Alternative Minimization Research Articles

Joint image registration and point spread function estimation for the super-resolution of satellite images

Image registration and point spread function (PSF) estimation are both key steps in image super-resolution (SR). Traditionally, these two steps are treated independently, which is adequate for natural images. However, for satellite images, which commonly suffer from focal plane distortion and unrecorded spacecraft jitter, it is always difficult to achieve satisfactory image registration or PSF estimation. Consequently, the errors brought by these two processes significantly affect each other and degrade the quality of the subsequent high-resolution (HR) reconstruction. In this paper, a novel joint image registration and PSF estimation method is proposed to produce HR images from a set of degraded low-resolution (LR) satellite images. The joint SR approach is formulated as a convex optimization problem which minimizes the combination of these two parts. It is aimed at achieving PSF estimation and registration simultaneously and progressively, to handle the error in different levels. In addition, the proposed method adopts a more generic observation model, including both geometric motion and radiation difference, which makes the model more universal. Moreover, an iterative scheme based on alternating minimization (AM) is developed to solve the presented cost function via simultaneous low-rank and total variation (LRTV) regularizations. The experimental results confirm the effectiveness of the proposed method on both simulated data and real satellite images.

Recurrent networks with soft-thresholding nonlinearities for lightweight coding

A long-standing and influential hypothesis in neural information processing is that early sensory networks adapt themselves to produce efficient codes of afferent inputs. Here, we show how a nonlinear recurrent network provides an optimal solution for the efficient coding of an afferent input and its history. We specifically consider the problem of producing lightweight codes, ones that minimize both ℓ1 and ℓ2 constraints on sparsity and energy, respectively. When embedded in a linear coding paradigm, this problem results in a non-smooth convex optimization problem. We employ a proximal gradient descent technique to develop the solution, showing that the optimal code is realized through a recurrent network endowed with a nonlinear soft thresholding operator. The training of the network connection weights is readily achieved through gradient-based local learning. If such learning is assumed to occur on a slower time-scale than the (faster) recurrent dynamics, then the network as a whole converges to an optimal set of codes and weights via what is, in effect, an alternative minimization procedure. Our results show how the addition of thresholding nonlinearities to a recurrent network may enable the production of lightweight, history-sensitive encoding schemes.

On the convergence of alternating minimization methods in variational PGD

The approximation of solutions to partial differential equations by tensorial separated representations is one of the most efficient numerical treatment of high dimensional problems. The key step of such methods is the computation of an optimal low-rank tensor to enrich the obtained iterative tensorial approximation. In variational problems, this step can be carried out by alternating minimization (AM) technics, but the convergence of such methods presents a real challenge. In the present work, the convergence of rank-one AM algorithms for a class of variational linear elliptic equations is studied. More precisely, we show that rank-one AM-sequences are in general bounded in the ambient Hilbert tensor space and are compact if a uniform non-orthogonality condition between iterates and the reaction term is fulfilled. In particular, if a rank-one AM-sequence is weakly convergent then it converges strongly and the common limit is a solution of the rank-one optimization problem.

A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

Abstract We consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic L 2 {L^{2}} -penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of W 1 , 2 ⁢ ( 0 , T ; L 2 ) {W^{1,2}(0,T;L^{2})} .

Stochastic Multidimensional Scaling

Multidimensional scaling (MDS) is a popular dimensionality reduction technique that has been widely used for network visualization and cooperative localization. However, the traditional stress minimization formulation of MDS necessitates the use of batch optimization algorithms that are not scalable to large-sized problems. This paper considers an alternative stochastic stress minimization framework that is amenable to incremental and distributed solutions. A novel linear-complexity stochastic optimization algorithm is proposed that is provably convergent and simple to implement. The applicability of the proposed algorithm to localization and visualization tasks is also expounded. Extensive tests on synthetic and real datasets demonstrate the efficacy of the proposed algorithm.

Convergence of alternate minimization schemes for phase-field fracture and damage

We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of parametrized [Formula: see text]-evolutions, introducing a suitable family of “intrinsic energy norms”. Further, we show that the limit evolution satisfies Griffith’s criterion, for a phase-field energy release, and that the irreversibility constraint is thermodynamically consistent.

MRI reconstruction using a joint constraint in patch-based total variational framework

Compressed sensing (CS) as an efficient means has been widely applied in magnetic resonance imaging (MRI). As a regularization term to enforce the sparsity in the finite difference domain, the conventional total variation (TV) has been introduced in this field, where the staircase effect is presented. To overcome this issue, a new framework in the difference domain called joint constraint patch-based total variation (JCTV) is proposed. First, the image patch is utilized as the unit for TV norm to improve the adaptativity. Second, JCTV introduces a new nonlocal constraint term that exploits the estimated coefficients of the fully sampled image via linear minimum mean square error (LMMSE) criterion to improve the reconstruction performance. Finally, an alternative minimization algorithm is developed to seek the solution. Extensive experiments on a set of in vivo MR images demonstrate that the proposed algorithm outperforms the state-of-the-art approaches in terms of peak signal-to-noise ratio and visual quality.

Multi-GPU Acceleration of Branchless Distance Driven Projection and Backprojection for Clinical Helical CT

Abstract Model-based image reconstruction (MBIR) techniques have the potential to generate high quality images from noisy measurements and a small number of projections which can reduce the x-ray dose in patients. These MBIR techniques rely on projection and backprojection to refine an image estimate. One of the widely used projectors for these modern MBIR based technique is called branchless distance driven (DD) projection and backprojection. While this method produces superior quality images, the computational cost of iterative updates keeps it from being ubiquitous in clinical applications. In this paper, we provide several new parallelization ideas for concurrent execution of the DD projectors in multi-GPU systems using CUDA programming tools. We have introduced some novel schemes for dividing the projection data and image voxels over multiple GPUs to avoid runtime overhead and inter-device synchronization issues. We have also reduced the complexity of overlap calculation of the algorithm by eliminating the common projection plane and directly projecting the detector boundaries onto image voxel boundaries. To reduce the time required for calculating the overlap between the detector edges and image voxel boundaries, we have proposed a pre-accumulation technique to accumulate image intensities in perpendicular 2D image slabs (from a 3D image) before projection and after backprojection to ensure our DD kernels run faster in parallel GPU threads. For the implementation of our iterative MBIR technique we use a parallel multi-GPU version of the alternating minimization (AM) algorithm with penalized likelihood update. The time performance using our proposed reconstruction method with Siemens Sensation 16 patient scan data shows an average of 24 times speedup using a single TITAN X GPU and 74 times speedup using 3 TITAN X GPUs in parallel for combined projection and backprojection.

Multi-GPU Acceleration of Branchless Distance Driven Projection and Backprojection for Clinical Helical CT.

Model-based image reconstruction (MBIR) techniques have the potential to generate high quality images from noisy measurements and a small number of projections which can reduce the x-ray dose in patients. These MBIR techniques rely on projection and backprojection to refine an image estimate. One of the widely used projectors for these modern MBIR based technique is called branchless distance driven (DD) projection and backprojection. While this method produces superior quality images, the computational cost of iterative updates keeps it from being ubiquitous in clinical applications. In this paper, we provide several new parallelization ideas for concurrent execution of the DD projectors in multi-GPU systems using CUDA programming tools. We have introduced some novel schemes for dividing the projection data and image voxels over multiple GPUs to avoid runtime overhead and inter-device synchronization issues. We have also reduced the complexity of overlap calculation of the algorithm by eliminating the common projection plane and directly projecting the detector boundaries onto image voxel boundaries. To reduce the time required for calculating the overlap between the detector edges and image voxel boundaries, we have proposed a pre-accumulation technique to accumulate image intensities in perpendicular 2D image slabs (from a 3D image) before projection and after backprojection to ensure our DD kernels run faster in parallel GPU threads. For the implementation of our iterative MBIR technique we use a parallel multi-GPU version of the alternating minimization (AM) algorithm with penalized likelihood update. The time performance using our proposed reconstruction method with Siemens Sensation 16 patient scan data shows an average of 24 times speedup using a single TITAN X GPU and 74 times speedup using 3 TITAN X GPUs in parallel for combined projection and backprojection.

Redefining short bowel syndrome in the 21st century.

In 1968, Wilmore and Dudrick reported an infant sustained by parenteral nutrition (PN) providing a potential for survival for children with significant intestinal resections. Increasing usage of TPN over time led to some patients developing Intestinal Failure Associated Liver Disease (IFALD), a leading cause of death and indication for liver/intestinal transplant. Over time, multidisciplinary teams called Intestinal Rehabilitation Programs (IRPs) began providing meticulous and innovative management. Usage of alternative lipid emulsions and lipid minimization strategies have resulted in the decline of IFALD and an increase in long-term and transplant-free survival, even in the setting of ultrashort bowel (< 20 cm). Autologous bowel reconstructive surgeries, such as the serial tapering enteroplasty procedure, have increased the likelihood of achieving enteral autonomy. Since 2007, the number of pediatric intestinal transplants performed has sharply declined and likely attributed to the newer innovations healthcare. Recent data support the need for changes in the listing criteria for intestinal transplantation given the overall improvement in outcomes. Over the last 50 y, the diagnosis of short bowel syndrome has changed from a death sentence to one of hope with a vast improvement of quality of life and survival.

An overview of the modelling of fracture by gradient damage models

The paper is devoted to gradient damage models which allow us to describe all the process of degradation of a body including the nucleation of cracks and their propagation. The construction of such model follows the variational approach to fracture and proceeds into two stages: (1) definition of the energy; (2) formulation of the damage evolution problem. The total energy of the body is defined in terms of the state variables which are the displacement field and the damage field in the case of quasi-brittle materials. That energy contains in particular gradient damage terms in order to avoid too strong damage localizations. The formulation of the damage evolution problem is then based on the concepts of irreversibility, stability and energy balance. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Moreover, the variational formulation leads to a natural numerical method based on an alternate minimization algorithm. Several numerical examples illustrate the ability of this approach to account for all the process of fracture including a 3D thermal shock problem where the crack evolution is very complex.

Simultaneous tumor segmentation, image restoration, and blur kernel estimation in PET using multiple regularizations

Accurate tumor segmentation from PET images is crucial in many radiation oncology applications. Among others, partial volume effect (PVE) is recognized as one of the most important factors degrading imaging quality and segmentation accuracy in PET. Taking into account that image restoration and tumor segmentation are tightly coupled and can promote each other, we proposed a variational method to solve both problems simultaneously in this study. The proposed method integrated total variation (TV) semi-blind de-convolution and Mumford–Shah segmentation with multiple regularizations. Unlike many existing energy minimization methods using either TV or L2 regularization, the proposed method employed TV regularization over tumor edges to preserve edge information, and L2 regularization inside tumor regions to preserve the smooth change of the metabolic uptake in a PET image. The blur kernel was modeled as anisotropic Gaussian to address the resolution difference in transverse and axial directions commonly seen in a clinic PET scanner. The energy functional was rephrased using the Γ-convergence approximation and was iteratively optimized using the alternating minimization (AM) algorithm. The performance of the proposed method was validated on a physical phantom and two clinic datasets with non-Hodgkin's lymphoma and esophageal cancer, respectively. Experimental results demonstrated that the proposed method had high performance for simultaneous image restoration, tumor segmentation and scanner blur kernel estimation. Particularly, the recovery coefficients (RC) of the restored images of the proposed method in the phantom study were close to 1, indicating an efficient recovery of the original blurred images; for segmentation the proposed method achieved average dice similarity indexes (DSIs) of 0.79 and 0.80 for two clinic datasets, respectively; and the relative errors of the estimated blur kernel widths were less than 19% in the transversal direction and 7% in the axial direction.

A New Algorithm for Optimizing TV-Based PolSAR Despeckling Model

The Wishart fidelity and total variation (TV) based variational model (WisTV) with the positive definite (PD) constraint has shown to be effective for the whole PolSAR covariance data speckle reduction. However, the existing algorithms for solving the WisTV model only give approximation solutions by projecting the results onto the set of PD matrices, and their parameters depend strongly on the data. The purpose of this letter is to propose a new optimization algorithm to address the issues. To keep the uniformity of the parameters for different PolSAR data, a sigmoid function-based normalization method is designed, which ensures the applicability of the WisTV model for the normalized data. By using the orthogonal decomposition of the PD variables, the WisTV model is converted into an unconstrained optimization problem which is further transformed into a multivariable problem based on the equivalent representations of the trace and logdet functions. The alternative minimization technique is then utilized to solve the final optimization problem. The subproblems for each individual variable are convex and their solutions have explicit expressions. Moreover, the computational complexity of the algorithm is discussed. Experimental results on both synthetic and real PolSAR data demonstrate the validity of the proposed algorithm.

Linear and nonlinear solvers for variational phase-field models of brittle fracture

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5–6× faster on a majority of the test cases considered.

TU‐H‐CAMPUS‐IeP3‐01: Simultaneous PET Restoration and PET/CT Co‐Segmentation Using a Variational Method

Purpose:PET images are usually blurred due to the finite spatial resolution, while CT images suffer from low contrast. Segment a tumor from either a single PET or CT image is thus challenging. To make full use of the complementary information between PET and CT, we propose a novel variational method for simultaneous PET image restoration and PET/CT images co‐segmentation.Methods:The proposed model was constructed based on the Γ‐convergence approximation of Mumford‐Shah (MS) segmentation model for PET/CT co‐segmentation. Moreover, a PET de‐blur process was integrated into the MS model to improve the segmentation accuracy. An interaction edge constraint term over the two modalities were specially designed to share the complementary information. The energy functional was iteratively optimized using an alternate minimization (AM) algorithm. The performance of the proposed method was validated on ten lung cancer cases and five esophageal cancer cases. The ground truth were manually delineated by an experienced radiation oncologist using the complementary visual features of PET and CT. The segmentation accuracy was evaluated by Dice similarity index (DSI) and volume error (VE).Results:The proposed method achieved an expected restoration result for PET image and satisfactory segmentation results for both PET and CT images. For lung cancer dataset, the average DSI (0.72) increased by 0.17 and 0.40 than single PET and CT segmentation. For esophageal cancer dataset, the average DSI (0.85) increased by 0.07 and 0.43 than single PET and CT segmentation.Conclusion:The proposed method took full advantage of the complementary information from PET and CT images.This work was supported in part by the National Cancer Institute Grants R01CA172638. Shan Tan and Laquan Li were supported in part by the National Natural Science Foundation of China, under Grant Nos. 60971112 and 61375018.

A Non-Local Low-Rank Approach to Enforce Integrability.

We propose a new approach to enforce integrability using recent advances in non-local methods. Our formulation consists in a sparse gradient data-fitting term to handle outliers together with a gradient-domain non-local low-rank prior. This regularization has two main advantages: 1) the low-rank prior ensures similarity between non-local gradient patches, which helps recovering high-quality clean patches from severe outliers corruption and 2) the low-rank prior efficiently reduces dense noise as it has been shown in recent image restoration works. We propose an efficient solver for the resulting optimization formulation using alternate minimization. Experiments show that the new method leads to an important improvement compared with previous optimization methods and is able to efficiently handle both outliers and dense noise mixed together.

On the Implementation of Collaborative TV Regularization: Application to Cartoon+Texture Decomposition

This paper deals with the analysis, implementation, and comparison of several vector-valued total variation (TV) methods that extend the Rudin-Osher-Fatemi variational model to color images. By considering the discrete gradient of a multichannel image as a 3D structure matrix with dimensions corresponding to the spatial extend, the differences to other pixels and the color channels, we introduce in [J. Duran, M. Moeller, C. Sbert, and D. Cremers, 'Collaborative Total Variation: A General Framework for Vectorial TV Models', SIAM Journal on Imaging Sciences, 9(1), pp.116-151, 2016] collaborative sparsity enforcing norms for penalizing the resulting tensor. We call this class of regularizations collaborative total variation (CTV). We first analyze the denoising properties of each collaborative norm for suppressing color artifacts while preserving image features and aligning edges. We then describe the primal-dual hybrid gradient method for solving the minimization problem in detail. The resulting CTV–L2 variational model can successfully be applied to many image processing tasks. On the one hand, an extensive performance comparison of several collaborative norms for color image denoising is provided. On the other hand, we analyze the ability of different CTV methods for decomposing a multichannel image into a cartoon and a textural part. Finally, we also include a short discussion on alternative minimization methods and compare their computational efficiency.

Binary tomography reconstruction from few projections with Total Variation regularization for bone microstructure studies.

Discrete tomography refers to a class of reconstruction methods adapted to discrete-valued images. Many different approaches have been investigated to address the binary case, when a two-phase object is considered. This reconstruction problem is very important in medical or material applications where it is crucial to reduce the number of projections. In this paper, we address the problem of binary image reconstruction for X-ray CT imaging from a small number of projections. We propose a TV (Total Variation) regularization approach and compare the results obtained with or without an additional box convex constraint. The schemes are applied to a simple disk image and to more complex bone cross-sections for various noise levels. The minimization of the regularization functional is performed with the state-of-the-art ADMM (Alternate Direction Minimization Method) algorithm. The methods perform equally well on a simple disk image. The additional box convex constraints improves the reconstruction results for complex structures with fine details.

Efficient AM Algorithms for Stochastic ML Estimation of DOA

The estimation of direction-of-arrival (DOA) of signals is a basic and important problem in sensor array signal processing. To solve this problem, many algorithms have been proposed, among which the Stochastic Maximum Likelihood (SML) is one of the most concerned algorithms because of its high accuracy of DOA. However, the estimation of SML generally involves the multidimensional nonlinear optimization problem. As a result, its computational complexity is rather high. This paper addresses the issue of reducing computational complexity of SML estimation of DOA based on the Alternating Minimization (AM) algorithm. We have the following two contributions. First using transformation of matrix and properties of spatial projection, we propose an efficient AM (EAM) algorithm by dividing the SML criterion into two components. One depends on a single variable parameter while the other does not. Second when the array is a uniform linear array, we get the irreducible form of the EAM criterion (IAM) using polynomial forms. Simulation results show that both EAM and IAM can reduce the computational complexity of SML estimation greatly, while IAM is the best. Another advantage of IAM is that this algorithm can avoid the numerical instability problem which may happen in AM and EAM algorithms when more than one parameter converges to an identical value.

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

This paper proposes a spatial-Radon domain computed tomography (CT) image reconstruction model based on data-driven tight frames (SRD-DDTF). The proposed SRD-DDTF model combines the idea of the joint image and Radon domain inpainting model of Dong, Li, and Shen [J. Sci. Comput., 54 (2013), pp. 333--349] and that of the data-driven tight frames for image denoising [J.-F. Cai, H. Ji, Z. Shen, and G.-B. Ye, Appl. Comput. Harmon. Anal., 37 (2014), p. 89--105]. It is different from existing models in that both the CT image and its corresponding high quality projection image are reconstructed simultaneously using sparsity priors by tight frames that are adaptively learned from the data to provide optimal sparse approximations. An alternative minimization algorithm is designed to solve the proposed model, which is nonsmooth and nonconvex. Convergence analysis of the algorithm is provided. Numerical experiments show that the SRD-DDTF model is superior to the model of Dong, Li, and Shen [J. Sci. Comput., 54 (2013)...