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Image Denoising Using Non Linear Diffusion Tensors 2011 8th International Multi-Conference on Systems, Signals & Devices IMAGE DENOISING USING NON LINEAR DIFFUSION TENSORS Benzarti Faouzi, Hamid Amiri Signal, Image Processing and Pattern Recognition: TSIRF (ENIT) Email: [email protected] , [email protected] ABSTRACT linear models. Many approaches have been proposed to remove the noise effectively while preserving the original Image denoising is an important pre-processing step for image details and features as much as possible. In the past many image analysis and computer vision system. It few years, the use of non linear PDEs methods involving refers to the task of recovering a good estimate of the true anisotropic diffusion has significantly grown and image from a degraded observation without altering and becomes an important tool in contemporary image changing useful structure in the image such as processing. The key idea behind the anisotropic diffusion discontinuities and edges. In this paper, we propose a new is to incorporate an adaptative smoothness constraint in approach for image denoising based on the combination the denoising process. That is, the smooth is encouraged of two non linear diffusion tensors. One allows diffusion in a homogeneous region and discourage across along the orientation of greatest coherences, while the boundaries, in order to preserve the natural edge of the other allows diffusion along orthogonal directions. The image. One of the most successful tools for image idea is to track perfectly the local geometry of the denoising is the Total Variation (TV) model [9] [8][10] degraded image and applying anisotropic diffusion and the anisotropic smoothing model [1] which has since mainly along the preferred structure direction. To been expanded and improved upon [3] [5][20]. Over the illustrate the effective performance of our model, we years, other very interesting denoising methods have been present some experimental results on a test and real emerged such as: Bilateral filter and its derivatives photographic color images. [6][11][12]. In our work, we address image denoising problem by using the so-called structure tensors [14] Index Terms— Image denoising, Non-linear which have proven their effectiveness in several areas diffusion, Structure tensors, PDEs. such as: texture segmentation [17], motion analysis [18] and corner detection [16][19][21]. The structure tensor provides a more powerful description of local pattern images better than a simple gradient. Based on its 1. INTRODUCTION eigenvalues and the corresponding eigenvectors, the tensor summarizes the predominant directions of the Image denoising has been one of the most important and gradient in a specified neighborhood of a point and the widely studied problems in image processing and degree to which those directions are coherent. Our computer vision. The need to have a very good image contribution in this work lies in the use of two coupled quality is increasingly required with the advent of the diffusion tensors which allow a complete and coherence new technologies in a various areas such as multimedia, regularization process which significantly improves the medical image analysis, aerospace, video systems and quality image. others. Indeed, the acquired image is often marred by This paper is organized as follows. In Section 2, we noise which may have a multiple origins such as: thermal introduce the non linear diffusion PDEs and discuss the fluctuations; quantify effects and properties of various options that have been implemented for the communication channels. It affects the perceptual quality anisotropic diffusion. In Section 3, we present the non of the image, decreasing not only the appreciation of the linear structure tensor formalism and its mathematical image but also the performance of the task for which the concept. Section 4 focuses on our proposed denoising image has been intended. The challenge is to design approach. Numerical experiments and results on test and methods, which can selectively smooth a degraded image real photographic images are shown in Section 5. without altering edges, losing significant features and producing reliable results. Traditionally, linear models 2. NON LINEAR DIFFUSION PDE: OVERVIEW have been commonly used to reduce noise. It is shown that these methods perform well in the flat regions of In this section, we review some basic mathematical images but do not preserve edges and discontinuities concepts of the nonlinear diffusion PDEs proposed by which are often smeared out. In contrast, nonlinear Perona and Malik [1]. Let be the grayscaled intensity image with a diffusion time t, for the models can handle edges in a much better way than those ͩ ʚͬ, ͭ, ͨʛ: Ŵ ͌ 978-1-4577-0411-6/11/$26.00 ©2011 IEEE image domain . The nonlinear PDE equation is An extension of the nonlinear diffusion filtering to given by: ͦ vector-valued image (e.g. color image) has been proposed Ǩ ͌ [20][3]. It evolves: under the diffusion equations: ) x (1) Ŵ ͌ (3) "/ͩ Ɣ ͪ͘͝ ʚ͛ʚ|ǧͩ|ʛǧͩʛ ͣ͢(e.g. Initialʚ0, ∞ʛCondition) ) ͦ ͩʚͬ, ͭ, 0ʛ Ɣ ͩͤ ͣ͢ (e.g. reflecting boundary) "/ͩ$ Ɣ ͪ͘͝ ʚ ͛ʚ∑&Ͱͥ |ǧͩ&| ʛǧͩ$ʛ ͝ Ɣ 1. ͢ Where :denotes the i th component channels of "ͩ) Ɣ 0 ͣ͢ " x ʚ0, ∞ʛ where denotes the first derivative regarding the Note thatͩ$ , represents the luminanceͩ function, which) coupled ͦall vectors channels taking the diffusion"/ timeͩ: t; : denotes the gradient modulus and ∑&Ͱͥ |ǧͩ&| g( ) is a non-increasing function, known as the diffusivity strong correlations among channels . However, this |ǧͩ| function which allow isotropic diffusion in flat regions luminance function is not being able to detect iso- and no diffusion near edges. luminance contours. By developing the divergence term of (1), we obtain: 3. NON LINEAR DIFFUSION TENSOR ɒ (2) ɑɑ " ʚ|O0 |ʛ The non linear diffusion PDE seen previously does not "/ͩ Ɣ ͛ ʚ|ͩ |ʛͩVV ƍ | O0 | ͩ ]] give reliable information in the presence of flow-like Ɣ ͗VͩVV ƍ ͗]ͩ ]] structures (e.g. fingerprints). It would be desirable to where: and are respectively rotate the flow towards the orientation of interesting ʹ ʹ features. This can be easily achieved by using the the secondͩVV spatialƔ ͂ derivatives ͩ ]] ofƔ in͂ the directions of the gradient , and its orthogonal ; H structure tensor, also referred to the second moment ͩ denotes the Hessian of u. According to these definʹitions, matrix. For a multivalued image, the structure tensor has Ɣ ǧͩ/|ǧͩ| Ɣ on the image discontinuities, we have the diffusion along the following form: (normal to the edge) weighted with and Μ ͦ Μ (4) ɑɑ ΗͰͥ ΗΦa ΗͰͥ ΗΦa ΗΧa a diffusion along (tangential to theV edge) weighted ) ͳ ∑ u ∑ u u ͗ Ɣ ͛ ʚ|ͩ |ʛ ͍a ƔƳ∑$Ͱͥ ǧͩ$a ǧͩ$a Ʒ Ɣ ʨ Μ Μ ͦ ʩ with . To understand the principle of ∑ΗͰͥ uΗΦauΗΧa ∑ΗͰͥ uΗΧa ɑ With : the smoothed the anisotropic ͗] Ɣ ͛ ʚ|ͩ |ʛdiffusion,/|ǧͩ| let represent a contour C (figure 1) separating two homogeneous regions of the version ǧͩof$a Ɣthe ͅ a dzgradient ǧͩ$ Ɣ ͅawhichdz ʚuix , uisiy ʛ obtained by a image, the isophote lines (e.g. level curves of equal gray- convolution with a Gaussian kernel . The structure levels) correspond to u(x,y) = c. In this case, the vector scale determines the size of the resultingͅa flow-like patterns. Increasing gives an increased distance is normal to the contour C, the set is then a moving orthonormal basis whose configuration depends on the between the resulting flow lines. ʚ, ʛ current coordinate point (x, y). In the neighborhood of a These new gradient features allow a more precise contour C, the image presents a strong gradient. To better description of the local gradient characteristics. However, preserve these discontinuities, it is preferable to diffuse it is more convenient to use a smoothed version of , only in the direction parallel to C (i.e. in the -direction). that is: ͍a In this case, we have to inhibit the coefficient of (5) ͥͥ͞ ͥͦ͞ (e.g. , and to suppose that the coefficient of VV ̈́` Ɣ ͅ` dz ͍a ƔƬ ư ͩ ͦͥ͞ ͦͦ͞ does not͗V Ɣ vanish. 0ʛ ͩ]] Where : a Gaussian kernel with standard deviation . The integrationͅ` scale averages orientation information. Therefore, it helps to stabilize the directional behavior of u(x,y)>c η the filter. In particular, it is possible to close interrupted (x o,y o) lines if is equal or larger than the gap size. In order to (x 1 , y1) ξ enhance coherent structures, the integration scale u(x,y)<c should be larger than the structure scale ( u(x,y)=c .In summary, the convolution with the Gaussian ͙ͬ: Ɣ 3ʛkernels , make the structure tensor measure more Figure 1 Image contour and its moving orthonormal basis coherent. ͅ aTo, ͅ `go further into the formalism, the structure tensor can be written over its eigenvalues ( , ) ʚ, ʛ So it appears the following conditions for the choice of and eigenvectors̈́` , that is: ͮ ͯ the g( ) functions to be edge preserving : ʚͮ, ͯʛ : avoids inverse diffusion λ θ ɑɑ ɑ θ θ = (6) ͛ ʚ0ʛ ƚ 0 ͕͘͢ ͛ ʚ0ʛ ƚ 0 : allows ͮ λ0 θͮ ̈́` Ɣ ʚ ͮ ͯʛ Ƭ ư Ƭ ư ͮͮͮ ƍͯͯͯ Ǝ isotropic lim |ǧ0| Ŵͤdiffusion lim for| ǧ0low|Ŵͤ gradient. ͙͗ͨ Ƙ 0 0 ͯ ͯ ͗ Ɣ ͗ Ɣ The eigenvectors of give the preferred local ͗ orientations, and the corresponding̈́` eigenvalues denote allowsƎ |ǧ0lim| ŴϦanisotropic͗ Ɣ |ǧ0lim |ŴϦdiffusion͗ Ɣ 0to ͕͘͢ preserve|ǧ0lim|ŴϦ discontinu͗ Ɣ 0: ities the local contrast along these directions. for the high gradient. The eigenvalues of are given by : ̈́` (7) In the coherence enhancing diffusion (CED) proposed by ͥ ͦ ͦ Weiker [4], the eigenvalues are assembled via: ͮ Ɣ ͦ ʚjͥͥ ƍ jͦͦ ƍ ǭʚjͥͥ Ǝ jͦͦ ʛ ƍ 4jͥͦ ʛ (8) ͥ ͦ ͦ (12) And the eigenvectorsͯ ͥͥ ( ͦͦ satisfyͥͥ :ͦͦ ͥͦ Ɣ ͦ ʚj ƍ j Ǝ ǭʚj Ǝ j ʛ ƍ 4j ʛ ͥ Ɣ ͗ͥ = ͮ, ͯʛ ͗ͥ ͚͝ ͮ Ɣ ͯ ͦ v ƥ ͥ ͥ v Ʃ ͦ%uv ͗ ƍ ʚ1 Ǝ ͗ ʛ exp ʠƎ ʚz~ͯzʛ ʡ ͙͙ͧ͠ v v v v ǯʚ%vv ͯ%uu ͮǭʚ%uu ͯ%vv ʛ ͮͨ%uv ʛ ͮͨ%uv (9) Where and .
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