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A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient

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Abstract

We propose a nonlinear partial differential equation (PDE) for regularizing a tensor which contains the first derivative information of an image such as strength of edges and a direction of the gradient of the image. Unlike a typical diffusivity matrix which consists of derivatives of a tensor data, we propose a diffusivity matrix which consists of the tensor data itself, i.e., derivatives of an image. This allows directional smoothing for the tensor along edges which are not in the tensor but in the image. That is, a tensor in the proposed PDE is diffused fast along edges of an image but slowly across them. Since we have a regularized tensor which properly represents the first derivative information of an image, the tensor is useful to improve the quality of image denoising, image enhancement, corner detection, and ramp preserving denoising. We also prove the uniqueness and existence of solution to the proposed PDE.

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physics and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, Singapore

    Jooyoung Hahn

  2. Department of Mathematical Sciences, KAIST, Daejeon, South Korea

    Chang-Ock Lee

Authors
  1. Jooyoung Hahn
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  2. Chang-Ock Lee
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Corresponding author

Correspondence to Jooyoung Hahn.

Additional information

This work was supported by KRF-2006-311-C00015. The research is supported by MOE (Ministry of Education) Tier II project T207N2202 and IDM project NRF2007IDMIDM002-010. In addition, support from SUG 20/07 is also gratefully acknowledged.

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Hahn, J., Lee, CO. A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient. J Math Imaging Vis 34, 137–151 (2009). https://doi.org/10.1007/s10851-009-0138-1

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  • DOI: https://doi.org/10.1007/s10851-009-0138-1

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