A singular K-space model for fast reconstruction of magnetic resonance images from undersampled data.

Reconstructing magnetic resonance images from undersampled k-space data is a challenging problem. This paper introduces a novel method of image reconstruction from undersampled k-space data based on the concept of singularizing operators and a novel singular k-space model. Exploring the sparsity of an image in the k-space, the singular k-space model (SKM) is proposed in terms of the k-space functions of a singularizing operator. The singularizing operator is constructed by combining basic difference operators. An algorithm is developed to reliably estimate the model parameters from undersampled k-space data. The estimated parameters are then used to recover the missing k-space data through the model, subsequently achieving high-quality reconstruction of the image using inverse Fourier transform. Experiments on physical phantom and real brain MR images have shown that the proposed SKM method constantly outperforms the popular total variation (TV) and the classical zero-filling (ZF) methods regardless of the undersampling rates, the noise levels, and the image structures. For the same objective quality of the reconstructed images, the proposed method requires much less k-space data than the TV method. The SKM method is an effective method for fast MRI reconstruction from the undersampled k-space data. Graphical abstract Two Real Images and their sparsified images by singularizing operator.

Wideband 2-D Array Design Optimization With Fabrication Constraints for 3-D US Imaging.

Ultrasound (US) 2-D arrays are of increasing interest due to their electronic steering capability to investigate 3-D regions without requiring any probe movement. These arrays are typically populated by thousands of elements that, ideally, should be individually driven by the companion scanner. Since this is not convenient, the so-called microbeamforming methods, yielding a prebeamforming stage performed in the probe handle by suitable custom integrated circuits, have so far been implemented in a few commercial high-end scanners. A possible approach to implement relatively cheap and efficient 3-D US imaging systems is using 2-D sparse arrays in which a limited number of elements can be coupled to an equal number of independent transmit/receive channels. In order to obtain US beams with adequate characteristics all over the investigated volume, the layout of such arrays must be carefully designed. This paper provides guidelines to design, by using simulated annealing optimization, 2-D sparse arrays capable of fitting specific applications or fabrication/implementation constraints. In particular, an original energy function based on multidepth 3-D analysis of the beam pattern is also exploited. A tutorial example is given, addressed to find the N e elements that should be activated in a 2-D fully populated array to yield efficient acoustic radiating performance over the entire volume. The proposed method is applied to a 32 ×32 array centered at 3 MHz to select the 128, 192, and 256 elements that provide the best acoustic performance. It is shown that the 256-element optimized array yields sidelobe levels even lower (by 5.7 dB) than that of the reference 716-element circular and (by 10.3 dB) than that of the reference 1024-element array.

Global Diffusion Tractography by Simulated Annealing.

OBJECTIVE: Our goal is to develop a robust global tractography method for cardiac diffusion imaging. METHODS: A graph is stretched over the whole myocardium to represent the fiber structure, and the solutions are minima of a graph energy measuring the fidelity to the data along with the fiber density and curvature. The optimization is performed by a variant of simulated annealing that offers increased design freedom without sacrificing theoretical convergence guarantees. RESULTS: Numerical experiments on synthetic and real data demonstrate the capability of our tractography algorithm to deal with low angular resolution, highly noisy data. In particular, our algorithm outperforms the Bayesian model-based algorithm of Reisert et al. (NeuroImage, vol. 54, no. 2, 2011) and the graph-based algorithm of Frindel et al. (Magn. Reson. Med., vol. 64, no. 4, 2010) at the noise levels typical of in vivo imaging. CONCLUSION: The proposed algorithm avoids the drawbacks of local techniques and is very robust to noise, which makes it a promising tool for in vivo diffusion imaging of moving organs. SIGNIFICANCE: Our approach is global in terms of both the fiber structure representation and the minimization problem. It also allows us to adjust the trajectory density by simply changing the vertex-lattice spacing in the graph model, a desirable feature for multiresolution tractography analysis.

2-D Ultrasound Sparse Arrays Multidepth Radiation Optimization Using Simulated Annealing and Spiral-Array Inspired Energy Functions.

Full matrix arrays are excellent tools for 3-D ultrasound imaging, but the required number of active elements is too high to be individually controlled by an equal number of scanner channels. The number of active elements is significantly reduced by the sparse array techniques, but the position of the remaining elements must be carefully optimized. This issue is faced here by introducing novel energy functions in the simulated annealing (SA) algorithm. At each iteration step of the optimization process, one element is freely translated and the associated radiated pattern is simulated. To control the pressure field behavior at multiple depths, three energy functions inspired by the pressure field radiated by a Blackman-tapered spiral array are introduced. Such energy functions aim at limiting the main lobe width while lowering the side lobe and grating lobe levels at multiple depths. Numerical optimization results illustrate the influence of the number of iterations, pressure measurement points, and depths, as well as the influence of the energy function definition on the optimized layout. It is also shown that performance close to or even better than the one provided by a spiral array, here assumed as reference, may be obtained. The finite-time convergence properties of SA allow the duration of the optimization process to be set in advance.

Mitotic spindle asymmetry in rodents and primates: 2D vs. 3D measurement methodologies.

Recent data have uncovered that spindle size asymmetry (SSA) is a key component of asymmetric cell division (ACD) in the mouse cerebral cortex (Delaunay et al., 2014). In the present study we show that SSA is independent of spindle orientation and also occurs during cortical progenitor divisions in the ventricular zone (VZ) of the macaque cerebral cortex, pointing to a conserved mechanism in the mammalian lineage. Because SSA magnitude is smaller in cortical precursors than in invertebrate neuroblasts, the unambiguous demonstration of volume differences between the two half spindles is considered to require 3D reconstruction of the mitotic spindle (Delaunay et al., 2014). Although straightforward, the 3D analysis of SSA is time consuming, which is likely to hinder SSA identification and prevent further explorations of SSA related mechanisms in generating ACD. We therefore set out to develop an alternative method for accurately measuring spindle asymmetry. Based on the mathematically demonstrated linear relationship between 2D and 3D analysis, we show that 2D assessment of spindle size in metaphase cells is as accurate and reliable as 3D reconstruction provided a specific procedure is applied. We have examined the experimental accuracy of the two methods by applying them to different sets of in vivo and in vitro biological data, including mouse and primate cortical precursors. Linear regression analysis demonstrates that the results from 2D and 3D reconstructions are equally powerful. We therefore provide a reliable and efficient technique to measure SSA in mammalian cells.

Cardiac fiber unfolding by semidefinite programming.

Diffusion-tensor imaging allows noninvasive assessment of the myocardial fiber architecture, which is fundamental in understanding the mechanics of the heart. In this context, tractography techniques are often used for representing and visualizing cardiac fibers, but their output is only qualitative. We introduce here a new framework toward a more quantitative description of the cardiac fiber architecture from tractography results. The proposed approach consists in taking three-dimensional (3-D) fiber tracts as inputs, and then unfolding these fibers in the Euclidean plane under local isometry constraints using semidefinite programming. The solution of the unfolding problem takes the form of a Gram matrix which defines the two-dimensional (2-D) embedding of the fibers and whose spectrum provides quantitative information on their organization. Experiments on synthetic and real data show that unfolding makes it easier to observe and to study the cardiac fiber architecture. Our conclusion is that 2-D embedding of cardiac fibers is a promising approach to supplement 3-D rendering for understanding the functioning of the heart.

A 3-D spatio-temporal deconvolution approach for MR perfusion in the brain.

We propose an original spatio-temporal deconvolution approach for perfusion-weighted MRI applied to cerebral ischemia. The regularization of the underlying inverse problem is achieved with spatio-temporal priors and the resulting optimization problem is solved by half-quadratic minimization. Our approach offers strong convergence guarantees, including when the spatial priors are non-convex. Moreover, experiments on synthetic data and on real data collected from subjects with ischemic stroke show significant performance improvements over the standard approaches-namely, temporal deconvolution based on either truncated singular-value decomposition or ℓ2-regularization-in terms of various performance measures.

A stochastic continuation approach to piecewise constant reconstruction.

We address the problem of reconstructing a piecewise constant 3-D object from a few noisy 2-D line-integral projections. More generally, the theory developed here readily applies to the recovery of an ideal n-D signal (n > or =1) from indirect measurements corrupted by noise. Stabilization of this ill-conditioned inverse problem is achieved with the Potts prior model, which leads to a challenging optimization task. To overcome this difficulty, we introduce a new class of hybrid algorithms that combines simulated annealing with deterministic continuation. We call this class of algorithms stochastic continuation (SC). We first prove that, under mild assumptions, SC inherits the finite-time convergence properties of generalized simulated annealing. Then, we show that SC can be successfully applied to our reconstruction problem. In addition, we look into the concave distortion acceleration method introduced for standard simulated annealing and we derive an explicit formula for choosing the free parameter of the cost function. Numerical experiments using both synthetic data and real radiographic testing data show that SC outperforms standard simulated annealing.

Stochastic nonlinear image restoration using the wavelet transform.

The dominant methodology for image restoration is to stabilize the problem by including a roughness penalty in addition to faithfulness to the data. Among various choices, concave stabilizers stand out for their boundary detection capabilities, but the resulting cost function to be minimized is generally multimodal. Although simulated annealing is theoretically optimal to take up this challenge, standard stochastic algorithms suffer from two drawbacks: i) practical convergence difficulties are encountered with second-order prior models and ii) it remains computationally demanding to favor the formation of smooth contour lines by taking the discontinuity field explicitly into account. This work shows that both weaknesses can be overcome in a multiresolution framework by means of the 2-D discrete wavelet transform (DWT). We first propose to improve convergence toward global minima by single-site updating on the wavelet domain. For this purpose, a new restricted DWT space is introduced and a theoretically sound updating mechanism is constructed on this subspace. Next, we suggest to incorporate the smoothness of the discontinuity field via an additional penalty term defined on the high frequency subbands. The resulting increase in complexity is small and the approach requires the specification of a unique extra parameter for which an explicit selection formula is derived.