Chapter

Scale Space and Variational Methods in Computer Vision

Volume 4485 of the series Lecture Notes in Computer Science pp 461-472

Nonlinear Diffusion on the 2D Euclidean Motion Group

  • Erik FrankenAffiliated withEindhoven University of Technology, Dept. of Biomedical Engineering
  • , Remco DuitsAffiliated withEindhoven University of Technology, Dept. of Biomedical Engineering
  • , Bart ter Haar RomenyAffiliated withEindhoven University of Technology, Dept. of Biomedical Engineering

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Abstract

Linear and nonlinear diffusion equations are usually considered on an image, which is in fact a function on the translation group. In this paper we study diffusion on orientation scores, i.e. on functions on the Euclidean motion group SE(2). An orientation score is obtained from an image by a linear invertible transformation. The goal is to enhance elongated structures by applying nonlinear left-invariant diffusion on the orientation score of the image. For this purpose we describe how we can use Gaussian derivatives to obtain regularized left-invariant derivatives that obey the non-commutative structure of the Lie algebra of SE(2). The Hessian constructed with these derivatives is used to estimate local curvature and orientation strength and the diffusion is made nonlinearly dependent on these measures. We propose an explicit finite difference scheme to apply the nonlinear diffusion on orientation scores. The experiments show that preservation of crossing structures is the main advantage compared to approaches such as coherence enhancing diffusion.