Em Erik Franken

Left-Invariant Diffusions on the Space of Positions and Orientations and their Application to Crossing-Preserving Smoothing of HARDI images

HARDI (High Angular Resolution Diffusion Imaging) is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. In this article we study left-invariant diffusion on the group of 3D rigid body movements (i.e. 3D Euclidean motion group) SE(3) and its application to crossing-preserving smoothing of HARDI images. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space of positions and orientations in 3D embedded in SE(3) and can be solved by ℝ3⋊S2-convolution with the corresponding Green’s functions. We provide analytic approximation formulas and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (nonlinear) convection-diffusions on HARDI data we explain the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan connection. This Cartan connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI data containing crossing-fibers.

Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part I: Linear left-invariant diffusion equations on SE(2)

We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R^2 x T. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Greens functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Greens functions (i.e., the heat kernels on SE(2)) of the left-invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the left-invariant vector fields on SE(2) by left-invariant generators of a Heisenberg group, and we derive suitable approximations of the Greens functions. The exact Greens functions are used in so-called collision distributions on SE(2), which are the product of two left-invariant resolvent diffusions given an initial distribution on SE(2). We use the left-invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.

Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores

Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherence-enhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence-enhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore, the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings.

An efficient method for tensor voting using steerable filters

In many image analysis applications there is a need to extract curves in noisy images. To achieve a more robust extraction, one can exploit correlations of oriented features over a spatial context in the image. Tensor voting is an existing technique to extract features in this way. In this paper, we present a new computational scheme for tensor voting on a dense field of rank-2 tensors. Using steerable filter theory, it is possible to rewrite the tensor voting operation as a linear combination of complex-valued convolutions. This approach has computational advantages since convolutions can be implemented efficiently. We provide speed measurements to indicate the gain in speed, and illustrate the use of steerable tensor voting on medical applications.

Nonlinear diffusion on the 2D Euclidean motion group

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.

Line Enhancement and Completion via Linear Left Invariant Scale Spaces on SE(2)

From an image we construct an invertible orientation score, which provides an overview of local orientations in an image. This orientation score is a function on the group SE (2) of both positions and orientations. It allows us to diffuse along multiple local line segments in an image. The transformation from image to orientation score amounts to convolutions with an oriented kernel rotated at multiple angles. Under conditions on the oriented kernel the transform between image and orientation score is unitary. This allows us to relate operators on images to operators on orientation scores in a robust way such that we can deal with crossing lines and orientation uncertainty. To obtain reasonable Euclidean invariant image processing the operator on the orientation score must be both left invariant and non-linear. Therefore we consider non-linear operators on orientation scores which amount to direct products of linear left-invariant scale spaces on SE (2). These linear left-invariant scale spaces correspond to well-known stochastic processes on SE (2) for line completion and line enhancement and are given by group convolution with the corresponding Greens functions. We provide the exact Greens functions and approximations, which we use together with invertible orientation scores for automatic line enhancement and completion.

From stochastic completion fields to tensor voting

Several image processing algorithms imitate the lateral interaction of neurons in the visual striate cortex V1 to account for the correlations along contours and lines. Here we focus on two methodologies: tensor voting by Guy and Medioni, and stochastic completion fields by Mumford, Williams and Jacobs. The objective of this article is to compare these two methods and to place them into a common mathematical framework. As a consequence we obtain a sound stochastic foundation of tensor voting, a new tensor voting field, and an analytic approximation of the stochastic completion kernel.

Scale Spaces on the 3D Euclidean Motion Group for Enhancement of HARDI Data

In previous work we studied left-invariant diffusion on the 2D Euclidean motion group for crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study the equivalent three-dimensional case. This is particularly useful for processing High Angular Resolution Diffusion Imaging (HARDI) data, which can be considered as 3D orientation scores directly. A complicating factor in 3D is that all practical 3D orientation scores are functions on a coset space of the 3D Euclidean motion group instead of on the entire group. We show that, conceptually, we can still apply operations on the entire group by requiring the operations to be ***-right-invariant . Subsequently, we propose to describe the local structure of the 3D orientation score using left-invariant derivatives and we smooth 3D orientation scores using left-invariant diffusion. Finally, we show a number of results for linear diffusion on artificial HARDI data.

Curvature Estimation for Enhancement of Crossing Curves

In this paper we describe a method for estimating curvature of elongated structures in images. The curvature estimation is performed on an invertible orientation score, which is a 3D entity obtained from a 2D image by convolution with a rotating kernel. By considering the group structure we can define left-invariant derivatives, which are essential to construct operations on the orientation score that amount to rotationally invariant operations on the corresponding image. The problem of estimating curvature of an oriented structure is stated as a minimization problem, which can be solved by eigenvector analysis of a matrix constructed from the non-symmetric Hessian matrix. The experiments show the method performs well for a wide range of curvatures and noise levels. The method clearly outperforms a related curvature estimation method by Van Ginkel et al. that tends to give estimates that are too small. We show how we can incorporate the curvature estimate in our method for coherence-enhancing diffusion in orientation scores. This method has superior performance in enhancing crossing contours, which is demonstrated on medical images.