361 research outputs found
A vector equilibrium problem for the two-matrix model in the quartic/quadratic case
We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and
(q_{k,n})_k related to the Hermitian two-matrix model with potentials V(x) =
x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients
in the recurrence relation satisfied by these polynomials, we obtain the
limiting distribution of the zeros of the polynomials p_{n,n} as n tends to
infinity. The limiting zero distribution is characterized as the first measure
of the minimizer in a vector equilibrium problem involving three measures which
for the case t=0 reduces to the vector equilibrium problem that was given
recently by two of us. A novel feature is that for t < 0 an external field is
active on the third measure which introduces a new type of critical behavior
for a certain negative value of t. We also prove a general result about the
interlacing of zeros of biorthogonal polynomials.Comment: 60 pages, 9 figure
Nonlinear Diffusion on the 2D Euclidean Motion Group
Linear and nonlinear diffusion equations are usually considered on an image, which is in fact a function on the translation group. In this paper we study diffusion on orientation scores, i.e. on functions on the Euclidean motion group SE(2). An orientation score is obtained from an image by a linear invertible transformation. The goal is to enhance elongated structures by applying nonlinear left-invariant diffusion on the orientation score of the image. For this purpose we describe how we can use Gaussian derivatives to obtain regularized left-invariant derivatives that obey the non-commutative structure of the Lie algebra of SE(2). The Hessian constructed with these derivatives is used to estimate local curvature and orientation strength and the diffusion is made nonlinearly dependent on these measures. We propose an explicit finite difference scheme to apply the nonlinear diffusion on orientation scores. The experiments show that preservation of crossing structures is the main advantage compared to approaches such as coherence enhancing diffusion
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). Thereby we improve the FOD as input for
subsequent tractography. Secondly, we introduce the fiber to bundle coherence
(FBC), a measure for quantification of fiber alignment. The FBC is computed
from a tractography result using the same PDE framework and provides a
criterion for removing the spurious fibers. We validate the proposed
combination of CSD and enhancement on phantom data and on human data, acquired
with different scanning protocols. On the phantom data we find that PDE
enhancements improve both local metrics and global metrics of tractography
results, compared to CSD without enhancements. On the human data we show that
the enhancements allow for a better reconstruction of crossing fiber bundles
and they reduce the variability of the tractography output with respect to the
acquisition parameters. Finally, we show that both the enhancement of the FODs
and the use of the FBC measure on the tractography improve the stability with
respect to different stochastic realizations of probabilistic tractography.
This is shown in a clinical application: the reconstruction of the optic
radiation for epilepsy surgery planning
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. Conventional
approaches for rotation invariance rely mostly on data augmentation, but this
does not guarantee the robustness of the output when the input is rotated. At
that, trained conventional CNNs may require test-time rotation augmentation to
reach their full capability. This study is focused on histopathology image
analysis applications for which it is desirable that the arbitrary global
orientation information of the imaged tissues is not captured by the machine
learning models. The proposed framework is evaluated on three different
histopathology image analysis tasks (mitosis detection, nuclei segmentation and
tumor classification). We present a comparative analysis for each problem and
show that consistent increase of performances can be achieved when using the
proposed framework
Universality of a double scaling limit near singular edge points in random matrix models
We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr
V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining
potential V_{s,t} is such that the limiting mean density of eigenvalues (as
n\to\infty and s,t\to 0) vanishes like a power 5/2 at a (singular) endpoint of
its support. The main purpose of this paper is to prove universality of the
eigenvalue correlation kernel in a double scaling limit. The limiting kernel is
built out of functions associated with a special solution of the P_I^2
equation, which is a fourth order analogue of the Painleve I equation. In order
to prove our result, we use the well-known connection between the eigenvalue
correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal
polynomials, together with the Deift/Zhou steepest descent method to analyze
the RH problem asymptotically. The key step in the asymptotic analysis will be
the construction of a parametrix near the singular endpoint, for which we use
the model RH problem for the special solution of the P_I^2 equation.
In addition, the RH method allows us to determine the asymptotics (in a
double scaling limit) of the recurrence coefficients of the orthogonal
polynomials with respect to the varying weights e^{-nV_{s,t}} on \mathbb{R}.
The special solution of the P_I^2 equation pops up in the n^{-2/7}-term of the
asymptotics.Comment: 32 pages, 3 figure
Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent
In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous
fluid develops cusp-like singularities. In recent papers [1, 2] we have showed
that singularities trigger viscous shocks propagating through the viscous
fluid. Here we show that the weak solution of the Hele-Shaw problem describing
viscous shocks is equivalent to a semiclassical approximation of a special real
solution of the Painleve I equation. We argue that the Painleve I equation
provides an integrable deformation of the Hele-Shaw problem which describes
flow passing through singularities. In this interpretation shocks appear as
Stokes level-lines of the Painleve linear problem.Comment: A more detailed derivation is include
Latent class growth analyses reveal overrepresentation of dysfunctional fear conditioning trajectories in patients with anxiety-related disorders compared to controls
Recent meta-analyses indicated differences in fear acquisition and extinction between patients with anxiety related disorders and comparison subjects. However, these effects are small and may hold for only a subsample of patients. To investigate individual trajectories in fear acquisition and extinction across patients with anxiety-related disorders (N = 104; before treatment) and comparison subjects (N = 93), data from a previous study (Duits et al., 2017) were re-analyzed using data-driven latent class growth analyses. In this explorative study, subjective fear ratings, shock expectancy ratings and startle responses were used as outcome measures. Fear and expectancy ratings, but not startle data, yielded distinct fear conditioning trajectories across participants. Patients were, compared to controls, overrepresented in two distinct dysfunctional fear conditioning trajectories: impaired safety learning and poor fear extinction to danger cues. The profiling of individual patterns allowed to determine that whereas a subset of patients showed trajectories of dysfunctional fear conditioning, a significant proportion of patients (?50 %) did not. The strength of trajectory analyses as opposed to group analyses is that it allows the identification of individuals with dysfunctional fear conditioning. Results suggested that dysfunctional fear learning may also be associated with poor treatment outcome, but further research in larger samples is needed to address this question
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