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1. Blind polychromatic X-ray CT reconstruction from poisson measurements

   https://doi.org/10.1109/ICASSP.2016.7471805  

   Gu, Renliang ; Dogandzic, Aleksandar ( March 2016 , 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   We develop a sparse image reconstruction method for Poisson-distributed polychromatic X-ray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. We employ our mass-attenuation spectrum parameterization of the noiseless measurements for single-material objects and express the mass-attenuation spectrum as a linear combination of B-spline basis functions of order one. A block coordinate-descent algorithm is developed for constrained minimization of a penalized Poisson negative log-likelihood (NLL) cost function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density-map image; the image sparsity is imposed using a convex total-variation (TV) norm penalty term. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step for estimating the density-map image and a limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) step for estimating the incident-spectrum parameters. We establish conditions for biconvexity of the penalized NLL objective function, which, if satisfied, ensures monotonicity of the NPG-BFGS iteration. We also show that the penalized NLL objective satisfies the Kurdyka-Łojasiewicz property, which is important for establishing local convergence of block-coordinate descent schemes in biconvex optimization problems. Simulation examples demonstrate the performance of the proposed scheme. 

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2. Polychromatic sparse image reconstruction and mass attenuation spectrum estimation via B-spline basis function expansion

   https://doi.org/10.1063/1.4914792  

   Gu, Renliang ; Dogandzic, Aleksandar ( January 2015 , AIP conference proceedings)

   We develop a sparse image reconstruction method for polychromatic tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. To obtain a parsimonious measurement model parameterization, we first rewrite the measurement equation using our mass attenuation parameterization, which has the Laplace integral form. The unknown mass-attenuation spectrum is expanded into basis functions using a B-spline basis of order one. We develop a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and sparsity of the density map image in the wavelet domain. This algorithm alternates between a Nesterov’s proximal-gradient step for estimating the density map image and an active-set step for estimating the incident spectrum parameters. Numerical simulations demonstrate the performance of the proposed scheme. 

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3. Sparse regularization with the ℓ0 norm

   https://doi.org/10.1142/S0219530522500105  

   Xu, Yuesheng ( July 2023 , Analysis and Applications)

   We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter [Formula: see text] multiple of the [Formula: see text] norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction. 

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4. Projected Nesterov's Proximal-Gradient Algorithm for Sparse Signal Recovery

   https://doi.org/10.1109/TSP.2017.2691661  

   Gu, Renliang ; Dogandzic, Aleksandar ( July 2017 , IEEE Transactions on Signal Processing)

   We develop a projected Nesterov’s proximal-gradient (PNPG) approach for sparse signal reconstruction that combines adaptive step size with Nesterov’s momentum acceleration. The objective function that we wish to minimize is the sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a convex regularization term. We apply sparse signal regularization where the signal belongs to a closed convex set within the closure of the domain of the NLL; the convex-set constraint facilitates flexible NLL domains and accurate signal recovery. Signal sparsity is imposed using the ℓ₁-norm penalty on the signal’s linear transform coefficients. The PNPG approach employs a projected Nesterov’s acceleration step with restart and a duality-based inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. Thanks to step-size adaptation, PNPG converges faster than the methods that do not adjust to the local curvature of the NLL. We present an integrated derivation of the momentum acceleration and proofs of O(k⁻²) objective function convergence rate and convergence of the iterates, which account for adaptive step size, inexactness of the iterative proximal mapping, and the convex-set constraint. The tuning of PNPG is largely application independent. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach. 

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5. Parameter choices for sparse regularization with the ℓ1 norm *

   https://doi.org/10.1088/1361-6420/acad22  

   Liu, Qianru ; Wang, Rui ; Xu, Yuesheng ; Yan, Mingsong ( January 2023 , Inverse Problems)

   Abstract We consider a regularization problem whose objective function consists of a convex fidelity term and a regularization term determined by the ℓ 1 norm composed with a linear transform. Empirical results show that the regularization with the ℓ 1 norm can promote sparsity of a regularized solution. The goal of this paper is to understand theoretically the effect of the regularization parameter on the sparsity of the regularized solutions. We establish a characterization of the sparsity under the transform matrix of the solution. When the objective function is block-separable or an error bound of the regularized solution to a known function is available, the resulting characterization can be taken as a regularization parameter choice strategy with which the regularization problem has a solution having a sparsity of a certain level. When the objective function is not block-separable, we propose an iterative algorithm which simultaneously determines the regularization parameter and its corresponding solution with a prescribed sparsity level. Moreover, we study choices of the regularization parameter so that the regularization term can alleviate the ill-posedness and promote sparsity of the resulting regularized solution. Numerical experiments demonstrate that the proposed algorithm is effective and efficient, and the choices of the regularization parameters can balance the sparsity of the regularized solution and its approximation to the minimizer of the fidelity function. 

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