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Digital Breast Tomosynthesis (DBT) is a modern 3D Computed Tomography X-ray technique for the early detection of breast tumors, which is receiving growing interest in the medical and scientific community. Since DBT performs incomplete sampling of data, the image reconstruction approaches based on iterative methods are preferable to the classical analytic techniques, such as the Filtered Back Projection algorithm, providing fewer artifacts. In this work, we consider a Model-Based Iterative Reconstruction (MBIR) method well suited to describe the DBT data acquisition process and to include prior information on the reconstructed image. We propose a gradient-based solver named Scaled Gradient Projection (SGP) for the solution of the constrained optimization problem arising in the considered MBIR method. Even if the SGP algorithm exhibits fast convergence, the time required on a serial computer for the reconstruction of a real DBT data set is too long for the clinical needs. In this paper we propose a parallel SGP version designed to perform the most expensive computations of each iteration on Graphics Processing Unit (GPU). We apply the proposed parallel approach on three different GPU boards, with computational performance comparable with that of the boards usually installed in commercial DBT systems. The numerical results show that the proposed GPU-based MBIR method provides accurate reconstructions in a time suitable for clinical trials. Digital Breast Tomosynthesis (DBT) is a quite recent 3D Computed Tomography (CT) technique providing a vol- umetric breast reconstruction as a stack of 2D images, each representing a cross-sectional slice of the breast itself1. When compared with traditional 2D mammography, DBT has the advantage of separating the breast anatomical tissues, which can be overlapped in a mammography, and this generally reduces false negative diagnosis. At the same time, since the DBT source emits X-rays only from a small number of angles in an arc trajectory, DBT pro- vides a low radiation dose comparable to the radiation dose used in a standard mammography. For this reason, DBT is appealing to the medical and scientific community as a breast screening routine2,3. Until a few years ago, the introduction of DBT in a clinical setting has been slowed down by the absence of effi- cient algorithms for image reconstruction from a limited number of projections. It is well known that the tradi- tional Filtered Back Projection (FBP)4 analytic algorithm amplifies noise and artifacts in the case of low-sampled data3. Iterative algorithms are known to perform better than FBP in the case of tomosynthesis data, but they require computational times incompatible with a clinical use3,5, since in clinical DBT examinations the breast reconstruction must be provided in 40 to 60 seconds. Nevertheless, the recent advent of low cost parallel architec- tures, such as Graphics Processing Units (GPUs), has provided the chance for a remarkable reduction of the image reconstruction time, making iterative methods a realistic alternative to analytic algorithms. Among the different iterative approaches in X-ray CT (see5 for a detailed classification of iterative meth- ods in CT), the so called Model-Based Iterative Reconstruction (MBIR) methods are now getting growing attention. In general, they try to model the acquisition process as accurately as possible, since they take into account system geometry, physical interactions of photons in the projections and prior information about the acquired volume. This approach produces better results than the traditional FBP in terms of quality of the reconstructed image and artifacts reduction, especially for low-dose or limited data X-ray CT. The MBIR methods can be described in a unifying framework as constrained or unconstrained minimization problems5–7, involving a fit-to-data function and a regularizing function, acting as a prior on the solution. A widely used regularization term, proposed in sparse tomography by8 and then used by many authors9–14

We present a discrepancy-like stopping criterium for iterative regularization methods for the solution of linear discrete ill-posed problems. The presented criterium terminates the iterations of the iterative method when the residual norm of the computed solution becomes less or equal to the residual norm of a regularized Truncated Singular Value Decomposition (TSVD) solution. We present two algorithms for the automatic computation of the TSVD residual norm using the Discrete Picard Condition. The first algorithm uses the SVD coefficients while the second one uses the Fourier coefficients. In this work, we mainly focus on the Conjugate Gradient Least Squares method, but the proposed criterium can be used for terminating the iterations of any iterative regularization method. Many numerical tests on some selected one dimensional and image deblurring problems are presented and the results are compared with those obtained by state-of-the-art parameter selection rules. The numerical results show the efficiency and robustness of the proposed criterium.

In this paper we propose a new fast splitting algorithm to solve the Weighted Split Bregman minimization problem in the backward step of an accelerated Forward–Backward algorithm. Beside proving the convergence of the method, numerical tests, carried out on different imaging applications, prove the accuracy and computational efficiency of the proposed algorithm.

We propose a scaled gradient projection algorithm for the reconstruction of 3D X-ray tomographic images from limited data. The problem arises from the discretization of an ill-posed integral problem and, due to the incompleteness of the data, has infinite possible solutions. Hence, by following a regularization approach, we formulate the reconstruction problem as the nonnegatively constrained minimization of an objective function given by the sum of a fit-to-data term and a smoothed differentiable Total Variation function. The problem is challenging for its very large size and because a good reconstruction is required in a very short time. For these reasons, we propose to use a gradient projection method, accelerated by exploiting a scaling strategy for defining gradient-based descent directions and generalized Barzilai–Borwein rules for the choice of the step-lengths. The numerical results on a 3D phantom are very promising since they show the ability of the scaling strategy to accelerate the convergence in the first iterations.

Objective:Immunofluorescence diagnostic systems cost is often dominated by high-sensitivity, low-noise CCD-based cameras which are used to acquire the fluorescence images. In this paper we investigate the use of low-cost CMOS sensors in a point-of-care immunofluorescence diagnostic application for the detection and discrimination of four different serotypes of the Dengue virus in a set of human samples. Methods: A two-phase post-processing software pipeline is proposed which consists in a first image enhancement stage for resolution increasing and segmentation, and a second diagnosis stage for the computation of the output concentrations. Results: We present a novel variational coupled model for the joint super-resolution and segmentation stage, and an automatic innovative image analysis for the diagnosis purpose. A specially designed Forward Backward-based numerical algorithm is introduced and its convergence is proved under mild conditions. We present results on a cheap prototype CMOS camera compared with the results of a more expensive CCD device, for the detection of the Dengue virus with a low-cost OLED light source. The combinationoftheCMOSsensorandthedevelopedpost-processingsoftwareallowstocorrectlyidentifythe different Dengue serotype using an automatized procedure. Conclusions: The results demonstrate that our diagnostic imaging system enables camera cost reduction up to 99%, at an acceptable diagnostic accuracy, with respect to the reference CCD-based camera system. The correct detection and identification of the Dengue serotypes has been confirmed by standard diagnostic methods (RT-PCR and ELISA)

Image reconstruction in spectral digital breast tomosynthesis (DBT) requires solving a large-scale nonlinear inverse problem. Most numerical approaches on real data used a simplified linear (and hence incorrect) mathematical model to reduce the computational costs. The aim of this paper is to consider the use of a nonlinear conjugate gradient method for very large-scale nonlinear least squares problems, and apply it to spectral DBT. Numerical experiments on 3-dimensional phantom images illustrate the effectiveness and efficiency of the proposed scheme.

This paper introduces an efficient method for solving nonconvex penalized minimization problems. The topic is relevant in many imaging problems characterized by sparse data. The proposed method originates from the iterative reweighting l 1 scheme, modified by the automatic update of the regularization parameter on the basis of the behavior of the objective function. Besides proving the convergence of the method, a modified algorithm is obtained and the performance is tested on two different sparse imaging problems. The proposed method can be viewed as a general framework which can be adapted to different one-parameter nonconvex penalty functions and applied to problems characterized by sparse data.

Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.

Digital breast tomosynthesis (DBT) is an imaging technique that allows the reconstruction of a pseudo three-dimensional image of the breast from a finite number of low-dose two-dimensional projections obtained by different x-ray tube angles. An issue that is often ignored in DBT is the fact that an x-ray beam is polyenergetic, i.e. it is composed of photons with different levels of energy. The polyenergetic model requires solving a large-scale, nonlinear inverse problem, which is more expensive than the typically used simplified, linear monoenergetic model. However, the polyenergetic model is much less susceptible to beam hardening artifacts, which show up as dark streaks and cupping (i.e. background nonuniformities) in the reconstructed image. In addition, it has been shown that the polyenergetic model can be exploited to obtain additional quantitative information about the material of the object being imaged. In this paper we consider the multimaterial polyenergetic DBT model, and solve the nonlinear inverse problem with a limited memory BFGS quasi-Newton method. Regularization is enforced at each iteration using a diagonally modified approximation of the Hessian matrix, and by truncating the iterations.

In this paper, we consider the problem of restoring blurred noisy vectorial images where the blurring model involves contributions from the different image channels (cross-channel blur). The proposed method restores the images by solving a sequence of quadratic constrained minimization problems where the constraint is automatically adapted to improve the quality of the restored images. In the present case, the constraint is the Total Variation extended to vectorial images, and the objective function is the $$\ell _2$$l2 norm of the residual. After proving the convergence of the iterative method, we report the results obtained on a wide set of test images, showing that this approach is efficient for recovering nearly optimal results.

The aim of this paper is to present a computational study on scaling techniques in gradient projectiontype (GP-type) methods for deblurring of astronomical images corrupted by Poisson noise. In this case, the imaging problem is formulated as a non-negatively constrained minimization problem in which the objective function is the sum of a fit-to-data term, the Kullback–Leibler divergence, and a Tikhonov regularization term. The considered GP-type methods are formulated by a common iteration formula, where the scaling matrix and the step-length parameter characterize the different algorithms. Within this formulation, both first-order and Newton-like methods are analysed, with particular attention to those implementation features and behaviours relevant for the image restoration problem. The numerical experiments show that suited scaling strategies can enable the GP methods to quickly approximate accurate reconstructions and then are useful for designing effective image deblurring algorithms.

Medical images obtained with emission processes are corrupted by Poisson noise. Aim of the paper is to show that the denoising problem can be efficiently modelled in a Bayesian statistical setting by a nonnegatively constrained minimization problem. The objective function is constituted by a data fitting term, the Kullback-Leibler divergence, plus a regularization term, the Total Variation function, weighted by a regularization parameter. We propose an efficient numerical method for the solution of the constrained problem. The method is a Newton projection method, where the inner system is solved by the Conjugate Gradient method, preconditioned and implemented in an efficient way for this specific application. The numerical results on simulated and real medical images prove the effectiveness of the method, both for the accuracy and the computational cost.

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 reduced-encoding imaging by generalized-series reconstruction (RIGR) method used leads to ill-conditioned linear systems with Hermitian Toeplitz matrix and noisy right-hand side. We analyze the behavior of some regularization methods such as the truncated singular value decomposition (TSVD), the Lavrent'yev regularization method and conjugate gradients (CG) type iterative methods. For what concerns the choice of the regularization parameter, we use some known methods and we propose new heuristic criteria for iterative regularization methods. The simulations are carried on test problems obtained from real data acquired on a 1.5 T Phillips system. © 2002 Elsevier Science Inc. All rights reserved.

Summary Objective:Immunofluorescence diagnostic systems cost is often dominated by high-sensitivity, low-noise CCD-based cameras which are used to acquire the fluorescence images. In this paper we investigate the use of low-cost CMOS sensors in a point-of-care immunofluorescence diagnostic application for the detection and discrimination of four different serotypes of the Dengue virus in a set of human samples. Methods: A two-phase post-processing software pipeline is proposed which consists in a first image enhancement stage for resolution increasing and segmentation, and a second diagnosis stage for the computation of the output concentrations. Results: blackWe present a novel variational coupled model for the joint super-resolution and segmentation stage, and an automatic innovative image analysis for the diagnosis purpose. A specially designed Forward Backward-based numerical algorithm is introduced and its convergence is proved under mild conditions. We present results on a cheap prototype CMOS camera compared with the results of a more expensive CCD device, for the detection of the Dengue virus with a low-cost OLED light source. The combination of the CMOS sensor and the developed post-processing software allows to correctly identify the different Dengue serotype using an automatized procedure. Conclusions: The results demonstrate that our diagnostic imaging system enables camera cost reduction up to 99%, at an acceptable diagnostic accuracy, with respect to the reference CCD-based camera system. The correct detection and identification of the Dengue serotypes has been confirmed by standard diagnostic methods (RT-PCR and ELISA). This article is protected by copyright. All rights reserved.

We consider in this paper the problem of reconstructing 3D Computed Tomography images from limited data. The problem is modeled as a nonnegatively constrained minimization problem of very large size. In order to obtain an acceptable image in short time, we propose a scaled gradient projection method, accelerated by exploiting a suitable scaling matrix and efficient rules for the choice of the step-length. In particular, we select the step-length either by alternating Barzilai-Borwein rules or by exploiting a limited number of back gradients for approximating second-order information. Numerical results on a 3D Shepp-Logan phantom are presented and discussed.

In this paper we propose an ε-subgradient method for solving a constrained minimization problem arising in super-resolution imaging applications. The method, compared to the state-of-the-art methods for single image super-resolution on some test problems, proves to be very efficient, both for the reconstruction quality and the computational time.

In this paper we present a parallel application for image restoration. The problem is relevant in some application fields, such as medicine or astronomy, and has a very high computational complexity so that it is difficult to solve it on scalar computers. The algorithm is based on data parallelism, that is realized with an adaptive decomposition of the image spatial domain for a class of degradation functions. We discuss the implementation on a cluster of six workstations connected through Ethernet network and on a Cray T3E with 128 processors. The results obtained with different images and different number of tasks show good scalability and speed-up.

The aim of this article is to present some numerical methods for the reconstruction of dynamic magnetic resonance images using a priori information. The methods are presented through a unifying algorithmic framework based on a generalized series representation of the dynamic image. We propose, in particular, the use of B-spline basis functions for the image representation and the use of regularization methods for the solution of the systems arising in the reconstruction process. Numerical experiments on both simulated and real data show that the proposed methods improve the quality of the reconstructed dynamic images over the traditional methods.

In this work we propose the use of B-spline functions for the parametric representation of high resolution images from low sampled data in the Fourier domain. Traditionally, exponential basis functions are employed in this situation, but they produce artifacts and amplify the noise on the data. We present the method in an algorithmic form and carefully consider the problem of solving the ill-conditioned linear system arising from the method by an efficient regularization method. Two applications of the proposed method to dynamic Magnetic Resonance images are considered. Dynamic Magnetic Resonance acquires a time series of images of the same slice of the body; in order to fasten the acquisition, the data are low sampled in the Fourier space. Numerical experiments have been performed both on simulated and real Magnetic Resonance data. They show that the B-splines reduce the artifacts and the noise in the representation of high resolution Magnetic Resonance images from low sampled data.

This paper presents a regularization method combined with bivariate spline functions for approximating experimental data affected by degradation errors. Some issues concerning the implementation of the method are given. Numerical experiments on two test problems from different fields of application are presented and analized.

This work addresses the problem of Magnetic Resonance Image Reconstruction from highly sub-sampled measurements in the Fourier domain. It is modeled as a constrained minimization problem, where the objective function is a non-convex function of the gradient of the unknown image and the constraints are given by the data fidelity term. We propose an algorithm, Fast Non Convex Reweighted (FNCR), where the constrained problem is solved by a reweighting scheme, as a strategy to overcome the non-convexity of the objective function, with an adaptive adjustment of the penalization parameter. We propose a fast iterative algorithm and we can prove that it converges to a local minimum because the constrained problem satisfies the Kurdyka-Lojasiewicz property. Moreover the adaptation of non convex l0 approximation and penalization parameters, by means of a continuation technique, allows us to obtain good quality solutions, avoiding to get stuck in unwanted local minima. Some numerical experiments performed on MRI sub-sampled data show the efficiency of the algorithm and the accuracy of the solution.

In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of different incident angles in order to reconstruct a pseudo-3D representation (i.e., a set of 2D slices) of the object. In this paper we describe some mathematical models for polyenergetic digital breast tomosynthesis image reconstruction that explicitly takes into account various materials composing the object and the polyenergetic nature of the x-ray beam. A polyenergetic model helps to reduce beam hardening artifacts, but the disadvantage is that it requires solving a large-scale nonlinear ill-posed inverse problem. We formulate the image reconstruction process (i.e., the method to solve the ill-posed inverse problem) in a nonlinear least squares framework, and use a Levenberg-Marquardt scheme to solve it. Some implementation details are discussed, and numerical experiments are provided to illustrate the performance of the methods.

This paper is focused on the solution of the blind deconvolution problem, here modeled as a separable nonlinear least squares problem. The well known ill-posedness, both on recovering the blurring operator and the true image, makes the problem really difficult to handle. We show that, by imposing appropriate constraints on the variables and with well chosen regularization parameters, it is possible to obtain an objective function that is fairly well behaved. Hence, the resulting nonlinear minimization problem can be effectively solved by classical methods, such as the Gauss-Newton algorithm.