 Open Access
 Total Downloads : 113
 Authors : Raminoson Tsiriniaina, Ralaivao Harinaivo Hajasoa, Randriamitantsoa Paul Auguste
 Paper ID : IJERTV6IS060460
 Volume & Issue : Volume 06, Issue 06 (June 2017)
 Published (First Online): 28062017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Image Restoration by Total Variation Image in Painting Case
Raminoson Tsiriniaina Telecommunication Automatic Signal Image Research Laboratory/Doctoral School in Science and Technology of
Engineering and Innovation/University of Antananarivo Antananarivo, Madagascar
Ralaivao Harinaivo Hajasoa Telecommunication Automatic Signal Image Research Laboratory/Doctoral School in Science and Technology of
Engineering and Innovation/University of Antananarivo Antananarivo, Madagascar
Randriamitantsoa Paul Auguste Telecommunication Automatic Signal Image Research Laboratory/Doctoral School in Science and Technology of
Engineering and Innovation/University of Antananarivo Antananarivo, Madagascar
AbstractThis paper describes algorithms for minimizing the total variation of an image. Many regularization models are presented : Tychonov model, RudinOsherFatemi (ROF) model and OsherSoleVese (OSV) model. We show applications to image inpainting.

DISCRETIZATION
The size of the processed image . We denote and with the usual scalar product in
Keywords Image restoration; regularization; total variation; image inpainting
I. INTRODUCTION
Image restoration, including image denoising, deblurring, inpainting, computed tomography, etc., plays an important
and the associate Euclidian norm :

Definition 1
(3)
role in numerous areas of applied sciences, such as medical and astronomical imaging, film restoration, and image/video coding. Its major purpose is to enhance the quality of giving image that is corrupted in various ways during the process of imaging, acquisition and communication.
Considering an original image , supposed that it was degraded by an additive noise , and evently by a fuzzy operator . The operator is modelling by a convolution product. From the observed image a degraded image, we have to . If we suppose that the additive noise is Gaussian, the method of Maximum likelihood conducts us to find as a solution of minimization problem.
(1)
design the norm in . This is an inverse illposed problem : the operator is not necessary inversible (and if it is inversible, the inverse is difficult to calculate). In other terms, the existence and/or the unicity of the solutions is not ensured or the solution is not stable. To solve numerically, we have to introduce a regularization term , and have to consider the following problem:
(2)
Let ; then the discrete gradient of , written
, is defined by
(4)
with

Definition 2
Let , we define the numerical divergence
operator such that the adjoint operator of by the following:
(5)
Laplacian (6)

Total variation
In the discrete case, the total variation can be written by :
(7)
Let us observe here that this functional is a discretization of the standard total variation, defined in the continuous setting for a function ( open subset of ) by
(8)
B. RudinOsherFatemi model
They [6] introduced in regularization the total variation, the problem is obtained after discretization of :
(10)
The discrete version of total variation is given, similarly in continuous case, by
with
Hence the solution of problem (10) is simply given by
(11)


VARIATIONNAL METHODS
A. Tychonov regularization
It is a process of regularization most classical and too short for image processing.
Let and , for all the problem is :
(9)
The Euler equation to find the solution is shown below: And to , we have:
Algorithm 1 Tychonov model
Input
Number of iteration N Image to be
Output
estimed solution Initialization
The restored image is very smoothed, because the Laplacian is an operator of isotropic diffusion.
with is the nonlinear projection on
Algorithm 2 Projections algorithm of Chambolle [3]
Input
Number of iteration N Image to be
Output
estimed solution
Initialization
Theorem 1 [3]
Let . Then converges to as .
The solution of the problem (10) is given by :
(12)
with
Algorithm 3 Projections algorithm of Chambolle with
Input
Number of iteration N Image to be
Output
estimed solution Initialization
with
C. OsherSoleVese model
We will present here a proposed model by OsherSolÃ© Vese stated as below :
(13)
is a discrete norm in which is the dual space of Sobolev space
Algorithm 4 OsherSoleVese model
Input
Number of iteration N Image to be
Output
estimed solution Initialization

INPAINTING REGULARIZATION
The object of inpainting is to reconstitute the missing or damaged regions in images, in order to make it more legible and to restore its unity. Mathematically speaking, inpainting is essentially an interpolation problem.
Considering an image defined on a domain but missing or damaged on a subset . Then it has to find methods or models to resolve
with is a masking operator. It is in fact a projection operator on :
The idea of variational methods is to minimize the quantity while adding a regularization term
In general, we consider the following problem :
(14)
In the Total Variation regularization, the norm L1 is used :
A necessary and sufficient condition for to be a solution of (14) is :
Then, we obtain the following concept :
Algorithm 5 TV algorithm for inpainting
Input
Number of iteration N Image to be
Output
estimed solution Initialization

NUMERICAL RESULTS
In this section we present some of the results obtained with the regularization models sush as Tychonov, ROF and OSV. For the numerical examples, we use mandril image, Fig. 1.
Fig. 1. Mandril image : original and mandril with mask
Fig. 2. Tychonov model : Top left to right bottom
Fig. 2 shows the smoothed images, implementation of the algorithm number 1 : Tychonov regularization.
TABLE I. VARIATION OF PSNR PER MODEL
No iterations
PSNR (db)
Tychonov model
ROF
model
OSV
model
0
11.73
11.73
11.73
10
15.36
13.77
14.64
20
17.09
16.56
18.15
50
19.4
18.378
20.26
75
20.89
20.7
22.74
100
22.72
23.87
25.85
150
25.1
28.59
29.94
200
28.5
37.48
36.47

CONCLUSION AND PERSPECTIVES
This paper focuses on the theory and on the implementation of minimizing the total variation on images. We have implemented many algorithms and we have compared the numerical results applied on image inpainting. The algorithms ROF and OSV are the most models but Tychonov model is very smoothed.
The one perspective is to apply the total variation on wavelet decomposition to restore an image inpainting. And it is also possible to implement thesealgorithms on sequence images or on videos.
Fig. 3. On the left ROF model, on the right OSV model : 1st line : 10 iterations
2nd line : 20 iterations 3rd line : 50 iterations 4th line : 100 iterations
We use Peak signaltonoise ratio (PSNR) to evaluate the performance of these algorithms.
(14)
with the Mean Square Error
REFERENCES

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J.F. Aujol, Traitement dimages par approches variationnelles et Ã©quations aux dÃ©rivÃ©es partielles, ENIT Tunis., 2005, pp.55.

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision 20, 2004, pp. 8997.

S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM, Journal on scientific computing, 24 (5), 2003, pp.17541767.

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