Additive Operator Splitting Research Articles

Detecting Water Bodies on RADARSAT Imagery

This paper presents a novel geodesic active contour (GAC) model based on an edge detector for rapid detection of water bodies from spaceborne synthetic aperture radar (SAR) imagery with high speckle noise. The original edge indicator function based on gradients is replaced by an edge indicator function based on the ratio of exponentially weighted averages (ROEWA) operator. Thus, the capability of edge detection and the accuracy of locating edges are greatly improved, which makes the model more appropriate for SAR images. In addition, an enhancing term is added to the original model's energy function in order to boost the strength for the contour's evolution. An unconditionally stable additive operator splitting (AOS) scheme and a fast algorithm for re-initialization of the level set function are adopted, which not only enhances the model's stability, but also speeds up the model's convergence remarkably. The experimental results on simulated and real RADARSAT-1/-2 images show its efficiency and accuracy.

On Semi-implicit Splitting Schemes for the Beltrami Color Image Filtering

The Beltrami flow is an efficient nonlinear filter, that was shown to be effective for color image processing. The corresponding anisotropic diffusion operator strongly couples the spectral components. Usually, this flow is implemented by explicit schemes, that are stable only for very small time steps and therefore require many iterations. In this paper we introduce a semi-implicit Crank-Nicolson scheme based on locally one-dimensional (LOD)/additive operator splitting (AOS) for implementing the anisotropic Beltrami operator. The mixed spatial derivatives are treated explicitly, while the non-mixed derivatives are approximated in an implicit manner. In case of constant coefficients, the LOD splitting scheme is proven to be unconditionally stable. Numerical experiments indicate that the proposed scheme is also stable in more general settings. Stability, accuracy, and efficiency of the splitting schemes are tested in applications such as the Beltrami-based scale-space, Beltrami denoising and Beltrami deblurring. In order to further accelerate the convergence of the numerical scheme, the reduced rank extrapolation (RRE) vector extrapolation technique is employed.

On the Acceleration of AOS Schemes for Nonlinear Diffusion Filtering

Based on the renowned additive operator splitting (AOS) schemes that ensure equal treatment of all coordinate axis, accelerated AOS schemes for solving regularized Perona-Malik (P-M) equation are presented. These ameliorated AOS schemes are stable unconditionally, consistent with nonlinear parabolic equations under certain circumstance and demonstrate a remarkably economical CPU time. The paper is intended to discuss the properties of the modified AOS schemes in several aspects. Finally, experiments show that as well as the edges and detailed information of images are preserved, the proposed AOS schemes are at least twice more efficient than the conventional AOS schemes.

A two-stage method for oil slick segmentation

In this paper we propose a two-stage algorithm for oil slick segmentation in synthetic aperture radar (SAR) images. In the first stage, we propose a new variational model to reduce speckles in non-textured SAR images. Applications to simulated and real SAR images show that the method is well balanced in the quality of the conventional criteria. Then, in the second stage, we use the fast Chan–Vese (CV) model and the level set method to segment the oil slick in the de-speckled SAR image. The additive operator splitting (AOS) scheme is used in the numerical implementation to improve computational efficiency. Experimental results show that our two-stage algorithm is effective for oil slick segmentation in SAR images.

Orientation-Matching Minimization for Image Denoising and Inpainting

In this paper, we propose an orientation-matching functional minimization for image denoising and image inpainting. Following the two-step TV-Stokes algorithm (Rahman et al. in Scale space and variational methods in computer vision, pp. 473---482, Springer, Heidelberg, 2007; Tai et al. in Image processing based on partial differential equations, pp. 3---22, Springer, Heidelberg, 2006; Bertalmio et al. in Proc. conf. comp. vision pattern rec., pp. 355---362, 2001), a regularized tangential vector field with zero divergence condition is first obtained. Then a novel approach to reconstruct the image is proposed. Instead of finding an image that fits the regularized normal direction from the first step, we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction. This functional yields a new nonlinear partial differential equation (PDE) for reconstructing denoised and inpainted images. The equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field, providing reconstructed images which have sharp edges and smooth regions. The additive operator splitting (AOS) scheme is used for discretizing Euler-Lagrange equations. We present the results of various numerical experiments that illustrate the improvements obtained with the new functional.

Moving objects segmentation based on piecewise constant Mumford-Shah model solving by additive operator splitting

We propose a moving objects segmentation method for color image sequences based on the piecewise constant Mumford-Shah model (also known as the C-V model) solving by the semi-implicit additive operator splitting (AOS) scheme, which is unconditionally stable, fast, and easy to implement. The method first uses the Gaussian mixture model for background modeling and then subtracts the background to obtain the moving regions that are the handling objects of our method. As a result of the introduction of the AOS scheme, we could use a rather large time step and still maintain the stability of the evolution process. Additionally, the method can easily be parallelized because the AOS scheme decomposes the equations into a sequence of one-dimensional (1-D) systems. The experimental results demonstrate that under real moving objects video tests, the AOS scheme accelerates the evolution of the curve and significantly reduces the number of iterations, and also demonstrates the validity of our method.

Anisotropic Diffusion for Hyperspectral Imagery Enhancement

Among all enhancement techniques being developed over the past two decades, anisotropic diffusion has received much attention and experienced significant developments, with promising results and applications in various specific domains. The elegant property of the technique is that it can enhance images by reducing undesirable intensity variability within the objects in the image, while improving SNR and enhancing the contrast of the edges in scalar and, more recently, vector-valued images, such as color, multispectral, and hyperspectral imagery. In this paper, we present an alternative hyperspectral anisotropic diffusion scheme that takes into account the recent advances and the specificities of hyperspectral remote sensing. In addition, the proposed anisotropic diffusion algorithm can improve the classification accuracy of hyperspectral imagery by reducing the spatial and spectral variability of the image, while preserving the edges of objects. It is also revealed that the additive operator splitting scheme of our method can increase computer efficiency. Qualitative experiments, based on a real hyperspectral remote sensing image, show significant improvements in visual effects when using our method. Quantitative analyses, based on classification accuracies, confirm the superiority and validity of the proposed diffusion algorithm.

On Two Multigrid Algorithms for Modeling Variational Multiphase Image Segmentation

In this paper, we present two related multigrid algorithms for multiphase image segmentation. Algorithm I solves the model by Vese-Chan. We first generalize our recently developed multigrid method to this multiphase segmentation model (MG1); we also give a local Fourier analysis for the local smoother which leads to a new and more effective smoother. Although MG1 is found many magnitudes faster than the fast method of additive operator splitting (AOS), both algorithms are not robust with regard to the initial guess. To overcome this dependence on the initial guess, we consider a hierarchical segmentation model which achieves multiphase segmentation by repeated use of the Chan-Vese two-phase model; our Algorithm II solves this model by a multigrid algorithm (MG2). Numerical experiments show that both algorithms are efficient and in particular MG2 is more robust than MG1 with respect to initial guesses. AMS subject classifications: 68U10, 65F10, 65K10.

Design of piezoelectric actuators using a multiphase level set method of piecewise constants

This paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi-material piezoelectric actuators with in-plane motion. First, an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain. Compared with standard level set methods using n scalar functions to represent 2 n phases, each constant value in the present method denotes one material phase and 2 n phases can be represented by 2 n pre-defined constants. Thus, only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities. In the context of designing smart actuators with in-plane motions, the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints. Thus, the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi-implicit scheme with additive operator splitting (AOS) algorithm. In such a way, the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized. The compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators. The major advantage of the present method is to remove numerical difficulties associated with the solution of the Hamilton–Jacobi equations in most conventional level set methods, such as the CFL condition, the regularization procedure to retain a signed distance level set function and the non-differentiability related to the Heaviside and the Delta functions. Two widely studied examples are chosen to demonstrate the effectiveness of the present method.

A new level set method for systematic design of hinge-free compliant mechanisms

This paper presents a new level set-based method to realize shape and topology optimization of hinge-free compliant mechanisms. A quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism. A semi-implicit scheme with an additive operator splitting (AOS) algorithm is employed to solve the Hamilton–Jacobi partial differential equation (PDE) in the level set method. The design of compliant mechanisms is mathematically represented as a general non-linear programming with a new objective function augmented by the high-order energy term. The structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints. In doing so, it is expected that numerical difficulties such as the Courant–Friedrichs–Lewy (CFL) condition and periodically applied re-initialization procedures in most conventional level set methods can be eliminated. In addition, new holes can be created inside the design domain. The final mechanism configurations consist of strip-like members suitable for generating distributed compliance, and solving the de-facto hinge problem in the design of compliant mechanisms. Two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms.

Geodesic active contour under geometrical conditions: theory and 3D applications

In this paper, we propose a new scheme for both detection of boundaries and fitting of geometrical data based on a geometrical partial differential equation, which allows a rigorous mathematical analysis. The model is a geodesic-active-contour-based model, in which we are trying to determine a curve that best approaches the given geometrical conditions (for instance a set of points or curves to approach) while detecting the object under consideration. Formal results concerning existence, uniqueness (viscosity solution) and stability are presented as well. We give the discretization of the method using an additive operator splitting scheme which is very efficient for this kind of problem. We also give 2D and 3D numerical examples on real data sets.

A semi-implicit level set method for structural shape and topology optimization

This paper proposes a new level set method for structural shape and topology optimization using a semi-implicit scheme. Structural boundary is represented implicitly as the zero level set of a higher-dimensional scalar function and an appropriate time-marching scheme is included to enable the discrete level set processing. In the present study, the Hamilton–Jacobi partial differential equation (PDE) is solved numerically using a semi-implicit additive operator splitting (AOS) scheme rather than explicit schemes in conventional level set methods. The main feature of the present method is it does not suffer from any time step size restriction, as all terms relevant to stability are discretized in an implicit manner. The semi-implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance. Hence, the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency. Resultantly, it allows enhanced relaxation on the time step size originally limited by the Courant–Friedrichs–Lewy (CFL) condition of the explicit schemes. The stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements. Furthermore, the present method avoids additional cost to globally reinitialize the level set function for regularization purpose. It is noted that the periodically applied reinitializations are time-consuming procedures. In particular, the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating, splitting and merging processes, which makes the final design independent of initial guess, and helps reduce the probability of converging to a local minimum. The availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs.

Additive and iterative operator splitting methods and their numerical investigation

As an alternative to the classical splitting methods, two new splitting schemes have been developed recently: the additive and the iterative splitting. In this paper we discuss the most important properties, the advantages and disadvantages of these schemes, and investigate their performance on simple examples as well as on more complex physical models.

Image Segmentation Using Some Piecewise Constant Level Set Methods with MBO Type of Projection

In this work, we are trying to propose fast algorithms for Mumford-Shah image segmentation using some recently proposed piecewise constant level set methods (PCLSM). Two variants of the PCLSM will be considered in this work. The first variant, which we call the binary level set method, needs a level set function which only takes values ±1 to identify the regions. The second variant only needs to use one piecewise constant level set function to identify arbitrary number of regions. For the Mumford-Shah image segmentation model with these new level set methods, one needs to minimize some smooth energy functionals under some constrains. A penalty method will be used to deal with the constraint. AOS (additive operator splitting) and MOS (multiplicative operator splitting) schemes will be used to solve the Euler-Lagrange equations for the minimization problems. By doing this, we obtain some algorithms which are essentially applying the MBO scheme for our segmentation models. Advantages and disadvantages are discussed for the proposed schemes.

Improved Mumford-Shah Functional for Coupled Edge-Preserving Regularization and Image Segmentation

An improved Mumford-Shah functional for coupled edge-preserving regularization and image segmentation is presented. A nonlinear smooth constraint function is introduced that can induce edge-preserving regularization thus also facilitate the coupled image segmentation. The formulation of the functional is considered from the level set perspective, so that explicit boundary contours and edge-preserving regularization are both addressed naturally. To reduce computational cost, a modified additive operator splitting (AOS) algorithm is developed to address diffusion equations defined on irregular domains and multi-initial scheme is used to speed up the convergence rate. Experimental results by our approach are provided and compared with that of Mumford-Shah functional and other edge-preserving approach, and the results show the effectiveness of the proposed method.

The Arithmetic Mean method for solving systems of nonlinear equations in finite differences

In this paper we consider the application of additive operator splitting methods for solving a finite difference nonlinear system of the form F ( u ) = ( I − τA( u )) u − w = 0 generated by the discretization of two dimensional diffusion–convection problems with Neumann boundary conditions. Existence and uniqueness of a solution of this system has been proved under standard assumptions on the matrix A( u ) and the source term w . Using the fact that the matrix A( u ) can be decomposed into different splittings, we develop a nonlinear Arithmetic Mean method and a two-stage iterative method (a fixed-point-Arithmetic Mean method) for solving the system above. The convergence of these methods has been analyzed. Numerical experiments show that the fixed-point-Arithmetic Mean method is rapidly convergent when the diffusion coefficient is weakly nonlinear.

Unsupervised hierarchical image segmentation with level set and additive operator splitting

This paper presents an unsupervised hierarchical segmentation method for multi-phase images based on a single level set (2-phase) method and the semi-implicit additive operator splitting (AOS) scheme which is stable, fast, and easy to implement. The method successively segments image subregions found at each step of the hierarchy using a decision criterion based on the variance of intensity across the current subregion. The segmentation continues until a specified number of levels has been reached. The segmentation information for sub-images at each stage is stored in a tree data structure, and is used for reconstructing the segmented images. The method avoids the complicated governing equations of the multi-phase segmentation approach, and appears to converge in fewer iterations. The method can easily be parallelized because the AOS scheme decomposes the equations into a sequence of one dimensional systems.

High performance cluster computing with 3-D nonlinear diffusion filters

This paper deals with parallelization and implementation aspects of partial differential equation (PDE)-based image processing models for large cluster environments with distributed memory. As an example we focus on nonlinear diffusion filtering which we discretize by means of an additive operator splitting (AOS). We start by decomposing the algorithm into small modules that shall be parallelized separately. For this purpose image partitioning strategies are discussed and their impact on the communication pattern and volume is analyzed. Based on the results we develop an algorithmic implementation with excellent scaling properties on massively connected low-latency networks. Test runs on two different high-end Myrinet clusters yield almost linear speedup factors up to 209 for 256 processors. This results in typical denoising times of 0.4 s for five iterations on a 256×256×128 data cube.

Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives. We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives. We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.

Efficient image segmentation using partial differential equations and morphology

The goal of this paper is to investigate segmentation methods that combine fast preprocessing algorithms using partial differential equations (PDEs) with a watershed transformation with region merging. We consider two well-founded PDE methods: a nonlinear isotropic diffusion filter that permits edge enhancement, and a convex nonquadratic variational image restoration method which gives good denoising. For the diffusion filter, an efficient algorithm is applied using an additive operator splitting (AOS) that leads to recursive and separable filters. For the variational restoration method, a novel algorithm is developed that uses AOS schemes within a Gaussian pyramid decomposition. Examples demonstrate that preprocessing by these PDE techniques significantly improves the watershed segmentation, and that the resulting segmentation method gives better results than some traditional techniques. The algorithm has linear complexity and it can be used for arbitrary dimensional data sets. The typical CPU time for segmenting a 256 2 image on a modern PC is far below 1 s.