definition 5. Let a, b ∈ R, n− 1 < α ≤ n ∈ N, u ∈ ACn[a, b]. The Caputo fractionalderivative of order α is defined by

CDα0xu(x) =

1

Γ(n− α)

t∫0

(x− s)n−(α+1)u(n)(s)ds. (3)

For 0 < α ≤ 1, the numerical approximation of (3) takes the form

CDα0xu(x) =

1

Γ(1− α)

∫ x

0

(x− ξ)−αu′(ξ)dξ

≈ 1

Γ(1− α)

N−1∑k=0

(k+1)·x

N∫k·x

N

(x− ξ)−αu′(ξk)dξ. (4)


For more details, see [19-32]. The remainder of this paper is organized as follows: insection 2, we construct a Prewitt Fractional Filter in Caputo (PFFC) sense, next, PrewittFractional Filter in Caputo-Fabrizio (PFFCF) sense is given in section 3. Section 4presents the experimental results of the proposed methods. A conclusion is consideredin section 5.

2. PREWITT FRACTIONAL FILTER IN THE CAPUTO SENSE

The goal of this section is to construct the PFFC sense. To achieve this goal, we firstdiscretize numerically the Caputo derivative in the interval [0;x] (analogously [0, y]).Let’s take a partition of N nodes of the interval [0;x], with step ∆x = x/N . Thus,there are N + 1 nodes. The N + 1 causal pixels can be given by

u0 = u(0),

u1 = u(x/N),...

uk = u(kx/N),...

uN = u(x).

By approximating the function u(x) using forward finite difference scheme, we obtain

(k+1)·x

N∫k·x

N

(x− ξ)−αu′(ξk)dξ,

≈u(kx+x

N)− u(kx

N)

∆x·

(kx+x)/N∫kx/N

(x− ξ)−αdξ

=u(kx+x

N)− u(kx

N)

−(1− α)(∆x)α·[(N − k − 1)1−α − (N − k)1−α]. (5)


Then replacing (5) into (4), we have

CDα0xu(x) ≈ 1

(∆x)αΓ(2− α)·N−1∑k=0

{[u((k + 1) · x/N)− u(k · x/N)

]·

·[(N − k − 1)1−α − (N − k)1−α]}

=1

(∆x)αΓ(2− α)

11−α · uN +(21−α − 2 · 11−α)uN−1

+[(N − j − 1)1−α + (N − j + 1)1−α

−2 · (N − j)1−α]uj + · · ·+[(N − 2)1−α

−2 · (N − 1)1−α +N1−α]u1

+[(N − 1)1−α −N1−α]u0

. (6)

From (6), we obtain N + 1 coefficients ci(i = 0, . . . , N), which depend on α:

c0 =11−α

(∆x)α · Γ(2− α),

c1 =21−α − 2 · 11−α

(∆x)α · Γ(2− α),

...

cj =(N − j − 1)1−α + (N − j + 1)1−α − 2 · (N − j)1−α

(∆x)α · Γ(2− α),

...

cN =(N − 1)1−α −N1−α

(∆x)α · Γ(2− α).

As in a digital 2-D image u(x, y), the shortest distance on x and y coordinates is onepixel, then we put ∆x = ∆y = 1. Thus from (6), we obtain the approximation ofCDα

0xu(x, y) and CDα0yu(x, y) given by

CDα0xu(x, y) ≈ 1

Γ(2− α)·

11−αu(x, y)

+(21−α − 2 · 11−α)u(x− 1, y)

+ · · ·+[(N − 1)1−α −N1−α]u(x− n, y)

, (7)


and

CDα0yu(x, y) ≈ 1

Γ(2− α)·

11−αu(x, y)

+(21−α − 2 · 11−α)u(x, y − 1)

+ · · ·+[(N − 1)1−α −N1−α]u(x, y − n)

, (8)

respectively. Using the expressions (7) and (8) for N = 2, we obtain

CDα0xu(x, y)

≈ 1

Γ(2− α)·[11−α · u(x, y) + (21−α − 2 · 11−α) · u(x− 1, y)

]=

21−α − 2 · 11−α

Γ(2− α)· u(x− 1, y) +

11−α

Γ(2− α)· u(x, y), (9)

and

CDα0yu(x, y)

≈ 1

Γ(2− α)·[11−α · u(x, y) + (21−α − 2 · 11−α) · u(x, y − 1)

]=

21−α − 2 · 11−α

Γ(2− α)· u(x, y − 1) +

11−α

Γ(2− α)· u(x, y), (10)

respectively. Taking into account the coefficients of the right-hand-side of theexpressions (9) and (10), we obtain horizontal[

21−α − 2 · 11−α

Γ(2− α)

11−α

Γ(2− α)

], (11)

and vertical 21−α − 2 · 11−α

Γ(2− α)

11−α

Γ(2− α)

, (12)

filters, respectively. From (11) and (12), we establish

21−α − 2 · 11−α

Γ(2− α)

21−α − 11−α

Γ(2− α)

11−α

Γ(2− α)

21−α − 2 · 11−α

Γ(2− α)

21−α − 11−α

Γ(2− α)

11−α

Γ(2− α)

21−α − 2 · 11−α

Γ(2− α)

21−α − 11−α

Γ(2− α)

11−α

Γ(2− α)

, (13)


and

21−α − 2 · 11−α

Γ(2− α)

21−α − 2 · 11−α

Γ(2− α)

21−α − 2 · 11−α

Γ(2− α)

21−α − 11−α

Γ(2− α)

21−α − 11−α

Γ(2− α)

21−α − 11−α

Γ(2− α)

11−α

Γ(2− α)

11−α

Γ(2− α)

11−α

Γ(2− α)

, (14)

respectively. The matrix (13) and (14) constitute the Prewitt fractional filters in theCaputo sense.

3. PREWITT FRACTIONAL FILTER IN THE CAPUTO-FABRIZIO SENSE

Following the idea as in the previous section, we obtain

(k+1)·x

N∫k·x

N

e−α

1−α (x−ξ)u′(ξk)dξ

≈u(kx+x

N)− u(kx

N)

∆x·

(kx+x)/N∫kx/N

e−α

1−α (x−ξ)dξ

=1− αα·u(kx+x

N)− u(kx

N)

∆x·[e−

α1−α (N−k−1)∆x − e−

α1−α (N−k)∆x

]. (15)

Inserting (15) into (2), we have

Dα0xu(x) ≈ 1

α·N−1∑k=0

{[u((k + 1) · x

N)− u(k · x

N)

x/N

]·

·[e−

α1−α [N−(k+1)] x

N − e−α

1−α [N−k] xN

]}

=1

α ·∆x

(1− e−

α1−α∆x

)uN +

(2e−

α·∆x1−α − e−2α·∆x

1−α − 1)uN−1+(

2e−2α·∆x1−α − e−3α·∆x

1−α − e−α·∆x1−α)uN−2 + · · ·+(

2e−α·(N−j)·∆x

1−α − e−α·(N−j−1)·∆x

1−α − e−α·(N−j+1)·∆x

1−α)uj

+ · · ·+(2e−

α·(N−1)·∆x1−α − e−

α·(N−2)·∆x1−α − e−

α·N·∆x1−α

)u1

+(2e−

α·N·∆x1−α − e−

α·(N−1)·∆x1−α − e−

α·(N+1)·∆x1−α

)u0

. (16)


From (16), we obtain N + 1 nonzero coefficients ci(i = 0, . . . , N) given by

c0 =1

α ·∆x(1− e−

α1−α∆x

),

c1 =1

α ·∆x(2e−

α1−α∆x − e−

2α1−α∆x − 1

),

c2 =1

α ·∆x(2e−2 α

1−α∆x − e−3 α1−α∆x − e−

α1−α∆x

),

...

cj =1

α ·∆x(2e−

α1−α (N−j)∆x − e−

α1−α (N−j−1)∆x − e−

α1−α (N−j+1)∆x

),

...

cN−1 =1

α ·∆x(2e−

α1−α (N−1)∆x − e−

α1−α (N−2)∆x − e−

α1−α ·N ·∆x

),

cN =1

α ·∆x(2e−

α1−α ·N ·∆x − e−

α1−α (N−1)∆x − e−

α1−α (N+1)∆x

).

Taking ∆x = ∆y = 1, as in the previous section, we obtain

Dα0yu(x, y)

≈ 1

α·

(1− e−

α1−α)u(x, y) +

(2e−

α1−α − e−2 α

1−α − 1)u(x− 1, y)

+(

2e−2 α1−α − e−3 α

1−α − e−α

1−α

)u(x− 2, y) + · · ·

+(2e−

α1−αN − e−

α1−α (N−1) − e−

α1−α (N+1)

)u(x− n, y)

, (17)

and

Dα0yu(x, y)

≈ 1

α·

(1− e−

α1−α)u(x, y) +

(2e−

α1−α − e−2 α

1−α − 1)u(x, y − 1)

+(

2e−2 α1−α − e−3 α

1−α − e−α

1−α

)u(x, y − 2) + · · ·

+(2e−

α1−αN − e−

α1−α (N−1) − e−

α1−α (N+1)

)u(x, y − n)

. (18)

Considering the expressions (17) and (18) for N = 2, we obtain

Dα0xu(x, y)

≈ 1

α·[(1− e−

α1−α ) · u(x, y) + (2e−

α1−α − e−2 α

1−α − 1) · u(x− 1, y)]

=2e−

α1−α − e−2 α

1−α − 1

α· u(x− 1, y) +

1− e−α

1−α

α· u(x, y), (19)


and

Dα0yu(x, y)

≈ 1

α·[(1− e−

α1−α ) · u(x, y) + (2e−

α1−α − e−2 α

1−α − 1) · u(x, y − 1)]

=2e−

α1−α − e−2 α

1−α − 1

α· u(x, y − 1) +

1− e−α

1−α

α· u(x, y), (20)

respectively. From the coefficients of the right-hand-side of the expressions (19) and(20), we obtain the horizontal[

2e−α

1−α − e−2 α1−α − 1

α

1− e−α

1−α

α

], (21)

and vertical 2e−

α1−α − e−2 α

1−α − 1

α

1− e−α

1−α

α

, (22)

filters, respectively. From the filters (21) and (22), we deduce the matrix

2e−α

1−α − e−2 α1−α − 1

α

e−α

1−α − e−2 α1−α

α

1− e−α

1−α

α

2e−α

1−α − e−2 α1−α − 1

α

e−α

1−α − e−2 α1−α

α

1− e−α

1−α

α

2e−α

1−α − e−2 α1−α − 1

α

e−α

1−α − e−2 α1−α

α

1− e−α

1−α

α

,

and

2e−α

1−α − e−2 α1−α − 1

α

2e−α

1−α − e−2 α1−α − 1

α

2e−α

1−α − e−2 α1−α − 1

α

e−α

1−α − e−2 α1−α

α

e−α

1−α − e−2 α1−α

α

e−α

1−α − e−2 α1−α

α

1− e−α

1−α

α

1− e−α

1−α

α

1− e−α

1−α

α

,

which are the Prewitt fractional filters in the Caputo-Fabrizio sense.


4. EXPERIMENTAL ANALYSIS

In this section, we give experimental results obtained by applying PFFC and PFFCF onthe test images given in Figure 5. The proposed fractional edge detectors are comparedwith the classical edge detection algorithms such as Sobel, Roberts, Prewitt, Canny,logarithm of Gaussian and zero crossing filters. We set ∆x = 1 and ∆y = 1, since theshortest distance of 2-D image on x and y coordinates is one pixel. To implement(7)-(8) and (17)-(18) numerically, we considered N = 2, which means that in theapproximation of fractional derivatives we only have taken into account the two firstterms. The performance of the two proposed filters has been assessed by using the PeakSignal to Noise Ratio (PSNR), defined by [34]

PSNR = 20 log10

( 255√MSE

),

where MSE is the mean square error, defined by

MSE =

N∑i=1

M∑k=1

‖xik − oik‖1

3NM.

Here, N andM denote the width and height of the image respectively, xik is the pixel inthe filtered image and oik is the pixel in the original image. ‖ · ‖1 denotes L1 norm and‖ · ‖2 denotes L2 norm (Euclidean distance). In the comparison of statistic parameters,it is important to note that the larger the PSNR value, the better the statistical result.Tables 1 to 4 describe the numerical analysis of the PSNR values on the resultant imagesobtained by using some traditional edge detection algorithms and the proposed methods.It is observed that the proposed filters PFFC and PFFCF are more efficient with higherPSNR values, while the traditional filters show the least efficiency by having lowerPSNR values. The PFFC brings the best results for = 0.9, while the PFFCF has thesecond best values, which are achieved for = 0.9. Thus, it is concluded that the twoproposed Prewitt fractional filters are performing better than the classical algorithms interms of edge detection.Figures 1 to 4, show a comparison of edge detection images obtained by using ourproposed techniques and traditional methods. From Figures 2 and 3: images of the leftand right of first row are original image and the processed by logarithm of Gaussianfilter, respectively. In the left and right of the second row there are images processed byCanny and zero-crossing methods, respectively. Images of the left and right of the thirdrow are obtained by PFFC and PFFCF, respectively. From Figures 1 and 4: in the leftand right of first row are original image and the processed by Sobel filter, respectively.In the left and right of the second row, there are images filtered by Roberts and Prewitt


methods, respectively. Images of the left and right of the third row are obtained by PFFCand PFFCF, respectively. After analysing visually all processed images, we observedthat those obtained by the proposed techniques present better edge definition.

a) b)

c) d)

e) f)

Figure 1: Comparison of different edge detection methods: a) original image; b), c)and d) are obtained by Sobel, Roberts and Prewitt filters, respectively; e) and f) are

obtained by using PFFC and PFFCF for α = 0.99, respectively.


a) b)

c) d)

e) f)

Figure 2: Comparison of different edge detection methods: a) original image; b), c)and d) are obtained by using Log of Laplacian, Canny and zero-crossing filters,

respectively; e) and f) are obtained by using PFFC and PFFCF for α = 0.99,respectively.


a) b)

c) d)

e) f)

Figure 3: Comparison of different edge detection methods: a) original image; b), c)and d) are obtained by using Log of Laplacian, Canny and zero-crossing filters,

respectively; e) and f) are obtained by using PFFC and PFFCF for α = 0.9,respectively.


a) b)

c) d)

e) f)

Figure 4: Comparison of different edge detection methods: a) original image; b), c)and d) are obtained by Sobel, Roberts and Prewitt filters, respectively; e) and f) are

obtained by using PFFC and PFFCF for α = 0.9, respectively.


a) b)

c) d)

Figure 5: Original images


Table 1: Comparative study of filters in terms of PSNR for kids image.

S.no. Filter Type PSNR

1 Sobel filter 5.2186

2 Roberts filter 5.2194

3 Prewitt filter 5.2495

4 Canny filter 5.1644

5 Zero-crossing filter 5.2576

6 LoG filter 5.2563

7 PFFC : α = 0.9 12.6799

8 PFFC : α = 0.99 6.7116

9 PFFCF : α = 0.9 6.7034

10 PFFCF : α = 0.99 6.6210

Table 2: Comparative study of filters in terms of PSNR for circuit image.







6 LoG filter 3.6845

7 PFFC : α = 0.9 10.8533

8 PFFC : α = 0.99 4.7372

9 PFFCF : α = 0.9 4.4775

10 PFFCF : α = 0.99 4.5469


Table 3: Comparative study of filters in terms of PSNR for boat image.







6 LoG filter 5.4045

7 PFFC : α = 0.9 11.5479

8 PFFC : α = 0.99 7.2739

9 PFFCF : α = 0.9 7.2860

10 PFFCF : α = 0.99 7.2271

Table 4: Comparative study of filters in terms of PSNR for camaraman image.







6 LoG filter 5.6656

7 PFFC : α = 0.9 12.1846

8 PFFC : α = 0.99 6.8332

9 PFFCF : α = 0.9 6.6153

10 PFFCF: α = 0.99 6.6069


5. CONCLUSION

In this paper, we proposed the construction of Prewitt fractional differential filters usingCaputo and Caputo-Fabrizio fractional derivatives. Experiments showed that filteredimages by the proposed methods have better edge definitions than those obtained bytraditional filters. The proposed techniques have demonstrated a good performance invisual quality and higher PSNR. The PFFC has given the best statistic results, whichhave been obtained for α = 0.9.

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