1,222 research outputs found
Curvature-driven PDE methods for matrix-valued images
Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edgelike structures in tensor fields, we first generalise Di Zenzo\u27s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts
A Convex Semi-Definite Positive Framework for DTI Estimation and Regularization
International audienceIn this paper we introduce a novel variational method for joint estimation and regularization of diffusion tensor fields from noisy raw data. To this end, we use the classic quadratic data fidelity term derived from the Stejskal-Tanner equation with a new smoothness term leading to a convex objective function. The regularization term is based on the assumption that the signal can be reconstructed using a weighted average of observations on a local neighborhood. The weights measure the similarity between tensors and are computed directly from the diffusion images. We preserve the positive semi-definiteness constraint using a projected gradient descent. Experimental validation and comparisons with a similar method using synthetic data with known noise model, as well as classification of tensors towards understanding the anatomy of human skeletal muscle demonstrate the potential of our method
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
PDE-based preprocessing of medical images
Medical imaging often requires a preprocessing step where filters are applied that remove noise while preserving semantically important structures such as edges. This may help to simplify subsequent tasks such as segmentation. One class of recent adaptive denoising methods consists of methods based on nonlinear partial differential equations (PDEs). In the present paper we survey our recent results on PDE-based preprocessing methods that may be applied to medical imaging problems. We focus on nonlinear diffusion filters and variational restoration methods. We explain the basic ideas, sketch some algorithmic aspects, illustrate the concepts by applying them to medical images such as mammograms, computerized tomography (CT), and magnetic resonance (MR) images. In particular we show the use of these filters as preprocessing steps for segmentation algorithms
PDEs for tensor image processing
Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. In this survey paper the most important PDEs for discontinuity-preserving denoising of tensor fields are reviewed such that the underlying design principles becomes evident. We consider isotropic and anisotropic diffusion filters and their corresponding variational methods, mean curvature motion, and selfsnakes. These filters preserve positive semidefiniteness of any positive semidefinite initial tensor field. Finally we discuss geodesic active contours for segmenting tensor fields. Experiments are presented that illustrate the behaviour of all these methods
Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging
Mumford-Shah and Potts functionals are powerful variational models for
regularization which are widely used in signal and image processing; typical
applications are edge-preserving denoising and segmentation. Being both
non-smooth and non-convex, they are computationally challenging even for scalar
data. For manifold-valued data, the problem becomes even more involved since
typical features of vector spaces are not available. In this paper, we propose
algorithms for Mumford-Shah and for Potts regularization of manifold-valued
signals and images. For the univariate problems, we derive solvers based on
dynamic programming combined with (convex) optimization techniques for
manifold-valued data. For the class of Cartan-Hadamard manifolds (which
includes the data space in diffusion tensor imaging), we show that our
algorithms compute global minimizers for any starting point. For the
multivariate Mumford-Shah and Potts problems (for image regularization) we
propose a splitting into suitable subproblems which we can solve exactly using
the techniques developed for the corresponding univariate problems. Our method
does not require any a priori restrictions on the edge set and we do not have
to discretize the data space. We apply our method to diffusion tensor imaging
(DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation
of the corpus callosum
Navier-Stokes Modelling of Non-Newtonian Blood Flow in Cerebral Arterial Circulation and its Dynamic Impact on Electrical Conductivity in a Realistic Multi-Compartment Head Model
Background and Objective: This study aims to evaluate the dynamic effect of
non-Newtonian cerebral arterial circulation on electrical conductivity
distribution (ECD) in a realistic multi-compartment head model. It addresses
the importance and challenges associated with electrophysiological modalities,
such as transcranial electrical stimulation, electro-magnetoencephalography,
and electrical impedance tomography. Factors such as electrical conductivity's
impact on forward modeling accuracy, complex vessel networks, data acquisition
limitations (especially in MRI), and blood flow phenomena are considered.
Methods: The Navier-Stokes equations (NSEs) govern the non-Newtonian flow model
used in this study. The solver comprises two stages: the first solves the
pressure field using a dynamical pressure-Poisson equation derived from NSEs,
and the second updates the velocity field using Leray regularization and the
pressure distribution from the first stage. The Carreau-Yasuda model
establishes the connection between blood velocity and viscosity. Blood
concentration in microvessels is approximated using Fick's law of diffusion,
and conductivity mapping is obtained via Archie's law. The head model used
corresponds to an open 7 Tesla MRI dataset, differentiating arterial vessels
from other structures. Results: The results suggest the establishment of a
dynamic model of cerebral blood flow for arterial and microcirculation. Blood
pressure and conductivity distributions are obtained through numerically
simulated pulse sequences, enabling approximation of blood concentration and
conductivity within the brain. Conclusions: This model provides an
approximation of dynamic blood flow and corresponding ECD in different brain
regions. The advantage lies in its applicability with limited a priori
information about blood flow and compatibility with arbitrary head models that
distinguish arteries.Comment: 13 pages; 8 figures; 2 tabl
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