Sub-Riemannian Fast Marching in SE(2)

We propose a Fast Marching based implementation for computing sub-Riemanninan (SR) geodesics in the roto-translation group SE(2), with a metric depending on a cost induced by the image data. The key ingredient is a Riemannian approximation of the SR-metric. Then, a state of the art Fast Marching solver that is able to deal with extreme anisotropies is used to compute a SR-distance map as the solution of a corresponding eikonal equation. Subsequent backtracking on the distance map gives the geodesics. To validate the method, we consider the uniform cost case in which exact formulas for SR-geodesics are known and we show remarkable accuracy of the numerically computed SR-spheres. We also show a dramatic decrease in computational time with respect to a previous PDE-based iterative approach. Regarding image analysis applications, we show the potential of considering these data adaptive geodesics for a fully automated retinal vessel tree segmentation.Comment: CIARP 201

Anisotropic diffusion of surface normals for feature preserving surface reconstruction

technical reportFor 3D surface reconstruction problems with noisy and incomplete range data measured from complex scenes with arbitrary topologies, a low-level representation, such as level set surfaces, is used. Such surface reconstruction is typically accomplished by minimizing a weighted sum of data-model discrepancy and model smoothness terms. This paper introduces a new nonlinear model smoothness term for surface reconstruction based on variations of the surface normals. A direct solution requires solving a fourth-order partial differential equation (PDE), which is very difficult with conventional numerical techniques. Our solution is based on processing the normals separately from the surface, which allows us to separate the problem into two second-order PDEs. The proposed method can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion

Anisotropic diffusion of surface normals for feature preserving surface reconstruction

Journal ArticleFor 3D surface reconstruction problems with noisy and incomplete range data measured from complex scenes with arbitrary topologies, a low-level representation, such as level set surfaces, is used. Such surface reconstruction is typically accomplished by minimizing a weighted sum of data-model discrepancy and model smoothness terms. This paper introduces a new onlinear model smoothness term for surface reconstruction based on variations of the surface normals. A direct solution requires solving a fourth-order partial differential equation (PDE), which is very difficult with conventional numerical techniques. Our solution is based on processing the normals separately from the surface, which allows us to separate the problem into two second-order PDEs. The proposed method can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion

Anisotropic diffusion of surface normals for feature preserving surface reconstruction

Journal ArticleFor 3D surface reconstruction problems with noisy and incomplete range data measure d from complex scenes with arbitrary topologies, a low-level representation, such as level set surfaces, is used. Such surface reconstruction is typically accomplished by minimizing a weighted sum of data-model discrepancy and model smoothness terms. This paper introduces a new nonlinear model smoothness term for surface reconstruction based on variations of the surface normals. A direct solution requires solving a fourth-order partial differential equation (PDE), which is very difficult with conventional numerical techniques. Our solution is based on processing the normals separately from the surface, which allows us to separate the problem into two second-order PDEs. The proposed method can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion

Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.Comment: 28 pages, 12 figure

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

Edge Aware Anisotropic Diffusion for 3D Scalar Data

Fig. 1: The left half of the figure demonstrates the consistency in smoothing of our method compared to the existing method. The right half of the figure demonstrates the de-noising capabilities of our method. All the images from (a-c) were obtained byrenderingan iso-surface of 153. (a) Diffused with an existing diffusion model proposed by Krissian et al. [20] with k = 40, and100 iterations (b) The original Sheep’s heart data. (c) Diffused with our method with σ = 1 and the same number of iterations. The yellow circle indicates aregionwheretheiso-surfacehasbothhighandmediumrangegradient magnitude, and the blue circle marks a region where the gradient magnitude is much lower. Note the inconsistent smoothing in (a) inside the yellow circle. (d) The tooth data contaminated with Poisson noise (SNR=12.8) (e)Theoriginaltoothdata(f)Diffusedwithourmethod(SNR=25.76) withσ = 1 and 25 iterations. We used the exact same transfer function to render all the images in(d-f). Abstract—Inthispaperwepresentanovelanisotropicdiffusionmodel targeted for 3D scalar field data. Our model preserves material boundaries as well as fine tubular structures while noise is smoothed out. One of the major novelties is the use of the directional second derivative to define material boundaries instead of the gradient magnitude for thresholding. This results in a diffusion model that has much lower sensitivity to the diffusion parameter and smoothes material boundaries consistently compared to gradient magnitude based techniques. We empirically analyze the stability and convergence of the proposed diffusion and demonstrate its de-noising capabilities for both analytic and real data. We also discuss applications in the context of volume rendering

Automatic liver vessel segmentation using 3D region growing and hybrid active contour model

This paper proposes a new automatic method for liver vessel segmentation by exploiting intensity and shape constraints of 3D vessels. The core of the proposed method is to apply two different strategies: 3D region growing facilitated by bi-Gaussian filter for thin vessel segmentation, and hybrid active contour model combined with K-means clustering for thick vessel segmentation. They are then integrated to generate final segmentation results. The proposed method is validated on abdominal computed tomography angiography (CTA) images, and obtains an average accuracy, sensitivity, specificity, Dice, Jaccard, and RMSD of 98.2%, 68.3%, 99.2%, 73.0%, 66.1%, and 2.56 mm, respectively. Experimental results show that our method is capable of segmenting complex liver vessels with more continuous and complete thin vessel details, and outperforms several existing 3D vessel segmentation algorithms