tschumperle-deriche_03 - Vector-Valued Image Regularization...

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Vector-Valued Image Regularization with PDE’s : A Common Framework for Different Applications D. Tschumperl´e R. Deriche INRIA, Odyss´ee Lab, 2004 Rte des Lucioles, BP 93, 06902 Sophia-Antipolis, France. { David.Tschumperle,Rachid.Deriche } @sophia.inria.fr Abstract 1 We address the problem of vector-valued image regu- larization with variational methods and PDE’s. From the study of existing formalisms, we propose a unifying frame- work based on a very local interpretation of the regulariza- tion processes. The resulting equations are then specialized into new regularization PDE’s and corresponding numeri- cal schemes that respect the local geometry of vector-valued images. They are finally applied on a wide variety of image processing problems, including color image restoration, in- painting, magnification and flow visualization. 1. Anisotropic regularization PDE’s raise a strong interest in the field of image processing. The ability to smooth data while preserving large global features such as contours and corners (discontinuities), has opened new ways to handle classical image-related issues (restoration, segmentation). Thus, many regularization schemes have been presented so far in the literature, particularly for the case of 2 D scalar images I : Ω R 2 R ( [1, 17, 18, 28] and references therein). Extensions of these algorithms to vector-valued images I : Ω R n have been recently proposed, leading to more elaborated diffusion PDE’s : a coupling between image channels appears in the equations, through the con- sideration of a local vector geometry , given pointwise by the spectral elements λ + , λ - (positive eigenvalues) and θ + , θ - (orthogonal eigenvectors) of the 2 × 2 symmetric and semi positive-definite matrix G = n j =1 I j I T j (also called structure tensor [25, 26, 28, 29]). The λ ± respectively de- fine the local min/max vector-valued variations of I in cor- responding spatial directions θ ± , i.e. the local configura- tion of the image discontinuities. (note that λ + = k∇ I k and θ + = I/ k∇ I k for scalar images , n = 1 ). Regulariza- tion schemes generally lie on one of these three following approaches, related to different interpretation levels : (1) Functional minimization : Regularizing an image I may be seen as the minimization of a functional E ( I ) mea- suring a global image variation. The idea is that minimizing this variation will flatten the image, then remove the noise : min I R n E ( I ) = R Ω φ ( N ( I )) d Ω (1) 1 This article was published in CVPR2003 - IEEE Conference on Com- puter Vision and Pattern Recognition , Madison/USA June 2003 where φ : R R is an increasing function and N ( I ) is a norm related to local image variations , for instance N ( I ) = p λ + + λ - = trace ( G ) 1 2 . The minimization of (1) is performed with a gradient descent (PDE) given by the Euler-Lagrange equations of E ( I ) . Useful references for
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tschumperle-deriche_03 - Vector-Valued Image Regularization...

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