68,766 research outputs found
Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator
We propose the eigenvalue problem of an anisotropic diffusion operator for
image segmentation. The diffusion matrix is defined based on the input image.
The eigenfunctions and the projection of the input image in some eigenspace
capture key features of the input image. An important property of the model is
that for many input images, the first few eigenfunctions are close to being
piecewise constant, which makes them useful as the basis for a variety of
applications such as image segmentation and edge detection. The eigenvalue
problem is shown to be related to the algebraic eigenvalue problems resulting
from several commonly used discrete spectral clustering models. The relation
provides a better understanding and helps developing more efficient numerical
implementation and rigorous numerical analysis for discrete spectral
segmentation methods. The new continuous model is also different from
energy-minimization methods such as geodesic active contour in that no initial
guess is required for in the current model. The multi-scale feature is a
natural consequence of the anisotropic diffusion operator so there is no need
to solve the eigenvalue problem at multiple levels. A numerical implementation
based on a finite element method with an anisotropic mesh adaptation strategy
is presented. It is shown that the numerical scheme gives much more accurate
results on eigenfunctions than uniform meshes. Several interesting features of
the model are examined in numerical examples and possible applications are
discussed
A Phase Field Model for Continuous Clustering on Vector Fields
A new method for the simplification of flow fields is presented. It is based on continuous clustering. A well-known physical clustering model, the Cahn Hilliard model, which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns-the actual clustering-during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters as a function of the underlying flow field. In addition, the model is expanded, involving elastic effects. In the early stages of the evolution shear layer type representation of the flow field can thereby be generated, whereas, for later stages, the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented drop-shaped appearance. Here, we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm.
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
Measuring cellular traction forces on non-planar substrates
Animal cells use traction forces to sense the mechanics and geometry of their
environment. Measuring these traction forces requires a workflow combining cell
experiments, image processing and force reconstruction based on elasticity
theory. Such procedures have been established before mainly for planar
substrates, in which case one can use the Green's function formalism. Here we
introduce a worksflow to measure traction forces of cardiac myofibroblasts on
non-planar elastic substrates. Soft elastic substrates with a wave-like
topology were micromolded from polydimethylsiloxane (PDMS) and fluorescent
marker beads were distributed homogeneously in the substrate. Using feature
vector based tracking of these marker beads, we first constructed a hexahedral
mesh for the substrate. We then solved the direct elastic boundary volume
problem on this mesh using the finite element method (FEM). Using data
simulations, we show that the traction forces can be reconstructed from the
substrate deformations by solving the corresponding inverse problem with a
L1-norm for the residue and a L2-norm for 0th order Tikhonov regularization.
Applying this procedure to the experimental data, we find that cardiac
myofibroblast cells tend to align both their shapes and their forces with the
long axis of the deformable wavy substrate.Comment: 34 pages, 9 figure
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
Finite element surface registration incorporating curvature, volume preservation, and statistical model information
We present a novel method for nonrigid registration of 3D surfaces and images. The method can be used to register surfaces by means of their distance images, or to register medical images directly. It is formulated as a minimization problem of a sum of several terms representing the desired properties of a registration result: smoothness, volume preservation, matching of the surface, its curvature, and possible other feature images, as well as consistency with previous registration results of similar objects, represented by a statistical deformation model. While most of these concepts are already known, we present a coherent continuous formulation of these constraints, including the statistical deformation model. This continuous formulation renders the registration method independent of its discretization. The finite element discretization we present is, while independent of the registration functional, the second main contribution of this paper. The local discontinuous Galerkin method has not previously been used in image registration, and it provides an efficient and general framework to discretize each of the terms of our functional. Computational efficiency and modest memory consumption are achieved thanks to parallelization and locally adaptive mesh refinement. This allows for the first time the use of otherwise prohibitively large 3D statistical deformation models
Semantic 3D Reconstruction with Finite Element Bases
We propose a novel framework for the discretisation of multi-label problems
on arbitrary, continuous domains. Our work bridges the gap between general FEM
discretisations, and labeling problems that arise in a variety of computer
vision tasks, including for instance those derived from the generalised Potts
model. Starting from the popular formulation of labeling as a convex relaxation
by functional lifting, we show that FEM discretisation is valid for the most
general case, where the regulariser is anisotropic and non-metric. While our
findings are generic and applicable to different vision problems, we
demonstrate their practical implementation in the context of semantic 3D
reconstruction, where such regularisers have proved particularly beneficial.
The proposed FEM approach leads to a smaller memory footprint as well as faster
computation, and it constitutes a very simple way to enable variable, adaptive
resolution within the same model
Colour image segmentation by the vector-valued Allen-Cahn phase-field model: a multigrid solution
We propose a new method for the numerical solution of a PDE-driven model for
colour image segmentation and give numerical examples of the results. The
method combines the vector-valued Allen-Cahn phase field equation with initial
data fitting terms. This method is known to be closely related to the
Mumford-Shah problem and the level set segmentation by Chan and Vese. Our
numerical solution is performed using a multigrid splitting of a finite element
space, thereby producing an efficient and robust method for the segmentation of
large images.Comment: 17 pages, 9 figure
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