Reconstructing Open Surfaces via Graph-Cuts Abstract-A novel graph-cuts-based method is proposed for reconstructing open surfaces from unordered point sets. Through a Boolean operation on the crust around the data set, the open surface problem is translated to a watertight surface problem within a restricted region. Integrating the variational model, Delaunay-based tetrahedral mesh and multiphase technique, the proposed method can reconstruct open surfaces robustly and effectively. Furthermore, a surface reconstruction method with domain decomposition is presented, which is based on the new open surface reconstruction method. This method can handle more general surfaces, such as nonorientable surfaces. The algorithm is designed in a parallel-friendly way and necessary measures are taken to eliminate cracks and conflicts between the subdomains. Numerical examples are included to demonstrate the robustness and effectiveness of the proposed method on watertight, open orientable, open nonorientable surfaces and combinations of such. R ECONSTRUCTING a surface from an unordered point data set has been a significant yet challenging problem in computer graphics for the last decade. As a critical step of creating computer graphics, surface reconstruction fills the gap between machine perception and machine understanding, i.e., the process from discrete scanned data to a continuous model. Due to the development of 3D scanners and the increasing demand of computer graphics, extensive research has been conducted in the surface reconstruction field, much of which was dedicated to the watertight surface reconstruction for its topological simplicity and desirable properties. Open surface reconstruction problems, however, occur often in real applications, such as incomplete scanned data. As a topic which has been overlooked, the open surface reconstruction problem, to some extent, has more significance than the watertight surface problem for its topological generality. The definitions of watertight and open surface are as follows:. A surface is defined as 2-manifold embedded in R 3 . In our study, we restrict a surface to be a compact 2-manifold, which we are referring to by saying a watertight surface. An open surface is defined as a 2-manifold with boundary embedded in R 3 . The boundary of a manifold S is denoted by @S. [1] Most surface reconstruction methods can be categorized into two groups, explicit methods and implicit methods. Explicit methods are mainly local geometric approaches based on Delaunay triangulation and dual Voronoi diagram such as Alpha shape and CRUST algorithm [2] , [3] , [4] , [5] , [6] . One advantage of these methods is their theoretical guarantee that there exists a subcomplex of Delaunay triangulation of the data set, which is homeomorphic to the ground-truth surface given a sufficient sampling. Since these methods are local approaches, the global topological characteristics such as watertight or open, will not affect their performances. Their target is the potential homeomorphic subcomplex embedded in the Delaunay triangulation. The topology of the subcomplex surface does not make any difference. Hence, the explicit method can handle quite a number of open surface cases.. However, the explicit methods are subject to many reconstruction difficulties such as nonuniformity, undersampling, and noises [3] , [7] , [8] . Hence, during the last decade, variational models were brought into the reconstruction field. The reconstruction problem is formulated as a minimization problem of an energy functional defined over surfaces. To minimize an energy functional with respect to the surface, a consistent parametrization of the surface is not always available during the optimization procedure. As a result, researchers turned to the implicit methods [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , such as the level set method, to gain flexibility of representation and mathematical facilities. One important such level set approach based on solving the underlying partial differential equations was proposed by Zhao in [7] , [11] . As an alternative, graph cuts can also minimize the energy functionals over implicitly defined surfaces, and has been successfully applied to the surface reconstruction problem in [20] , [21] , [22] . The main advantages of graph cuts are the efficiency and ability to find global minima. However, the competence of both the level set method and graph cuts is lost on more general topologies [23] . Some reconstruction methods have been proposed to handle open surfaces recently [18] , [19] , [22] , [24] , [25] , [26] which work successfully in practice. However, the robustness or the efficiency is still in study. To sum up, all these methods for open surfaces have some disadvantages and "it is not clear how to devise methods for curves and surfaces that have ends or edges (respectively) within the computational domain" [23] .. In this paper, a novel variational reconstruction method for open surfaces is proposed. Unlike previous methods, our method separates the two types of ill posedness in the open surface problem and handles them sequentially in different ways. The explicit methods are adapted to handle the uncertainty of the surface boundary. The medial axis frequently used in the explicit methods are applied in a different manner. And the implicit method is used to tackle the uncertainty of the data connectivity issue, which contains various reconstruction difficulties such as undersampling and noises. This new methodology not only grants the merits from both explicit and implicit methods, but also provides approaches to more general cases such as the combination of open and watertight surfaces. Following is a description of our algorithm.. In the proposed method, the data set points as well as the properly generated background points are inserted to an unstructured tetrahedral mesh framework in a Delaunay way. In the tetrahedral mesh, a crust is established as the vicinity domain around the data set. A graph dual to the whole mesh is then built according to the energy functional and the minimization is achieved by applying max-flow/ min-cut algorithms. To avoid trivial null result, it is necessary to specify boundary conditions before minimization. This can only be accomplished under the assumption that the domain can be separated into two or more subdomains by the watertight crust, which does not hold any more for an open surface problem.. To tackle this issue, a Boolean operation is proposed to restrict the region of interest within a narrow band. In the proposed method, two crusts with different thickness are built around the data set. The medial axis of the thick crust is to be obtained. One more crust is then built around the boundary of the medial axis. Subsequently, the two crusts around the data set are trimmed by the crust around the boundary. The trimmed thick crust can be separated by the trimmed thin crust. Hence, in the restricted region, i.e., the trimmed thick crust, the trimmed thin crust is watertight such that region growing algorithms and graph-cut techniques can be applied. More details and illustrations of this series of operations are provided in Section 3.. Furthermore, a surface reconstruction method based on domain decomposition is presented. The domain decomposition idea has been applied to computer vision [27] , [28] , [29] . Recently it is found also useful as a robust alternating minimization scheme between overlapped subspaces [30] , [31] . In recent study, the dual graph could be subdivided into subgraphs as well to gain extra efficiency [32] . In the decomposition method, the whole domain is decomposed into several subdomains. The input data are then decomposed into subproblems to be approached separately. Certain measure is taken to eliminate cracks and conflicts effectively. The parallel efficiency may be undermined due to the interaction between subdomains. However, it can be compensated largely by a proper decomposition scheme. The method proposed can handle more general surfaces such as nonorientable surfaces. To the best of our knowledge, this is the first attempt to approach the nonorientable surface reconstruction problem via graph cuts.. The remainder of this paper is organized as follows: In Section 2, a brief review of watertight surface reconstruction based on Delaunay triangulation and graph cuts is given. Section 3 deals with the open surface problem. The new method to tackle this problem is proposed and the algorithm is given in details. Section 4 gives an important application, the surface reconstruction based on domain decomposition, which can handle more general surfaces. In Section 5, various numerical examples are presented to demonstrate effectiveness and robustness of the proposed method on all kinds of surfaces. Finally, Section 6 concludes the paper. In this paper, a variational reconstruction method for open surface is proposed based on graph cuts. The proposed method could robustly reconstruct not only open surfaces but also more general surfaces such as the hybrids of open and watertight ones. Surface reconstruction based on domain decomposition, as an important application, is also presented. Certain measures are taken to eliminate cracks and conflicts on the interface. Parallel efficiency is discussed as well. Parallel implementation of this domain decomposition method and investigation of its efficiency is one of our future research interests. . For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Linear Intensity-based Image Registration by Markov Random Fields and Discrete Optimization We propose a framework for intensity-based registration of images by linear transformations, based on a discrete Markov Random Field (MRF) formulation. Here, the challenge arises from the fact that optimizing the energy associated with this problem requires a high-order MRF model. Currently, methods for optimizing such high-order models are less general, easy to use, and efficient, than methods for the popular second-order models. Therefore, we propose an approximation to the original energy by an MRF with tractable second-order terms. The approximation at a certain point p in the parameter space is the normalized sum of evaluations of the original energy at projections of p to two-dimensional subspaces. We demonstrate the quality of the proposed approximation by computing the correlation with the original energy, and show that registration can be performed by discrete optimization of the approximated energy in an iteration loop. A search space refinement strategy is employed over iterations to achieve sub-pixel accuracy, while keeping the number of labels small for efficiency. The proposed framework can encode any similarity measure, is robust to the settings of the internal parameters, and allows an intuitive control of the parameter ranges. We demonstrate the applicability of the framework by intensity-based registration, and 2D-3D registration of medical images. The evaluation is performed by random studies and real registration tasks. The tests indicate increased robustness and precision compared to corresponding standard optimization of the original energy, and demonstrate robustness to noise. Finally, the proposed framework allows the transfer of advances in MRF optimization to linear registration problems. In the last years, the solution of computer vision problems by Markov Random Fields (MRFs) [1, 2] and discrete optimization has become increasingly popular. Different image analysis problems have been solved by this approach, such as segmentation [3] or non-linear registration [4] , just to name some. Recent advances [5, 6, 7, 8] in discrete optimization methods make this approach very attractive.. However, the MRF-based optimization has not yet been applied to the classical problem of intensity-based registration of images by linear transformations [9, 10, 11] . When analyzed closely, this problem turns out to be rather different in nature from many other problems which have so far been successfully addressed by MRF-based techniques. The differences are twofold.. For many MRF-based methods, the parameters can be assigned to certain points in the image domain at which the parameters are located, and the parameters have a limited local region of influence. In most cases, the parameters are directly identified with pixels and they only influence a direct neighborhood. That is, the parameters are localized in space and have. The second difference originates from the fact that the solution for the linear registration problem cannot be determined by optimizing the single parameters independently of each other. In consequence, the original problem of estimating the linear transformation with n degrees of freedom (DOF) requires an n th -order MRF model. Compared to the second-order models, the current higher-order optimization algorithms are often complex to use or pose additional constraints on the class of energies they can be applied to. Also, the optimization of higherorder terms was until now mostly only demonstrated for regularization terms -a setting that does not translate to the linear registration problem. And finally, as of now, the efficiency of recent methods for second-order models cannot be matched by methods for higher-order models. This is a major point, since the single similarity evaluations for the linear registration problem are computationally demanding. In order to be able to use the current efficient MRF optimization techniques for the intensity-based estimation of linear transformations, we present an approximation of the original energy by a second-order MRF model. This approximated MRF energy term is optimized by a suitable discrete optimization method inside an iteration loop. In every iteration, the range of the discrete search space for the single parameters is refined, eventually resulting in sub-pixel precision. In order to guarantee convergence, the descent of the original energy is ensured. The resulting algorithm is outlined in Figure 1 .. The remainder of this paper is organized in the following way. In Section 2, the actual method is presented. After formulating the general problem of intensity-based estimation of linear transformations, we introduce the basic terms of discrete MRF models in Section 2.1, and argue that the estimation of the transformation parameters of the original energy requires a higher-order MRF model. Following this, in Section 2.2 we introduce the approximation to the original energy by a tractable second-order model. In order to complete the description of the actual resulting registration method, the Sections 2.3 -2.6 provide details on parameterization, discretization of the search space, the employed discrete optimization method, and implementation. Section 2.7 summarizes and discusses the properties of the proposed method. In Section 3, we evaluate the quality of the proposed energy approximation, as well as the dependency of the method on the settings of the internal parameters. We demonstrate and evaluate an exemplary application to registration of 3D images in Section 4, and an application to 2D-3D registration of medical images in Section 5. Finally, Section 6 concludes the paper.. This paper extends and unifies our initial work from [12, 13] . We provide a revised presentation, including a new interpretation of the approximation, an evaluation of the approximation quality, as well as the influence of the internal parameters, extended experiments and a comparison to other optimization methods.

O O O O O B I I O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Robust PCA Methods for Complete and Missing Data Abstract: In this paper, we consider and introduce methods for robust principal component analysis (PCA), including also cases where there are missing values in the data. PCA is a widely applied standard statistical method for data preprocessing, compression, and analysis. It is based on the second-order statistics of the data and is optimal for Gaussian data, but it is often applied to data sets having unknown or other types of probability distributions. PCA can be derived from minimization of the mean-square representation error or maximization of variances under orthonormality constraints. However, these quadratic criteria are sensitive to outliers in the data and long-tailed distributions, which may considerably degrade the results given by PCA. We introduce robust methods for estimation of both the PCA eigenvectors directly or the PCA subspace spanned by them. Experimental results show that our methods provide often better results than standard PCA when outliers are present in the data. Furthermore, we extend our methods to incomplete data with missing values. The problems arising in such cases have several features typical for nonlinear models. Principal component analysis (PCA) is a well-known standard linear technique which is based on the second-order statistics of the data represented by the covariance matrix of the data. PCA is used widely, especially as a preprocessing method for whitening or decorrelating the data and compressing it optimally in the meansquare error sense before application of more involved nonlinear data processing or classification methods. But sometimes applying PCA alone is sufficient for getting adequate results if the problem considered is simple enough. PCA is discussed from various viewpoints for example in [7, 9, 18, 20] . In the next section, we consider the mathematical definition of PCA, the optimization problems leading to PCA, and methods for computing it.. Robustness is a particularly important and desirable property of any method, especially if the data contains exceptionally large values due to outliers, measurement errors, or impulsive noise [14, 15] . For example, the standard mean-square and least-squares error criteria are not robust against such atypical values, which may cause large errors to the results or in extreme cases even render them useless. This is easy to understand from a simple example. Assume for simplicity that we have at our disposal a data set for which the error e for the first sample to be considered is one measurement unit and for another sample 10 such measurement units. Using the mean-square or least-squares error criteria, the error for the latter sample gets a hundredfold (10 2 ) weight compared for the error for the first sample (1 2 ) in computing the optimal solution. Thus, in an extreme case, a few very large errors due to outliers can determine the optimal solution almost solely, with many small errors corresponding to appropriate measurements having a small effect to it only. This is clearly an undesirable situation.. Robustness is often achieved by using instead of standard squared criteria or loss functions robust criteria which grow less than quadratically, and we also apply this strategy later on. To this end, many different robust loss functions have been introduced, see for example Tab. 2.1 in [9] . Absolute error | e | is in principle a simple and natural first choice for a robust criterion function, but there are two problems associated with using it. First, it is much more difficult to handle mathematically than quadratic criteria, whose optimal solutions obtained using differentiation lead typically to simple linear equations. Another problem is that the derivative of the absolute error has a discontinuity at origin. Several of the robust loss or criterion functions listed in Tab. 2.1 in [9] suffer from the same problem.. Many authors have studied robust PCA using various approaches. Robust PCA methods are briefly reviewed in the introductory parts of the papers [12, 31] , in which one can find more references on the topic. In particular, in [4, 12, 38] Bayesian probabilistic approaches have been developed for the problem. These methods require making assumptions on the distribution of the noise and outliers. In the papers [4] and [38] , the authors assume Student's t-distribution, having longer tails than the Gaussian distribution, and derive an EM (expectation maximization) algorithm for this model. Gao [12] replaces Student's t-distribution with double exponential (Laplace) distribution, and uses a variational Bayesian approach leading to an approximative EM algorithm.. Torre and Black [31] use robust M-estimation developed in statistics to elaborate the theory of robust subspace learning for linear models. In the early paper by Xu and Yuille [35] , entire data vectors are regarded as outlier vectors. In many applications it is, however, more realistic to assume that only some components in a data vector are contaminated (that is, outliers), while most of their components are not outliers. Ding et al. [11] develop for robust PCA so-called R 1 -norm, which is a kind of intermediate form between the standard Euclidean norm and L 1 -norm.. Another problem often appearing in practical data sets is that there are missing values in the data vectors. General methods for handling missing values are presented in [2, 21] . Methods for treating missing values in PCA are discussed in Section 13.6 in [18] . We shall discuss them in more detail in Section 4. We have developed probabilistic methods for missing values in PCA, and applied them successfully to the huge Netflix data set in [26, 27] . This work is summarized in the long journal paper [17] . There one can find also a general discussion of the problems arising with missing values in PCA, demonstrating several features which are usually associated with nonlinear models, such as ovefitting and poor local minima.. There exist many papers on robust PCA methods and, on the other hand, on treating missing values in PCA, but very few authors have considered both these problems simultaneously. The only paper that we know is our own one [22] . There a variational Bayesian methods is developed for this problem, and applied to a difficult real-world data set with very good results. This Bayesian method is, however, mathematically involved and quite demanding. In this paper, we develop simpler neural approaches for this combined problem.. The remainder of this paper is organized as follows. In the next section, we discuss PCA from various viewpoints, which are then used in Section 3 for deriving several robust methods. In Section 4, we discuss how missing values can be handled in context with PCA and robust PCA. In the following section, we present experimental results for both artificially generated data for which the theoretically correct results are known, and for real-world forest fires data. The paper ends with conclusions in Section 6. Principal component analysis (PCA) is a well-known standard technique based on the second-order statistics of the data represented by its covariance matrix. PCA is commonly used for preprocessing and compressing data because of its optimal properties, namely maximization of variance and minimization of the mean-square representation error. PCA is also used for whitening the data, which normalizes the data with respect to its first-order (mean vector subtracted) and second-order statistics (covariance matrix becomes unit matrix). This is useful and necessary for example in many methods designed for independent component analysis, allowing them to concentrate on the higher-order statistics of the data.. However, problems arise in practical situations when the data contain outliers and/or missing values. Several methods have been introduced for robust PCA when the data contain outliers and, on the other hand, for handling missing values, but both these cases have usually not been considered simultaneously. In this paper, we introduce several methods for robust PCA, of which the robust version of the least-squares based PAST algorithm performs the best. Also a simple method for processing the components of the data vectors directly for increasing their robustness is useful, while generalization of variance maximization does not lead to useful methods.. For handling missing values, we consider two methods, an imputation algorithm and nearest neighbor method. The performance of these methods varies for the simulated data set and real-world Forest Fires data set that we have used for testing our methods. In general, the nearest neighbor method seems to be more useful for real-world data.

O O O O O O O O O O O O O O O O O O O B I I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B I O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation Abstract-This paper analyzes the asymptotic exponent of the weight spectrum for irregular doubly-generalized LDPC (D-GLDPC) codes. In the process, an efficient numerical technique for its evaluation is presented, involving the solution of a 4 × 4 system of polynomial equations. The expression is consistent with previous results, including the case where the normalized weight or stopping set size tends to zero. The spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree, a case which includes Tanner codes; for this case it is also shown how certain symmetry properties of the local weight distribution at the CNs induce a symmetry in the overall weight spectral shape function. Finally, using these new results, weight and stopping set size spectral shapes are evaluated for some example generalized and doubly-generalized LDPC code ensembles. Recently, the design and analysis of coding schemes representing generalizations of Gallager's low-density parity-check (LDPC) codes [1] has gained increasing attention. This interest is motivated above all by the search for coding schemes which offer a better compromise between waterfall and error floor performance than is currently offered by state-of-the-art LDPC codes.. In the Tanner graph of an LDPC code, any degree-q variable node (VN) may be interpreted as a length-q repetition code, i.e., as a (q, 1) linear block code. Similarly, any degree-s check node (CN) may be interpreted as a length-s single paritycheck (SPC) code, i.e., as a (s, s − 1) linear block code. The first proposal of a class of linear block codes generalizing LDPC codes may be found in [2] , where it was suggested to replace each CN of a regular LDPC code with a generic linear block code, to enhance the overall minimum distance. The corresponding coding scheme is known as a regular generalized LDPC (GLDPC) code, or Tanner code, and a CN that is not a SPC code as a generalized CN. More recently, irregular GLDPC codes were considered (see for instance [3] ). For such codes, the VNs exhibit different degrees and the CN set is composed of a mixture of different linear block codes.. A further generalization step is represented by doublygeneralized LDPC (D-GLDPC) codes [4] . In a D-GLDPC code, not only the CNs but also the VNs may be represented by generic linear block codes. The VNs which are not repetition codes are called generalized VNs. The main motivation for introducing generalized VNs is to overcome some problems connected with the use of generalized CNs, such as an overall code rate loss which makes GLDPC codes interesting mainly for low code rate applications, and a loss in terms of decoding threshold (for a discussion on drawbacks of generalized CNs and on beneficial effects of generalized VNs we refer to [5] and [6] , respectively).. A useful tool for analysis and design of LDPC codes and their generalizations is represented by the asymptotic exponent of the weight distribution. As usual in the literature, this exponent will be referred to as the growth rate of the weight distribution or the weight spectral shape of the ensemble, the two expressions being used interchangeably throughout this paper. The growth rate of the weight distribution was introduced in [1] to show that the minimum distance of a randomly generated regular LDPC code with a VN degree of at least three is a linear function of the codeword length with high probability. The same approach was taken in [7] and [8] to obtain related results on the minimum distance of subclasses of Tanner codes.. The growth rate of the weight distribution has been subsequently investigated for unstructured ensembles of irregular LDPC codes. Works in this area are [9] - [12] . In particular, in [12] a technique for evaluation of the growth rate of any (eventually expurgated) irregular LDPC ensemble has been developed, based on Hayman's formula. The nonbinary weight distribution of nonbinary LDPC codes was analyzed in [13] and [14] , while the binary weight distribution of nonbinary LDPC codes was derived in [15] and [16] . Asymptotic weight enumerators of ensembles of irregular LDPC codes based on protographs and on multiple edge types have been derived in [17] and [18] , [19] , respectively. The approach proposed in [17] has then been extended to protograph GLDPC codes and to protograph D-GLDPC codes in [20] and [21] , respectively. In contrast to the present work, the evaluation of the weight enumerators in [17] - [21] require numerical solution of a highdimensional optimization problem.. In this paper, an analytical expression for the growth rate of the weight distribution of a general unstructured irregular ensemble of D-GLDPC codes is developed. The present work also extends to the fully-irregular case an expression for the growth rate obtained in [22] assuming a CN set composed of linear block codes all of the same type. In the process of this development, we obtain an efficient evaluation tool for computing the growth rate exactly. This tool always requires the solution of a 4 × 4 polynomial system of equations, regardless of the number of VN types and CN types in the D-GLDPC ensemble. As shown through numerical examples, the proposed tool allows to obtain a precise plot of the growth rate with a low computational effort. The derived result may be regarded as a generalization to the D-GLDPC case of the corresponding result for LDPC codes which was derived in [12] .. In [23] , the growth rate of the weight distribution, G(α), was analyzed in the region of small (fractional) codeword weight α. A precise characterization of the class of code ensembles with good growth rate behavior (i.e., having a negative initial slope of the growth rate curve -typical codes from ensembles without this property have high error floors, even under maximum a posteriori (MAP) decoding) was deduced in [23] from the analysis therein. In contrast to the formula developed in [23] , the present paper proves an exact analytical expression for G(α) for any α. Note that a good weight spectral shape will not only have a negative initial slope, but the value α * where it crosses the horizontal axis, called the critical exponent codeword weight ratio, will also be large. For any code chosen from the ensemble, we expect "exponentially few" codewords of weight smaller than α * n, and "exponentially many" codewords of weight larger than α * n. Crucially, the results in the present paper allow us to evaluate α * numerically for any given ensemble; this parameter then provides a firstorder characterization of the weight distribution behavior (a reasonably large α * is generally desirable). As explained in Section II, assuming transmission over the binary erasure channel (BEC) and iterative decoding, the developed formula is also valid for the asymptotic exponent of the stopping set size distribution upon replacing the local weight enumerating function (WEF) of each VN and CN type with an appropriate polynomial function. This also allows for the evaluation of the critical exponent stopping set size ratio, which is a good indicator of the error floor performance when using belief propagation decoding over the BEC.. A compact formula for the spectral shape is derived for the special case of a GLDPC code ensemble with a regular VN set and a hybrid CN set. Symmetry properties of the weight spectral shape are also investigated for this case; in particular, it is proved that the weight spectral shape function of such a GLDPC ensemble is symmetric w.r.t. normalized weight α = 1/2 if the local WEF of each CN is a symmetric polynomial. This result establishes a connection between symmetry properties at a "microscopic" level (i.e., at the nodes of the Tanner graph) and symmetry of the "macroscopic" growth rate function.. The paper is organized as follows. Section II defines the D-GLDPC ensemble of interest, and introduces some definitions and notation pertaining to this ensemble. Section III presents the main result of the paper regarding the evaluation of the weight and stopping set size spectral shapes. Section IV derives the spectral shapes of a family of check-hybrid GLDPC code ensembles as a corollary to the main result, and also identifies a sufficient condition for symmetry of the weight spectral shape for such ensembles. Section V provides a proof of the main result of the paper, and Section VI provides additional proofs for other results in the paper. Section VII provides some examples of spectral shapes of GLDPC and D-GLDPC codes, and Section VIII concludes the paper.

B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Frequency domain metamodelling of a feedback queue ABSTRACTSchruben and Cogliano [I9871 introduced Frequency Domain Experiments as a tool for metamodel identification. To design a frequency domain experiment, the experimenter must choose appropriate values for certain experimental variables such as oscillation frequency, window size, and range of oscillation. In this paper, we demonstrate that these experimental variables affect the outcome of frequency domain experiments and the magnitude of these effects are model dependent.The feedback queue used for the experiments described in this paper is shown in Figure 1 . Customers arrive according to a Poisson Process with mean interarrival time A, join a FIFO (first in-first out) queue before a server which has an exponentially distributed random service time with mean S. After being served, a customer leaves the system with probability 1 -p or rejoins the queue with probability p. We consider mean service time S as the input factor for the system and the amount of time a customer spends in the system as its response.One of the purposes of building a simulation model is to determine which (input) factors significantly affect the model (system) response. A commonly used approach to determine how the input factors affect the model's response is to develop a response surface model. This usually takes the form of a K-order polynomial where E(Y) is the expected value of tbe response Y, Zj is a term in the Korder polynomial, i.e., a particular product of non-negative integer powers of the input factors Xi, e.g., XfXz, where the sum of the exponents is not greater than K, pj is the co-efficient for the term Zj; and t is the number of potential terms in the prospective model. In subsequent discussion the K-order polynomial on the right hand side of (1) is referred to as a "metalrpdel" [Kleijnen, 19821. Output produced by a simulation model for the above system will typically consist of a series of values {Yi}, where Yi represents the time spent by the ith customer inside the system. In subsequent discussions, we assume that {Yi} has been ordered such that Yi represents time spent inside the system by the itb customer entering the system. Let the service time for the itb customer be sampled from an exponential distribution having mean Si. The series {pi} constitutes the Input series for the input parameter 'service rate'.Schruben and Cogliano I19871 describe how frequency domain experiments can be used for metamodel identification of linear systems. Contrary to conventional simulation runs, input factors of interest are oscilIated sinusoidally during a frequency domain experiment. The spectrum of the output series of the simulation run is then estimated and examined for peaks at corresponding indicator frequencies. In this paper, we highlight some important issues connected with frequency domain experiments and use the example of a feedback queue for this purpose. Our choice of feedback queue as an example is motivated by the fact that this is a common component of many useful systems, e.g., models of computer systems [Som and Sargent, 19871, and open queues with no feedback are special cases of feedback queues with zero feedback.All the experiments described in this paper use the input series (Si}, whereand Us and L, indicate lower and upper limits of the range of interest for input parameter S and ws indicates the oscillation frequency for parameter S. Each experiment consists of a pair of runs called a signal run and a control run. In a signal run, one uses the input parameter series {Si} with non-zero os. In a control run, one uses ws = 0, i.e., the input series does not oscillate during a control run. Spectra of the output series from the signal and control runs (?,(w) and &co), respectively) are then estimated and the spectral ratio R Concerning (iv), we shall always assume that the input parameter is oscillated over the entire range of interest (as in eqn (2) Frequency domain experiments for metamodel identification promises to be an attractive alternative to current statistical methods because of the potential economy in experimentation [Schruhen and Cogliano, 19871. In order to produce the best results, the values of certain experimental variables must he chosen appropriately. For further discussion of these issues, see Som, Sargent, and Schruben [19871. ACKNOWLEDGEMENTS

O O O O O B O O O O O O O O O O O B O O B O B O O O O O O O O O B O O O O O O O O O B O O O O O O B O B O O O O B O O O O O O O O O O O O O B O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O B O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O


Vertical drain consolidation analysis in one, two and three dimensions Abstract A new method, the η or ‘eta’ method, for modeling consolidation by vertical and horizontal drains is presented. The approach is applicable in one, two and three dimensional as well as axisymmetric cases. Material and geometry properties are familiar from unit cell vertical drain analysis and are consistent across dimensions. An uncoupled finite element method (FEM) program is used to test the efficacy of the new approach. Because drains are not explicitly modeled in the finite element mesh, mesh complexity and computational time are greatly reduced. Unlike existing plane strain matching methods there is no special transformation of permeability or drain properties. The analyses conducted indicate that the η method provides an efficient and consistent means of modeling drains in any dimension. A new method, the  or 'eta' method, for modeling drains in one, two, or three To test the  method, Equation (2) was discretized with the finite element method and a variety of analyses were conducted. Movement of the soil particles was assumed to be vertical only, so strain was related to volume compressibility, v m , and pore pressure change by:. The analysis was thus uncoupled (i.e. no displacement degrees of freedom only pore pressure degrees of freedom). The program used for the finite element analyses was a modified version of program "p86" from the book by Smith and Griffiths [36] . Originally program "p86" was for general two (plane) or three dimensional analysis of the consolidation equation using implicit time integration with the "theta" method. The author modified "p86" source code to allow input of volume compressibility, axisymmetric analysis as well as element integrations involving  and w . FEM meshes were first prepared in 'Gmsh' by Geuzaine and Remacle [37] , an open source 2D and 3D

O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Adaptive Object Tracking Based on an Effective Appearance Filter Abstract-We propose a similarity measure based on a Spatial-color Mixture of Gaussians (SMOG) appearance model for particle filters. This improves on the popular similarity measure based on color histograms because it considers not only the colors in a region but also the spatial layout of the colors. Hence, the SMOG-based similarity measure is more discriminative. To efficiently compute the parameters for SMOG, we propose a new technique with which the computational time is greatly reduced. We also extend our method by integrating multiple cues to increase the reliability and robustness. Experiments show that our method can successfully track objects in many difficult situations. A tracked object can be located by maximizing the similarity measure between a reference window and a candidate window. The maximum can be realized through either a deterministic way or a stochastic way.. Deterministic methods localize the tracked object in each frame by iteratively searching for a region which maximizes the similarity measure between this region and the target window. For example, Comaniciu et al. [3] employed the Mean Shift algorithm for object tracking. These methods are computationally efficient. However, the methods may converge to a local maximum: They are sensitive to background distractors, clutter, occlusions, and quick moving objects. These problems can be mitigated by stochastic methods which maintain multiple hypotheses in the state space and, in this way, achieve more robustness to the local maximum.. Among various stochastic methods, Particle Filters (PF) [4] , [6] are very successful. Particle filters are simple, robust, and effective and have obtained success in many challenging tasks. Particle filters simultaneously track multiple hypotheses and recursively approximate the posterior probability density function (pdf) in the state space with a set of random sampled particles. Both the appearance model and the similarity measure are very important to the performance of particle filters. The particles are weighted according to a similarity measure (i.e., the observation likelihood function) and the sampling of particles is also dependent on the similarity measure.. Indeed, the effectiveness and the robustness of the similarity measure greatly affect the performance of both deterministic and stochastic methods. This paper focuses on developing an effective observation model and similarity measure in the context of improving the performance of particle filters. The key contributions of this paper can be summarized as follows:. 1. an effective SMOG appearance model and a SMOG-based similarity measure, 2. a new technique for efficiently computing the parameters of SMOG, 3. a new shape similarity measure, and 4. a complete SMOG tracking algorithm. Experiments and comparisons with several other popular methods show that our method can achieve very promising performance in tracking objects in the presence of challenging situations including fast movements, clutter, scaling, changing appearance, color distractors, and occlusions. We have described an effective appearance model (SMOG) in a joint spatial-color space, and an associated similarity measure. The SMOG appearance model and similarity measure consider both the spatial distributions and the color information of objects. Hence, SMOG more effectively represents objects and the SMOGbased similarity measure is more discriminative than the color histogram-based appearance model and similarity measure. We also propose a new technique which greatly improves the computational efficiency of our method with which the number of particles and the size of target candidate region can be greatly increased without significant change in the processing time of the proposed method. We also propose a new shape similarity measure which considers the spatial distribution of the size of, and the corresponding gradient intensities of the edge points and we integrate this shape similarity measure with the SMOG-based color similarity measure.. Experiments on a variety of video sequences show SMOG (using color cue only or multiple cues) achieves very promising results in handling a variety of challenges. In Section 6.1, we compare the proposed method with two popular color histogram-based methods: the Mean Shift (MS) tracker and the Condensation (C) tracker. For the purpose of comparisons between the techniques, we evaluate the proposed method using the SMOG-based color cue only (which we refer to as SMOG1). In Section 6.2, we show the performance of the proposed SMOG tracker using multiple cues (called SMOG2). Quantitative experimental comparisons of the four methods (MS, C, SMOG1, and SMOG2) are given in Section 6.3.

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Refinement and Verification of Sequence Diagrams Using the Process Algebra CSP Sequence diagrams are often used in the modular design of softwares. In this paper, we propose a method to verify correctness of sequence diagrams. With this method, using the process algebra CSP, concurrent systems can be synthesized from a number of sequence diagrams. We define new CSP operators for the synthesis of sequence diagrams. We also report on a tool implementing our synthesis method and demonstrate how the tool analyzes sequence diagrams. The requirements for the software systems become more complicated and diversified in recent years. To implement such complex systems, component-based programming has spread.. UML diagrams are often used for designing software components. UML is a standardized modeling language developed by OMG. Especially in upstream development, UML sequence diagrams are frequently used to understand and verify the behavior of components.. However, the UML specification is complicated and flexible. So it is difficult to verify UML diagrams automatically. It has relied on manual review to find mistakes such as inconsistencies and insufficient refinements between sequence diagrams. If such mistakes are found in a late development stage, it may take a lot of time and cost to correct them.. In this paper, we define a subset of sequence diagrams with formal semantics and propose a method to verify correctness of the sequence diagrams. With this method, developers can clarify the specifications by using formal description and find bugs by using automatic verification.. Compared with the related works described in Sect. 7, the main advantage of this work is nondeterminism can be considered. It means that our approach can handle abstract sequence diagrams. Sequence diagrams are often abstract in early development stage. Our approach can be applied to such diagrams.. To verify sequence diagrams, we propose a synthesis method of a formal expression called CSP (Communicating Sequential Processes) [1] , [2] from sequence diagrams in order to find mistakes in the early design stage based on the process algebra CSP. This synthesis method consists of two steps: At first, an order of sending and receiving is extracted from a sequence diagram for each component and it is formally expressed as a CSP process. Next, two or more CSP processes extracted from a number of sequence diagrams for the component is combined to a CSP process which represents the whole behavior of the component. This synthesis method allows us to verify properties of the concurrent system consisting of the components by using CSP-tools, for example, the model checker FDR [3] . The paper is organized as follows: First, we briefly explain sequence diagram. In Sect. 3, we introduce CSP and give new operators • and $ for combining two or more sequence diagrams. Then, the synthesis method is presented. In Sects. 4 and 5, we report on a sequence diagram synthesizer which is an implementation of our synthesis method and demonstrates the tool by a shopping site example. Finally, in Sect. 6, we discuss related works. In this paper, we have extended the process algebra CSP with two new operators • and $ for formally describing behaviors of components in sequence diagrams and for verifying them. Furthermore, we have proved CSP laws for • and $ as shown in Theorems 1 and 2, and we have presented a method for transforming processes which contain • and $ into standard processes, according to the CSP laws. The method allows us to use standard CSP-tools such as FDR.. We have developed a tool SD2CSP which is an implementation of our method, thus it generates CSP processes with • and $ from sequence diagrams and then transforms them to standard CSP processes in FDR syntax. SD2CSP can detect errors by verifying whether the concrete sequence diagrams behaves like the abstract sequence diagrams. We have demonstrated a sequence of usage of SD2CSP by an example of a shopping site.. It is a future work to extend SD2CSP with datapassing. Currently, data-passing between components has not been supported in SD2CSP yet, but various data-types such as integer, list, set, and user-defined type are supported in FDR. Therefore we believe it is feasible to verify datapassing. Also, we are discussing the refinements of datatypes.. And in the future, we plan to improve the usability of our tool SD2CSP, by extending preprocesses of sequence diagrams. We plan following preprocesses:. • Automatic insertion of state invariants from additional scenarios • Automatic complement of abnormal sequence diagrams There are some other studies for synthesis of state-based model from scenario-based model.. A.W. Biermann has proposed a method for automatic program generation from execution step of sample calculation [6] . This method constructs state transition model from executed instructions and conditions satisfied in each step.. Steps could be combined to one state if they have the same instruction. To generate deterministic program, states are divided if they have two or more transitions with the same condition and different destinations. The generation tool outputs a deterministic program with the minimum number of states.. E. Mäkinen has proposed a tool named "MAS" [7] , in which a similar idea to [6] is applied to sequence diagrams. In this method, sending events are assigned to states and receiving events are used as transition conditions. MAS generates deterministic state transition model by a similar algorithm to [6] . When the generated model is different from the expectations, the model can be modified by entering counterexamples.. Biermann's method [6] and Makinen's method [7] generates compact deterministic state-based models that can execute input scenarios. However, these methods are not necessarily available in the early design stage because nondeterminism exists there. Our tool SD2CSP is available to sequence diagrams even if details in the components have not been decided because our method uses the failures model in CSP, where nondeterminism can be considered.. The approach using Live Sequence Chart has been proposed by D. Harel et al. [8] . Live Sequence Chart is an extension of sequence diagrams, it can describe a conditional scenario that must occur when the condition is satisfied. By explicitly giving such conditions in Live Sequence Chart, the meaning of the scenarios are clarified, the state transition model satisfying the conditions can be derived.. However, in the actual software development, to exactly define all the conditions of scenarios is not easy because the conditions can depend on internal states in objects which have not been fixed in the early design stage. Our method is available for normal sequence diagrams without such explicit conditions. R. Alur et al. have proposed a method using a chart named "HMSC" [9] . HMSC shows the execution order between message sequence charts. Based on this approach, Message Sequence Chart plugin (LTSA-MSC), which is a plugin for LTSA model checker, has been developed [10] . LTSA-MSC verifies message sequence charts and a HMSC on LTSA model checker.. When using LTSA-MSC, nondeterminism can be described in HMSC, but behaviors described in each sequence chart must be deterministic. Therefore, if sequence charts contain nondeterminism, they must be carefully divided at the nondeterministic points to create the HMSC model. Each component selects a message it sends, but the selection procedure may not be fixed in an abstract level at the early stage. In our approach, such selection can be nondeterministically expressed by the internal choice in CSP.

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Estimation of current-induced pile groups scour using a rule-based method Scour phenomenon around piles could endanger the stability of the structures placed on them.Therefore, an accurate estimation of the scour depth around piles is very important for engineers.Due to the complexity of the interaction between the current, seabed and pile group; prediction of the scour depth is a difficult task and the available empirical formulas have limited accuracy.Recently, soft computing methods such as artificial neural networks (ANN) and adaptive neuro-fuzzy inference system (ANFIS) have been used for the prediction of the scour depth. However, these methods do not give enough insight into the generated models and are not as easy to use as the empirical formulas. In this study, new formulas are given that are compact, accurate and physically sound. In comparison with the other soft computing methods, this approach is more transparent and robust. Comparison between the developed formulas and previous empirical formulas showed the superiority of the developed ones in terms of accuracy. In addition, the given formulas can be easily used by engineers to estimate the scour depth around pile groups.Moreover, in this study, design factors are given for different levels of acceptable risks, which can be useful for design purposes. Key words | model tree, pile group, probabilistic design, scour depth, soft computing ; Melville & Coleman ; Richardson & Davis ), comparatively limited studies are available on current-induced scour around pile groups. Hannah () investigated scour around different arrangements of pile groups. He studied two-pile tandem and two-pile side by side arrangements in steady current condition and proposed a relationship between the pile group scour depth and spacing between the piles. Breusers () reported hydraulic-model experiments on the scour at two-leg and three-leg offshore platforms due to wave and current. A series of experiments on current-induced Pile groups are widely used in practice to support hydraulic and marine structures. Generally, this type of structure can significantly reduce construction costs, compared with spread footer structures when sediment scour is a consideration. Scour around the piles is one of the most important aspects of the structure stability and the reliable design of structures requires accurate prediction of the maximum scour depth around them. The process of the scour around a group of piles is fundamentally complex because it depends on many influential variables such as the arrangement of the piles, pile geometry, the sediment and flow Breusers & Raudkivi ; Melville ; Melville & Chiew different configurations (two-pile tandem, two-pile side by side, 2 × 2 square group, 3 × 3 square group, 5 × 5 square group and circular group arrangements). They found that the scour depth increases with the number of piles in a square pile group with G/D ¼ 4, where G is the distance between piles, and D is the pile's diameter.. The above mentioned researchers examined different pile arrangements but did not propose any formula to predict the scour depth around pile groups. A series of experiments on scour around piles was conducted by the US Federal Highway Administration (FHWA), reported in HEC-18 (Richardson & Davis ) . They presented an empirical formula for estimation of the scour depth around single piles. They modified this formula to predict the scour depth around a group of piles. From these experiments, they found that the relative scour depth (S/D, where S is the equilibrium scour depth) is governed by Froude number (Fr) and h/D where h is the flow depth directly upstream of the pile. They also noticed the effect of pile shape, angle of attack and bed material characteristics on the scour depth.. Ataie-Ashtiani & Beheshti () also conducted a series of experiments on the scour around various arrangements of pile groups. They modified the formula presented by Richardson & Davis () by adding a correction factor. This factor was obtained for pile groups aligned to the flow using results from their experiments.. Their studies showed that S/D decreases as the gap to diameter ratio (G/D) increases. They also investigated the relationship between S/D and number of piles in line with flow (m) and number of piles normal to the flow (n).. In brief, due to the complexity of the scour process, a general model has not yet been presented to estimate the scour depth around a group of vertical piles. Hence, a transparent and robust model can be very useful for this purpose.. One of the most common soft computing methods, artificial neural networks (ANN) has been used for scour estimation used feed-forward multi-layer perceptron (MLP) for scour depth estimation around pile groups. Bateni & Jeng () also used adaptive neuro-fuzzy inference system (ANFIS) to predict the scour depth around pile groups. They compared their results with those of previous studies and showed that ANN was more accurate than existing empirical formulas. In their study, geometrical parameters, such as G/D and number of piles, were not considered and they Beheshti () and showed that the scour depth is mainly governed by G/D. However, the application of the model trees for the prediction of scour depth around a group of piles has not yet been investigated.. The main purpose of this study is to develop a robust and transparent model for the prediction of the scour depth around pile groups with different pile arrangements.. Hence, M5', as one of the algorithms of Model Tree, is used for developing predictive and simple formulas. approach to split the domain with the multivariate linear regression method to obtain the formulas. In this study we also used the multi linear regression (MLR) method for the prediction of pile group scour depth. This paper is outlined as follows: a background is given about the studies conducted on current-induced scour depth around pile groups. Model Tree and M5' algorithm are then described. The MLR method is described next. The used data set and modeling approach are then presented followed by the results and discussion section. Finally, the summary and conclusion are presented. The local scour around piles occurs due to the interaction of flow, sediment and structure (Figure 1 ). When the quantities of arriving sediments are equal to the quantities of leaving sediments, dynamic equilibrium occurs. This depth is called equilibrium scour depth (Melville & Sutherland ) . Scour around a single round pile depends on three classes of parameters, i.e. the flow characteristics, the sediment properties and the pile geometry. Thus, the functional relationship between equilibrium scour depth and its independent variable is (Ettema et al. ) :. where μ is the fluid dynamic viscosity, ρ is the fluid density, U is the average velocity of approach flow, U c is the critical value of U associated with initiation of motion of particles on the bed surface, h is the flow depth, g is the gravitational acceleration and d 50 is the sediment mean diameter. With the help of dimensional analysis, these independent variables can be presented in the following dimensionless form:. where Fr is Froude number and Re is the pile Reynolds number. These are defined as:. The dimensionless parameters used in Equation (2) account for the effects of different physical processes that occur during the scour, such as flow-structure interaction, flow-seabed interaction and sediment transport. In Equation. (2) the Reynolds number, Froude number and depth to diameter ratio account for the flow pattern around the piles. U/U c describes the flow effects on the seabed process and D/d 50 describes the interaction between pile and bed sediment.. For scour around a pile group, distance between piles, number of piles parallel to the flow and number of piles normal to the flow are also important. Thus, the dimensionless function for scour around a pile group can be written as:. Researchers have suggested several empirical methods to estimate the scour depth around a pile group. In the following these methods are briefly explained.. A commonly used empirical formula was provided by FHWA (Richardson & Davis ), which recommends:. where K 1 is the correction factor for pier nose shape, K 2 is the correction factor for angle of attack, K 3 is the coefficient based on the channel bed condition and K 4 is the correction factor for armoring by bed material size. In HEC-18, it is recommended that the effective width of an equivalent full depth pile group (D*) should be used in the above mentioned formula to predict the scour depth. D* is the product of the projected width of the piles onto a plane normal to the flow multiplied by a spacing factor and a number of aligned rows factor:. where D proj is sum of the non-overlapping projected widths of piles, K G is the coefficient for pile spacing and K m is the coefficient for the number of aligned rows (Richardson & Davis ).. Another method for the prediction of the pile groups scour depth was given by Salim & Jones () . In this method, the scour depth is calculated using Equation (6) to the flow direction. This correction factor is:. In this method, the maximum scour depth is calculated by multiplying the correction factor (K Gmn ) and scour depth obtained for the equivalent solid pier using Richardson

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