A Novel Variational PDE Technique for Image Denoising Tudor Barbu Institute of Computer Science of the Romanian Academy, Iaşi, Romania tudbar@iit.tuiasi.ro Abstract. A robust variational PDE model for image noise removal is proposed in this paper. One considers an energy functional to be minimized, based on a novel smoothing constraint. Then, the corresponding Euler-Lagrange equation is determined. The obtained PDE model is solved, by using a numerical discretization scheme. Some results of our image denoising experiments and method comparisons are also described in this article. Keywords: image denoising, variational technique, PDE models, Euler Lagrange equations, energy functional minimization, smoothing function, discretization scheme. 1 Introduction During the past two decades, the mathematical models have been increasingly used in some traditionally engineering domains like signal and image processing, analysis, and computer vision [1]. The variational and Partial Differential Equation (PDE) based techniques have been widely used and studied in this fields in the past few years because of their modelling flexibility and some advantages of their numerical implementation [2]. Thus, some important application areas of the variational PDE methods are image denoising, image reconstruction (inpainting), image segmentation (contour tracking), image registration and optical flow [1, 2]. We consider a variational approach for image denoising in this paper. Image noise removal with feature preservation is still a focus in the image processing area and serious challenge for the researchers. An efficient image denoising approach must not only substantially reduce the noise amount but also preserve the boundaries and other characteristics [3]. Conventional image filters, like averaging, median, or the classic 2D Gaussian filter succeed in noise reduction, but also have an edge-blurring effect [4]. The linear PDE-based denoising techniques are derived from the the use of the Gaussian filter in multiscale image analysis [4, 5]. The convolution of an image with a 2D Gaussian kernel amounts to solve the diffusion equation in two dimensions (heat equation). The nonlinear PDE-based approaches are able to smooth the images while preserving their edges, also avoiding the localization problems of linear filtering. The most popular nonlinear PDE denoising method is the influential nonlinear anisotropic M. Lee et al. (Eds.): ICONIP 2013, Part III, LNCS 8228, pp. 501–508, 2013. © Springer-Verlag Berlin Heidelberg 2013 502 T. Barbu diffusion scheme developed by P. Perona and J. Malik in 1987 [5, 6]. Numerous denoising techniques derived from their algorithm have been proposed since then [5]. There are many ways to get the nonlinear PDEs. In image processing and computer vision it is very common to obtain them from some variational problems. The basic idea of any variational PDE technique is the minimization of an energy functional [1, 7-9]. The variational techniques have important advantages in both theory and computation, compared with other methods. They can achieve high speed, accuracy, and stability using the extensive results of the numerical PDE approaches. An influential variational denoising and restoration model was developed by Rudin, Osher and Fetami in 1992. Their technique, named Total Variation (TV) denoising, is based on the minimization of the TV norm [7]. TV denoising is remarkably effective at simultaneously preserving boundaries whilst smoothing away noise in flat regions, but it also suffers from the staircasing effect and its corresponding EulerLagrange equation is highly nonlinear and difficult to compute. In recent years, many PDE approaches that improve this classical variational model have been proposed [1]. The novel PDE variational technique provided in this paper achieves an efficient smoothing result while preserving the image edges and also solves the staircase problem [8, 9, 12]. The main contribution of our denoising variational model is the robust smoothness term (regularizer) introduced in the energy functional that is described in the next section. Also, we provide a satisfactory discretization of the PDE model, a good approximation of the Euler-Lagrange equation being described in the third section of this article. Numerous image denoising experiments using this method and method comparisons have been performed. They are discussed in the fourth section. The conclusions are presented in the last section and the paper ends with a list of references. 2 Variational Model for Image Noise Reduction The general variational framework used in image processing and computer vision is characterized by an energy functional having the following form: E [u ( x ) ] =  (D ( u ) + S ( u ) )dx (1) Ω where D(u) represents the data component and S(u) is the smoothing term of the functional [7, 10]. So, one must determine the unknown function u(x) on the domain Ω ⊂ R 2 , that minimizes the above energy: u min = arg min E [u ( x ) ] (2) u∈U In the variational image denoising case, one considers an image u 0 affected by Gaussian noise. The general form of the energy functional used by variational image smoothing processes is: A Novel Variational PDE Technique for Image Denoising E [u ] =  (u − u 0 ) 2 + αψ Ω ( ∇ u ), 2 α >0 503 (3) where the function ψ represents the regularizer (penalizer) of the smoothing term and α is the regularization parameter or smoothness weight [10]. We develop an efficient smoothing component, based on a novel penalizer function and a proper value of the smoothness weight. Thus, we consider the following regularizer: ψ : [0, ∞ ) → [0, ∞ ) : ψ (s ) = η  ln  s + β  k s2 + γ β   + ν ⋅ s ; k > 0,η , β , γ ,ν ∈ ( 0,1)   (4) We consider some proper values for the penalizer’s parameters. The values of k ,η , β , γ ,ν and α which provide a successful denoising are specified in section 4, related to numerical experiments. Then, we compute a minimizer for the energy functional given by (3), using the function ψ given by (4): u min = arg min E ( u ) = arg min u ∈U u ∈U  (u − u 0 ) 2 + αψ Ω ( ∇ u )dxdy 2 (5) The minimization result u min will correspond to the denoised (smoothed) image. The minimization process is performed by solving the following Euler-Lagrange equation [7,10,11]: ( ( u − u 0 − α div ψ ' ∇ u 2 )∇ u ) = 0 ⇔ u −α u 0 ( ( − div ψ ' ∇ u 2 )∇ u ) = 0 (6) Thus, we obtain the following PDE equation: ( ( ∂u = div ψ ' ∇ u ∂t 2 )∇ u )− u −α u 0 (7) where the positive function ψ ' is obtained by computing the derivative of the function given by (4) as follows: ψ ' (s ) = 2 ν βs 2 + γ + η k βs 2 + γ Therefore, the partial differential equation (7) becomes (8) 504 T. Barbu 2     ∂u = div  η β ∇ u + γ + α k ⋅ ∇ u  − u − u 0    ∂t 2 α   β γ u ∇ +        u (0, x , y ) = u 0  (9) One can demonstrate the PDE model given by (9) converges to a unique strong solution, that is u * = u min . We propose a robust discretization scheme for solving it, which is described in the next section. 3 Discretization Scheme for the PDE Model We consider a proper numerical approximation of the proposed PDE model’s solution. Thus, our discretization scheme uses a 4-NN discretization of the Laplacian operator [6]. From (7) we have ∂ u = div ψ ' ∇ u 2 ∇ u − u − u 0 , which leads to ∂t α the following relation: ( ( ) ) ( ( u ( x , y , t + 1 ) ≅ u ( x , y , t ) + div ψ ' ∇ u 2 )∇ u ) − u −α u 0 (10) One can approximate (10) using the image gradient magnitudes in particular directions, as following: u t +1 = u t + λ ψ q∈ N ( p ) ( ' ∇u p ,q (t ) 2 )∇ u p ,q (t ) − u − u0 α (11) where λ ∈ (0, 1) and t = 1, …, N. In the equation above N (p) represents the the 4-neighborhood of the argument pixel, described by its coordinates, p = (x, y). Obviously, it represents a set of image pixels given by their coordinates: N ( p ) = {( x − 1, y ), ( x + 1, y ), ( x , y − 1), ( x , y + 1) } Also, ∇ u p ,q (12) is the image gradient magnitude in the direction given by pixel q at iteration t, being computed as follows: ∇ u p ,q (t ) = u ( q , t ) − u ( p , t ) (13) A Novel Variational PDE Technique for Image Denoising 505 The maximum number of iterations, N, is empirically chosen. The proposed iterative denoising scheme applies the operation given by (11) for each t value, from 0 to N. Our noise removal technique produces the smoothed image u N from the noised image u 0 = u0 in a relatively small number of steps, being characterized by a quite low N value. That means, the PDE model developed here converges fast to the solution u N ≅ u min . The effectiveness of the proposed PDE denoising approach and its discretization is proved by the satisfactory image smoothing results obtained from our experiments. These numerical experiments are discussed in the next section of the article. 4 Experiments and Method Comparisons The described variational PDE denoising technique have been applied on numerous image datasets. We have performed numerous image smoothing experiments, using the proposed technique, on hundreds noisy images, and obtained very good results. Thus, the original images have been corrupted with various level of Gaussian noise (various values for mean and variance). Then, the denoising model have been applied to them with some properly chosen parameters which provide best results. These empirically detected parameter values are: α = 9 , k = 25 ,η = 0 .7 , β = 0 .66 , γ = 0 .5,ν = 0 .2 , λ = 0 .3, N = 15 (14) We assess the performance of our noise reduction method using the norm of the error image measure [8, 9]. Thus, if u orig represents the original (noise-free) form of the image, then the norm of the error image is computed as: NE ( u ) = X Y   (u x =1 y =1 N ( x , y ) − u orig ( x , y )) 2 (15) where [X ×Y ] is the image dimension. Our denoising techniques provides low enough values for this performance measure. From the performed method comparisons we have found that our variational technique outperforms other noise removal approaches. Thus, we have compared it with some other PDE-based methods and also with some non-PDE denoising algorithms. Our approach provides considerable better image denoising and edge-preserving results than non-PDE image filters, like Gaussian, average and median filters. It also achieves a better smoothing and, given its lower time complexity, converges faster than other variational schemes, such as the quadratic variational model, characterized by a regularizer ψ s 2 = s 2 , or the Perona-Malik variational scheme, given by ( ) 506 T. Barbu    s  ψ (s 2 ) = λ 2  log  1 + 2   [10]. Because of its low execution time, this method λ    2 can be used for denoising large image sets, like those of social networks [12]. Several image denoising results and method comparisons are described in the next figures and tables. In Fig.1, there are displayed: a) the original [512 × 512 ] Lena image; b) the image corrupted with Gaussian noise given by μ = 0.211 and var = 0.023; c) the image smoothed by our variational model; d) quadratic denoising; e) PeronaMalik noise removal; f) – i) denoising results achieved by the 2D Gaussian, average, median and Wiener [3× 3] filter kernels. The corresponding norm of the error values are displayed in Table 1. Fig. 1. Lena image denoised using various smoothing techniques A Novel Variational PDE Technique for Image Denoising 507 Table 1. Norm-of-the-error values for several noise removal techniques Our alg. 4 . 9 × 10 Quadratic 3 6 . 2 × 10 3 P-M 5 . 9 × 10 3 Gaussian Average 7 . 4 × 10 6 . 4 × 10 3 3 Median Wiener 6 × 10 5 . 6 × 10 3 3 In Fig. 2 the same denoising models are applied on the Baboon image, while the corresponding values of the NE measure are registered in Table 2. As one can see in Fig. 1 and Fig. 2, the variational approach proposed here provides the best edge-preserving image smoothing. The staircasing effect, representing creation in the image of flat regions separated by artifact boundaries [9, 13], is also removed by our denoising technique. Fig. 2. Baboon image denoised using various smoothing techniques Table 2. Norm-of-the-error values for several noise removal techniques Our alg. Quadratic 5 × 10 6 × 10 3 3 P-M 5 . 9 × 10 3 Gaussian Average 7 . 3 × 10 6 . 5 × 10 3 Median 3 6 . 1 × 10 Wiener 3 5 . 8 × 10 3 508 5 T. Barbu Conclusions We have proposed a variational PDE denoising approach in this paper. This technique performs an efficient noise removal and also preserves the boundaries of the image. The main original contribution of this article is the efficient smoothing component introduced in the energy functional of the variational model. It is based on a novel regularizer function. Also, we propose a robust discretization of the PDE model given by the corresponding Euler-Lagrange equation. Our developed variational technique reduces also the staircasing effects and converges fast to the solution represented by the denoised image. It also outperforms many other variational PDE methods and non-PDE denoising techniques [1,13], as resulting from the performed experiments and the method comparison. We intend to further investigate this variational scheme and to provide more mathematical treatment of it in the future. Thus, the demonstration of the convergence of our PDE model to a unique strong solution will be the subject of our future work in this domain. PDE-based color image denoising [14] will also be a next research field. References 1. Chan, T., Shen, J., Vese, L.: Variational PDE Models in Image Processing. Notices of the AMS 50(1) (2003) 2. Song, B.: Topics in Variational PDE Image Segmentation, Inpainting and Denoising, University of California (2003) 3. Ning, H.E., Ke, L.U.A.: A Non Local Feature-Preserving Strategy for Image Denoising. Chinese Journal of Electronics 21(4) (2012) 4. Jain, A.K.: Fundamentals of Digital Image Processing. Prentice Hall, NJ (1989) 5. Weickert, J.: Anisotropic Diffusion in Image Processing. European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart (1998) 6. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. In: Proc. of IEEE Computer Society Workshop on Computer Vision, pp. 16–22 (November 1987) 7. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60(1), 259–268 (1992) 8. Barbu, T., Barbu, V., Biga, V., Coca, D.: A PDE variational approach to image denoising and restorations. Nonlinear Anal. RWA 10, 1351–1361 (2009) 9. Barbu, T.: Variational Image Denoising Approach with Diffusion Porous Media Flow. Abstract and Applied Analysis 2013, Article ID 856876, 8 pages (2013) 10. Popuri, K.: Introduction to Variational Methods in Imaging. In: CRV 2010 Tutorial Day (2010) 11. Fox, C.: An introduction to the calculus of variations. Courier Dover Publications (1987) 12. Alboaie, L., Vaida, M.F.: Trust and Reputation Model for Various Online Communities. Studies in Informatics and Control 20(2), 143–156 (2011) 13. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005) 14. Kim, S.: PDE-based image restoration: A hybrid model and color image denoising. IEEE Transactions on Image Processing 15(5), 1163–1170 (2006) View publication stats