22 research outputs found
Uniform Penalty inversion of two-dimensional NMR Relaxation data
The inversion of two-dimensional NMR data is an ill-posed problem related to
the numerical computation of the inverse Laplace transform. In this paper we
present the 2DUPEN algorithm that extends the Uniform Penalty (UPEN) algorithm
[Borgia, Brown, Fantazzini, {\em Journal of Magnetic Resonance}, 1998] to
two-dimensional data. The UPEN algorithm, defined for the inversion of
one-dimensional NMR relaxation data, uses Tikhonov-like regularization and
optionally non-negativity constraints in order to implement locally adapted
regularization. In this paper, we analyze the regularization properties of this
approach. Moreover, we extend the one-dimensional UPEN algorithm to the
two-dimensional case and present an efficient implementation based on the
Newton Projection method. Without any a-priori information on the noise norm,
2DUPEN automatically computes the locally adapted regularization parameters and
the distribution of the unknown NMR parameters by using variable smoothing.
Results of numerical experiments on simulated and real data are presented in
order to illustrate the potential of the proposed method in reconstructing
peaks and flat regions with the same accuracy
Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis
This paper is concerned with a novel regularisation technique for solving
linear ill-posed operator equations in Hilbert spaces from data that is
corrupted by white noise. We combine convex penalty functionals with
extreme-value statistics of projections of the residuals on a given set of
sub-spaces in the image-space of the operator. We prove general consistency and
convergence rate results in the framework of Bregman-divergences which allows
for a vast range of penalty functionals. Various examples that indicate the
applicability of our approach will be discussed. We will illustrate in the
context of signal and image processing that the presented method constitutes a
locally adaptive reconstruction method
Near-optimal Compressed Sensing Guarantees for Total Variation Minimization
Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for two-dimensional images x ∈ ℂN2 to signals x ∈ ℂNd of arbitrary dimension d ≥ 2 and to isotropic total variation problems. In this paper, we show that a multidimensional signal x ∈ ℂNd can be reconstructed from O(s dlog(Nd)) linear measurements y = Ax using total variation minimization to a factor of the best s-term approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension d
Near-optimal compressed sensing guarantees for anisotropic and isotropic total variation minimization
Consider the problem of reconstructing a multidimensional signal from partial information, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. Recently, guarantees for two-dimensional images were established. This paper extends these theoretical results to signals of arbitrary dimension and to both the anisotropic and isotropic total variation problems. To be precise, we show that a multidimensional signal can be reconstructed from a small number of linear measurements using total variation minimization to within a factor of the best approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension and a logarithmic factor in the signal dimension. The proof relies on bounds in approximation theory concerning the compressibility of wavelet expansions of bounded-variation functions
Multicontrast MRI reconstruction with structure-guided total variation
Magnetic resonance imaging (MRI) is a versatile imaging technique that allows
different contrasts depending on the acquisition parameters. Many clinical
imaging studies acquire MRI data for more than one of these contrasts---such as
for instance T1 and T2 weighted images---which makes the overall scanning
procedure very time consuming. As all of these images show the same underlying
anatomy one can try to omit unnecessary measurements by taking the similarity
into account during reconstruction. We will discuss two modifications of total
variation---based on i) location and ii) direction---that take structural a
priori knowledge into account and reduce to total variation in the degenerate
case when no structural knowledge is available. We solve the resulting convex
minimization problem with the alternating direction method of multipliers that
separates the forward operator from the prior. For both priors the
corresponding proximal operator can be implemented as an extension of the fast
gradient projection method on the dual problem for total variation. We tested
the priors on six data sets that are based on phantoms and real MRI images. In
all test cases exploiting the structural information from the other contrast
yields better results than separate reconstruction with total variation in
terms of standard metrics like peak signal-to-noise ratio and structural
similarity index. Furthermore, we found that exploiting the two dimensional
directional information results in images with well defined edges, superior to
those reconstructed solely using a priori information about the edge location.Engineering and Physical Sciences Research Council (Grant ID: EP/H046410/1)This is the final version of the article. It first appeared from Society for Industrial and Applied Mathematics via http://dx.doi.org/10.1137/15M1047325
Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics
In this paper we present a spatially-adaptive method for image reconstruction
that is based on the concept of statistical multiresolution estimation as
introduced in [Frick K, Marnitz P, and Munk A. "Statistical multiresolution
Dantzig estimation in imaging: Fundamental concepts and algorithmic framework".
Electron. J. Stat., 6:231-268, 2012]. It constitutes a variational
regularization technique that uses an supremum-type distance measure as
data-fidelity combined with a convex cost functional. The resulting convex
optimization problem is approached by a combination of an inexact alternating
direction method of multipliers and Dykstra's projection algorithm. We describe
a novel method for balancing data-fit and regularity that is fully automatic
and allows for a sound statistical interpretation. The performance of our
estimation approach is studied for various problems in imaging. Among others,
this includes deconvolution problems that arise in Poisson nanoscale
fluorescence microscopy