p-Laplace Variational Image Inpainting Model Using Riesz Fractional Differential Filter

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 7, No. 2, April 2017, pp. 850 – 857
ISSN: 2088-8708 850
Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
p-Laplace Variational Image Inpainting Model Using Riesz
Fractional Differential Filter
G Sridevi1
and S Srinivas Kumar2
1
Department of ECE, Aditya Engineering College, Kakinada, AP, India
2
Department of ECE, JNTUK, Kakinada, AP, India
Article Info
Article history:
Received May 25, 2016
Revised Mar 14, 2017
Accepted Mar 29, 2017
Keyword:
Fractional calculus
Image inpainting
Partial Differential Equations
Riesz fractional derivative
Variational models
ABSTRACT
In this paper, p-Laplace variational image inpainting model with symmetric Riesz fractional
differential filter is proposed. Variational inpainting models are very useful to restore many
smaller damaged regions of an image. Integer order variational image inpainting models
(especially second and fourth order) work well to complete the unknown regions. However,
in the process of inpainting with these models, any of the unindented visual effects such
as staircasing, speckle noise, edge blurring, or loss in contrast are introduced. Recently,
fractional derivative operators were applied by researchers to restore the damaged regions
of the image. Experimentation with these operators for variational image inpainting led to
the conclusion that second order symmetric Riesz fractional differential operator not only
completes the damaged regions effectively, but also reducing unintended effects. In this arti-
cle, The filling process of damaged regions is based on the fractional central curvature term.
The proposed model is compared with integer order variational models and also Grunwald-
Letnikov fractional derivative based variational inpainting in terms of peak signal to noise
ratio, structural similarity and mutual information.
Copyright c 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
G Sridevi
Department of Electronics and Communication Engineering
Aditya Engineering College
Surampalem, Andhra Pradesh, India
sridevi gamini@yahoo.com
1. INTRODUCTION
Image inpainting, is an art of implementing untraceable modifications on images. It is used to restore the
damaged regions of an image based on the pixel information from the known regions. It is not only used to recover the
damaged parts but also used to discard the overlaid text and undesired objects. Inpainting is most useful in recovering
the old photographs and images in fine art museums. It can be used as a pre-processing step for other image processing
problems like image segmentation, pattern recognition and image registration. In this work, image inpainting model
for text removal and scratch removal are demonstrated.
The image inpainting techniques are mainly classified into three categories: textural inpainting, structural
inapinting and hybrid inpainting (combination of two approaches). Textural inpainting is mainly connected with the
texture synthesis. Many texture inpainting methods have been proposed since a famous texture synthesis algorithm
was developed by Efros and Leung [1]. Many other texture synthesis algorithms are proposed with the improvement
in speed and effectiveness of the Efros-Leung method.
Structure inpainting is the process of introducing smoothness priors to diffuse (propagate) local structured
information from source regions to unknown regions along the isophote direction. It uses partial differential equations
(PDE) and variational reconstructions methods. Marcelo et al. [2] introduced first PDE based digital image inpainting.
These models produce good results in restoring the non-textured or relatively smaller unknown regions. Navier-stokes
equations of fluid dynamics were used by the same authors, to inpaint the unknown regions by considering the image
intensity as a stream and isophote lines as flow of streamlines. However, these are slow iterative processes. In order
to minimize the computational time a fast marching technique is described in [3], which fills the unknown region in
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Institute of Advanced Engineering and Science
w w w . i a e s j o u r n a l . c o m
, DOI: 10.11591/ijece.v7i2.pp850-857

IJECE ISSN: 2088-8708 851
single iteration using weighted means.
First variational approach to image completion was proposed by Masnou and Morel [4]. A famous variational
work was introduced by Chan and Shen [5] in 2001. Their method completes the missing regions by minimizing the
total variation (TV) norm. They retain the sharp edges for non-textured parts using curvature term in the corresponding
Euler-Lagrange equation. TV-norm converts smooth (flat) regions into piecewise constant levels (staircase effect).
Meanwhile, small details and textured regions are smoothed out. The TV inpainting model has been extended and
considerably improved in subsequent works, such as curvature driven diffusion [5], Euler’s elastica equation [6],
Gauss curvature driven diffusion [7], fractional curvature driven diffusion [8], fractional TV inpainting in spatial and
wavelet domain [9], and fractional order anisotropic diffusion [10].
Fractional differentiation[11], [12] finds an important role in the area of signal and image processing. Frac-
tional differentiation can be viewed as the generalization of integer differentiation. The definition of fractional dif-
ferentiation is not united. The commonly used definitions are proposed by the authors Grünwald-Letnikov (G-L),
Riemann-Liouville (R-L), Caputo and Riesz. Many researchers have applied these definitions to many image process-
ing applications. Yi et al. [8] proposed G-L fractional derivative based curvature driven diffusion for the minimization
of metal artifacts in computerized X-ray images. Benoˆıt et al. [13] implemented fractional derivative for the detection
of edges. Yi-Fei et al. [14] constructed the fractional differential masks to enhance the texture elements in the images.
Yi et al. [15] proposed two new non-linear PDE image inpainting models using R-L fractional order derivative with
4-directional masks. Stanislas and Roberto [16] proposed fractional order diffusion for image reconstruction inspired
by the work of [17]. Qiang et al. [18] applied symmetric Riesz fractional derivative for enhancing the textured images.
Yi et al. [9] applied fractional order derivative defined by the second definition of Yi-Fei et al. [14] and filling process
is achieved by the fractional curvature term.
In this article, fractional derivative is combined with integer order variational inpainting model. The sec-
ond order symmetric Riesz fractional differential operator is considered in this work, because it possesses non-local
and anti-roational characteristics. The proposed model gives good visual effects and superior objective performance
metrics viz., PSNR, SSIM, and MI with respect to integer order variational image inpainting models.
This article is organized in five sections. In section 2, fractional order variational inpainting model is pre-
sented and fractional central curvature term is also represented. In section 3, construction of symmetric Riesz fractional
differential filter is presented. Simulation results are explained in section 4. Conclusions are given in section 5.
2. PROPOSED MODEL
Given an image, f € L2
pΩq, with Ω € R2
an inpainting or missing domain having boundary fΩ, and E an
surrounding domain nearby fΩ. The problem is to reconstruct the original image u from the observed image f.
A fractional order variational model is proposed in this article, which provides not only an effective image
inpainting, but also visible reduction of unintended effects. The proposed variational model mnimizes an energy cost
functional J, containing a mask that specify known and unknown regions of the image. Therefore, the completed and
enhanced image is determined as a result of the next minimization
Jαpupx, yqq 1
p
M¸
x1
M¸
y1
| α
upx, yq|p
λΩ
2
M¸
x1
M¸
y1
|upx, yq¡fpx, yq|2
(1)
where α is any real number and p € r1, 2s, the mask is based on the characteristic function of the inpainting region,
which is represented as
λΩ
5
λ, px, yq € Ω
0, otherwise
and the Neumann boundary condition fu{fn 0 is applied. Where n is an unit vector outward perpendicular to fΩ.
The first term of (1) is the fractional regularization term, which is used to inpaint the damaged parts based
on the non-local characteristics of the image. The second term of (1) is fidelity term, which is used to preserve the
important features like edges and λΩ is a scaling parameter in the inpainting region Ω, which is used to tune the weight
of two terms in the inpainting region only. According to the fractional calculus of variations, the Euler-Lagrange
equation is
p¡1qαdivα
¢ α
upx, yq
| αupx, yq|p2¡pq

λΩpupx, yq¡fpx, yqq 0 (2)
The computation of numerical algorithm is based on the gradient descent approach. and the following fractional vari-
atinal model is obtained.
p-Laplace Variational Image Inpainting Model Using Riesz Fractional Differential Filter (G Sridevi)

852 ISSN: 2088-8708
fupx, yq
ft
p¡1qαcurvα
upx, yq λΩpupx, yq¡fpx, yqq (3)
The result of minimization (1), representing the restored image, will be determined by solving (3). The frac-
tional central curvature is introduced to increase the performance of image reconstruction. The discrete representation
of the fractional central curvature term curvα
upx, yq is represented as
curvα
upx, yq divα
¢ α
upx, yq
| αupx, yq|p2¡pq

α
x¡
¤
¥
α
x upx, yq

| α
x upx, yq|2 0.5 ¦| α
ycupx, yq|2
¨2¡p
2

α
y¡
¤
¥
α
y upx, yq

| α
y upx, yq|2 0.5 ¦| α
xcupx, yq|2
¨2¡p
2

(4)
where, is a small constant to stay away divide by zero. The proposed symmetric Riesz fractional differential filter
coefficients are used to diffuse the pixel information in the inpainting region based on fractional central curvature term.
The construction of the Riesz fractional differential coefficients will be explained in the next section.
3. CONSTRUCTION OF RIESZ FRACTIONAL DIFFERENTIAL FILTER
The second order symmetric fractional order derivative of upxqfor the infinite interval p¡V x Vqbased
on the Riesz definition is represented as a combination of the right and left sided R-L fractional derivatives
fα
f|x|α
upxq ¡cv
¢
fα
fp¡xqα
fα
fxα

upxq (5)
where cv

2cos
πα
2
¨¨¡1
with α $ 1, pm ¡1q α m 2 for m € N
fα
fp¡xqα
upxq 1
Γpm ¡αq
¢
¡ f
fx
m
V»
x
upmqpζq
pζ ¡xqα¡m 1
dζ (6)
fα
fxα
upxq 1
Γpm ¡αq
fm
fxm
x»
¡V
upζq
px ¡ζqα¡m 1
dζ (7)
The symmetric Riesz fractional order derivative is represented based on second order fractional centered difference
method [19] with step h,
fα
upxq
f|x|α
¡ 1
hα
V¸
k¡V
p¡1qk
Γpα 1q
Γ
α
2 ¡k 1
¨
Γ
α
2 k 1
üpx ¡khq (8)
By noting Euler’s reflection formula for Gamma function, Γ
α
2
¨
Γ

1 ¡ α
2
¨
π
sinpπα
2 q and
Γ pαqΓ p1 ¡αq π
sinpπαq π
2sinpπα
2 qcospπα
2 q gives
Γ pαqΓ p1 ¡αq
Γ
α
2
¨
Γ

1 ¡ α
2
¨ 1
2cos
πα
2
¨ (9)
By substituting equ. (9) in equ.(8), one can has
fα
upxq
f|x|α
¡ 1
2cos
πα
2
¨
hα

V¸
k0
wα
k upk ¡xhq
0¸
k¡V
wα
k upk ¡xhq
'
(10)
where
wα
0 Γ

1 ¡ α
2
¨
αΓ

1 α
2
¨
Γp¡αq (11)
IJECE Vol. 7, No. 2, April 2017: 850 – 857

IJECE ISSN: 2088-8708 853
(a) (b) (c)
(d) (e) (f)
Figure 1. Fractional differential ﬁlters paqWα
x¡ pbqWα
x pcqWα
y¡ pdqWα
y peqWα
xc pfqWα
yc
wα
k p¡1qk 1
Γ
α
2
¨
Γ

1 ¡ α
2
¨
Γ
α
2 ¡k 1
¨
Γ
α
2 k 1
¨
Γp¡αq, k ¨1, ¨2, ... (12)
In the perspective of images, the smallest distance between the two pixels in x-direction and y-direction is
one. For a 2-D image, upx, yq at a pixel px1, y1q, in the positive x-direction N 1 pixels are considered. Therefore,
ukpx1, y1q upx1 ¡ kh, y1q, where h x1{N, 0 ¤ k ¤ N, and N is the number of divisions. Similar procedure
is considered in other directions, like positive y-direction, negative x-direction, negative y-direction. Consider h 1
and the anterior forward N 1 equivalent fractional order difference of the fractional partial differentiation in the
positive x-direction is
fα
upxq
f|x|α
! ¡ 1
2cos
πα
2
¨
hα
N¸
k0
wα
k upk ¡xhq, 0 α ¤ 2, α $ 1 (13)
For the central difference in x-direction of the image upx, yq at a pixel px1, y1q, N 1 pixels are considered in the
positive x-direction and N pixels are considered in the negative x-direction. Therefore, ukpx1, y1q upx1 ¡kh, y1q¡
upx1 kh, y1q. Similar procedure is considered for central y-direction. So, the anterior 2N 1 equivalent fractional
order centeral difference of the fractional partial differentiation in the central x-direction is
fα
upxq
f|x|α
! ¡ 1
2cos
πα
2
¨
hα
N¸
k¡N
wα
k upk ¡xhq, 0 α ¤ 2, α $ 1 (14)
The fractional differential ﬁlters along symmetric directions, the positive x-axis pWα
x q, negative x-axis
pWα
x¡q, positive y-axis pWα
y q, negative y-axis pWα
y¡q, central x-axis pWα
xcq, central y-axis pWα
ycq are constructed

854 ISSN: 2088-8708
(a) (b) (c) (d) (e)
(f) (g) (h) (i) (j)
Figure 2. Comparison of variational inpainting models for text removal (a) Ground truth image, (f) Image with overlaid
text (PSNR = 17.75 dB), (b) Inpainted image using TV model [20] (PSNR = 30.45 dB), (c) Inpainted image using
fourth order PDE model [21] (PSNR=31.6 dB), (d) Inpainted image using Yi et al. model [9] (PSNR = 31.6 dB), (e)
Inpainted image using proposed model (PSNR = 32.8 dB), (g) Enlargement of parrot’s face of (b), (h) Enlargement of
parrot’s face of (c), (i) Enlargement of parrot’s face of (d), (j) Enlargement of parrot’s face of (e)
and shown in Figure 1. These fractional differential filters possess non-local and anti-rotational properties. In Figure
1, Cα
u0
is the filter coefficient corresponding with the interested pixel. The size of the filter is 2N 1, where N is any
positive integer and, one implements p2N 1q ¢ p2N 1q fractional differential filter. Airspace filtering technique
is performed on the symmetric directions with p2N 1q ¢ p2N 1q fractional differential filter. The usage of the
airspace filter is to move the window pixel by pixel and these are computed using
α
l upx, yq
N¸
i¡N
N¸
j¡N
Wα
l pi, jqupx i, y jq (15)
where l x , x¡, y , y¡, xc, yc
The fractional differential filter coefficients are
Cα
u0
1
2cos
πα
2
¨
pα ¡1qΓ

1 ¡ α
2
¨
Γ

1 α
2
¨
Γp2 ¡αq (16)
Cα
uk
1
2cos
πα
2
¨
p¡1qk
αpα ¡1qΓ
α
2
¨
Γ

1 ¡ α
2
¨
Γ
α
2 ¡k 1
¨
Γ
α
2 k 1
¨
Γp2 ¡αq, k ¨1, ¨2, ... (17)
4. RESULTS AND DISCUSSION
The proposed technique described here has been tested on large collections of images affected by missing
regions. The USC-SIPI database is used in our experiments. The proposed technique provides an effective restoration
of the degraded image, completing successfully the missing zones. It also preserves the image details, like edges, and
reduces the unintended effects, such as image blurring, staircasing and speckle effects. The optimal image reconstruc-
tion results are achieved by the proper selection of fractional order. This value is detected by trial and error, through
emprical observation. In this work, when α 1.4 the proposed model produces optimal reconstruction result.
The performance of this fractional order vartional model has been quantified by using well-known measures,
such as peak Signal to Noise Ratio (PSNR), Structural Similarity (SSIM) [22], and Mutual Information (MI)[23].
This approach outperforms numereous state of the art inpainting methods. This fractional order variational image

IJECE ISSN: 2088-8708 855
inpainting technique is able to restore multiple missing regions. For this reason, it can be successfully used for some
important tasks, such as removing the superimposed text, removing the scratches, or removing the watermarks from
the digital images.
A text removing example using proposed technique described in Figure 2, wheresome method comparison
results are displayed. The images of that figure depict the inpainting results achieved by various inpainting techniques
on the parrots color image collected from LIVE image database and cropped to [256 X 256] . The text is superimposed
on the image and the inpainting techiniques are are applied. These inpainting techniques are carried out in YCbCr
color space. The text is almost removed by all the models. However, one can observe that, the texture part near the
parrot’s eye is not restored well by state of the art methods, such as TV inpainting b), fourth order PDE model c), and
Yi et al. model d) [9]. These models do not preserve edges and produce loss in contrast. The zoomed version of these
techniques are shown in Figure 2(g)-(j). The inpainting regions after applying the proposed model are filled effectively
than the other three models. Inpainting models for text removal with the same damaged mask are applied on different
images and registerd in Table 1. As one could observe in that table, the performance measures of proposed inpainting
technique achieve the highest values. One more observation is that, the proposed model works well even if the image
having partially textured regions, but the other three models are not. The logic is that, the total variation model is
second order PDE, Yi et al. model (α=1.8) is closed to fourth order PDE, where as the proposed model (α=1.4) is
closed to third order PDE.
(a) (b) (c) (d) (e)
(f) (g) (h) (i) (j)
Figure 3. Inpainting of artificial lines on pepper’s image (a) Ground truth image, (f) Damaged image (PSNR=16.60dB)
(b) Inpainted image using TV model [20] (PSNR = 33.97 dB, SSIM = 0.9001, MI = 3.6850), (c) Inpainted image using
fourth order PDE model [21] (PSNR = 33.89 dB, SSIM = 0.9312, MI = 3.8402), (d) Inpainted image using Yi et al.
model [9] (PSNR = 35.9 dB, SSIM = 0.9497, MI = 4.3149) (e) Inpainted image using proposed model (PSNR = 36.16
dB, SSIM = 0.9712, MI = 5.4789), (g) Residual image of (b), (h) Residual image of (c), (i) Residual image of (d), (j)
Residual image of (e)
Table 1. Comparison of inpainting models for text removal on different images
Image I/P
PSNR
TV [20] Fourth order PDE [21] Yi et al. [9] Proposed model
PSNR SSIM MI PSNR SSIM MI PSNR SSIM MI PSNR SSIM MI
Cameraman 16.16 30.38 0.9374 3.74 30.58 0.9379 3.75 31.05 0.9396 3.79 32.94 0.9473 5.23
Elaine 18.69 38.16 0.9420 4.89 38.56 0.9478 4.92 38.72 0.9587 4.94 39.60 0.9694 5.95
Lena 19.60 34.21 0.9288 4.51 34.32 0.9327 4.57 34.52 0.9369 4.64 35.02 0.9532 5.75
Mandrill 19.53 31.18 0.8275 3.03 31.20 0.8349 3.04 31.23 0.8455 3.06 33.53 0.9516 4.80
The proposed model is also applied to remove the unwanted scratches from the image. The simulation results
on pepper’s image are shown in Figure 3. This inpainting technique outperforms the TV inpainting, fourth order PDE
model, Yi et al. model. The experiment shows the loss of contrast after applying the inpainting techniques. In order

856 ISSN: 2088-8708
Table 2. Comparison of inpainting models for scratch removal on different images
Image I/P
PSNR
TV [20] Fourth order PDE [21] Yi et al. [9] Proposed model
PSNR SSIM MI PSNR SSIM MI PSNR SSIM MI PSNR SSIM MI
Cameraman 17.47 26.68 0.8020 3.05 26.55 0.8532 3.22 27.64 0.9290 3.82 29.20 0.9234 5.28
Man 18.24 32.21 0.9050 3.48 31.76 0.9145 3.42 32.10 0.9277 3.64 33.52 0.9493 5.11
Lena 16.64 31.84 0.8762 3.94 32.42 0.9124 3.60 32.68 0.9381 3.96 33.26 0.9590 5.15
House 16.97 34.74 0.8522 3.08 34.94 0.8865 3.22 34.87 0.9043 3.42 35.91 0.9211 4.93
to understand the loss of contrast, the residual images pf ¡u 100q are shown in Figure 3(g)-(j). Figure 3(g), shows
the result of total variation inpainting. It produces loss in contrast and edges are also blurred. Fourth order PDE model
fills the damaged regions effectively than TV model. However, edges are smoothed. Yi et al. model preserves the
contrast to some extent only because the fractional curvature term is applied, which is based on forward and backward
fractional differences. The proposed model uses fractional central differences. Hence, there is no loss in contrast
and edges are also not blurred by the proposed model. When the fractional order is 1.4, the proposed model gives
higher results than other models in terms of PSNR, SSIM, and MI also in visual quality. The inpainting techniques
on different images with the same mask are applied and the simulation results are registered in Table 2. One could
observe that, the performance measures of proposed inpainting technique achieve the highest values.
5. CONCLUSIONS
In this article, symmetric Riesz fractional differential filter is applied to p-Laplace variational image in-
painting. Fractional order variational inpainting models restored superior to integer order variational models. The
symmetric Riesz filter possesses non-local property, anti-rotational property, and inpainting region is filled based on
the fractional central curvature term. It uses forward, backward, and fractional central differences. Therefore, this
model provides the effective image inpainting and overcomes the unintended visual effects. The simulation results
display that the performance of the proposed model is exceeding integer order variational models and Yi et al. model
[9].
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BIOGRAPHIES OF AUTHORS
G Sridevi received B.Tech degree in Electronics and Communication Engineering from Nagarjuna
University, Andhra Pradesh, India and Masters degree from Jawaharlal Nehru Technological Uni-
versity, Kakinada, Andhra Pradesh, India in 2000 and 2009 respectively. She is currently pursuing
her Ph.D in Jawaharlal Nehru Technological University, Kakinada. Her areas of interest are Digital
image processing and Digital signal processing. She has more than 14 years of teaching expe-
rience. She is presently working as an Associate professor in the department of Electronics and
Communication Engineering in Aditya Engineering College, Surampalem, Andhra Pradesh. She is
the member of Institution of Electronics and Telecommunication Engineers.
S Srinivas Kumar is working as a Professor in the department of Electronics and Communication
Engineering and Director (Research and Development), JNTU College of Engineering, Kakinada,
India. He received his M.Tech. from Jawaharlal Nehru Technological University, Hyderabad, India.
He received his Ph.D. from E ECE Department, IIT Kharagpur. He has twenty eight years of
experience in teaching and research. He has published more than 50 research papers in National
and International journals. Five Research scholars have completed their Ph. D and presently 11
research scholars are working under his supervision in the areas of Image processing and Pattern
recognition. His research interests are Digital image processing, Computer vision, and application
of Artiﬁcial neural networks and Fuzzy logic to engineering problems.

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