Convex Energy Function Research Articles

Fractional differentiation-based variational level set model for noisy image segmentation without contour initialization

Abstract Noisy image segmentation is a hot topic in image analysis. In this paper, we present a novel methodology for tackling this issue through the integration of fractional differentiation in the frequency domain with a variational level set model (VLSM), which eliminates user-selected initial contours by incorporating the convex energy function. Additionally, the fractional differentiation reduces noises while preserving more detail information. Experiments on synthetic and real noisy images demonstrate that our proposed model surpasses other denoising VLSMs in terms of noise reduction, segmentation accuracy, and efficiency.

Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

Drone routing with energy function: Formulation and exact algorithm

Drone delivery is known as a potential contributor in improving efficiency and alleviating last-mile delivery problems. For this reason, drone routing and scheduling has become a highly active area of research in recent years. Unlike the vehicle routing problem, however, designing drones’ routes is challenging due to multiple operational characteristics including multi-trip operations, recharge planning, and energy consumption calculation. To fill some important gaps in the literature, this paper solves a multi-trip drone routing problem, where drones’ energy consumption is modeled as a nonlinear function of payload and travel distance. We propose adding logical cuts and subgradient cuts in the solution process to tackle the more complex nonlinear (convex) energy function, instead of using the linear approximation method as in the literature, which can fail to detect infeasible routes due to excess energy consumption. We use a 2-index formulation to model the problem and develop a branch-and-cut algorithm for the formulation. Benchmark instances are first generated for this problem. Numerical tests indicate that even though the original model is nonlinear, the proposed approach can solve large problems to optimality. In addition, in multiple instances, the linear approximation model yields routes that under the nonlinear energy model would be energy infeasible. Use of a linear approximation for drone energy leads to differences in energy consumption of about 9% on average compared to the nonlinear energy model.

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.

Factorization-Based Active Contour for Water-Land SAR Image Segmentation via the Fusion of Features

The segmentation of synthetic aperture radar (SAR) water-land images is a very difficult task not only because of strong multiplicative noise but also due to the blurred boundary, irregular shape, and together with diminished contrast. In this paper, we propose a matrix factorization active contour model based on fused features for SAR image segmentation. First, to enhance the robustness, multiple features are utilized. For each pixel location, feature maps (matrix) are constructed by combining wavelet textual features, Gaussian (DoG) filter features, and Gabor filter features via local spectral histogram, which improves spatial pattern and express image structure. Second, the energy function is constructed based on region information and edge information of SAR image. Region information is obtained via matrix factorization theory on the feature matrix. Edge information is obtained by modified the ratio of exponentially weighted averages operator. Then, a convex energy function is proposed to avoid the local minima. A fast dual formulation is introduced for the evolution of the contour. Finally, synthetic and real SAR data are used for verification. The experimental results demonstrate the proposed algorithm is effective for water/land segmentation in SAR images.

Flower image segmentation with PCA fused colored covariance and gabor texture features based level sets

Abstract This paper presents a framework for segmenting flower images captured with a digital camera. Segmenting flowers from images is a complex problem attributed to translation, scaling, rotation with variable backgrounds in each captured image. We propose to solve this problem using principle component analysis based color texture fusion as a prior parameter for level set evolution (FCTAC). First, Color Gabor textures (CGT) and Color Level Covariance Matrix (CLCM) texture features are extracted. Principle component analysis based fusion constructs a color discriminative texture as a knowledge base with convex energy function for active contours without edges. The proposed global segmentation framework with fused textures will avoid the local minimums during curve evolution. We test the proposed segmentation model on the benchmark oxford flower image dataset and our own dataset. The results of FCTAC were tested against the state-of-the-art methods in accuracy and efficiency.

Homogenization of Cahn–Hilliard-type equations via evolutionary $$\varvec{\Gamma }$$-convergence

In these notes we discuss two approaches to evolutionary \(\Gamma \)-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ \(\Gamma \)-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals \(\Gamma \)-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

Shape Optimization for a Class of Semilinear Variational Inequalities with Applications to Damage Models

The present contribution investigates shape optimization problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are first derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimization problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from other optimization problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimization problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system fo...

O.1 - Mass-specific threshold values of global minima for convex energy functionals with an interactive breast mass segmentation model

O.1 - Mass-specific threshold values of global minima for convex energy functionals with an interactive breast mass segmentation model

An active contour model based on fused texture features for image segmentation

Texture image segmentation plays an important role in various computer vision tasks. In this paper, a convex texture image segmentation model is proposed. First, the texture features of Gabor and GLCM (gray level co-occurrence matrix) are extracted for original image. Then, the two kinds of texture features are fused together to effectively construct a discriminative feature space by concatenating with each other. In the image segmentation step, a convex energy function is defined by taking the non-convex vector-valued model of Active Contour without Edges (ACWE) into a global minimization framework (GMAC). The proposed global minimization energy function with fused textures (GMFT) can avoid the existence of local minima in the minimization of the vector-valued ACWE model. In addition, a fast dual formulation is adopted to achieve the efficient contour evolution. The experimental results on synthetic and natural animal images demonstrate that the proposed GMFT model obtains more satisfactory segmentation results compared to two state-of-the-art methods in terms of segmentation accuracy and efficiency.

Nonlocal variational image segmentation models on graphs using the Split Bregman

Variational functionals such as Mumford-Shah and Chan-Vese methods have a major impact on various areas of image processing. After over 10 years of investigation, they are still in widespread use today. These formulations optimize contours by evolution through gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima. In this paper, we propose an image segmentation model in a variational nonlocal means framework based on a weighted graph. The advantages of this model are twofold. First, the convexity global minimum (optimum) information is taken into account to achieve better segmentation results. Second, the proposed global convex energy functionals combine nonlocal regularization and local intensity fitting terms. The nonlocal total variational regularization term based on the graph is able to preserve the detailed structure of target objects. At the same time, the modified local binary fitting term introduced in the model as the local fitting term can efficiently deal with intensity inhomogeneity in images. Finally, we apply the Split Bregman method to minimize the proposed energy functional efficiently. The proposed model has been applied to segmentation of real medical and remote sensing images. Compared with other methods, the proposed model is superior in terms of both accuracy and efficient.

Fast image segmentation by convex minimisation and split Bregman method

A convex minimisation model for image segmentation is proposed. The basic idea of this model is that objects will be detected automatically if background is removed. The local information of every pixel is used to make the model applicable to images with intensity inhomogeneity. Also, by using a convex approximation of the Heaviside function, the convex energy function of the proposed model is obtained. Then it is minimised by applying the split Bregman method, which is a fast technique to obtain the global minimiser. The experimental results demonstrate that the proposed model is powerful in efficiency and accuracy.

A Novel Dictionary Based Computer Vision Method for the Detection of Cell Nuclei

Cell nuclei detection in fluorescent microscopic images is an important and time consuming task in a wide range of biological applications. Blur, clutter, bleed through and partial occlusion of nuclei make individual nuclei detection a challenging task for automated image analysis. This paper proposes a novel and robust detection method based on the active contour framework. Improvement over conventional approaches is achieved by exploiting prior knowledge of the nucleus shape in order to better detect individual nuclei. This prior knowledge is defined using a dictionary based approach which can be formulated as the optimization of a convex energy function. The proposed method shows accurate detection results for dense clusters of nuclei, for example, an F-measure (a measure for detection accuracy) of 0.96 for the detection of cell nuclei in peripheral blood mononuclear cells, compared to an F-measure of 0.90 achieved by state-of-the-art nuclei detection methods.

Efficient Algorithm for Nonconvex Minimization and Its Application to PM Regularization

In image processing, nonconvex regularization has the ability to smooth homogeneous regions and sharpen edges but leads to challenging computation. We propose some iterative schemes to minimize the energy function with nonconvex edge-preserving potential. The schemes are derived from the duality-based algorithm proposed by Bermúdez and Moreno and the fixed point iteration. The convergence is proved for the convex energy function with nonconvex potential and the linear convergence rate is given. Applying the proposed schemes to Perona and Malik's nonconvex regularization, we present some efficient algorithms based on our schemes, and show the approximate convergence behavior for nonconvex energy function. Experimental results are presented, which show the efficiency of our algorithms, including better denoised performance of nonconvex regularization, faster convergence speed, higher calculation precision, lower calculation cost under the same number of iterations, and less implementation time under the same peak signal noise ratio level.

Efficient globally optimal segmentation of cells in fluorescence microscopy images using level sets and convex energy functionals

In high-throughput applications, accurate and efficient segmentation of cells in fluorescence microscopy images is of central importance for the quantification of protein expression and the understanding of cell function. We propose an approach for segmenting cell nuclei which is based on active contours using level sets and convex energy functionals. Compared to previous work, our approach determines the global solution. Thus, the approach does not suffer from local minima and the segmentation result does not depend on the initialization. We consider three different well-known energy functionals for active contour-based segmentation and introduce convex formulations of these functionals. We also suggest a numeric approach for efficiently computing the solution. The performance of our approach has been evaluated using fluorescence microscopy images from different experiments comprising different cell types. We have also performed a quantitative comparison with previous segmentation approaches.

Global optimization of wavelet-domain hidden Markov tree for image segmentation

This work presents a global energy minimization method for multiscale image segmentation using convex optimization theory. The construction of energy function is motivated by the intuition that the larger the entropy, the less a priori information one has on the value of the random variables. First, we represent the wavelet-domain hidden Markov tree (WHMT) model of the original image as a structured energy function, which is proved convex in marginal distributions. Next, we derive the maximum lower bound of the energy function through Lagrange dual transform for the purpose of incorporating marginal constraints into optimization. Finally, a modified belief propagation optimization algorithm is used to perform global energy minimization of the dual convex energy function. Experiments on real image segmentation problems demonstrate the superior performance of this new algorithm when compared with nonconvex ones.

A New Projection-Based Neural Network for Constrained Variational Inequalities

This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples.

Conformal equivalence of triangle meshes

We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of discrete conformal equivalence for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing a convex energy function, whose Hessian turns out to be the well known cot-Laplace operator. This method can also be used to map a surface mesh to a parameter domain which is flat except for isolated cone singularities, and we show how these can be placed automatically in order to reduce the distortion of the parameterization. We present the salient features of the theory and elaborate the algorithms with a number of examples.

A Novel Neural Network for a Class of Convex Quadratic Minimax Problems

Based on the inherent properties of convex quadratic minimax problems, this article presents a new neural network model for a class of convex quadratic minimax problems. We show that the new model is stable in the sense of Lyapunov and will converge to an exact saddle point in finite time by defining a proper convex energy function. Furthermore, global exponential stability of the new model is shown under mild conditions. Compared with the existing neural networks for the convex quadratic minimax problem, the proposed neural network has finite-time convergence, a simpler structure, and lower complexity. Thus, the proposed neural network is more suitable for parallel implementation by using simple hardware units. The validity and transient behavior of the proposed neural network are illustrated by some simulation results.

Min-energy voltage allocation for tree-structured tasks

We study job scheduling on processors capable of running at variable voltage/speed to minimize energy consumption. Each job in a problem instance is specified by its arrival time and deadline, together with required number of CPU cycles. It is known that the minimum energy schedule for n jobs can be computed in O(n3) time, assuming a convex energy function. We investigate more efficient algorithms for computing the optimal schedule when the job sets have certain special structures. When the time intervals are structured as trees, the minimum energy schedule is shown to have a succinct characterization and is computable in time O(P) where P is the tree’s total path length. We also study an on-line average-rate heuristics AVR and prove that its energy consumption achieves a small constant competitive ratio for nested job sets and for job sets with limited overlap. Some simulation results are also given.