- Email: [email protected]
Contribution to Image and Contours Restoration
Real-Time Imaging 7, 315–326 (2001) doi:10.1006/rtim.2001.0238, available online at http://www.idealibrary.com on
Contribution to Image and Contours Restoration
D
igital images are generally degraded by different sources during their acquisition. This is due of two types of phenomena: the deterministic phenomenon of blur which is introduced by relative motion between a camera and the object, and the stochastic phenomena such as atmospheric turbulence, noise and other factors. So, it becomes very difficult for high level processing systems (object detection, three-dimensional reconstruction, characters recognize . . .) to extract reliable features from the incomplete edges. Our objective is to reduce the effect of this degradation and recover the original image from the degraded image with better edge detection. The Markov Random Field (MRF) modelization allows us to restore images with taking into account some constraints such as the smoothing constraint and the edge preserving. Our approach is focused on a new deterministic algorithm that permits approaching the global optimum and reduces computational time. We will present the semi-quadratic regularization model adapted to discontinuities in order to model smoothing constraints of homogeneous zones and to preserve contours. The obtained results on real images are satisfying since we reached our goal of a smoothed homogenous area with preserved edge. # 2001 Academic Press
K. Achour, N. Zenati and H. Laga Robotics and Artificial Intelligence Laboratory, Advanced Technologies Development Center, 128, Chemin Mohamed Gacem, El-Madania, 16075, Algiers, Algeria. E-mail: [email protected], [email protected]
Introduction The problem of image restoration is very important in a large number of applications in early vision processing. For many years researchers have worked on this problem, since noisy image restoration is a very important preliminary step for various high level vision processing. Many solutions have been proposed, among them being the classical approach which is based on filtering techniques. In this case, the image is considered as being the composition of two signals: the real signal which is condensed in low frequencies and the noise signal. The filtering operation consists then of separating these two signals by applying a low pass filter. However, this approach does not consider the possibility of the
1077-2014/01/040315+12 $35.00/0
signal discontinuity but rather looks to find smoothed solutions, that causes a loss of edges (thin details) which are primordial elements in image processing. During the last few years, this approach has been quickly abandoned in favour to a new method based on the Markovian random field (MRF) which has been successfully introduced in many fundamental issues of image analysis and computer vision such as image restoration [1–5], edge detection [6], image segmentation [7,8], surface reconstruction [7], stereovision [8] and motion analysis [9,10]. Markovian random field (MRF) is an efficient and powerful framework for specifying nonlinear interactions between features of the same nature or of different one. When they are associated to
#
2001 Academic Press
316
K. ACHOUR ETAL.
the maximum a posteriori criterion, they lead to the minimization of a global energy function which may exhibit local minima. This minimization is generally performed using deterministic [3] or stochastic algorithms [5]. Stochastic algorithms may be time consuming while deterministic schemes often get ‘‘stuck’’ in local minima of the energy function. Among stochastic methods, we can mention the ones using three line processes, first introduced by Geman and Geman [5]. They explicitly take into account the cost of creating edges in the form of a line-process. The problem is then to minimize the energy which is nonconvex with respect to the image intensity and the line processes. One way to obtain the minimum is to use the well-known stochastic technique called ‘‘simulating annealing’’. This method is optimal but very time consuming. Another way to deal with the problem is to search for optimal deterministic algorithms. From a theoretical point of view, the main draw back of these techniques is that they are not guaranteed to reach the global minimum. But from a practical point of view, the shape of the energy function obtained for images (many minima) and the results are usually good enough if not optimal. Deterministic algorithms yield solutions at a much lower computational cost. One of the most interesting deterministic techniques is the Graduate Non-Convexity (GNC) algorithm introduced by Blacke and Zissermann [11]. An extension of this technique using a Compound Gauss-Markov Field (CGMRF) [12], have been derived in [13] and [14] for image restoration. The random Markovian field (MRF) approach allows to restore the image under the constraint of preserving edges: An image f is considered as the realization of a random variable field defined on a set of sites S¼{si} [3,14]. On S, we define a neighborhood system V and a set of cliques.The neighborhood Vsk of a site sk is defined as follows: Vsk ¼ fsj 2 S=j 6¼ k et sk 2 Vsj The clique c of n neighbors.
th
ð1Þ
order is a set of n sites of all
A field f defined on a set of sites S is a Markovian random field (MRF) according to the neighbor system V if:
8 2 S; 8 2 S;
Pð fs Þ40 Pð fs =f ÿ f fs gÞ ¼ Pð fs =Vs Þ;
ð2Þ
with Vs the neighborhood of the site s and f7{fs} the image field without the site s. The relation (2) shows that the probability of the attribute apparition on a site is higher than zero and the attribute of the site depends only on its neighbors. In order to quantify the properties of the image, we associate to every clique c a function of potential ’c. We define then the local energy function of a site s as:
UðsÞ ¼
X
’c ðsÞ:
ð3Þ
c2C=s2S
The sum of the potential of all the cliques defines the global energy of the Markov random field. In order to exploit the statistical aspect of the MRF, we introduced the global aspect which allows us to encode the likelihood of the image taken as a whole. It is defined by the Hammersley-Clifford theorem, given as follows: If f is a Markov Random Field (MRF) and the probability of any configuration is nonzero, then its probability is given by the Gibbs law: probð f Þ ¼ ð1=ZÞ expðÿUð f ÞÞ
ð4Þ
As for the local aspect, it models the local constraints of the image. In applying the laws on the join probabilities, we obtain the following result: eÿ’ð fs Þ Probð fs =f s Þ ¼ P ÿ’ð fs Þ ¼ Probð fs =Vs Þ; e
ð5Þ
fs 2
Where ’ is the local energy , Vs the neighborhood of the site s and f s¼(f7fs) The organization of this paper is as follows: Different regularization models and the model used in order to smoothen constraints and preserve edges are given in the next section. Following this we present an overview of different restoration algorithms. Next, the GNC strategy (Graduated-Non-Convexity) and our approach for realizing the series E(p) is presented. Then, a new deterministic restoration algorithm is proposed and analyzed. Experimental results are then presented and finally, the paper concludes with a discussion and some perspectives.
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION
Regularization Digital images are generally degraded during their acquisition, although the degradation processes affecting images are generally nonlinear. The distortions are often adequately modeled by a linear blur and an additive Gaussian white noise, giving the observation equation d ¼ Rf þ B;
ð6Þ
where f, d and B are respectively the real image, the acquired image and the additive white Gaussian noise which defines the random phenomenon. R is the ‘‘blur matrix’’ characterizing the acquisition system and corresponds to a blur function representing the linear spatially invariant or spatially varying distortion. In our case R is the identity matrix. The restoration of the degraded image d consists of resolving the following equation: f ¼ arg max pð f =dÞ f
ð7Þ
Since f is an MRF defined on a set of sites S and according to the Hammersley-Clifford theorem, its probability follows the Gibbs law given as follows [15, 16]. pð f Þ ¼
1 ÿUð f Þ e ; Z
ð8Þ
where U( f) is the global energy function involved in the relation between the attributes of different sites and Z is a constant called ‘‘partition function’’. By applying the probability law of Bayes, and the relationships (4), (6) and (7), we can show that a problem of restoration is just a problem of a function minimization E( f) defined as follows: Eð f Þ ¼ ðd ÿ Rf Þ2 þ 22 Uð f Þ ¼ E1 þ E2
ð9Þ
where E1 is the attachment term to the data and E2 the smoothness term.
317
which consists of choosing energy functions in order to represent simultaneously the smoothing and contour preserving constraints. This problem is called the regularization. Standard or semi-quadratic regularization The local energy function used in this model is the quadratic function gq(u)¼u2, where u is the gradient. This model is easier to implement because the quadratic function is convex. For its minimization we can use the classical method based on the gradient descent. However, this function imposes the global smoothing of the solution, therefore it does not preserve the discontinuities [17, 18]. Model with line process. The Boolean line process was introduced to take into account the strong gradient and thus avoid the global smoothing . However, its inconvenience is the fact that it does not assure the stability of the value and besides, its Boolean nature does not preserve the soft edges. We define the truncated quadratic by the energy function given as follows: GBZ ðuÞ ¼ Minðu2 ; Þ: This function is similar pffiffito the quadratic function with the values less than a and constant for pffiffi the higher values, but, it is not differentiable at u ¼ a. We have presented different energy functions which allows us to model the constraints imposed on the solution. Each of these models presents a set of advantages and disadvantages. The model adapted to discontinuities, contrary to the quadratic model and to the model with line process, allows an image restoration with the possibility of considering the signal discontinuities [17–19]. Model adapted to discontinuities It is defined from the energy function g(u) an interaction function h(u) given as follows:
Regularization models hðuÞ
g0 ðuÞ : 2u
ð10Þ
The purpose of digital image restoration is to operate on the degraded images to obtain an improved image f * (which minimizes E) that is as close as possible to the original image.
Thus, for an interactive function h to be adapted to discontinuities, the following conditions have to be verified [5, 20]:
The definition of the regularization model consists of defining the term E2 (the smoothness term) of Eqn (9),
a) h [ C1 b) h(u) ¼ h(7u)
318
K. ACHOUR ETAL.
Table 1. Action functions/potential adapted to discontinuities h(u)
g(u)
(a)
exp(7u )
7exp(7u2)
(b)
1 ð1þu2 Þ2 1 1þu2 1 1þjuj
u2 1þu2
(c) (d)
2
The simulated annealing, which is an analogy with the process of annealing used in metallurgy, optimizes the Gibbs sampler. It introduces, in the energy function, a new parameter T called temperature. The probability P(f/d) is given by the following equation:
ln(1+u2)
Pð f =dÞ ¼
2(|u|+ln(1+|u|)
c) h(u) 4 0 d) h decreasing function e) Limu??h(u) ¼ 0 According to the last two conditions, we can deduct the following result: So that a given model is adapted to discontinuities, it is necessary that: lim jg0 ðuÞj ¼ C:
1 uðf =dÞ e T Z
ð11Þ
The parameter T is initialized to a very important value, then decreases gradually until reaching a very low value (generally 1) that allows quitting the local optima and controls the length of the jumps. The algorithms based on the principle of the simulated annealing converge to the global optimum and therefore they give very good results. The processing time is reduced in regard to the Gibbs sampler but it remains long. The deterministic algorithms
juj!1
If C ¼ 0, the model prevents smoothing, otherwise (C40) it allows smoothing. The following Table 1 gives us some interaction functions/potential adapted to discontinuities.
Restoration Algorithms When the model adapted to discontinuities is established, we define the maximization algorithm of the probability P(f/d). Thus, we distinguish two types of algorithms according to whether we directly use the probability P(f/d) or minimize the global energy function E(f). The stochastic algorithms The stochastic algorithms allow us to find the configuration f* which maximizes the probability P(f/d), according to Eqn (7) [15,16,21]. The Gibbs sampler leads towards the global optimum and therefore gives the best results. However, in practice, its implementation is quasi-impossible for two reasons: – The processing of the partition function Z requires a sum on the set of all possible configurations of an MRF. – The computational time is very important.
Contrary to the stochastic algorithms, deterministic algorithms are very fast but converge toward some local minima [15,22]. In order to exploit the fastness of these algorithms, new strategies were developed such as Graduated Non-Convexity (GNC) and Mean Field Annealing (MFA). The graduated non-convexty Principle The aim of this algorithm is to break down the nonconvexity of the criterion to be minimized, E(f). Its principle is to make series of functions E(p)(f), where p varies from one to zero with E (1)¼convex, and E (0)=E. Then, the procedure is to minimize successively these terms on resetting at each minimization the result by the precedent minimization. This algorithm uses the line process as the regularization model. In fact: Eð f ; l; mÞ ¼ E1 þ E2 þ E3 ;
ð12Þ
E1 ¼ ðd ÿ Rf Þ2
ð13Þ
with
E2 ¼ 2
X
ð fi; j ÿ fiÿ1; j Þð1 ÿ li; j Þ þ ðfi; j ÿ fi; jþ1 Þð1 ÿ mi; j Þ
i; j
ð14Þ E3 ¼
X
ðli; j ÿ mi; j Þ
i;j
Where a is the cost to pay to introduce an edge.
ð15Þ
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION Its principle is then to minimize E ( f, l, m) according to the line process when the field image is fixed, then we minimize according to the field image when the process line is fixed. The minimization according to the line process gives us the following series E(p): X ðpÞ ðg ; ðfi; j ÿ fiÿ1; j Þ þ gðpÞ E ðpÞ ðf Þ ¼ E1 þ ; ðfi; j ÿ fi; jþ1 ÞÞ
Since g is the energy function adapted to discontinuities,h(u)40. Suppose now, that h(u) is constant (w40), then Eqn (13) yields: g00 ðu; wÞ ¼ 2w:
ð22Þ
Integrating this function twice we have: g ðu; wÞ ¼ wu2 þ ðuÞ
ð23Þ
with,
i; j
ðuÞ ¼ c u þ b;
ð16Þ
ð24Þ
where c and b are constant. If we suppose u is constant , we obtain:
with: 8 <
2 sijtj5q l ÿ cðjtj ÿ r2 Þ=2 if qjtj5r : else
ð17Þ
c ¼ 1=4p; r2 ¼ ð2=c þ 1=2 Þ; q ¼ =2 r
ð18Þ
gðpÞ ; ðtÞ ¼
319
Our work then consists of developing an algorithm based on the principle of the GNC , which exploits the model adapted to discontinuities and the convexity of the quadratic model; it is called the semi-quadratic regularization. Semi-quadratic regularization Let g(u) represent the energy function adapted to discontinuities. The semi-quadratic regularization consists of introducing the auxiliary variable w, so that: ð19Þ
Different transformations are performed on g(u) in order to obtain the function g*convex in u i.e. g 00 ðuÞ40
Differentiating this equation with respect to u, we obtain: g0 ðuÞ ¼ 2wu u:
ð27Þ
Then, wu
g0 ðuÞ ¼ hðuÞ: 2u
ð28Þ
From this study and the results, we can deduce the following theorem: Let g be an interaction function adapted to discontinuities there exists a convex and decrease function C such that: gðuÞ ¼ inf ðwu2 þ ðwÞÞ w
ð29Þ
Whatever u fixed, wu ¼ arg min ðwu2 þ ðwÞÞ ¼ w
g0 ðuÞ ¼ hðuÞ: 2u
ð30Þ
ð20Þ
By differentiating a second time with respect to u, we obtain: g00 ðuÞ ¼ 2uh0 ðuÞ þ 2hðuÞg00 ðuÞ
ð26Þ
Note that in practice, the knowledge of C(w) is not necessary because it does not intervene in the future state calculation.
According to Equation (10), we have: g0 ðuÞ ¼ 2uhðuÞ:
From Eqns (23) and (25) we obtain: gðuÞ ¼ g ðu; wu Þ ¼ wu u2 þ ðwu Þ:
– it is based on the line process which is unstable. – it brings back to the quadratic model which risks generating a global smoothing of the solution.
gðuÞ ¼ inf w ðg ðu; wÞÞ:
ð25Þ
wu is the value of w which minimizes g*(u, w) according to w.
However, this model has two inconveniences:
wu ¼ arg minw g ðu; wÞ ¼ hðuÞ
ð21Þ
The Proposed Restoration Algorithm Now, we present our proposed restoration algorithm. We consider in this case the neighborhood system of the
320
K. ACHOUR ETAL.
fourth order. The following results are valid for any system. Let E(f) represent the global energy function to minimize. We have: Eð f Þ ¼ inf E ð f ; wx ; wy Þ; ðwx;wyÞ
ð31Þ
where (wx,wy) are the auxiliary variables : wx represents horizontal orientations, and wy the vertical orientations. X ½ðwx Þi; j ðDx f Þ2i; j E ð f ; wx ; wy Þ ¼ ðd ÿ Rf Þ2 þ 22 i; j
þ ðwx Þ þ 2
2
X
Estimation of future states The estimation of future states consists of resolving the following equation of Algorithm 2: Algorithm 2. The improved Restoration Algorithm f (0) ¼ d; d (0) ¼ d; n¼0; Repeat: ðwx ; wy Þðnþ1Þ ¼ arg min E ð f ðnÞ ; wx ; wy Þ: ðwx ;wy Þ
½ðwy Þi; j ðDy f Þ2i; j
þ ðwy Þ;
ð32Þ
i; j
f ðnþ1Þ ¼ arg min ð f ðnÞ ; ðwx ; wy Þðnþ1Þ Þ: f
where Dx f and Dy f represent the calculated gradients according to these two directions x and y. The algorithm of restoration consists of minimizing E*( f,wx,wy) according to the auxiliary variable where the image field is fixed, then we minimize according to the image field where the auxiliary variable is fixed. We reiterate this process until the limit is reached. The associated algorithm is given by Algorithm 1. Algorithm 1. Algorithm of Restoration
Until Stability Until Stability f nþ1 ¼ arg min E ð f n ; wÞ f
ð33Þ
Let us recall that the function E*( f,w) is convex in f, therefore for its minimization we used a classical method. For this application, we choose the method of cyclic direction which has been adapted for characteristics of the Markovian random field. This method is illustrated by Algorithm 3: Algorithm 3. Estimation algorithm of future states
f (0)¼d; n¼0; repeat: nþ1
ðwx ; wy Þ
¼ arg min E ð f ðwx ;wy Þ
ðnÞ
; wx ; wy Þ:
f nþ1 ¼ arg min ð f ðnÞ ; ðwx ; wy Þðnþ1Þ Þ: f
Until stability The first optimization is obtained directly from Eqn (28), the second optimization can be made by the iterative method such as the gradient method or the method of cyclic direction. We know that after each optimization the obtained image is better than the input image, then we consider that the acquired image for (n+1) optimizations is just the restored image for the optimization n. The algorithm 1 becomes as follows:
n¼0; f (0) ¼ d; For each site s do: fs ¼ arg min ðE ð f n =fs Þ ¼ fsn þ Þ; wÞ ¼ Eð ÞÞ;
replace fs by fs+a; n¼n+1; end for; Until stability;
To calculate a, we only have to resolve the following equation: dEð Þ ¼ 0: d
ð34Þ
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION We use the local expression E*( f, w), and consider a set of cliques to which the site s belongs as an elementary image. This choice is justified mathematically by differentiating E(a)with respect to a. Thus, the resolution of Eqn (34) comes down to resolving the first order polynomial equation. Note that when we resolve Eqn (21), the term c(w) is equal to zero, because it does not depend on a. Then in practice, the knowledge of c(w) is not necessary.
321
Experimental Results In this section, we present some results of applying our technique to restore real and degraded images. In order to evaluate the efficiency of the implemented restoration model, different tests are applied to real images shown in Figures 1, 2 and 3 and a degraded image with 20db Gaussian noise respectively shown on Figure 4. All images are of size 2566256 pixels and coded on 256 grey levels (8 bits encoded).
Stability criterion In preceding paragraphs, we have developed iterative algorithms which converge after an infinite number of iterations. However, in practice it is necessary to define a criterion called the stability criterion, which is the halt test for different algorithms. A good halt criterion has to study: . Changes brought to the image which are quantified by the variation of global energy DE from one iteration to another . . The interval of the variation of DE during a certain number of iterations is called stability period PS. So as to take into account these two constraints, the stability criterion considers that the system is stable if the variation of DE is less than a given threshold SS during PS iterations. This criterion is illustrated by Algorithm 4 given below: Algorithm 4. Algorithm of stability criterion Stability criterion (DE, SS, Ps nb) if DE5SS NB++; If nb¼SS stability criterion ¼ true; if not stability criterion ¼ false end
For the implemented model of restoration, we have to define two thresholds of stability: . Global stability threshold SSG for algorithm 2. . Local stability threshold SS for algorithm 3 and a period of stability PS.
After several experiments, we obtained a set of standard values for some parameters like the smoothness parameter l2, global stability threshold SSG and local stability threshold SSO. The influence of these parameters on the expected results was well tested during our experiments. For images shown in Figures 1, 2 and 3, a low value of l2(l2 ¼ 50) is efficacy to obtain a restored image close to the original one. For a large value of l2 the restored image is too smoothed. For a degraded image with a white Gaussian noise shown on Figure 4 a high value of l2 is necessary to obtain a good restored images. However, for other parameters used such as SSG and SSO, a judicious choice of the values is necessary to obtain good results. Finally , the restoration is performed with the following parameters: . 2 ¼ 0; SSG ¼ 0.02; SSO ¼ 0.05; PS ¼ 3 for Figures 1(a) and 2(a). . 2 ¼ 50; SSG ¼ 0.03; SSO ¼ 0.05; PS ¼ 3 for Figure 3(a). . 2 ¼ 250; SSG ¼ 0.01; SSO ¼ 0.06; PS ¼ 4 for Figure 4(a). To show the results of edge contours detection, we opted for an optimal edge detection operator called CannyDeriche [15]. The filter parameter is equal to 1.5 in all cases which show a compromise between precision and localization of edge detection. In Figures 1(b), 2(b) and 3(b), we can see the resultant edges on the original desk image (before processing). Results of restoration with their corresponding obtained edges by our algorithm are shown respectively by Figures 1(c) and 1(d), Figures 2(c) and 2(d) and Figures 3(c) and 3(d). We can clearly appreciate a more rich and closed edges contour. The best restoration, are obtained after eight optimizations and 26 iterations for the image shown in Figure 2(c) and after five optimizations and 28 iterations for the image shown in Figure 3(c). This
322
K. ACHOUR ETAL.
Figure 1. (a) Original image; (b) the corresponding contour image detected by the Deriche-Canny operator; (c) restored image; (d) the corresponding edges of restored image by Deriche-Canny operator.
illustrates the fast convergence of our algorithm and the quality of the results. We have also tested our algorithm on an image degraded with a Gaussian noise shown in Figure 4. The result of restoration with corresponding edges contours are illustrated in Figures 4(c) and 4(d). We can appreciate the quality of the restoration with closed
edges but the time consumption is rather more important.
Conclusion We have presented in this paper the result of our work on image restoration in order to obtain a best edge
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION
323
Figure 2. (a) Original image; (b) the contour image generated by the Deriche-Canny operator ; (c) restored image after 8 optimizations and 26 iterations (d) edges of the restored image by the Deriche-Canny operator.
contours detection. The adopted approach is based on the use of Markovian random field modeling to resolve the inverse problem, taking into account the smoothing constraint and the edge preserving. To implement our algorithm for image restoration and edges contours
closing, we have used a deterministic method which exploits the efficiency of the regularization model adapted to discontinuities in order to model the constraints of the image (preserving edges and smoothness of homogeneous zones), and while relating to the
324
K. ACHOUR ETAL.
Figure 3. (a) Original image; (b) the contour image generated by the Deriche-Canny operator; (c) restored image after 5 optimizations and 28 iterations ; (d) edges of the restored image by the Deriche-Canny operator.
quadratic model (that is convex) during the minimization of the global energy. The efficiency of the proposed model in this study shown by the good results obtained on several
images and on contours images. However, for highly degraded images a good restoration result can be obtained by using a multi-resolution treatment. This will make the object of our future contribution in this area.
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION
325
Figure 4. (a) Image degraded with 20Db Gaussian noise; (b) the corresponding contour image detected by the Deriche-Canny operator; (c) restored image; (d) the corresponding edges of the restored image by Deriche-Canny operator.
References 1. Achour, K., Benkhelif, M. & Seddini, M. (1996) Contribution to Contours restoration by Markovian Approach. Proceedings ICSPAT ’96, vol. 1, Oct 1996, p. 1017. 2. Achour, K., Zenati, N. & Laga, H. (1998) Contribution to restoration of degraded images, Proceedings of CARI’98, Dakar, Se´ne´gal.
3. Besak, J. (1974) On the statistical analysis of dirty pictures. J. Royal Statist. Soc. 48: 192–236. 4. Chalmond, B. (1988) Image restoration using an estimated Markov model. Signal Processing 15: 115–129. 5. Geman, S. & Geman, D. (1984) Stochastic relaxation Gibbs distribution and the bayesian restoration of images. IEEE Trans. Pattern Anal. Machine Intell. 6: 721–741.
326
K. ACHOUR ETAL.
6. Geman, S., Geman, D., Graffigne, C. & Pong, D. (1990) Bondary detection by constrained optimization. IEEE Trans. Pattern Anal. Machine Intell. 12: 609–628. 7. Chou, P.B. & Brown, C.M. (1990) The theory and practice of bayesian image modeling. Int. J. Comput. Vision 4: 185–210. 8. Barnard, S.T. (1989) Stochastic stereo matching over scale. Int. J. Comput. Vision 3: 17–32. 9. Bouthemy, P. & Franc¸ois, E. (1993) Motion segmentation and qualitative dynamic scene analysis from an image and sequence. Int. J. Comput. Vision 10: 57–182. 10. Bouthemy, P. & Lalande, P. (1993) Recovery of moving object masks in an image sequence using local spatiotemporal contextual information. Opt. Eng. 32: 1205– 1212. 11. Blacke, A. & Zisserman, A. (1987) Visual reconstruction. Cambridge, MA: MIT Press. 12. Jeng, F.C. & Wood, J.W. (1988) Image estimation by stochastic relaxation in the compound gaussian case. Proceedings of ICASSP, New York, Apr 1988, pp 1016– 1019. 13. Simchony, T., Chellappa, R. & Lichtenstein, Z. (1989) The Graduate Non-convexity algorithm for image estimation using Compound Gauss-Markov Field model. Proceedings of ICASSP, Glasgow, May 1989, pp. 1417–1420. 14. Deriche, R. (1987) Using Canny’s criteria to derive a recursively implemented optimal edge detector. Int. J. Comput. Vision 1.
15. Bernard, M. & Sigelle, M. (1996) Champs de Gibbs et champs Markoviens. Rapport de Recherche, TELECOM. Paris, Departement Image. 16. Geman, S., Geman, D. & Gibbs. (1984). Distribution and Bayesian Restoration of Images. IEEE Transaction on Pattern Analysis and Machine Intelligence (PAMI), 6: 721. 17. Charbonnier, P., Feraud, L.B., Aubert, G. & Barlaud, M. (1997) Deterministic edge processing regularization in computed imaging. IEEE Transaction on Image Processing 6: 298–318. 18. Li, S.Z. (1995) On Discontinuity Adaptive Smoothness Priors in Computer Vision. IEEE Transaction on PAMI 17, 576–595. 19. Zerubia, J. & Polyette, F. (1990) Detection de contours et restauration d’images par des algorithmes deterministes de relaxation. Mise en oeuvre sur la machine a` connexions CM2. Rapport de recherche no 1291, Oct 1990, INRIA, France. 20. Kang, M.G. & Katsaggelos, A.K. (1995) General Choice of the Regularization Functional in image restoration. IEEE Transaction on Image Processing 4: 594–612. 21. Geman, D. & Reynolds, G. (1992) Restoration and Recovery of Discontinuities. IEEE Transaction on Pattern Analysis and Machine Intelligence (PAMI) 14: 267–284. 22. Berthod, M., Kato, Z. & Zerubia, J. (1996) A deterministic Approach to the MAP problem. IEEE Transaction on Image Processing 2: 1312–1328.
Contribution to Image and Contours Restoration
D
igital images are generally degraded by different sources during their acquisition. This is due of two types of phenomena: the deterministic phenomenon of blur which is introduced by relative motion between a camera and the object, and the stochastic phenomena such as atmospheric turbulence, noise and other factors. So, it becomes very difficult for high level processing systems (object detection, three-dimensional reconstruction, characters recognize . . .) to extract reliable features from the incomplete edges. Our objective is to reduce the effect of this degradation and recover the original image from the degraded image with better edge detection. The Markov Random Field (MRF) modelization allows us to restore images with taking into account some constraints such as the smoothing constraint and the edge preserving. Our approach is focused on a new deterministic algorithm that permits approaching the global optimum and reduces computational time. We will present the semi-quadratic regularization model adapted to discontinuities in order to model smoothing constraints of homogeneous zones and to preserve contours. The obtained results on real images are satisfying since we reached our goal of a smoothed homogenous area with preserved edge. # 2001 Academic Press
K. Achour, N. Zenati and H. Laga Robotics and Artificial Intelligence Laboratory, Advanced Technologies Development Center, 128, Chemin Mohamed Gacem, El-Madania, 16075, Algiers, Algeria. E-mail: [email protected], [email protected]
Introduction The problem of image restoration is very important in a large number of applications in early vision processing. For many years researchers have worked on this problem, since noisy image restoration is a very important preliminary step for various high level vision processing. Many solutions have been proposed, among them being the classical approach which is based on filtering techniques. In this case, the image is considered as being the composition of two signals: the real signal which is condensed in low frequencies and the noise signal. The filtering operation consists then of separating these two signals by applying a low pass filter. However, this approach does not consider the possibility of the
1077-2014/01/040315+12 $35.00/0
signal discontinuity but rather looks to find smoothed solutions, that causes a loss of edges (thin details) which are primordial elements in image processing. During the last few years, this approach has been quickly abandoned in favour to a new method based on the Markovian random field (MRF) which has been successfully introduced in many fundamental issues of image analysis and computer vision such as image restoration [1–5], edge detection [6], image segmentation [7,8], surface reconstruction [7], stereovision [8] and motion analysis [9,10]. Markovian random field (MRF) is an efficient and powerful framework for specifying nonlinear interactions between features of the same nature or of different one. When they are associated to
#
2001 Academic Press
316
K. ACHOUR ETAL.
the maximum a posteriori criterion, they lead to the minimization of a global energy function which may exhibit local minima. This minimization is generally performed using deterministic [3] or stochastic algorithms [5]. Stochastic algorithms may be time consuming while deterministic schemes often get ‘‘stuck’’ in local minima of the energy function. Among stochastic methods, we can mention the ones using three line processes, first introduced by Geman and Geman [5]. They explicitly take into account the cost of creating edges in the form of a line-process. The problem is then to minimize the energy which is nonconvex with respect to the image intensity and the line processes. One way to obtain the minimum is to use the well-known stochastic technique called ‘‘simulating annealing’’. This method is optimal but very time consuming. Another way to deal with the problem is to search for optimal deterministic algorithms. From a theoretical point of view, the main draw back of these techniques is that they are not guaranteed to reach the global minimum. But from a practical point of view, the shape of the energy function obtained for images (many minima) and the results are usually good enough if not optimal. Deterministic algorithms yield solutions at a much lower computational cost. One of the most interesting deterministic techniques is the Graduate Non-Convexity (GNC) algorithm introduced by Blacke and Zissermann [11]. An extension of this technique using a Compound Gauss-Markov Field (CGMRF) [12], have been derived in [13] and [14] for image restoration. The random Markovian field (MRF) approach allows to restore the image under the constraint of preserving edges: An image f is considered as the realization of a random variable field defined on a set of sites S¼{si} [3,14]. On S, we define a neighborhood system V and a set of cliques.The neighborhood Vsk of a site sk is defined as follows: Vsk ¼ fsj 2 S=j 6¼ k et sk 2 Vsj The clique c of n neighbors.
th
ð1Þ
order is a set of n sites of all
A field f defined on a set of sites S is a Markovian random field (MRF) according to the neighbor system V if:
8 2 S; 8 2 S;
Pð fs Þ40 Pð fs =f ÿ f fs gÞ ¼ Pð fs =Vs Þ;
ð2Þ
with Vs the neighborhood of the site s and f7{fs} the image field without the site s. The relation (2) shows that the probability of the attribute apparition on a site is higher than zero and the attribute of the site depends only on its neighbors. In order to quantify the properties of the image, we associate to every clique c a function of potential ’c. We define then the local energy function of a site s as:
UðsÞ ¼
X
’c ðsÞ:
ð3Þ
c2C=s2S
The sum of the potential of all the cliques defines the global energy of the Markov random field. In order to exploit the statistical aspect of the MRF, we introduced the global aspect which allows us to encode the likelihood of the image taken as a whole. It is defined by the Hammersley-Clifford theorem, given as follows: If f is a Markov Random Field (MRF) and the probability of any configuration is nonzero, then its probability is given by the Gibbs law: probð f Þ ¼ ð1=ZÞ expðÿUð f ÞÞ
ð4Þ
As for the local aspect, it models the local constraints of the image. In applying the laws on the join probabilities, we obtain the following result: eÿ’ð fs Þ Probð fs =f s Þ ¼ P ÿ’ð fs Þ ¼ Probð fs =Vs Þ; e
ð5Þ
fs 2
Where ’ is the local energy , Vs the neighborhood of the site s and f s¼(f7fs) The organization of this paper is as follows: Different regularization models and the model used in order to smoothen constraints and preserve edges are given in the next section. Following this we present an overview of different restoration algorithms. Next, the GNC strategy (Graduated-Non-Convexity) and our approach for realizing the series E(p) is presented. Then, a new deterministic restoration algorithm is proposed and analyzed. Experimental results are then presented and finally, the paper concludes with a discussion and some perspectives.
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION
Regularization Digital images are generally degraded during their acquisition, although the degradation processes affecting images are generally nonlinear. The distortions are often adequately modeled by a linear blur and an additive Gaussian white noise, giving the observation equation d ¼ Rf þ B;
ð6Þ
where f, d and B are respectively the real image, the acquired image and the additive white Gaussian noise which defines the random phenomenon. R is the ‘‘blur matrix’’ characterizing the acquisition system and corresponds to a blur function representing the linear spatially invariant or spatially varying distortion. In our case R is the identity matrix. The restoration of the degraded image d consists of resolving the following equation: f ¼ arg max pð f =dÞ f
ð7Þ
Since f is an MRF defined on a set of sites S and according to the Hammersley-Clifford theorem, its probability follows the Gibbs law given as follows [15, 16]. pð f Þ ¼
1 ÿUð f Þ e ; Z
ð8Þ
where U( f) is the global energy function involved in the relation between the attributes of different sites and Z is a constant called ‘‘partition function’’. By applying the probability law of Bayes, and the relationships (4), (6) and (7), we can show that a problem of restoration is just a problem of a function minimization E( f) defined as follows: Eð f Þ ¼ ðd ÿ Rf Þ2 þ 22 Uð f Þ ¼ E1 þ E2
ð9Þ
where E1 is the attachment term to the data and E2 the smoothness term.
317
which consists of choosing energy functions in order to represent simultaneously the smoothing and contour preserving constraints. This problem is called the regularization. Standard or semi-quadratic regularization The local energy function used in this model is the quadratic function gq(u)¼u2, where u is the gradient. This model is easier to implement because the quadratic function is convex. For its minimization we can use the classical method based on the gradient descent. However, this function imposes the global smoothing of the solution, therefore it does not preserve the discontinuities [17, 18]. Model with line process. The Boolean line process was introduced to take into account the strong gradient and thus avoid the global smoothing . However, its inconvenience is the fact that it does not assure the stability of the value and besides, its Boolean nature does not preserve the soft edges. We define the truncated quadratic by the energy function given as follows: GBZ ðuÞ ¼ Minðu2 ; Þ: This function is similar pffiffito the quadratic function with the values less than a and constant for pffiffi the higher values, but, it is not differentiable at u ¼ a. We have presented different energy functions which allows us to model the constraints imposed on the solution. Each of these models presents a set of advantages and disadvantages. The model adapted to discontinuities, contrary to the quadratic model and to the model with line process, allows an image restoration with the possibility of considering the signal discontinuities [17–19]. Model adapted to discontinuities It is defined from the energy function g(u) an interaction function h(u) given as follows:
Regularization models hðuÞ
g0 ðuÞ : 2u
ð10Þ
The purpose of digital image restoration is to operate on the degraded images to obtain an improved image f * (which minimizes E) that is as close as possible to the original image.
Thus, for an interactive function h to be adapted to discontinuities, the following conditions have to be verified [5, 20]:
The definition of the regularization model consists of defining the term E2 (the smoothness term) of Eqn (9),
a) h [ C1 b) h(u) ¼ h(7u)
318
K. ACHOUR ETAL.
Table 1. Action functions/potential adapted to discontinuities h(u)
g(u)
(a)
exp(7u )
7exp(7u2)
(b)
1 ð1þu2 Þ2 1 1þu2 1 1þjuj
u2 1þu2
(c) (d)
2
The simulated annealing, which is an analogy with the process of annealing used in metallurgy, optimizes the Gibbs sampler. It introduces, in the energy function, a new parameter T called temperature. The probability P(f/d) is given by the following equation:
ln(1+u2)
Pð f =dÞ ¼
2(|u|+ln(1+|u|)
c) h(u) 4 0 d) h decreasing function e) Limu??h(u) ¼ 0 According to the last two conditions, we can deduct the following result: So that a given model is adapted to discontinuities, it is necessary that: lim jg0 ðuÞj ¼ C:
1 uðf =dÞ e T Z
ð11Þ
The parameter T is initialized to a very important value, then decreases gradually until reaching a very low value (generally 1) that allows quitting the local optima and controls the length of the jumps. The algorithms based on the principle of the simulated annealing converge to the global optimum and therefore they give very good results. The processing time is reduced in regard to the Gibbs sampler but it remains long. The deterministic algorithms
juj!1
If C ¼ 0, the model prevents smoothing, otherwise (C40) it allows smoothing. The following Table 1 gives us some interaction functions/potential adapted to discontinuities.
Restoration Algorithms When the model adapted to discontinuities is established, we define the maximization algorithm of the probability P(f/d). Thus, we distinguish two types of algorithms according to whether we directly use the probability P(f/d) or minimize the global energy function E(f). The stochastic algorithms The stochastic algorithms allow us to find the configuration f* which maximizes the probability P(f/d), according to Eqn (7) [15,16,21]. The Gibbs sampler leads towards the global optimum and therefore gives the best results. However, in practice, its implementation is quasi-impossible for two reasons: – The processing of the partition function Z requires a sum on the set of all possible configurations of an MRF. – The computational time is very important.
Contrary to the stochastic algorithms, deterministic algorithms are very fast but converge toward some local minima [15,22]. In order to exploit the fastness of these algorithms, new strategies were developed such as Graduated Non-Convexity (GNC) and Mean Field Annealing (MFA). The graduated non-convexty Principle The aim of this algorithm is to break down the nonconvexity of the criterion to be minimized, E(f). Its principle is to make series of functions E(p)(f), where p varies from one to zero with E (1)¼convex, and E (0)=E. Then, the procedure is to minimize successively these terms on resetting at each minimization the result by the precedent minimization. This algorithm uses the line process as the regularization model. In fact: Eð f ; l; mÞ ¼ E1 þ E2 þ E3 ;
ð12Þ
E1 ¼ ðd ÿ Rf Þ2
ð13Þ
with
E2 ¼ 2
X
ð fi; j ÿ fiÿ1; j Þð1 ÿ li; j Þ þ ðfi; j ÿ fi; jþ1 Þð1 ÿ mi; j Þ
i; j
ð14Þ E3 ¼
X
ðli; j ÿ mi; j Þ
i;j
Where a is the cost to pay to introduce an edge.
ð15Þ
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION Its principle is then to minimize E ( f, l, m) according to the line process when the field image is fixed, then we minimize according to the field image when the process line is fixed. The minimization according to the line process gives us the following series E(p): X ðpÞ ðg ; ðfi; j ÿ fiÿ1; j Þ þ gðpÞ E ðpÞ ðf Þ ¼ E1 þ ; ðfi; j ÿ fi; jþ1 ÞÞ
Since g is the energy function adapted to discontinuities,h(u)40. Suppose now, that h(u) is constant (w40), then Eqn (13) yields: g00 ðu; wÞ ¼ 2w:
ð22Þ
Integrating this function twice we have: g ðu; wÞ ¼ wu2 þ ðuÞ
ð23Þ
with,
i; j
ðuÞ ¼ c u þ b;
ð16Þ
ð24Þ
where c and b are constant. If we suppose u is constant , we obtain:
with: 8 <
2 sijtj5q l ÿ cðjtj ÿ r2 Þ=2 if qjtj5r : else
ð17Þ
c ¼ 1=4p; r2 ¼ ð2=c þ 1=2 Þ; q ¼ =2 r
ð18Þ
gðpÞ ; ðtÞ ¼
319
Our work then consists of developing an algorithm based on the principle of the GNC , which exploits the model adapted to discontinuities and the convexity of the quadratic model; it is called the semi-quadratic regularization. Semi-quadratic regularization Let g(u) represent the energy function adapted to discontinuities. The semi-quadratic regularization consists of introducing the auxiliary variable w, so that: ð19Þ
Different transformations are performed on g(u) in order to obtain the function g*convex in u i.e. g 00 ðuÞ40
Differentiating this equation with respect to u, we obtain: g0 ðuÞ ¼ 2wu u:
ð27Þ
Then, wu
g0 ðuÞ ¼ hðuÞ: 2u
ð28Þ
From this study and the results, we can deduce the following theorem: Let g be an interaction function adapted to discontinuities there exists a convex and decrease function C such that: gðuÞ ¼ inf ðwu2 þ ðwÞÞ w
ð29Þ
Whatever u fixed, wu ¼ arg min ðwu2 þ ðwÞÞ ¼ w
g0 ðuÞ ¼ hðuÞ: 2u
ð30Þ
ð20Þ
By differentiating a second time with respect to u, we obtain: g00 ðuÞ ¼ 2uh0 ðuÞ þ 2hðuÞg00 ðuÞ
ð26Þ
Note that in practice, the knowledge of C(w) is not necessary because it does not intervene in the future state calculation.
According to Equation (10), we have: g0 ðuÞ ¼ 2uhðuÞ:
From Eqns (23) and (25) we obtain: gðuÞ ¼ g ðu; wu Þ ¼ wu u2 þ ðwu Þ:
– it is based on the line process which is unstable. – it brings back to the quadratic model which risks generating a global smoothing of the solution.
gðuÞ ¼ inf w ðg ðu; wÞÞ:
ð25Þ
wu is the value of w which minimizes g*(u, w) according to w.
However, this model has two inconveniences:
wu ¼ arg minw g ðu; wÞ ¼ hðuÞ
ð21Þ
The Proposed Restoration Algorithm Now, we present our proposed restoration algorithm. We consider in this case the neighborhood system of the
320
K. ACHOUR ETAL.
fourth order. The following results are valid for any system. Let E(f) represent the global energy function to minimize. We have: Eð f Þ ¼ inf E ð f ; wx ; wy Þ; ðwx;wyÞ
ð31Þ
where (wx,wy) are the auxiliary variables : wx represents horizontal orientations, and wy the vertical orientations. X ½ðwx Þi; j ðDx f Þ2i; j E ð f ; wx ; wy Þ ¼ ðd ÿ Rf Þ2 þ 22 i; j
þ ðwx Þ þ 2
2
X
Estimation of future states The estimation of future states consists of resolving the following equation of Algorithm 2: Algorithm 2. The improved Restoration Algorithm f (0) ¼ d; d (0) ¼ d; n¼0; Repeat: ðwx ; wy Þðnþ1Þ ¼ arg min E ð f ðnÞ ; wx ; wy Þ: ðwx ;wy Þ
½ðwy Þi; j ðDy f Þ2i; j
þ ðwy Þ;
ð32Þ
i; j
f ðnþ1Þ ¼ arg min ð f ðnÞ ; ðwx ; wy Þðnþ1Þ Þ: f
where Dx f and Dy f represent the calculated gradients according to these two directions x and y. The algorithm of restoration consists of minimizing E*( f,wx,wy) according to the auxiliary variable where the image field is fixed, then we minimize according to the image field where the auxiliary variable is fixed. We reiterate this process until the limit is reached. The associated algorithm is given by Algorithm 1. Algorithm 1. Algorithm of Restoration
Until Stability Until Stability f nþ1 ¼ arg min E ð f n ; wÞ f
ð33Þ
Let us recall that the function E*( f,w) is convex in f, therefore for its minimization we used a classical method. For this application, we choose the method of cyclic direction which has been adapted for characteristics of the Markovian random field. This method is illustrated by Algorithm 3: Algorithm 3. Estimation algorithm of future states
f (0)¼d; n¼0; repeat: nþ1
ðwx ; wy Þ
¼ arg min E ð f ðwx ;wy Þ
ðnÞ
; wx ; wy Þ:
f nþ1 ¼ arg min ð f ðnÞ ; ðwx ; wy Þðnþ1Þ Þ: f
Until stability The first optimization is obtained directly from Eqn (28), the second optimization can be made by the iterative method such as the gradient method or the method of cyclic direction. We know that after each optimization the obtained image is better than the input image, then we consider that the acquired image for (n+1) optimizations is just the restored image for the optimization n. The algorithm 1 becomes as follows:
n¼0; f (0) ¼ d; For each site s do: fs ¼ arg min ðE ð f n =fs Þ ¼ fsn þ Þ; wÞ ¼ Eð ÞÞ;
replace fs by fs+a; n¼n+1; end for; Until stability;
To calculate a, we only have to resolve the following equation: dEð Þ ¼ 0: d
ð34Þ
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION We use the local expression E*( f, w), and consider a set of cliques to which the site s belongs as an elementary image. This choice is justified mathematically by differentiating E(a)with respect to a. Thus, the resolution of Eqn (34) comes down to resolving the first order polynomial equation. Note that when we resolve Eqn (21), the term c(w) is equal to zero, because it does not depend on a. Then in practice, the knowledge of c(w) is not necessary.
321
Experimental Results In this section, we present some results of applying our technique to restore real and degraded images. In order to evaluate the efficiency of the implemented restoration model, different tests are applied to real images shown in Figures 1, 2 and 3 and a degraded image with 20db Gaussian noise respectively shown on Figure 4. All images are of size 2566256 pixels and coded on 256 grey levels (8 bits encoded).
Stability criterion In preceding paragraphs, we have developed iterative algorithms which converge after an infinite number of iterations. However, in practice it is necessary to define a criterion called the stability criterion, which is the halt test for different algorithms. A good halt criterion has to study: . Changes brought to the image which are quantified by the variation of global energy DE from one iteration to another . . The interval of the variation of DE during a certain number of iterations is called stability period PS. So as to take into account these two constraints, the stability criterion considers that the system is stable if the variation of DE is less than a given threshold SS during PS iterations. This criterion is illustrated by Algorithm 4 given below: Algorithm 4. Algorithm of stability criterion Stability criterion (DE, SS, Ps nb) if DE5SS NB++; If nb¼SS stability criterion ¼ true; if not stability criterion ¼ false end
For the implemented model of restoration, we have to define two thresholds of stability: . Global stability threshold SSG for algorithm 2. . Local stability threshold SS for algorithm 3 and a period of stability PS.
After several experiments, we obtained a set of standard values for some parameters like the smoothness parameter l2, global stability threshold SSG and local stability threshold SSO. The influence of these parameters on the expected results was well tested during our experiments. For images shown in Figures 1, 2 and 3, a low value of l2(l2 ¼ 50) is efficacy to obtain a restored image close to the original one. For a large value of l2 the restored image is too smoothed. For a degraded image with a white Gaussian noise shown on Figure 4 a high value of l2 is necessary to obtain a good restored images. However, for other parameters used such as SSG and SSO, a judicious choice of the values is necessary to obtain good results. Finally , the restoration is performed with the following parameters: . 2 ¼ 0; SSG ¼ 0.02; SSO ¼ 0.05; PS ¼ 3 for Figures 1(a) and 2(a). . 2 ¼ 50; SSG ¼ 0.03; SSO ¼ 0.05; PS ¼ 3 for Figure 3(a). . 2 ¼ 250; SSG ¼ 0.01; SSO ¼ 0.06; PS ¼ 4 for Figure 4(a). To show the results of edge contours detection, we opted for an optimal edge detection operator called CannyDeriche [15]. The filter parameter is equal to 1.5 in all cases which show a compromise between precision and localization of edge detection. In Figures 1(b), 2(b) and 3(b), we can see the resultant edges on the original desk image (before processing). Results of restoration with their corresponding obtained edges by our algorithm are shown respectively by Figures 1(c) and 1(d), Figures 2(c) and 2(d) and Figures 3(c) and 3(d). We can clearly appreciate a more rich and closed edges contour. The best restoration, are obtained after eight optimizations and 26 iterations for the image shown in Figure 2(c) and after five optimizations and 28 iterations for the image shown in Figure 3(c). This
322
K. ACHOUR ETAL.
Figure 1. (a) Original image; (b) the corresponding contour image detected by the Deriche-Canny operator; (c) restored image; (d) the corresponding edges of restored image by Deriche-Canny operator.
illustrates the fast convergence of our algorithm and the quality of the results. We have also tested our algorithm on an image degraded with a Gaussian noise shown in Figure 4. The result of restoration with corresponding edges contours are illustrated in Figures 4(c) and 4(d). We can appreciate the quality of the restoration with closed
edges but the time consumption is rather more important.
Conclusion We have presented in this paper the result of our work on image restoration in order to obtain a best edge
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION
323
Figure 2. (a) Original image; (b) the contour image generated by the Deriche-Canny operator ; (c) restored image after 8 optimizations and 26 iterations (d) edges of the restored image by the Deriche-Canny operator.
contours detection. The adopted approach is based on the use of Markovian random field modeling to resolve the inverse problem, taking into account the smoothing constraint and the edge preserving. To implement our algorithm for image restoration and edges contours
closing, we have used a deterministic method which exploits the efficiency of the regularization model adapted to discontinuities in order to model the constraints of the image (preserving edges and smoothness of homogeneous zones), and while relating to the
324
K. ACHOUR ETAL.
Figure 3. (a) Original image; (b) the contour image generated by the Deriche-Canny operator; (c) restored image after 5 optimizations and 28 iterations ; (d) edges of the restored image by the Deriche-Canny operator.
quadratic model (that is convex) during the minimization of the global energy. The efficiency of the proposed model in this study shown by the good results obtained on several
images and on contours images. However, for highly degraded images a good restoration result can be obtained by using a multi-resolution treatment. This will make the object of our future contribution in this area.
CONTRIBUTION TO IMAGE AND-CONTOURS RESTORATION
325
Figure 4. (a) Image degraded with 20Db Gaussian noise; (b) the corresponding contour image detected by the Deriche-Canny operator; (c) restored image; (d) the corresponding edges of the restored image by Deriche-Canny operator.
References 1. Achour, K., Benkhelif, M. & Seddini, M. (1996) Contribution to Contours restoration by Markovian Approach. Proceedings ICSPAT ’96, vol. 1, Oct 1996, p. 1017. 2. Achour, K., Zenati, N. & Laga, H. (1998) Contribution to restoration of degraded images, Proceedings of CARI’98, Dakar, Se´ne´gal.
3. Besak, J. (1974) On the statistical analysis of dirty pictures. J. Royal Statist. Soc. 48: 192–236. 4. Chalmond, B. (1988) Image restoration using an estimated Markov model. Signal Processing 15: 115–129. 5. Geman, S. & Geman, D. (1984) Stochastic relaxation Gibbs distribution and the bayesian restoration of images. IEEE Trans. Pattern Anal. Machine Intell. 6: 721–741.
326
K. ACHOUR ETAL.
6. Geman, S., Geman, D., Graffigne, C. & Pong, D. (1990) Bondary detection by constrained optimization. IEEE Trans. Pattern Anal. Machine Intell. 12: 609–628. 7. Chou, P.B. & Brown, C.M. (1990) The theory and practice of bayesian image modeling. Int. J. Comput. Vision 4: 185–210. 8. Barnard, S.T. (1989) Stochastic stereo matching over scale. Int. J. Comput. Vision 3: 17–32. 9. Bouthemy, P. & Franc¸ois, E. (1993) Motion segmentation and qualitative dynamic scene analysis from an image and sequence. Int. J. Comput. Vision 10: 57–182. 10. Bouthemy, P. & Lalande, P. (1993) Recovery of moving object masks in an image sequence using local spatiotemporal contextual information. Opt. Eng. 32: 1205– 1212. 11. Blacke, A. & Zisserman, A. (1987) Visual reconstruction. Cambridge, MA: MIT Press. 12. Jeng, F.C. & Wood, J.W. (1988) Image estimation by stochastic relaxation in the compound gaussian case. Proceedings of ICASSP, New York, Apr 1988, pp 1016– 1019. 13. Simchony, T., Chellappa, R. & Lichtenstein, Z. (1989) The Graduate Non-convexity algorithm for image estimation using Compound Gauss-Markov Field model. Proceedings of ICASSP, Glasgow, May 1989, pp. 1417–1420. 14. Deriche, R. (1987) Using Canny’s criteria to derive a recursively implemented optimal edge detector. Int. J. Comput. Vision 1.
15. Bernard, M. & Sigelle, M. (1996) Champs de Gibbs et champs Markoviens. Rapport de Recherche, TELECOM. Paris, Departement Image. 16. Geman, S., Geman, D. & Gibbs. (1984). Distribution and Bayesian Restoration of Images. IEEE Transaction on Pattern Analysis and Machine Intelligence (PAMI), 6: 721. 17. Charbonnier, P., Feraud, L.B., Aubert, G. & Barlaud, M. (1997) Deterministic edge processing regularization in computed imaging. IEEE Transaction on Image Processing 6: 298–318. 18. Li, S.Z. (1995) On Discontinuity Adaptive Smoothness Priors in Computer Vision. IEEE Transaction on PAMI 17, 576–595. 19. Zerubia, J. & Polyette, F. (1990) Detection de contours et restauration d’images par des algorithmes deterministes de relaxation. Mise en oeuvre sur la machine a` connexions CM2. Rapport de recherche no 1291, Oct 1990, INRIA, France. 20. Kang, M.G. & Katsaggelos, A.K. (1995) General Choice of the Regularization Functional in image restoration. IEEE Transaction on Image Processing 4: 594–612. 21. Geman, D. & Reynolds, G. (1992) Restoration and Recovery of Discontinuities. IEEE Transaction on Pattern Analysis and Machine Intelligence (PAMI) 14: 267–284. 22. Berthod, M., Kato, Z. & Zerubia, J. (1996) A deterministic Approach to the MAP problem. IEEE Transaction on Image Processing 2: 1312–1328.