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.

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.

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.

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.

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 N-direction box spline is generally a (nonuniform) polynomial spline of degree N-1. 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.

We define partial cognitive relay channel as a two-transmitter, two-receiver interference channel with a relay node such that one of the transmitters is a cognitive transmitter. As a more feasible assumption we assume that the cognitive transmitter has non-causally access to a part of the other transmitter's encoded message. The cognitive radio transmitter transmits simultaneously over the same channel instead of waiting for an idle channel which is traditional for a cognitive radio. We calculate the achievable rate region for partial cognitive relay channel based on the combination of partial decode-compress and forward for the relay, rate-splitting and Gel'fand and Pinsker coding. Numerical evaluation of the region in the practical Gaussian case is performed. We show that by defining a proper cooperation strategy among the transmitters, equipping some of them with the cognition capability and exploiting the relay nodes, the effective bandwidth of each transmitter will enhance greatly.