We study the convergence of a generic half-quadratic algorithm for minimizing a wide class of $C^{1}$ objectives that occur in inverse imaging problems; this algorithm amounts to solving a sequence of positive definite systems (the inner systems) and has the advantages of simplicity and versatility. Half-quadratic optimization has been meticulously studied, both theoretically and experimentally, but two difficulties remain: first, the practical solutions of the inner systems are approximate, which may hamper convergence; and second, convergence to a stationary point of the objective is not guaranteed if the set of such points contains a continuum. We present new results that do not suffer from these limitations and hence extend our work in [M. Robini and Y. Zhu, SIAM J. Imaging Sci., 8 (2015), pp. 1752--1797]. We consider the inexact process in which the inner systems are solved to a fixed arbitrary accuracy defined in terms of the energy norm of the error. We show that this process converges to a stationary point of the objective under minimal conditions ubiquitous in regularized reconstruction and restoration. Our main results are based on the assumption that the objective has the Kurdyka--Łojasiewicz property, for which we provide constructing rules using the concept of tameness from the theory of o-minimal structures. We also propose an implementation using a truncated conjugate gradient method that controls the accuracy at negligible additional cost. Experiments on three different inverse problems show that the resulting algorithm performs well in various nonconvex scenarios and converges to solutions accurate to full machine precision.

Purpose The robustness of a recently introduced globally convergent deconvolution algorithm with temporal and edge‐preserving spatial regularization for the deconvolution of dynamic susceptibility contrast perfusion magnetic resonance imaging is assessed in the context of ischemic stroke. Theory and Methods Ischemic tissues are not randomly distributed in the brain but form a spatially organized entity. The addition of a spatial regularization term allows to take into account this spatial organization contrarily to the sole temporal regularization approach which processes each voxel independently. The robustness of the spatial regularization in relation to shape variability, hemodynamic variability in tissues, noise in the magnetic resonance imaging apparatus, and uncertainty on the arterial input function selected for the deconvolution is addressed via an original in silico validation approach. Results The deconvolution algorithm proved robust to the different sources of variability, outperforming temporal Tikhonov regularization in most realistic conditions considered. The limiting factor is the proper estimation of the arterial input function. Conclusion This study quantified the robustness of a spatio‐temporal approach for dynamic susceptibility contrast‐magnetic resonance imaging deconvolution via a new simulator. This simulator, now accessible online, is of wide applicability for the validation of any deconvolution algorithm. Magn Reson Med 78:1981–1990, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

We consider the problem of reconstructing an image from noisy projection data. The problem is formulated as the minimization of a general objective function that can be particularized to the standard objectives used in medical applications (the most common choice is the sum of a convex data-fidelity term and a gradient-based regularization term). Our main focus is on half-quadratic (HQ) optimization, which amounts to solving a sequence of positive definite systems, or inner systems, and has the advantages of simplicity and versatility. HQ algorithms have been meticulously studied, but two difficulties remain: first, the inner systems are solved approximately in practice, which may hamper convergence; second, pointwise convergence is not guaranteed when the set of stationary points of the objective contains a continuum. We present new results that do not suffer from these limitations and we propose an efficient implementation based on conjugate gradient. The resulting inexact HQ algorithm sets a threshold on the accuracy of the computed solutions to the inner systems; it converges globally to a stationary point of the objective under a minimal set of conditions ubiquitous in image reconstruction. We investigate the behavior of the inexact HQ algorithm in the context of limited-data tomography. Our experiments show that it performs well in various nonconvex scenarios and converges to solutions accurate to full machine precision.

We present new global convergence results for half-quadratic optimization in the context of image reconstruction. In particular, we do not assume that the inner optimization problem is solved exactly and we include the problematic cases where the objective function is nonconvex and has a continuum of stationary points. The inexact algorithm is modeled by a set-valued map defined from the majorization-minimization interpretation of half-quadratic optimization, and our main convergence results are based on the Kurdyka-Lojasiewicz inequality. We also propose a practical implementation that uses the conjugate gradient method and whose efficiency is illustrated by numerical experiments.

Cardiac tractography consists in generating fiber paths for representing and visualizing cardiac fiber architecture. In this process, a large redundant number of fiber paths are usually generated in order to cover the whole myocardium. However, it is difficult to render and visualize dense fibers, and this poses problems for further qualitative and quantitative analysis of cardiac fiber structure. We propose a sparse cardiac fiber structure representation to highlight the main cardiac architecture and perform effective visualization and quantitative analysis. To this end, a fiber oriented approach, called fiber merging, is developed that allows us to obtain simplified cardiac fiber architecture. Fiber merging is achieved by averaging and connecting fiber segments according to a cell grid. Experiments on real cardiac data show that the produced simplified fiber architecture can highlight the cardiac fiber configuration.

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.

We develop a globally convergent deconvolution algorithm for perfusion dynamic susceptibility contrast MRI applied to stroke. This algorithm includes, in addition to temporal and spatial regularization terms, a non-negativity constraint. Experiments on real data show performance improvements with the non-negativity constraint in the temporal regularization context.

The role of 2D matrix arrays in the realization of real time 3D ultrasound imaging is crucial as the latter permit to acquire a complete volume. The lack of control systems for these arrays containing thousands of elements and the dimension of connection cables for such a number of elements are the main obstacles to the use of these technologies. The sparse array techniques present a solid candidate to solve these technological limitations but give rise to beam pattern deteriorations in terms of energy loss and unwanted lobes apparition. The irregular positioning of the array elements permits to drastically decrease the unwanted lobes while the energy loss may be compensated by increasing the element size. However, as elements get larger, the directivity decreases. To solve this directivity reduction, we propose to combine elements of different sizes in such a manner that a broad directivity is maintained using small elements whereas the energy loss may be compensated by wide ones. Additionally, an innovative simulated annealing based algorithm is used to refine the array beam profile when the active element number is set to 256. The results compared to those obtained by the basic non-grid array show significant improvement in the beam pattern.

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.

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.

This paper studies and evaluates the feasibility and the performance of different level interpolations for improving spatial resolution of diffusion tensor magnetic resonance imaging (DT-MRI or DTI). In particular, the following techniques are investigated: anisotropic interpolation operating on scalar gray-level images, log-Euclidean interpolation method, and the quaternion interpolation method, which operate on diffusion tensor fields. The performance is evaluated both qualitatively and quantitatively using criteria such as tensor determinant, fractional anisotropy (FA), mean diffusivity (MD), fiber length, etc. We conclude that tensor field interpolations allow avoiding undesirable swelling effect in DTI, which is not the case with scalar gray-level interpolation, and that scalar gray-level image interpolation and log-Euclidean tensor field interpolation suffer from decrease in FA and MD, which may mislead the interpretation of the clinical parameters FA and MD. In contrast, the quaternion tensor field interpolation avoids such FA and MD decrease, which suggests its use for clinical applications.

One of the next challenges of DTI in cardiac domain is to assess moving fibers. An important bottleneck is the acquisition time due to the angular resolution required to extract accurate myocardial fibers. Therefore, the creation of a method able to extract reliable fibers from low angular resolution is a key point to accelerate DTI acquisition, consequently a considerable step toward temporal fibers. In this study we introduce a graph-based method for fiber tracking and evaluate its performances.

Diffusion tensor magnetic resonance imaging (DT-MRI) is becoming a prospective imaging technique in clinical applications because of its potential for in vivo and non-invasive characterization of tissue organization. However, the acquisition of diffusion-weighted images (DWIs) is often corrupted by noise and artifacts, and the intensity of diffusion-weighted signals is weaker than that of classical magnetic resonance signals. In this paper, we propose a new denoising method for DT-MRI, called structure-adaptive sparse denoising (SASD), which exploits self-similarity in DWIs. We define a similarity measure based on the local mean and on a modified structure-similarity index to find sets of similar patches that are arranged into three-dimensional arrays, and we propose a simple and efficient structure-adaptive window pursuit method to achieve sparse representation of these arrays. The noise component of the resulting structure-adaptive arrays is attenuated by Wiener shrinkage in a transform domain defined by two-dimensional principal component decomposition and Haar transformation. Experiments on both synthetic and real cardiac DT-MRI data show that the proposed SASD algorithm outperforms state-of-the-art methods for denoising images with structural redundancy. Moreover, SASD achieves a good trade-off between image contrast and image smoothness, and our experiments on synthetic data demonstrate that it produces more accurate tensor fields from which biologically relevant metrics can then be computed.

Ultrasound imaging is one of the least expensive and safest diagnostic modalities routinely used. An attractive recent development in this field is three-dimensional (3D) imaging with two-dimensional (2D) matrix probes. The difficulty in implementing these probes comes from their large number of elements; for instance, the probe considered in this paper is composed of 1024 elements, whereas the number of channels of most current beamformers ranges from 64 to 256. To reduce the number of active elements, we propose a new sparse array design technique based on simulated annealing. Our method is capable of significantly reducing the number of probe elements as well as the side lobe level in a reasonable amount of computing time. Experiments in the context of hepatic biopsy show that good imaging performance can be obtained with only 177 active elements out of the total of 1024.

Simulated annealing (SA) is a well-known optimal approach to global optimization which is often used in inverse treatment planning. However, SA generally converges very slowly and many acceleration techniques have been proposed at the expense of a loss of theoretical convergence properties. In this paper, we investigate a recently proposed generalization of SA for dose optimization. This class of algorithms, called stochastic continuation (SC), is theoretically grounded and introduces substantial flexibility in the design of annealing-based methods; simply speaking, SC is a variant SA in which both the generation mechanism and the energy function are allowed to be time-dependent. We propose an SC approach to particle therapy that can be easily applied to a large class of inverse treatment planning problems. Numerical experiments indicate that it outperforms SA both qualitatively and quantitatively.

Simulated annealing (SA) is a generic optimization method that is quite popular because of its ease of implementation and its optimal convergence properties. Still, SA is widely reported to converge very slowly and it is common practice to allow extra freedom in its design at the expense of losing global convergence guarantees.In this paper, we derive simple sufficient conditions for the global convergence of SA when the cost function and the candidate solution generation mechanism are temperature-dependent. These conditions are surprisingly weak - they do not involve the variations of the cost function with temperature - and exponential cooling makes it possible to be arbitrarily close to the best possible convergence exponent of standard SA. (C) 2011 Elsevier B.V. All rights reserved.

The standard approach to image reconstruction is to stabilize the problem by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. The existing attempts to favor the formation of smooth contour lines take the edge field explicitly into account; they either are computationally expensive or produce disappointing results. In this paper, we propose to incorporate the smoothness of the edge field in an implicit way by means of an additional penalty term defined in the wavelet domain. We also derive an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data fidelity term is non-quadratic and the cost function is nonconvex. Numerical experiments show that our technique preserves edge sharpness while smoothing contour lines; it produces visually pleasing reconstructions which are quantitatively better than those obtained without wavelet-domain constraints.

Simulated annealing (SA) and deterministic continuation are well-known generic approaches to global optimization. Deterministic continuation is computationally attractive but produces suboptimal solutions, whereas SA is asymptotically optimal but converges very slowly. In this paper, we introduce a new class of hybrid algorithms which combines the theoretical advantages of SA with the practical advantages of deterministic continuation. We call this class of algorithms stochastic continuation (SC). In a nutshell, SC is a variation of SA in which both the energy function and the communication mechanism are allowed to be time-dependent. We first prove that SC inherits the convergence properties of generalized SA under weak assumptions. Then, we show that SC can be successfully applied to optimization issues raised by the Bayesian approach to signal reconstruction. The considered class of energy functions arises in maximum a posteriori estimation with a Markov random field prior. The associated minimization task is NP-hard and beyond the scope of popular methods such as loopy belief propagation, tree-reweighted message passing, and graph cuts and its extensions. We perform numerical experiments in the context of three-dimensional reconstruction from a very limited number of projections; our results show that SC can substantially outperform both deterministic continuation and SA.

Cardiac diffusion tensor magnetic resonance imaging (DT-MRI) is noise sensitive, and the noise can induce numerous systematic errors in subsequent parameter calculations. This paper proposes a sparse representation-based method for denoising cardiac DT-MRI images. The method first generates a dictionary of multiple bases according to the features of the observed image. A segmentation algorithm based on nonstationary degree detector is then introduced to make the selection of atoms in the dictionary adapted to the image's features. The denoising is achieved by gradually approximating the underlying image using the atoms selected from the generated dictionary. The results on both simulated image and real cardiac DT-MRI images from ex vivo human hearts show that the proposed denoising method performs better than conventional denoising techniques by preserving image contrast and fine structures.