Solving an Inverse Diffusion Problem for Magnetic Resonance Dosimetry by a Fast Regularization Method (original) (raw)

Optimal regularization for MR diffusion signal reconstruction

2012 9th IEEE International Symposium on Biomedical Imaging (ISBI), 2012

In this paper we address two problems related to the parametric reconstruction of the diffusion signal in the complete 3D Q-space. We propose a modified Spherical Polar Fourier (mSPF) basis to naturally impose the continuity of the diffusion signal on the whole space. This mathematical constraint results in a dimension reduction with respect to the original SPF basis. In addition, we derive the expression of a Laplace regularization operator in this basis, and compute optimal regularization weight using generalized cross validation (GCV). Experiments on synthetic and real data show that this regularization leads to a more accurate reconstruction than the commonly used low-pass filters.

Diffusion tensor magnetic resonance image regularization

Medical Image Analysis, 2004

As multi-dimensional complex data become more common, new regularization schemes tailored to those data are needed. In this paper we present a scheme for regularising diffusion tensor magnetic resonance (DT-MR) data, and more generally multi-dimensional data defined by a direction map and one or several magnitude maps. The scheme is divided in two steps. First, a variational method is proposed to restore direction fields with preservation of discontinuities. Its theoretical aspects are presented, as well as its application to the direction field that defines the main orientation of the diffusion tensors. The second step makes use of an anisotropic diffusion process to regularize the magnitude maps. The main idea is that for a range of data it is possible to use the restored direction as a prior to drive the regularization process in a way that preserves discontinuities and respects the local coherence of the magnitude map. We show that anisotropic diffusion is a convenient framework to implement that idea, and define a regularization process for the magnitude maps from our DT-MR data. Both steps are illustated on synthetic and real diffusion tensor magnetic resonance data.

Diffusion MRI signal reconstruction with continuity constraint and optimal regularization

Medical Image Analysis, 2012

In diffusion MRI, the reconstruction of the full Ensemble Average Propagator (EAP) provides new insights in the diffusion process and the underlying microstructure. The reconstruction of the signal in the whole Q-space is still extremely challenging however. It requires very long acquisition protocols, and robust reconstruction to cope with the very low SNR at large b-values. Several reconstruction methods were proposed recently, among which the Spherical Polar Fourier (SPF) expansion, a promising basis for signal reconstruction. Yet the reconstruction in SPF is still subject to noise and discontinuity of the reconstruction. In this work, we present a method for the reconstruction of the diffusion attenuation in the whole Q-space, with a special focus on continuity and optimal regularization. We derive a modified Spherical Polar Fourier (mSPF) basis, orthonormal and compatible with SPF, for the reconstruction of a signal with continuity constraint. We also derive the expression of a Laplace regularization operator in the basis, together with a method based on generalized cross validation for the optimal choice of the parameter. Our method results in a noticeable dimension reduction as compared with SPF. Tested on synthetic and real data, the reconstruction with this method is more robust to noise and better preserves fiber directions and crossings.

Regularization methods in dynamic MRI

Applied Mathematics and Computation, 2002

In this work we consider an inverse ill{posed problem coming from the area of dynamic Magnetic Resonance Imaging (MRI), where high resolution images must be reconstructed from incomplete data sets collected in the Fourier domain. The RIGR (Reduced{encoding Imaging by Generalized{series Reconstruction) method used leads to ill{ conditioned linear systems with noisy right hand side. We analyze the behaviour of three regularization methods, the Truncated Singular Value Decomposition, the Tikhonov regularization method and the Conjugate Gradients, together with some methods for the choice of the regularization parameter. The numerical results obtained on real data show that the solutions given by the three methods are comparable in terms of errors, but the Conjugate Gradients is the most e cient in terms of computational complexity.

Iterative Regularization Methods for a Discrete Inverse Problem in MRI

2008

We propose and investigate efficient numerical methods for inverse problems related to Magnetic Resonance Imaging (MRI). Our goal is to extend the recent convergence results for the Landweber-Kaczmarz method obtained in , in order to derive a convergent iterative regularization method for an inverse problem in MRI.

Maximum Entropy Technique and Regularization Functional for Determining the Pharmacokinetic Parameters in DCE-MRI

Journal of Digital Imaging

This paper aims to solve the arterial input function (AIF) determination in dynamic contrast-enhanced MRI (DCE-MRI), an important linear ill-posed inverse problem, using the maximum entropy technique (MET) and regularization functionals. In addition, estimating the pharmacokinetic parameters from a DCE-MR image investigations is an urgent need to obtain the precise information about the AIF–the concentration of the contrast agent on the left ventricular blood pool measured over time. For this reason, the main idea is to show how to find a unique solution of linear system of equations generally in the form of y=Ax+b,$$ y = A x + b , named an ill-conditioned linear system of equations after discretization of the integral equations, which appear in different tomographic image restoration and reconstruction issues. Here, a new algorithm is described to estimate an appropriate probability distribution function for AIF according to the MET and regularization functionals for the contrast...

Multichannel compressed sensing MR image reconstruction using statistically optimized nonlinear diffusion

Magnetic resonance in medicine, 2017

Eliminate the need for parametric tuning in total variation (TV) based multichannel compressed-sensing image reconstruction using statistically optimized nonlinear diffusion without compromising image quality. Nonlinear diffusion controls the denoising process using a contrast parameter that separates the gradients corresponding to noise and true edges in the image. This parameter is statistically estimated from the variance of combined image gradient to yield minimum steady-state reconstruction error. In addition, it uses acquired k-space data to bias the diffusion process toward an optimal solution. The proposed method is compared with TV using a set of noisy spine and brain data sets for three, four, and five-fold undersampling. It is observed that the choice of regularization parameter (step size) of TV-based methods requires prior tuning based on an extensive search procedure. In contrast, statistical estimation of contrast parameter removes this need for tuning by adapting to ...

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

Journal of Imaging, 2021

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

Codomain scale space and regularization for high angular resolution diffusion imaging

2008

Regularization is an important aspect in high angular resolution diffusion imaging (HARDI), since, unlike with classical diffusion tensor imaging (DTI), there is no a priori regularity of raw data in the co-domain, i.e. considered as a multispectral signal for fixed spatial position. HARDI preprocessing is therefore a crucial step prior to any subsequent analysis, and some insight in regularization paradigms and their interrelations is compulsory. In this paper we posit a codomain scale space regularization paradigm that has hitherto not been applied in the context of HARDI. Unlike previous (first and second order) schemes it is based on infinite order regularization, yet can be fully operationalized. We furthermore establish a closedform relation with first order Tikhonov regularization via the Laplace transform.

Analysis of Some Optimization Techniques for Regularization of Inverse Problems

2016

The main objective in inverse problems is to approximate some unknown parameters or attributes of interest, given some measurements that are only indirectly related to these parameters. This type of problem appears in many areas of science, engineering and industry. Examples can be found in medical computerized tomography, groundwater flow modeling, etc. In the process of solving these problems often appears an instability phenomenon known as ill-posedness which requires regularization. Ill-posedness is related to the fact that the presence of even a small amount of noise in the data can lead to enormous errors in the approximated solution. Different regularization techniques have been proposed in the literature. In this thesis our focus is put on Total Variation regularization. We study the total variation regularization for both image denoising and image deblurring problems. Three algorithms for total variation regularization will be analysed, namely the split Bregman algorithms, ...