50Publications

3,012Citation Statements Received

3,556Citation Statements Given

How they've been cited

2,100

32

2,976

4

How they cite others

1,750

34

3,519

3

Publications

Order By: Most citations

Abstract. 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 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 Green's 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 Green's 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 Green's functions. The exact Green's 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.

Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherenceenhancing 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 coherenceenhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore,

This paper presents a robust and fully automatic filter-based approach for retinal vessel segmentation. We propose new filters based on 3D rotating frames in so-called orientation scores, which are functions on the Lie-group domain of positions and orientations [Formula: see text]. By means of a wavelet-type transform, a 2D image is lifted to a 3D orientation score, where elongated structures are disentangled into their corresponding orientation planes. In the lifted domain [Formula: see text], vessels are enhanced by means of multi-scale second-order Gaussian derivatives perpendicular to the line structures. More precisely, we use a left-invariant rotating derivative (LID) frame, and a locally adaptive derivative (LAD) frame. The LAD is adaptive to the local line structures and is found by eigensystem analysis of the left-invariant Hessian matrix (computed with the LID). After multi-scale filtering via the LID or LAD in the orientation score domain, the results are projected back to the 2D image plane giving us the enhanced vessels. Then a binary segmentation is obtained through thresholding. The proposed methods are validated on six retinal image datasets with different image types, on which competitive segmentation performances are achieved. In particular, the proposed algorithm of applying the LAD filter on orientation scores (LAD-OS) outperforms most of the state-of-the-art methods. The LAD-OS is capable of dealing with typically difficult cases like crossings, central arterial reflex, closely parallel and tiny vessels. The high computational speed of the proposed methods allows processing of large datasets in a screening setting.

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 R 3 S 2 -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 leftinvariant 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 leftinvariant diffusion takes place. We provide experiments of R. Duits ( )

Abstract. By means of a special type of wavelet unitary transform we construct an orientation score from a grey-value image. This orientation score is a complex-valued function on the 2D Euclidean motion group SE(2) and gives us explicit information on the presence of local orientations in an image. As the transform between image and orientation score is unitary we can relate operators on images to operators on orientation scores in a robust manner. Here we consider nonlinear adaptive diffusion equations on these invertible orientation scores. These nonlinear diffusion equations lead to clear improvements of the celebrated standard "coherence enhancing diffusion" equations on images as they can enhance images with crossing contours. Here we employ differential geometry on SE(2) to align the diffusion with optimized local coordinate systems attached to an orientation score, allowing us to include local features such as adaptive curvature in our diffusions.

Abstract. We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green's functions which we derive in explicit form. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant basis of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green's functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, second order stochastic evolution equations.

£ he xetherlnds yrgniztion for ienti eserh is grtefully knowledged for nnil supportF I P R. Duits, M. Felsberg, G.H. Granlund , B.M. ter Haar Romeny Abstract snspired y the erly visul system of mny mmmlins we onsider the onstrution ofE nd reonstrution fromE n orienttion sore U f X R P ¢ S I 3 C s lol orienttion representtion of n imgeD f X R P 3 RF he mpping f U 3 U f is wvelet trnsform orresponding to reduile representtion of the iuliden motion group onto L P @R P A nd oriented wvelet P L P @R P AF his wvelet trnsform is speil se of reently developed generliztion of the stndrd wvelet theory nd hs the prtil dvntge over the usul wvelet pprohes in imge nlysis @onstruted y irreduile representtions of the similitude groupA tht it llows stle reonstrution from one @single sleA orienttion soreF ine our wvelet trnsform is unitry mpping with stle inverseD we diretly relte opertions on orienttion sores to opertions on imges in roust mnnerF purthermoreD y geometril exmintion of the iuliden motion group G a R P o TD whih is the domin of our orienttion soresD we dedue tht n opertor¨on orienE ttion sores must e left invrint to ensure tht the orresponding opertor I ¨ on imges is iuliden invrintF es n exmple we onsider ll liner seond order left invrint evolutions on orienttion sores orresponding to stohsti proesses on GF es n pplition we detet elongted strutures in @medilA imges nd utomtilly lose the gps etween themF pinllyD we onsider roust orienttion estimtes y mens of hnnel representtionsD where we omine roust orienttion estimtion nd lerning of wvelets resulting in n utoEssoitive proessing of orienttion feturesF rere liner verging of the hnnel repE resenttion is equivlent to roust orienttion estimtion nd n dpttion of the wvelet to the sttistis of the onsidered imge lss leds to n utoEssoitive ehvior of the systemF

This paper presents a method for retinal vasculature extraction based on biologically inspired multi-orientation analysis. We apply multi-orientation analysis via so-called invertible orientation scores, modeling the cortical columns in the visual system of higher mammals. This allows us to generically deal with many hitherto complex problems inherent to vessel tracking, such as crossings, bifurcations, parallel vessels, vessels of varying widths and vessels with high curvature. Our approach applies tracking in invertible orientation scores via a novel geometrical principle for curve optimization in the Euclidean motion group SE(2). The method runs fully automatically and provides a detailed model of the retinal vasculature, which is crucial as a sound basis for further quantitative analysis of the retina, especially in screening applications.

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

hi@scite.ai

334 Leonard St

Brooklyn, NY 11211

Copyright © 2024 scite LLC. All rights reserved.

Made with 💙 for researchers

Part of the Research Solutions Family.