Image Restoration by Total Variation Image in Painting Case

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, Rudin-Osher-Fatemi (ROF) model and Osher-Sole-Vese (OSV) model. We show applications to image inpainting.

1. 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 :

   1. 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 ill-posed 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

   2. Definition 2

      Let , we define the numerical divergence

      operator such that the adjoint operator of by the following:

      (5)

      Laplacian (6)

   3. 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. Rudin-Osher-Fatemi 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)

2. 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. Osher-Sole-Vese model

   We will present here a proposed model by Osher-Solé- Vese stated as below :

   (13)

   is a discrete norm in which is the dual space of Sobolev space

   Algorithm 4 Osher-Sole-Vese model

   Input

   Number of iteration N Image to be

   Output

   estimed solution Initialization

3. 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

4. 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

5. 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 signal-to-noise ratio (PSNR) to evaluate the performance of these algorithms.

(14)

with the Mean Square Error

REFERENCES

1. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing:Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences Vol. 147, in Springer.

2. J.F. Aujol, Traitement dimages par approches variationnelles et équations aux dérivées partielles, ENIT Tunis., 2005, pp.55.

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

4. S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM, Journal on scientific computing, 24 (5), 2003, pp.1754-1767.

5. S. N. Ghate, S. Achaliya and S. Raveendran, An algorithm of total variation for image inpainting,, International Journal of Computer and Electronics Research, vol. 1, issue 3, October 2012, pp. 124-130.

6. L. I. Rudin. S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, Volume 60, Issue 1-4, 1992, pp. 259-268

7. T. Zeng, X. Li and M. Ng, Alternating minimization method for total variation based wavelet shrinkage model, Commun. Comput. Phys., vol. 8, no. 5, November 2010, pp. 976-994.