Roto-translation equivariant convolutional networks: Application to histopathology image analysis.

Rotation-invariance is a desired property of machine-learning models for medical image analysis and in particular for computational pathology applications. We propose a framework to encode the geometric structure of the special Euclidean motion group SE(2) in convolutional networks to yield translation and rotation equivariance via the introduction of SE(2)-group convolution layers. This structure enables models to learn feature representations with a discretized orientation dimension that guarantees that their outputs are invariant under a discrete set of rotations.

Lesion Segmentation in Ultrasound Using Semi-Pixel-Wise Cycle Generative Adversarial Nets.

Breast cancer is the most common invasive cancer with the highest cancer occurrence in females. Handheld ultrasound is one of the most efficient ways to identify and diagnose the breast cancer. The area and the shape information of a lesion is very helpful for clinicians to make diagnostic decisions.

Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs.

Fokker-Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on S E ( 2 ) . Here, we extend these approaches to 3D using Fourier transform on the Lie group S E ( 3 ) of rigid body motions.

Optimal Paths for Variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis.

We present a PDE-based approach for finding optimal paths for the Reeds-Shepp car. In our model we minimize a (data-driven) functional involving both curvature and length penalization, with several generalizations. Our approach encompasses the two- and three-dimensional variants of this model, state-dependent costs, and moreover, the possibility of removing the reverse gear of the vehicle.

Design and Processing of Invertible Orientation Scores of 3D Images.

The enhancement and detection of elongated structures in noisy image data are relevant for many biomedical imaging applications. To handle complex crossing structures in 2D images, 2D orientation scores were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores .

Stability metrics for optic radiation tractography: Towards damage prediction after resective surgery.

Background: An accurate delineation of the optic radiation (OR) using diffusion MR tractography may reduce the risk of a visual field deficit after temporal lobe resection. However, tractography is prone to generate spurious streamlines, which deviate strongly from neighboring streamlines and hinder a reliable distance measurement between the temporal pole and the Meyer's loop (ML-TP distance).

New Method: Stability metrics are introduced for the automated removal of spurious streamlines near the Meyer's loop.

Retrieving challenging vessel connections in retinal images by line co-occurrence statistics.

Natural images contain often curvilinear structures, which might be disconnected, or partly occluded. Recovering the missing connection of disconnected structures is an open issue and needs appropriate geometric reasoning. We propose to find line co-occurrence statistics from the centerlines of blood vessels in retinal images and show its remarkable similarity to a well-known probabilistic model for the connectivity pattern in the primary visual cortex.

Template Matching via Densities on the Roto-Translation Group.

We propose a template matching method for the detection of 2D image objects that are characterized by orientation patterns. Our method is based on data representations via orientation scores, which are functions on the space of positions and orientations, and which are obtained via a wavelet-type transform. This new representation allows us to detect orientation patterns in an intuitive and direct way, namely via cross-correlations.

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in .

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional for a curve on a sphere with fixed boundary points and directions. The total length is free, denotes the spherical arclength, and denotes the geodesic curvature of  . Here the smooth external cost is obtained from spherical data.

Robust Retinal Vessel Segmentation via Locally Adaptive Derivative Frames in Orientation Scores.

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.

Improving Fiber Alignment in HARDI by Combining Contextual PDE Flow with Constrained Spherical Deconvolution.

We propose two strategies to improve the quality of tractography results computed from diffusion weighted magnetic resonance imaging (DW-MRI) data. Both methods are based on the same PDE framework, defined in the coupled space of positions and orientations, associated with a stochastic process describing the enhancement of elongated structures while preserving crossing structures. In the first method we use the enhancement PDE for contextual regularization of a fiber orientation distribution (FOD) that is obtained on individual voxels from high angular resolution diffusion imaging (HARDI) data via constrained spherical deconvolution (CSD).

Crossing-preserving multi-scale vesselness.

The multi-scale Frangi vesselness filter is an established tool in (retinal) vascular imaging. However, it cannot properly cope with crossings or bifurcations since it only looks for elongated structures. Therefore, we disentangle crossings/bifurcations via (multiple scale) invertible orientation scores and apply vesselness filters in this domain.

Evaluating contextual processing in diffusion MRI: application to optic radiation reconstruction for epilepsy surgery.

Diffusion MRI and tractography allow for investigation of the architectural configuration of white matter in vivo, offering new avenues for applications like presurgical planning. Despite the promising outlook, there are many pitfalls that complicate its use for (clinical) application. Amongst these are inaccuracies in the geometry of the diffusion profiles on which tractography is based, and poor alignment with neighboring profiles.

Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2).

To model association fields that underly perceptional organization (gestalt) in psychophysics we consider the problem of minimizing [Formula: see text] for a planar curve having fixed initial and final positions and directions. Here () is the curvature of the curve with free total length . This problem comes from a model of geometry of vision due to Petitot (in J.

A Sobolev norm based distance measure for HARDI clustering: a feasibility study on phantom and real data.

Dissimilarity measures for DTI clustering are abundant. However, for HARDI, the L2 norm has up to now been one of only few practically feasible measures. In this paper we propose a new measure, that not only compares the amplitude of diffusion profiles, but also rewards coincidence of the extrema.