Oriented half Gaussian kernels and anisotropic diffusion Baptiste Magnier, Philippe Montesinos To cite this version: Baptiste Magnier, Philippe Montesinos. Oriented half Gaussian kernels and anisotropic diffusion. 9th International Conference on Computer Vision Theory and Applications, VISAPP 2014; Lisbon; Portugal; 5 January 2014 through 8 January 2014; Code 107286, Jan 2014, Lisbonne, Portugal. pp.2945-2956. hal-01940388 HAL Id: hal-01940388 https://hal.archives-ouvertes.fr/hal-01940388 Submitted on 30 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Oriented Half Gaussian Kernels and Anisotropic Diffusion Baptiste Magnier and Philippe Montesinos Ecole des Mines dALES, LGI2P, Parc Scientifique G.Besse, 30035 Nˆımes Cedex fbaptiste.magnier, [email protected] Keywords: Half anisotropic Gaussian kernel, diffusion PDEs. Abstract: Nonlinear PDEs (partial differential equations) offer a convenient formal framework for image regularization and are at the origin of several efficient algorithms. In this paper, we present a new approach which is based (i) on a set of half Gaussian kernel filters, and (ii) a nonlinear anisotropic PDE diffusion. On one hand, half Gaussian kernels provide oriented filters whose flexibility enables to detect edges with great accuracy. On the other hand, a nonlinear anisotropic diffusion scheme offers a means to smooth images while preserving fine structures or details, e.g. lines, corners and junctions. Based on the calculus of the gradient magnitude and two diffusion directions, we construct a diffusion control function able to achieve precise image regulariza- tion. Some quantified experimental results compared to existing PDEs approaches and a discussion about the parameterizing of the method are presented. 1 A Framework of Anisotropic It should be noted that Koenderink (Koenderink, Diffusion with PDE 1984) was the first to underline the equivalence be- tween the convolutionp with a Gaussian kernel of stan- Obtain regularized versions of noisy, corrupted or dard deviation 2t and the solution of the PDE de- degraded images caused for example by compression scribing the heat diffusion, at a time t. This smooth- artifacts is a difficult task in image processing. How- ing process, called isotropic diffusion, is known to ever, preserving significant internal structures is a smooth noise but blur edges, leading to loose image field that has largely benefited from techniques of Par- structures. In order to regularize images by control- tial Differential Equations (PDE) (Aubert and Korn- ling the diffusion, Perona and Malik (Perona and Ma- probst, 2006; Magnier and Montesinos, 2013). PDEs lik, 1990) have proposed a model described by the belong to one of the most important part of mathe- following equation: matical analysis and are closely related to the physi- ¶I (x;y;t) = div(g(k∇Ik) · k∇Ik) (2) cal world. In this context, images are considered as ¶t evolving functions of time and a regularized image where div represents the divergence operator and can be seen as a version of the original image at a g(s) : [0;+¥[!]0;+¥[ a decreasing function satisfy- special stage. Thereby, PDEs methods smooth locally ing g(0) = 1 and g(+¥) = 0, this function could be the image following one or several directions which 2 − k∇Ik are different in each point of the image. In this paper, chosen as g(k∇Ik) = e K , with K 2 R a con- let us note I : W ! R; (W ⊂ R2) a grey level image stant that can be assimilated to a gradient threshold or with I(x;y) corresponding to the pixel intensity of co- a diffusion barrier. ordinates (x;y). Considering I0 the original image, The decomposition of the eq. 2 with the second the general evolution model can be formally written derivatives of I in orthogonal directions (x ? h) re- in the following form: spectively in the edge direction called x and in the I ¶I (x;y;t) = F (I(x;y;t)) gradient direction labelled h = ∇ enables to under- ¶t (1) k∇Ik I(x;y;0) = I0(x;y) stand the diffusion behavior (Kornprobst et al., 1997): where F is a given image processing algorithm, pre- ¶I (x;y;t) = c · I + c · I (3) serving edges having high gradient. F represents a ¶t x xx h hh function of the original image I0 and its first and 2 2 where (I ;I ) = ¶ I ; ¶ I , c and c are coeffi- second order spatial derivatives (Caselles and Morel, xx hh ¶x2 ¶h2 x h 1998). cients tuning the diffusion (diagrammed in Fig. 1(a)). When cxx = chh = 1, the eq. 3 is equivalent to the heat equation (Koenderink, 1984). Choosing cx = g(k∇Ik), a gradient function and ch = g(k∇Ik) + 0 k∇Ik·g (k∇Ik), the diffusion process described in eq. Object 3 can be interpreted as two directional heat flows1 with different diffusion intensities in the h and x di- rections to preserve discontinuities: • Inside homogeneous regions, the gradient mag- (a) (b) nitude k∇Ik is small and the diffusion becomes Figure 1: Diagrams of edge diffusion. (a) An image contour isotropic. and its moving vector basis (x;h) and diffusion representa- tion with ellipsoids. The more the gradient is high, the more • On edges, the diffusion becomes anisotropic, be- the ellipse is elongated. Gradient and tangential direction ing attenuated by the function g, and is inhibited denoted (h;x) and diffusion representation with ellipsoids, when the two coefficients (cxx;chh) tend to zero. note that ellipsoids are not always oriented in the x direction Diffusion control is done with finite differences so using tensorial methods. (b) Our desired diffusion represen- tation with half ellipsoids, the more the edge is sharped and that many contours of details are preserved. However, the angle is acute, the more the half ellipses are thin. within images corrupted by a heavy noise, generally, this noise is not totally removed because the diffusion process is inhibited. ing the divergence (Weickert, 1999) or the trace Gaussian filtering for gradient estimation has been (Tschumperle´ and Deriche, 2005), the smoothing used in a number of works to elaborate the model pre- along a contour in inversely proportional to the con- sented in eq. 3 less sensitive to noise and more sta- tour strength in the direction of the eigenvector asso- ble. We can mention here the approach of Alvarez ciated to the higher eigenvalue. Inside homogeneous et al. (Alvarez et al., 1992) which induces for each regions, eigenvalues are close to zero and the diffu- pixel either an adaptive unidirectional tangential dif- sion becomes isotropic. fusion I at level of edges or an efficient isotropic xx As demonstrated in (Tschumperle´ and Deriche, smoothing for noise removal inside homogeneous re- 2005), trace based PDE is best suited to understand gions. Nevertheless, this smoothing model does not the local smoothing geometry behavior and these dif- allow a progressive diffusion in the gradient direc- fusion scheme ensure coherence smoothing direc- tion h because it depends on two diffusion barriers. tions but the Gaussian behavior on curved struc- Consequently, in the presence of a high noise, even in tures or corners results in a ”mean curvature flow homogeneous regions, this diffusion scheme behaves effect” leading to round small structures or corners. like the Mean Curvature Motion (Catte´ et al., 1992) In order to compensate this drawback, the author of (MCM) method which consists in performing the dif- (Tschumperle,´ 2006) proposed a curvature-preserving fusion only along the tangential direction x or along smoothing PDE that diffuses the image I along a field isophote lines. Although the MCM scheme regular- of vectors w issued by the eigenvectors of J . Despite izes the image in edge directions, this approach tends r the fact that the author of this method (Tschumperle,´ to round corners after a certain number of iterations 2006) has demonstrated that it better preserves cor- and can create stripes inside noisy homogeneous re- ners and small details in the image, as the other ten- gions. sorial approaches, when the anisotropic coefficient is Instead of considering only the gradient mag- too large, the diffusion of a high noise brings a fiber nitude to drive the diffusion, tensorial approaches effect in homogeneous regions. To avoid this unde- (Weickert, 1999; Tschumperle´ and Deriche, 2005; sired diffusion effect, it is preferable to use a higher Tschumperle,´ 2006) contribute to another image dif- standard deviation of the Gaussian s, however this fusion formalism. From a structure tensor Jr = T leads to delocate even so the corners, diffuse small Gr ∗ ∇Is∇Is , where Gr denotes a Gaussian ker- objects and also blur edges. nel of standard deviation r, authors of (Weickert, 1999; Tschumperle´ and Deriche, 2005; Tschumperle,´ In this paper, we propose a new PDE scheme 2006) elaborate a tensor field which specifies the that regularizes images considering two contour di- local smoothing geometry defined from the spec- rections. This diffusion process correctly preserves tral elements of the structure tensor. Then, us- corners as well as small objects and becomes isotropic inside homogeneous regions without generating un- 1Note that if k∇Ik> pK , then c <0 and the diffusion desired fiber effect of artifacts. Thanks to a rotat- 2 h equation behaves locally like an inverse diffusion equation ing Gaussian derivative half-filter, we extract a gra- which is an unstable process enhancing features.