94 research outputs found
Aggregated motion estimation for real-time MRI reconstruction
Real-time magnetic resonance imaging (MRI) methods generally shorten the
measuring time by acquiring less data than needed according to the sampling
theorem. In order to obtain a proper image from such undersampled data, the
reconstruction is commonly defined as the solution of an inverse problem, which
is regularized by a priori assumptions about the object. While practical
realizations have hitherto been surprisingly successful, strong assumptions
about the continuity of image features may affect the temporal fidelity of the
estimated images. Here we propose a novel approach for the reconstruction of
serial real-time MRI data which integrates the deformations between nearby
frames into the data consistency term. The method is not required to be affine
or rigid and does not need additional measurements. Moreover, it handles
multi-channel MRI data by simultaneously determining the image and its coil
sensitivity profiles in a nonlinear formulation which also adapts to
non-Cartesian (e.g., radial) sampling schemes. Experimental results of a motion
phantom with controlled speed and in vivo measurements of rapid tongue
movements demonstrate image improvements in preserving temporal fidelity and
removing residual artifacts.Comment: This is a preliminary technical report. A polished version is
published by Magnetic Resonance in Medicine. Magnetic Resonance in Medicine
201
Adaptive minimax optimality in statistical inverse problems via SOLIT -- Sharp Optimal Lepskii-Inspired Tuning
We consider statistical linear inverse problems in separable Hilbert spaces
and filter-based reconstruction methods of the form , where is the available data, the forward
operator, an ordered filter,
and a regularization parameter. Whenever such a method is used in
practice, has to be appropriately chosen. Typically, the aim is to
find or at least approximate the best possible in the sense that mean
squared error (MSE)
w.r.t.~the true solution is minimized. In this paper, we introduce
the Sharp Optimal Lepski\u{\i}-Inspired Tuning (SOLIT) method, which yields an
a posteriori parameter choice rule ensuring adaptive minimax rates of
convergence. It depends only on and the noise level as well as the
operator and the filter and
does not require any problem-dependent tuning of further parameters. We prove
an oracle inequality for the corresponding MSE in a general setting and derive
the rates of convergence in different scenarios. By a careful analysis we show
that no other a posteriori parameter choice rule can yield a better performance
in terms of the order of the convergence rate of the MSE. In particular, our
results reveal that the typical understanding of Lepski\u\i-type methods in
inverse problems leading to a loss of a log factor is wrong. In addition, the
empirical performance of SOLIT is examined in simulations.Comment: Some technical parts are polished, and a comparison with classical
Lepskii is included in the simulation sectio
Quick Adaptive Ternary Segmentation: An Efficient Decoding Procedure For Hidden Markov Models
Hidden Markov models (HMMs) are characterized by an unobservable (hidden)
Markov chain and an observable process, which is a noisy version of the hidden
chain. Decoding the original signal (i.e., hidden chain) from the noisy
observations is one of the main goals in nearly all HMM based data analyses.
Existing decoding algorithms such as the Viterbi algorithm have computational
complexity at best linear in the length of the observed sequence, and
sub-quadratic in the size of the state space of the Markov chain. We present
Quick Adaptive Ternary Segmentation (QATS), a divide-and-conquer procedure
which decodes the hidden sequence in polylogarithmic computational complexity
in the length of the sequence, and cubic in the size of the state space, hence
particularly suited for large scale HMMs with relatively few states. The
procedure also suggests an effective way of data storage as specific cumulative
sums. In essence, the estimated sequence of states sequentially maximizes local
likelihood scores among all local paths with at most three segments. The
maximization is performed only approximately using an adaptive search
procedure. The resulting sequence is admissible in the sense that all
transitions occur with positive probability. To complement formal results
justifying our approach, we present Monte-Carlo simulations which demonstrate
the speedups provided by QATS in comparison to Viterbi, along with a precision
analysis of the returned sequences. An implementation of QATS in C++ is
provided in the R-package QATS and is available from GitHub
Detection and inference of changes in high-dimensional linear regression with non-sparse structures
For data segmentation in high-dimensional linear regression settings, the regression parameters are often assumed to be sparse segment-wise, which enables many existing methods to estimate the parameters locally via -regularised maximum likelihood-type estimation and then contrast them for change point detection. Contrary to this common practice, we show that the sparsity of neither regression parameters nor their differences, a.k.a. differential parameters, is necessary for consistency in multiple change point detection. In fact, both statistically and computationally, better efficiency is attained by a simple strategy that scans for large discrepancies in local covariance between the regressors and the response. We go a step further and propose a suite of tools for directly inferring about the differential parameters post-segmentation, which are applicable even when the regression parameters themselves are non-sparse. Theoretical investigations are conducted under general conditions permitting non-Gaussianity, temporal dependence and ultra-high dimensionality. Numerical results from simulated and macroeconomic datasets demonstrate the competitiveness and efficacy of the proposed methods
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