Masih Nilchian

Fast iterative reconstruction of differential phase contrast X-ray tomograms

Differential phase-contrast is a recent technique in the context of X-ray imaging. In order to reduce the specimens exposure time, we propose a new iterative algorithm that can achieve the same quality as FBP-type methods, while using substantially fewer angular views. Our approach is based on 1) a novel spline-based discretization of the forward model and 2) an iterative reconstruction algorithm using the alternating direction method of multipliers. Our experimental results on real data suggest that the method allows to reduce the number of required views by at least a factor of four.

Sparse Stochastic Processes and Discretization of Linear Inverse Problems

We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuous-domain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the well-studied methods of Tikhonov and l1-type regularizations as particular cases, but also open the door to a broader class of sparsity-promoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, magnetic resonance imaging, and X-ray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.

A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems

B-splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of box splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported box splines (tensor-product B-splines, separable or not) in any number of dimensions. The proposed box spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an -direction box spline is generally a (nonuniform) polynomial spline of degree . The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a box spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms.

A Forward Regridding Method With Minimal Oversampling for Accurate and Efficient Iterative Tomographic Algorithms

Reconstruction of underconstrained tomographic data sets remains a major challenge. Standard analytical techniques frequently lead to unsatisfactory results due to insufficient information. Several iterative algorithms, which can easily integrate a priori knowledge, have been developed to tackle this problem during the last few decades. Most of these iterative algorithms are based on an implementation of the Radon transform that acts as forward projector. This operator and its adjoint, the backprojector, are typically called few times per iteration and represent the computational bottleneck of the reconstruction process. Here, we present a Fourier-based forward projector, founded on the regridding method with minimal oversampling. We show that this implementation of the Radon transform significantly outperforms in efficiency other state-of-the-art operators with O(N2log2N) complexity. Despite its reduced computational cost, this regridding method provides comparable accuracy to more sophisticated projectors and can, therefore, be exploited in iterative algorithms to substantially decrease the time required for the reconstruction of underconstrained tomographic data sets without loss in the quality of the results.

Optimized Kaiser–Bessel Window Functions for Computed Tomography

Kaiser-Bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties: 1) their short support leads to a low computational cost and 2) their rotational symmetry makes their imaging transform independent of the direction. In this paper, we aim at optimizing the parameters of these basis functions. We present a formalism based on the theory of approximation and point out the importance of the partition-of-unity condition. While we prove that, for compact-support functions, this condition is incompatible with isotropy, we show that minimizing the deviation from the partition of unity condition is highly beneficial. The numerical results confirm that the proposed tuning of the Kaiser-Bessel window functions yields the best performance.

Variational Phase Imaging Using the Transport-of-Intensity Equation

We introduce a variational phase retrieval algorithm for the imaging of transparent objects. Our formalism is based on the transport-of-intensity equation (TIE), which relates the phase of an optical field to the variation of its intensity along the direction of propagation. TIE practically requires one to record a set of defocus images to measure the variation of intensity. We first investigate the effect of the defocus distance on the retrieved phase map. Based on our analysis, we propose a weighted phase reconstruction algorithm yielding a phase map that minimizes a convex functional. The method is nonlinear and combines different ranges of spatial frequencies - depending on the defocus value of the measurements - in a regularized fashion. The minimization task is solved iteratively via the alternating-direction method of multipliers. Our simulations outperform commonly used linear and nonlinear TIE solvers. We also illustrate and validate our method on real microscopy data of HeLa cells.

Constrained regularized reconstruction of X-ray-DPCI tomograms with weighted-norm

In this paper we introduce a new reconstruction algorithm for X-ray differential phase-contrast Imaging (DPCI). Our approach is based on 1) a variational formulation with a weighted data term and 2) a variable-splitting scheme that allows for fast convergence while reducing reconstruction artifacts. In order to improve the quality of the reconstruction we take advantage of higher-order total-variation regularization. In addition, the prior information on the support and positivity of the refractive index is considered, which yields significant improvement. We test our method in two reconstruction experiments involving real data; our results demonstrate its potential for in-vivo and medical imaging.

Differential phase-contrast X-ray computed tomography: From model discretization to image reconstruction

Our contribution in this paper is two fold. First, we propose a novel discretization of the forward model for differential phase-contrast imaging that uses B-spline basis functions. The approach yields a fast and accurate algorithm for implementing the forward model, which is based on the first derivative of the Radon transform. Second, as an alternative to the FBP-like approaches that are currently used in practice, we present an iterative reconstruction algorithm that remains more faithful to the data when the number of projections dwindles. Since the reconstruction is an ill-posed problem, we impose a total-variation (TV) regularization constraint. We propose to solve the reconstruction problem using the alternating direction method of multipliers (ADMM). A specificity of our system is the use of a preconditioner that improves the convergence rate of the linear solver in ADMM. Our experiments on test data suggest that our method can achieve the same quality as the standard direct reconstruction, while using only one-third of the projection data. We also find that the approach is much faster than the standard algorithms (ISTA and FISTA) that are typically used for solving linear inverse problems subject to the TV regularization constraint.

Fast 3D Reconstruction Method for Differential Phase Contrast x-Ray CT

We present a fast algorithm for fully 3D regularized X-ray tomography reconstruction for both traditional and differential phase contrast measurements. In many applications, it is critical to reduce the X-ray dose while producing high-quality reconstructions. Regularization is an excellent way to do this, but in the differential phase contrast case it is usually applied in a slice-by-slice manner. We propose using fully 3D regularization to improve the dose/quality trade-off beyond what is possible using slice-by-slice regularization. To make this computationally feasible, we show that the two computational bottlenecks of our iterative optimization process can be expressed as discrete convolutions; the resulting algorithms for computation of the X-ray adjoint and normal operator are fast and simple alternatives to regridding. We validate this algorithm on an analytical phantom as well as conventional CT and differential phase contrast measurements from two real objects. Compared to the slice-by-slice approach, our algorithm provides a more accurate reconstruction of the analytical phantom and better qualitative appearance on one of the two real datasets.

Joint absorption and phase retrieval in grating-based x-ray radiography

Given the raw absorption and differential phase-contrast images obtained from a grating-based x-ray radiography, we formulate the joint denoising of the absorption image and retrieval of the non-differential phase image as a regularized inverse problem. The choice of the regularizer is driven by the existing correlation between absorption and differential phase; it leads to the linear combination of a total-variation norm with a total-variation nuclear norm. We then develop the corresponding algorithm to efficiently solve this inverse problem. We evaluate our method using different experiments, including mammography data. We conclude that our method provides useful information in the context of mammography screening and diagnosis.