Uncertainty Quantification in Image Segmentation Using the Ambrosio–Tortorelli Approximation of the Mumford–Shah Energy

The quantification of uncertainties in image segmentation based on the Mumford–Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio–Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings.

     

A Weakly Supervised Multi-task Ranking Framework for Actor–Action Semantic Segmentation

International Journal of Computer Vision - Modeling human behaviors and activity patterns has attracted significant research interest in recent years. In order to accurately model human behaviors,...     

Accelerated Bregman Method for Linearly Constrained \ell _1–\ell _2 Minimization

In this paper, we consider the linearly constrained $\ell _1$ – $\ell _2$ minimization, and we propose an accelerated Bregman method for solving this minimization problem. The proposed method is based on the extrapolation technique, which is used in accelerated proximal gradient methods proposed by Nesterov and others, and on the equivalence between the Bregman method and the augmented Lagrangian method. A convergence rate of $\mathcal{O }(\frac{1}{k^2})$ is proved for the proposed method when it is applied to solve a more general linearly constrained nonsmooth convex minimization problem. We numerically test our proposed method on a synthetic problem from compressive sensing. The numerical results confirm that our accelerated Bregman method is faster than the original Bregman method.     

Convergence Analysis of Primal–Dual Based Methods for Total Variation Minimization with Finite Element Approximation

We consider a minimization model with total variational regularization, which can be reformulated as a saddle-point problem and then be efficiently solved by the primal–dual method. We utilize the consistent finite element method to discretize the saddle-point reformulation; thus possible jumps of the solution can be captured over some adaptive meshes and a generic domain can be easily treated. Our emphasis is analyzing the convergence of a more general primal–dual scheme with a combination factor for the discretized model. We establish the global convergence and derive the worst-case convergence rate measured by the iteration complexity for this general primal–dual scheme. This analysis is new in the finite element context for the minimization model with total variational regularization under discussion. Furthermore, we propose a prediction–correction scheme based on the general primal–dual scheme, which can significantly relax the step size for the discretization in the time direction. Its global convergence and the worst-case convergence rate are also established. Some preliminary numerical results are reported to verify the rationale of considering the general primal–dual scheme and the primal–dual-based prediction–correction scheme.     

Energy stable and large time-stepping methods for the Cahn–Hilliard equation

We present the numerical methods for the Cahn–Hilliard equation, which describes phase separation phenomenon. The goal of this paper is to construct high-order, energy stable and large time-stepping methods by using Eyre's convex splitting technique. The equation is discretized by using a fourth-order compact difference scheme in space and first-order, second-order or third-order implicit–explicit Runge–Kutta schemes in time. The energy stability for the first-order scheme is proved. Numerical experiments are given to demonstrate the performance of the proposed methods.     

Energy stable compact scheme for Cahn–Hilliard equation with periodic boundary condition

We present a compact scheme to solve the Cahn–Hilliard equation with a periodic boundary condition, which is fourth-order accurate in space. We introduce schemes for two and three dimensions, which are derived from the one-dimensional compact stencil. The energy stability is completely proven for the proposed scheme based on the application of the compact method and well-known convex splitting methods. Detailed proofs of the mass conservation and unique solvability are also established. Numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods.     

Guaranteed Energy Error Estimators for a Modified Robust Crouzeix–Raviart Stokes Element

This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix–Raviart finite element method for the Stokes equations. The modification from Linke (Comput Methods Appl Mech Eng 268:782–800, 2014) is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.     

Nonmonotone Barzilai–Borwein Gradient Algorithm for \ell _{1}-Regularized Nonsmooth Minimization in Compressive Sensing

This study aims to minimize the sum of a smooth function and a nonsmooth \(\ell _{1}\) -regularized term. This problem as a special case includes the \(\ell _{1}\) -regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, and so on. However, the non-differentiability of the \(\ell _{1}\) -norm causes more challenges especially in large problems encountered in many practical applications. This study proposes, analyzes, and tests a Barzilai–Borwein gradient algorithm. At each iteration, the generated search direction demonstrates descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the \(\ell _{1}\) -norm. A nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where each iteration requiring the values of the objective function and the gradient of the smooth term. Under some conditions, the proposed algorithm appears globally convergent. The limited experiments using some nonconvex unconstrained problems from the CUTEr library with additive \(\ell _{1}\) -regularization illustrate that the proposed algorithm performs quite satisfactorily. Extensive experiments for \(\ell _{1}\) -regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms that have been specifically designed in recent years.     

A package-SFERCB-“Segmentation,feature extraction,reduction and classification analysis by both SVM and ANN for brain tumors”

The objective of this experimentation is to develop an interactive CAD system for assisting radiologists in multiclass brain tumor classification. The study is performed on a diversified dataset of 428 post contrast T1-weighted MR images of 55 patients and publically available dataset of 260 post contrast T1-weighted MR images of 10 patients. The first dataset includes primary brain tumors such as Astrocytoma (AS), Glioblastoma Multiforme (GBM), childhood tumor-Medulloblastoma (MED) and Meningioma (MEN), along with secondary tumor-Metastatic (MET). The second dataset consists of Astrocytoma (AS), Low Grade Glioma (LGL) and Meningioma (MEN). The tumor regions are marked by content based active contour (CBAC) model. The regions are than saved as segmented regions of interest (SROIs). 71 intensity and texture feature set is extracted from these SROIs. The features are specifically selected based on the pathological details of brain tumors provided by the radiologist. Genetic Algorithm (GA) selects the set of optimal features from this input set. Two hybrid machine learning models are implemented using GA with support vector machine (SVM) and artificial neural network (ANN) (GA-SVM and GA-ANN) and are tested on two different datasets. GA-SVM is proposed for finding preliminary probability in identifying tumor class and GA-ANN is used for confirmation of accuracy. Test results of the first dataset show that the GA optimization technique has enhanced the overall accuracy of SVM from 79.3% to 91.7% and of ANN from 75.6% to 94.9%. Individual class accuracies delivered by GA-SVM are: AS-89.8%, GBM-83.3%, MED-95.6%, MEN-91.8%, and MET-97.1%. Individual class accuracies delivered by GA-ANN classifier are: AS-96.6%, GBM-86.6%, MED-93.3%, MEN-96%, MET-100%. Similar results are obtained for the second dataset. The overall accuracy of SVM has increased from 80.8% to 89% and that of ANN has increased from 77.5% to 94.1%. Individual class accuracies delivered by GA-SVM are: AS-85.3%, LGL-88.8%, MEN-93%. Individual class accuracies delivered by GA-ANN classifier are: AS-92.6%, LGL-94.4%, MED-95.3%. It is observed from the experiments that GA-ANN classifier has provided better results than GA-SVM. Further, it is observed that along with providing finer results, GA-SVM provides advantage in speed whereas GA-ANN provides advantage in accuracy. The combined results from both the classifiers will benefit the radiologists in forming a better decision for classifying brain tumors.     

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

This paper presents an algorithm for the Quillen–Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k [ x₀, . . . ,x_n ] / I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .     

Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

In this paper, we investigate numerical solution of the diffuse interface model with Peng–Robinson equation of state, that describes real states of hydrocarbon fluids in the petroleum industry. Due to the strong nonlinearity of the source terms in this model, how to design appropriate time discretizations to preserve the energy dissipation law of the system at the discrete level is a major challenge. Based on the “Invariant Energy Quadratization” approach and the penalty formulation, we develop efficient first and second order time stepping schemes for solving the single-component two-phase fluid problem. In both schemes the resulted temporal semi-discretizations lead to linear systems with symmetric positive definite spatial operators at each time step. We rigorously prove their unconditional energy stabilities in the time discrete sense. Various numerical simulations in 2D and 3D spaces are also presented to validate accuracy and stability of the proposed linear schemes and to investigate physical reliability of the target model by comparisons with laboratory data.     

Robust Food–Energy–Water–Environmental Security Management: Stochastic Quasigradient Procedure for Linkage of Distributed Optimization Models under Asymmetric Information and Uncertainty

Cybernetics and Systems Analysis - The paper presents a consistent algorithm for regional and sectoral distributed models’ linkage and optimization under asymmetric information based on...     

Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems

We give a survey on recent developments of stabilization methods based on local projection type. The considered class of problems covers scalar convection–diffusion equations, the Stokes problem and the linearized Navier–Stokes equations. A new link of local projection to the streamline diffusion method is shown. Numerical tests for different type of boundary layers arising in convection–diffusion problems illustrate the stabilizing properties of the method.     

Compact quantum gates for hybrid photon–atom systems assisted by Faraday rotation

We present some compact circuits for a deterministic quantum computing on the hybrid photon–atom systems, including the Fredkin gate and SWAP gate. These gates are constructed by exploiting the optical Faraday rotation induced by an atom trapped in a single-sided optical microcavity. The control qubit of our gates is encoded on the polarization states of the single photon, and the target qubit is encoded on the ground states of an atom confined in an optical microcavity. Since the decoherence of the flying qubit with atmosphere for a long distance is negligible and the stationary qubits are trapped inside single-sided microcavities, our gates are robust. Moreover, ancillary single photon is not needed and only some linear-optical devices are adopted, which makes our protocols efficient and practical. Our schemes need not meet the condition that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity, which is different from the quantum computation with a double-sided optical microcavity. Our calculations show that the fidelities of the two hybrid quantum gates are high with the available experimental technology.     

Bilevel model for production–distribution planning solved by using ant colony optimization

This paper addresses a hierarchical production–distribution planning problem. There are two different decision makers controlling the production and the distribution processes, respectively, that do not cooperate because of different optimization strategies. The distribution company, which is the leader of the hierarchical process, controls the allocation of retailers to each depot and the routes which serve them. In order to supply items to retailers, the distribution company orders from the manufacturing company the items which have to be available at the depots. The manufacturing company, which is the follower of the hierarchical process, reacts to these orders deciding which manufacturing plants will produce them. A bilevel program is proposed to model the problem and an ant colony optimization based approach is developed to solve the bilevel model. In order to construct a feasible solution, the procedure uses ants to compute the routes of a feasible solution of the associated multi-depot vehicle route problem. Then, under the given data on depot needs, the corresponding production problem of the manufacturing company is solved. Global pheromone trail updating is based on the leader objective function, which involves costs of sending items from depots to retailers and costs of acquiring items from manufacturing plants and unloading them into depots. A computational experiment is carried out to analyze the performance of the algorithm.     

A two-parameter model for crack growth simulation by combined FEM–DBEM approach

This paper describes the application of a two-parameters crack growth model, based on the usage of two threshold material parameters (ΔK_th and K_max,th) and on the allowance for residual stresses, introduced at the crack tip by a fatigue load spectrum or by material plastic deformations. The coupled usage of finite element method (FEM) and dual boundary element method (DBEM) is proposed in order to take advantage of the main capabilities of the two methods.The procedure is validated by comparison with available experimental results, in order to assess its capability to predict the retardation phenomena, introduced by a variable load spectrum or by a plastic deformation introduced with a tool on the panel (indentation).In particular two different tests are made: the first test involve a CT specimen undergoing a load spectrum and the second one involve a dented panel undergoing a constant amplitude fatigue load. In both cases a satisfactory numerical–experimental correlation will be proved.The main advantages of the aforementioned procedure are: the simplicity of the crack growth law calibration (few constant amplitude tests are sufficient without the need for any non-physical calibration parameters), and the possibility to simulate residual stress effects on crack propagation with a simplified approach, based on linear elastic fracture mechanics.