Nonlinear Diffusion on the 2D Euclidean Motion Group

Abstract

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.

The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group

“…For linear and non-linear stochastic processes for contour enhancement and their applications in image processing, see [27], [14], [12]. In this article we show that the solutions of all left-invariant linear evolution equations are given by convolution with the corresponding Green's function, which we explicitly derive.…”

“…, so we are considering only diffusion, where D ij may even depend on U , we refer to [27], [12]. Furthermore in Section 5 we discuss an efficient method to compute the exact Green's functions in all cases (with periodic boundary conditions), where we explicitly put the connection with the exact solutions (with periodic boundary conditions) in the special cases above.…”

“…The result of such an image sampling is a function U f ∈ L 2 (G) on the Euclidean motion group manifold G = R 2 T, which is and contour completion (right) via left-invariant evolution equations on invertible orientation scores. For details on (medical) image analysis applications, see [27], [15], [47], [26] and [19].…”

Curvature Estimation for Enhancement of Crossing Curves

“…For some choices of K there exists a stable inverse transformation [2], which is obtained by either convolving U (·, θ) with the mirrored conjugate kernel of K followed by integration over θ, or simply by f = 2π 0 U (x, θ)dθ. Our choice for K, which has the reconstruction property using the second method, is (see [5])…”

“…To circumvent the problem of crossing structures we propose a method for coherence enhancing diffusion in orientation scores in [5]. In an orientation score (sometimes called orientation space) the local orientation information in the * The project was financially supported by the Dutch BSIK program entitled Molecular Imaging of Ischemic heart disease (project number BSIK 03033).…”