An operator-difference scheme for abstract Cauchy problems An abstract Cauchy problem for second-order hyperbolic differential equations containing the unbounded self-adjoint positive linear operator A(t) with domain in an arbitrary Hilbert space is considered. A new second-order difference scheme, generated by integer powers of A(t), is developed. The stability estimates for the solution of this difference scheme and for the first-and second-order difference derivatives are established in Hilbert norms with respect to space variable. To support the theoretical statements for the solution of this difference scheme, the numerical results for the solution of one-dimensional wave equation with variable coefficients are presented. Hyperbolic partial differential equations with both constant/variable coefficients are of common occurrence in many branches of physics and several other areas of science and engineering, e.g., electromagnetic, electrodynamics, thermodynamics, hydrodynamics, elasticity, fluid dynamics, wave propagation, materials science and biological systems [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] . In numerical techniques for solving these equations, the problem of stability in various functional spaces has received a great deal of importance and attention (see [13] [14] [15] [16] [17] [18] [19] [20] [21] ). Especially, a proper difference scheme with a time dependent unbounded operator provides a suitable model for analyzing the stability.. In this paper, we consider a second-order difference scheme for the following initial-value problem. where A(t) is an unbounded self-adjoint positive linear operator with domain D(A(t)) in an arbitrary Hilbert space H. It is known (see [22, 23] ) that various initial-boundary value problems for hyperbolic equations can be reduced to the initialvalue problem (1.1). Note that (1.1) is the well-known wave equation in the special case when A(t) is equal to the Laplace operator ∆. There is a huge variety of works on the finite difference schemes for numerical solutions of linear hyperbolic partial differential equations and there are several approaches to study the stabilities of these difference schemes (see [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] and the references therein). The stability analysis of these difference schemes are performed using operator splitting method and certain energy inequalities involving some assumptions that the magnitudes of the grid step sizes τ and h with respect to the time and space variables respectively are connected by a rule. For example, in [33] , the stability of a new threelevel difference scheme for solving the second-order linear hyperbolic equation with constant coefficients is based on an assumption between τ and h. Fourier method is one of the techniques to study the stability of finite difference schemes; however, it cannot be applied for problem (1.1) because the operator depends on t.. The study of difference schemes for hyperbolic equations without using any necessary condition concerning the grid step sizes τ and h is of great interest. Such difference schemes for approximately solving the initial-value problem (1.1) were studied for the first time in [14] . In this work, the stability estimates for the solution of the following first-order difference scheme. and for the first-and second-order difference derivatives were established.. Further, when A(t) = A, the stability estimates for the solution of the first-order difference scheme, a simple case of the above difference scheme, and for the solutions of the following two types of second-order difference schemes. have been established in [16] . In [17, 18] , for the same problem, the high-order two-step difference methods generated by an exact difference scheme, and by the Taylor expansion on three points have been discussed; here the stability estimates for approximate solutions by these difference methods are also discussed. In this paper, we shall study high-order two-step difference methods for approximately solving the main problem (1.1) without using any assumption between the grid step sizes τ and h. In [19] , a second-order modified Crank-Nicholson difference scheme was developed, and the stability estimates of the solution of the difference method and its first-and second-order difference derivatives were established. In [20] , another modified second-order explicit difference scheme was developed, and the stability estimate involving. ‖ C τ of the second-order difference derivative was obtained though the stability estimates for the solution of the difference scheme and for the first-order difference derivative were not obtained. However, the difference methods in [19, 20] are generated by the square roots of A(t). Thus, for a practical realization of these difference methods it is necessary to first construct an operator A 1/2 (t), which obviously is not easy. Hence, in spite of theoretical results, the application of these methods for numerically solving an initial-value problem is not very practical.. We also note that in [34] [35] [36] [37] . has been constructed. It is clear that the above difference scheme is a generalization of the second-order difference scheme (1.3). In this paper, the stability estimates of the solution of the difference scheme and its first-and second-order difference derivatives were also established. In the present paper, a new second-order difference method generated by integer powers of A(t) for approximately solving the initial-value problem (1.1) is developed. This difference scheme is a generalization of the second-order difference scheme (1.4). The stability estimates for the solution of this difference scheme and for the first-and second-order difference derivatives are established. The theoretical statements for this difference method are supported by the numerical experiments for one-dimensional hyperbolic partial differential equation with Dirichlet boundary conditions. In this work, a new second-order difference scheme generated by the integer powers of unbounded self-adjoint positive linear operator A(t) with domain D(A(t)) in an arbitrary Hilbert space H for approximately solving the initial-value problem (1.1) is constructed. The stability estimates of the solution of this difference scheme and its first-and second-order difference derivatives are established using the techniques and tools of Hilbert space. The numerical illustration shows that the secondorder difference scheme is stable and more accurate compared to the first-order difference scheme.

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An Efficient Algorithm for Statistical Multiple Alignment on Arbitrary Phylogenetic Trees ABSTRACT The importance of incorporating statistical analysis into biological studies has become abundantly clear over the years. In particular, many interesting problems which arise in the eld of bioinformatics have been successfully addressed using statistical models. For instance, the evolution of biological sequences, which is the focus of the present paper, has been formulated in a statistical framework in which mutation events are seen as stochastic processes. A closer examination of the past progress reveals, however, that although substitution processes have been modeled as continuous-time evolutionary processes for more than three decades (Jukes and Cantor, 1969) and have been widely used in phylogenetics and genealogy (Felsenstein, 2001) , modeling insertion and deletion processes based on evolution has only recently been generalized to an arbitrary number of sequences (Steel and Hein, 2001; Hein, 2001; Miklós, 2002; Hein et al., 2002) . Furthermore, implementations of such generalizations have hitherto required quite a large running time. Alternative probabilistic approaches to sequence alignment exist (for example, see Mitchison [1999] ), but insertion and deletion processes in such models are not explicitly based on evolution.. In the present paper, we have constructed several algorithmic improvements which make multiple statistical alignment computationally tractable. More precisely, we have proved that the one-state recursion exists for an arbitrary number of sequences, thus reducing the space complexity by a factor of O.. where k is the number of sequences. Furthermore, we have developed a reverse traversal algorithm which calculates, with time complexity linear in the number of sequences, each transition factor appearing in the one-state dynamic programming algorithm. This reduces the time complexity of the entire algorithm to O.k2 k L k /, where L is the geometric mean sequence length. In fact, most of the transition factors appearing in the recursion share a common part, which hence needs to be calculated only once. Therefore, with a more careful implementation, the running time of the nal algorithm can be reduced to O.2 k L k /. The achieved improvements are not only theoretical, but make multiple statistical alignment on trees possible in practice, as we have shown in the globin example. Compared to the ordinary hidden Markov model implementation on a four-sequence tree, our method gives rise to a speedup of about a factor 50, and as well as reduces the number of states from 45 to 1, with an associated reduction in memory usage.. Our algorithm is useful mainly for trees with small number of leaves and may be used for hypothesis testing or as a basis for the quartet method. Even for small trees, however, corner cutting techniques are necessary in practical applications. The example of Section 5 requires computing a table of size 4:8 £ 10 8 in a straightforward implementation, while corner cutting can reduce this by a factor of about 500. Because the probability mass is highly concentrated around the alignment region, we lose almost no contribution to the total probability, even though only 0:2% of the entire table is actually visited. For more than, say, six sequences, our method will cease to be practical, and MCMC methods will be more useful. Some promising attempts have already been made in this direction (Holmes and Bruno, 2001; Jensen and Hein, 2002) . Our algorithm can be used to compare exact likelihood calculations and approximations, and hence it might still be useful for developing new sampling techniques.. The underlying sequence evolution model has the drawback of treating only single nucleotide insertions and deletions, as was already mentioned in the original paper (Thorne et al., 1991) . A model that can handle insertion and deletion events of whole subsequences is expected to perform better (Thorne et al., 1992) , just as af ne gaps improve alignments in score-based methods. Similarly, models that incorporate some structure information will probably result in better multiple alignments as well. We hope to incorporate these ideas in the future. Using the TKF model, it is easy to compute the likelihood of a speci c evolutionary history, given as a sequence of events. In effect, this integrates out all possible evolutionary histories on the edges of the phylogenetic tree, while xing the nucleotide sequences and their mutual alignment at internal nodes. The likelihood of observing the given nucleotide sequences at the leaves is obtained by summing this likelihood over all possible sequences of events. Since we usually do not have information about nucleotide sequences at the internal nodes, such a computation should sum out the internal nucleotide sequences as well. This is what our algorithm computes.. In order to write down the main theorem, we need some notation. Set T is the set of nodes of the phylogenetic tree relating the sequences, and r is its root. If n 2 T is a node, then n l and n r denote the left and right immediate descendants of n. For a node n, ¿ .n/ denotes the length of the incoming branch, and we abbreviate B n :D B ¿ .n/ etc. For the root, ¿ .r/ D 1 by assumption. Finally, p n .® !°/ is the probability that nucleotide ® evolves into°in time ¿ .n/. Theorem 1. Suppose A 1 ; : : : ; A k are sequences related by a phylogenetic tree T with k leaves and root r. Let K D .K 1 ; : : : ; K k / and let P .K/ be the probability of producing, under the TKF91 model, the pre xes of A 1 ; : : : ; A k up to position K 1 ; : : : ; K k , respectively, where the numbering starts at 0 with the immortal link. The following algorithm computes P .K/:. If n is an internal node, then. If n is a leaf node, then. By K n , we mean the component of the vector K corresponding to leaf n, and similarly for v n . By a. we denote the character at position K n ¡ 1 in the sequence at leaf n.. The numbers G v K .r; ¡/ are the transition factors and play the same role as the transition factors in Fig. 2 . They are sums and differences of probabilities, designed to exactly cancel all contributions of illegal transitions due to the merging of states into a single probability, analogous to the two-sequence case described in Section 3. Since we now have k leaves, the dynamic programming recursion for P refers back to 2 k ¡ 1 entries, instead of just three. Note that G 0 K does not depend on K, so a small acceleration can be achieved by computing the prefactor 1=G 0 K only once. A more substantial acceleration is achieved by noting that G v K .n; ®/ and G v K .n; ¡/ depend only on the components of v which correspond to the leaves contained in the subtree of n. Therefore, if the tree with root n has s leaves, it is enough to compute G v K .n; ®/ for 2 s values of v. Once these are computed, any entry for the ancestral node can be computed in constant time. Therefore, the overall running time of the algorithm is 2 k L k , where k is the number of sequences and L is the geometric mean of the sequence lengths.

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Relational information gain We introduce relational information gain, a refinement scoring function measuring the informativeness of newly introduced variables. The gain can be interpreted as a conditional entropy in a well-defined sense and can be efficiently approximately computed. In conjunction with simple greedy general-to-specific search algorithms such as FOIL, it yields an efficient and competitive algorithm in terms of predictive accuracy and compactness of the learned theory. In conjunction with the decision tree learner TILDE, it offers a beneficial alternative to lookahead, achieving similar performance while significantly reducing the number of evaluated literals. Many ILP or relational learning systems build discriminative models by a stepwise refinement of logical-relational features. For example, in general-to-specific rule learners like FOIL (Quinlan and Cameron-Jones 1993) , features are the bodies of Horn clauses that are constructed by adding one literal at a time. In models that adopt a decision-tree style design (in a wide sense), like TILDE (Blockeel and De Raedt 1998) , Multi-Relational Decision Trees (Knobbe et al. 1999) , Relational Probability Trees (Neville et al. 2003) , or Type Extension Trees (TETs) (Frasconi et al. 2008) , features are represented by branches in the tree structure, which are constructed in an iterative top-down process.. A distinguishing characteristic of incremental feature construction in relational learning is the possibility to refine a current feature for a given set of entities X by introducing new entities Y and their attributes via relations r (X, Y ) .. The search for the best feature refinement is typically directed by some scoring function that evaluates its usefulness for discriminating the class label of X. A refinement that does not introduce any new entities can be scored in a relatively straightforward manner using standard information gain metrics. A refinement introducing new entities is more difficult to evaluate, however: standard metrics can measure the direct informativeness of such a refinement, i.e., the direct improvement in the feature's discriminative power. However, it is widely recognized that the main benefit of introducing Y is not always its direct informativeness, but the possibility it opens up to construct, in further refinement steps, informative features for X by imposing suitable conditions on Y . Two main approaches have been used to take into account this potential informativeness of a literal introducing new entities. A first approach evaluates determinate literals (Quinlan 1991) , which are literals where for each X there exists exactly one Y with r(X, Y ). Determinate literals are not directly informative, but their inclusion in the clause is computationally relatively inexpensive, which is why, e.g., FOIL adds all possible determinate literals to a clause in order to exploit their potential informativeness. A second approach consists of lookahead techniques (Blockeel and De Raedt 1997; Castillo and Wrobel 2004; Struyf et al. 2006) , where for the scoring of the literal r(X, Y ) already further possible refinement steps using Y are considered. Both determinate literals and lookahead have severe limitations: the former represents only a very special kind of potentially informative literals, and the latter is subject to a combinatorial search space explosion when performing lookahead over multiple refinement steps (which is why, in practice, lookahead may need to be constrained to certain user-defined refinement patterns).. The goal of this paper is to develop a notion of relational information gain (RIG) for scoring candidate literals that introduce new variables, such that both direct and potential informativeness can be measured. Specifically, we have the following desiderata for RIG:. 1. RIG captures a sound and general information theoretic concept of reduction in conditional entropy of the class label distribution. It thereby is widely applicable, and not a specialized heuristic scoring function for a specific model or search strategy. Relational information gain is a refinement scoring function that has a sound informationtheoretic justification. We have introduced an algorithm for calculating approximate RIG scores and implemented it in conjunction with two popular ILP systems that use literal scoring heuristics: TILDE and FOIL. RIG, however, is not specifically conceived for these systems and its scope and applicability is more general. In conjunction with both learners, our experiments on synthetic slotchain data clearly show the ability of RIG in discovering potential informativeness of a literal, without requiring a lookahead. In the experiments with real data RIG was competitive in terms of accuracy and speed with other state of the art methods. However, no evidence was found that long slotchain like dependencies played a major role for the prediction tasks in these datasets. We can therefore conclude that RIG scoring is a widely applicable approach that is generally viable, with the potential of giving superior results in domains that are characterized by probabilistic dependencies transmitted over chains of several relations. The experiments performed with TILDE show that the use of RIG greatly lowers the number of queries being evaluated throughout the search, with respect to lookahead: RIG-TILDE evaluates a number of queries which is usually of the same order of magnitude of TILDE without lookahead, and about 1% (from 0.6% for s.cerev to 20% for Financial) and 0.5% (from 0.2% for ADL 1 to 20% for Financial) of the ones evaluated by TILDE-L1 and -L2, respectively. Note that our savings in the number of evaluated queries does not directly translate into faster learning, because of the difference between computing (10) and (11) which is highly problem-dependent. Using the naive implementation reported in (12), we obtained comparable computing times on average between RIG-TILDE and TILDE-L1, whereas TILDE-L2 was generally one or two orders of magnitude slower. RIG-TILDE achieves results which are on average comparable to the best ones obtained by varying the amount of lookahead, while generating simpler models in terms of number of literals in most cases.. The gene essentiality task shows that the use of RIG can be of great impact also in real-world complex problems, in which the use of deep lookahead can be very expensive, and sometimes prohibitive. The relational approach in this data set outperforms results in Gustafson et al. (2006) that are based on a simple propositional Naive Bayes.. The additional FOIL experiments demonstrate the applicability of RIG scoring in different learning systems. However, since the new WRIG score differs from the native FOIL scoring function also in other aspects than RIG's implicit lookahead capabilities, it is not directly clear to what extent the observed improvement of WRIG-FOIL over FOIL is due to this main novel feature of WRIG. Experiments with the modified version WRIG-FOIL noLA provide inconclusive results, with only the experiments on Carcinogenesis indicating a major impact of the lookahead feature. Most closely related to the RIG score in terms of purpose and applicability are the already mentioned lookahead strategies for relational learners.. The most basic form of lookahead is an exhaustive fixed-depth lookahead, where all possible refinements consisting of d or fewer literals are scored for their direct informativeness. Due to the exponential search space explosion, this approach is typically not feasible in practice even for small d. To alleviate the problem, one can reduce the search space by imposing user-defined constraints on admissible multi-literal refinements (Blockeel and De Raedt 1997) , or use purely syntactic conditions on admissible refinement macros (Castillo and Wrobel 2004) . Seeing that even depth-1 lookahead can be computationally very costly, Struyf et al. (2006) propose an efficient approximation to exhaustive depth-1 lookahead, called feature based estimation (FBE) . In FBE the space of admissible single-literal refinements is reduced, and information needed for scoring the candidate literals is pre-computed. None of these approaches tries to score the potential informativeness of a candidate literal without explicitly computing direct information scores obtainable by further refinement steps. A distinguishing feature of RIG score is that it measures the potential informativeness of a single-literal VI-refinement only based on the current refinement scenario consisting of the pseudo-iid example set and current query, but independent of all relations in the data not yet occurring in the query. The downside of this approach is that it can lead to overly optimistic estimates of potential informativeness, since the maximization over B sets in (8) and (9) is based on the implicit expectation that the data contains sufficiently many and informative attributes and relations that will allow to characterize the elements in B in subsequent refinement steps.. Also related to RIG scoring is the ACORA system described by Perlich and Provost (2006) . This system constructs features for relational learning by measuring to what extent positive and negative examples are connected to different entities via a given candidate chain of relations. This is related to the RIG approach in that the identities of entities reached by a relation are considered, and a relation is considered informative if positive and negative examples are connected to different entities. In a crucial difference to the RIG approach, ACORA directly uses this kind of informativeness to construct features that refer to the identities of the objects reached by the chain. RIG, in contrast, is based on the assumption that object identifiers are not available for feature construction, and that therefore objects associated with negative and positive examples, respectively, need to be indirectly characterized by their attributes. The ACORA approach, thus, is applicable only when test and training examples are connected to the same set of potentially related objects (as e.g., in collaborative filtering scenarios, where test and training customers are connected to the same set of books that they might potentially be interested in). RIG, on the other hand, is applicable for the construction of models for domains that are not "stationary" in this sense (in molecular data, for example, no atom in a training molecule is connected to the same atoms as any atom in a test molecule).

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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O


Managing Networks and Services of the Future Abstract Today, we see several emerging environments, with a large diversity of devices, networks, providers and service domains, characterized as the networks and services of the future. This special issue contains four papers on how to manage important topics of these environments: policy model to support ontology-driven reasoning for autonomic computing, SOA-based NGM management, metrics to distinguish application traffic, and SLA design and service provisioning for outsourced services.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A uniform approach to constraint-solving for lists, multisets, compact lists, and sets Lists, multisets, and sets are well-known data structures whose usefulness is widely recognized in various areas of computer science. They have been analyzed from an axiomatic point of view with a parametric approach in Dovier et al. [1998], where the relevant unification algorithms have been developed. In this article, we extend these results considering more general constraints, namely, equality and membership constraints and their negative counterparts. Programming and specification languages allow us to represent information by means of data structures, each of them characterized by a specific This research has been partially supported by MURST project PRIN 2005015491. Author's addresses: A. Dovier and C. Piazza, Università di Udine, Dip. di Matematica e Informatica, Via delle Scienze 206, 33100 Udine, Italy; email: {dovier,piazza}@dimi.uniud.it; G. Rossi, Università di Parma, Dip. di Matematica, Parco Area delle Scienze 53/A, 43100 Parma, Italy; email: gianfranco.rossi@unipr.it Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or permissions@acm.org. C 2008 ACM 1529 -3785/2008 /06-ART15 $5.00 DOI 10.1145 /1352582.1352583 http://doi.acm.org/ 10.1145 organization of the elements involved and by a corresponding access policy. In this article we consider the following structures, which represent distinct though strongly related abstractions: lists, multisets, compact lists, and sets.. Each of these four data structures contains an arbitrary (possibly empty) collection of elements of any type, where each element can be either an elementary data object or a composite object. Let us define an aggregate as a data structure with this property. The basic difference among the four considered aggregates lies in the specific handling of order and/or repetitions of elements. Lists are ordered collections of elements, where duplicates are allowed. Multisets, often called bags in the literature, are lists in which the ordering is irrelevant. Compact lists are lists in which contiguous occurrences of the same element are collapsed into a single element. Finally, in sets both ordering and duplicates are not relevant.. The importance of these data structures is widely recognized in various areas of computer science. Lists are the classical example in use to introduce dynamic data structures in imperative programming languages. They are the fundamental data structure in functional and logic languages. Sets are the main data structure used in specification languages (e.g., in Z [Potter et al. 1996 ]) and in high-level declarative programming languages [Beeri et al. 1991; Dovier et al. 1996; Gervet 1997; Hill and Lloyd 1994] ; moreover, imperative programming languages may take advantage from the set data abstraction (e.g., SETL [Schwartz et al. 1986] ). Multisets emerge as the most natural data structure in several areas, ranging from coordination languages [Banatre and Métayer 1993] to Database theory [Grumbach and Milo 1996] , from membrane and DNA computing modeling [Pāun 2000 ] to linear logic [Tzouvaras 1998 ]. The notion of compact list is much less developed and some examples of its application are suggested in Dovier et al. [1998] .. Lists, multisets, compact lists, and sets have been analyzed from an axiomatic point of view. In Dovier et al. [1998] , they have been studied in the context of Constraint Logic Programming languages, where these aggregates are represented as terms by means on different constructors. Each aggregate is associated with a theory which specifies the properties of the aggregate constructor symbol.. In Dovier et al. [1998] , equalities between terms in each of the four theories were studied. In particular, the unification problems in the equational theories, which describe the properties of the four aggregates, were solved by providing unification algorithms for all of them. NP-unification algorithms for sets and multisets are also presented in Aliffi et al. [1999] and Dantsin and Voronkov [1999] .. In this article we extend the results presented in Dovier et al. [1998] to the case of more general constraints. The constraints we consider are conjunctions of literals based on both equality and membership predicate symbols. For each aggregate, we introduce a first-order theory and we investigate the problem of deciding whether a constraint is satisfiable in each model of the theory. We base our decidability results on the introduction of a standard model and a solved form for each aggregate. These results allow us to solve the constraint satisfiability problems by applying rewriting procedures which map satisfiable constraints into solved-form constraints.. The article is organized as follows. In Section 2 we briefly discuss the existing results for similar problems. After the preliminary definitions of Section 3, in Section 4 we recall from Dovier et al. [1998] the first-order and equational theories of the four aggregates. In Section 5 we define the notion of constraint and we identify the standard models for the theories used to describe the considered aggregates. To ease the presentation, we choose the multiset theory as the working theory and we briefly point out the differences with the other theories. We show that satisfiability of constraints in standard models is equivalent to satisfiability in any model. Then we define the notion of solved form for our constraints, and we prove that solved-form constraints are satisfiable in the proposed standard models. In Section 6 we describe the constraint rewriting procedures used to eliminate all constraints not in solved form. We use these procedures in Section 7 to solve the general satisfiability problem for the considered constraints. Some conclusions are drawn in Section 8. We have extended the results of Dovier et al. [1998] studying the constraint solving problem for four different theories, namely, the theories of lists, multisets, compact lists, and sets. The analyzed constraints are conjunctions of literals based on equality and membership predicate symbols. We have identified the standard models for these theories by showing that they correspond with the theories on the class of considered constraints. We have developed a notion of solved form (proved to be satisfiable) and presented the rewriting algorithms which allow this notion to be used to decide the satisfiability problems for the four aggregates. In particular, we presented a constraint solving technique parametric with respect to these theories and we have pointed out the differences and similarities among the four kinds of aggregates.. An implementation of the results described in this article can be found in the Constraint Logic Programming language {log} ( http://prmat.math.unipr.it/ gianfr/setlog.Home.html). In this language the aggregate theories discussed in this paper (except that for compact lists) are combined all together to provide a general framework where to deal with several of the proposed forms of aggregates simultaneously. As a matter of fact, the choices made in the axiomatic definition of the theories, as well as the parametric definition of the relevant constraint rewriting procedures, make their combination into a single general framework immediately feasible, with only a very limited effort.. As further work it could be interesting to study the properties of the four aggregates in presence of append-like operators (append for lists, ∪ for sets, for multisets). These operators cannot be defined without using universal quantifiers (or recursion) with the languages analyzed in this article [Dovier et al. 2000b ]. The problem of set and multiset unification has been tackled by several authors, using different representations (see Dovier et al. [2001] for a survey on the set unification problem). These problems are often reduced to ACI and AC unification problems, respectively (see, e.g., Büttner [1986] and Livesey and Siekmann [1976] ). In these cases, a union-based representation is usually employed, where the union binary function symbols ∪ and are used as the set and multiset constructors, respectively. The operators ∪ and fulfill associativity (A) and commutativity (C). Moreover, ∪ is idempotent (I). In order to deal with nested sets and multisets, the unary function symbols {·} and {[ · ]} are also included. They act as singleton constructors for sets and multisets, respectively. Thus, the set {a, b, c} can be represented as a term of the form {a} ∪ {b} ∪ {c} and the multiset { [ a, b, b, c ] . Since {·} and {[ · ]} are free function symbols, they do not fulfill any particular axiom (see, e.g., Baader and Nipkow [1998] ). Equational theories which also allow one to deal with nested sets and multisets are called general ACI and general AC, respectively.. Unification and disunification algorithms for general ACI and AC theories can be obtained by exploiting both the results for simpler cases (unification with constants- Baader and Büttner [1988] ) and the combining approach developed in Schulz [1995, 1996] . This approach, however, due to its generality, tends to produce a huge number of failing nondeterministic computation branches, which can be pruned using more ad hoc procedures.. The general problem of solving disequations with respect to a given equational theory has also been addressed in Bückert [1988] , where a technique to transform disequations into universally quantified unification problems was presented. The method described in Bückert [1988] cannot be applied in the case of theories over sets, since it can generate undecidable formulas. In fact, existentially quantified formulas containing equations and disequations are decidable in the case of AC theories, as a corollary of the results presented in Comon [1993] . Unfortunately, the same results cannot be applied to ACI theories, and hence to sets. These theories were studied in Dovier et al. [2004] , where constraint solving procedures were developed.. As far as membership and notmembership are concerned, we are not aware of studies that extend those equational theories to encompass this kind of constraints. Actually, for sets, both membership and notmembership could be easily defined in terms of equality and disequality constraints: t ∈ s can be defined as {t} ∪ s = s and t / ∈ s as {t} ∪ s = s. Conversely, for multisets, membership t ∈ s can be defined as ∃X ({[ t ]} X = s), where X is a new variable, while t / ∈ s can be defined as ∀X ({[ t ]} X = s), that is, using a formula with universal quantification. Note that t / ∈ s could be simplified to. where R is a variable, is not reducible to a system of equalities and disequalities. For lists and compact lists of unknown length, both membership and notmembership cannot be defined in terms of equality and disequality constraints.. The union-based representation can also be used for lists and compact lists, where the union operator is associative for lists, and associative and partially idempotent for compact lists.. An alternative approach consists of considering a list-like representation based on an element insertion constructor for each of the four aggregates (see Section 4). In Dovier et al. [2000b] some comparisons between the union-like and list-like representations were presented and they highlight the different expressive powers. In particular, it turns out that the singleton operator is not expressible using existentially quantified formulas with union. Furthermore, the list-based representation was shown to be more natural for dealing with membership constraints. General constraint solving procedures based on this approach, though limited to the case of sets, were presented in [Dovier and Rossi 1993; Dovier et al. 2000b ]. In Dovier et al. [1998] we considered the four data structures considered in this article, using the list-like representation for all of them, but limitedly to the case of unification. Note that constraints on sets are particular cases of formulas of multilevel syllogistics, studied in Cantone et al. [2001] , where axioms for sets are not simply equational axioms. However, Cantone et al. [2001] were mainly concerned with decidability results rather than with constraint-solving procedures.. In this article we make use of the list-like representation constraint-solving procedures (that can be used as decision procedures, as well) for constraints involving equality and membership literals.

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


Chinese document layout analysis using an adaptive regrouping strategy In document layout analysis, the defining conditions for textlines and text regions involve certain numerical parameters (e.g. inter-character spacing and inter-textline spacing) whose values can only be estimated when textlines and text regions have already been formed. This seemingly chicken-and-egg problem can be solved through an adaptive regrouping strategy, which consists of three operations. First, we group basic ingredients into preliminary textlines and text regions according to crude parametric values. Second, we refine our estimate of the parametric values based on the groups thus formed. Third, we form new groups by splitting and merging existing groups based on the newly estimated values. This paper applies the above strategy to Chinese documents whose complexity derives from the coexistence of horizontal and vertical textlines. Successful results are obtained using this approach. The accuracy rates for identifying text regions and textlines are above 98% in a test database consisting of over one thousand document samples and various layout structures. The complexity of Chinese documents lies in the high degree of freedom allowed in the layout structure. Western documents have only a horizontal reading order, while Chinese documents have both horizontal and vertical orders. An article in a newspaper, for example, may contain horizontal headlines and vertical content textlines. This poses a challenge for both top-down and bottom-up analysis. When using a top-down approach, an area segmented according to certain spatial clues may contain textlines of different orders, leaving the seg-2 mentation work incomplete. On the other hand, by using a bottom-up approach both the scope and the reading order for each textline and text region must be determined at the same time as the textlines are constructed.. Because more than one reading order can appear on the same page and they cannot initially be distinguished, we purposely construct both horizontal and vertical textlines from the same regions. We then determine the reading order based on the fact that the inner spacing within incorrect textlines reflects the spacing between correct textlines, and is, in general, wider than the inner spacing within correct textlines. Thus, instead of viewing the solution to this problem as merely top-down or bottom-up, we can view the layout analysis as a multiple constraint problem (Chang [1] [2] ). Our adaptive regrouping strategy is a practical way to solve this type of problem.. The remainder of this paper is organized as follows. Section 2 provides the background of the problem and the outline of our approach. Section 3 describes the initial construction of textlines. Section 4 describes the method for determining the reading order of textlines. Section 5 contains the method for resolving remaining conflicts between textlines. In Section 6, we present the method for constructing text regions. Section 7 details our test results. Finally, in Section 8, we present our conclusions. Our approach to textual segmentation is a series of approximation schemes that gradually builds up knowledge about high-order objects. It refines constructed objects based on the acquired knowledge. The advantage of this process is that knowledge acquisition and object construction go hand in hand. In the beginning, when little is known about any textual objects, we construct textlines in all possible directions. From textlines that are compatible in 16 size, but conflicting in reading order, we are able to infer a normal space size for the possible text regions. Based on this knowledge, we are able to eliminate textlines that manifest incorrect reading directions. Normal space size information also provides a clue to separate textlines from groups to which they have been wrongly attributed. When conflicts between textlines no longer exist, we are able to combine all textlines that are compatible in size and direction. The purpose of doing this is not only to obtain text regions, but also to detect two-column structures in the resultant text regions and to connect broken pieces into full textlines. The effectiveness of this approach is confirmed by the high accuracy rates obtained in a test of over one thousand samples and seven major layout types. A parallel search is conducted first and we obtain the text region R 2 . Table 4 . The sizes of images whose processing times are shown in Table 3 .

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Resolving motion correspondence for densely moving points Abstract This paper studies the motion correspondence problem for which a diversity of qualitative and statistical solutions exist. We concentrate on qualitative modeling, especially for situations where assignment conflicts arise, either because multiple features compete for one detected point or because multiple detected points fit a single feature point. We leave out the possibility of point track initiation and termination, because that principally conflicts with allowing for temporary point occlusion. We introduce individual, combined, and global motion models and fit existing qualitative solutions in this framework. Additionally, we present a new efficient tracking algorithm that satisfies these -possibly constrained -models in a greedy matching sense, including an effective way to handle detection errors and occlusion. The performance evaluation shows that the proposed algorithm outperforms existing greedy matching algorithms. Finally, we describe an extension to the tracker that enables automatic initialization of the point tracks. Several experiments show that the extended algorithm is efficient, hardly sensitive to its few parameters, and qualitatively better than other algorithms, including the presumed optimal statistical multiple hypothesis tracker. Motion correspondence has a number of applications in computer vision, ranging from motion analysis, object tracking and surveillance to optical flow and structure from motion [11] , [24] , [25] , [26] . Motion correspondence must be solved when features are to be tracked that appear identical or that are retrieved with a simple at t kC 1 , and in (c) there is a missing measurement at t kC 1 feature detection scheme which loses essential information about their appearance. Hence, the motion correspondence problem deals with finding corresponding points from one frame to the next in the absence of significant appearance identification (see Fig. 1(a) ). The goal is to determine a path or track of the moving feature points from entry to exit from the scene, or from the start to the end of the sequence. During presence in the scene, a point may be temporarily occluded by some object. Additionally, a point may be missed and other points may be falsely detected because of a failing detection scheme, as in Fig. 1(b) and Fig. 1 (c) 1 .. A candidate solution to the correspondence problem is a set of tracks that describes the motion of each point from scene entry to exit. We adopt a uniqueness constraint, stating that one detected point uniquely matches one feature point. When 2-D projections from a 3-D scene are analyzed, this is not trivial, because one feature point may obscure another. If we further assume that all M points are detected in all n frames, the number of possible track sets is .M!/ n 1 . Among these solutions, there is a unique track set that describes the true motion of the M points. In order to identify the true motion track set, we need prior knowledge about the point motion, because otherwise all track sets are equally plausible. This knowledge can range from general physical properties like exact feature point locations, but, given the measurements and a number of corresponding probability density functions, it estimates these positions. The MHT attempts to match a variable number of feature points globally, while allowing for missing and false detections. Quite a few attempts have been made to restrain the consequent combinatorial explosion, such as [3] , [4] , [5] , [6] , [15] , [16] . More recently, the equivalent sliding window algorithms have been developed, which match points using a limited temporal scope. Then, these solve a multidimensional assignment problem, which is again NP-hard, but real-time approximations using Lagrangian relaxation techniques are available [7] , [8] , [17] , [18] , [23] .. A number of reasons make the statistical approaches less suitable as a solution to the motion correspondence problem. First, the assumptions that the points move independently and, more strongly, that the measurements are distributed normally around their predicted position may not hold. Second, since statistical techniques model all events as probabilities, these techniques typically have quite a number of parameters, such as the Kalman filter parameters, and a priori probabilities for false measurements, and missed detections. In general, it is certainly not trivial to determine optimal settings for these parameters. In the experiments section we. show that the best known statistical method (MHT) is indeed quite sensitive to its parameter setting. Moreover, the a priori knowledge used in the statistical models is not differentiated between the different points. As a consequence, the initialization may be severely hampered if the initial point speeds are widely divergent, because the state of the motion models only gradually adapts to the measurements. Finally, the statistical methods that optimize over several frames are despite their approximations computationally demanding, since the complexity grows exponentially with the number of points. In this paper we showed an adequate way to model the motion correspondence problem of tracking a fixed number of feature points in a non-statistical way. By fitting existing algorithms in this motion framework, we showed which approximations these algorithms make. An approximation that all described algorithms have in common is that they greedily match measurements to tracks. For this approximation we proposed an optimal algorithm, the Greedy Optimal Assignment (GOA) tracker, which obviously qualitatively outperforms all other algorithms. The way the proposed algorithm handles detection errors and occlusion turned out to be effective and more accurate than the other described algorithms. Moreover, the experiments show clearly that its computational performance is among the fastest. Also the self-initializing version of the GOA tracker turned out to be adequate and hardly sensitive to the maximum speed constraint (d max ) setting. Briefly, for the tracking of a fixed number of feature points the proposed tracker has proven to be efficient and qualitatively the best.. Among the described algorithms the R&S algorithm is completely surpassed because it operates under the same conditions, while the GOA tracker outperforms R&S both qualitatively and computationally. The S&S algorithm, which does not support track continuation, is computationally very demanding. The major drawbacks of the C&V algorithm are its relatively poor performance, especially with respect to the initialization, its restricted track continuation capability, and its sensitivity to the d max setting. Still, S&S and C&V may be considered because both support the tracking of a variable number of points and C&V can be very fast. In the previous section we indicated how their performance can be improved by incorporating GOA features in these algorithms. In a number of experiments we included the statistical multiple hypothesis tracker. Even though the MHT optimizes over several frames, which makes it computationally demanding, it turned out that it does not perform better than the GOA tracker. Possible causes are the effective initialization of the GOA tracker and the fact that the MHT models the tracking of a varying number of points, although we set the respective probabilities as to inform that the number of points is fixed. Most importantly, the MHT has quite a few parameters for which the tuning proved to be far from trivial.. In conclusion, the proposed qualitative motion framework has proven to be an adequate modeling of the motion correspondence problem. As such, it reveals a number of possibilities to achieve qualitative improvements, ranging from more specialized individual models to S.D/approximations with an extended temporal scope. Throughout this paper, we introduced a framework for motion modeling, and we presented the greedy optimal assignment (GOA) tracker that we extended with self-initialization. In this section we discuss some potential other extensions and improvements.. Although the tracking of a variable number of points conflicts with occlusion handling, it is certainly a feature that should be considered as an extension to the GOA tracker. Among the described algorithms we have seen two ways to approach this conflict of requirements, either by actually not implementing track continuation (S&S) or by only allowing occlusion during a very limited number of frames (C&V). First, the GOA tracker can support track initiation and termination by replacing the slave measurements with the phantom points as in S&S. Alternatively, the GOA scheme can be incorporated in the C&V algorithm. The idea is that at each time instance, the GOA scheme is applied first to find corresponding measurements for all point tracks that have been established so far. Then, the original C&V scheme links the remaining measurements if possible. As a result the tracking features of C&V still apply and its performance increases 6 .. Further, to deal more effectively with the underlying physical motion, the order of the individual motion models could be increased, e.g. by modeling point acceleration. Clearly, extending the scope of the individual models implies difficulties for the model initialization and the track continuation capabilities.. Finally, the scope over which the global matching is approximated can be extended. In this paper, we approximated S.D/in a greedy sense, i.e. we only minimized the combined model over two successive frames.. We already illustrated in Fig. 3 that extending the scope for this minimization would yield more plausible tracking results. However, extending the scope implies that we need to cope with an increasingly complex problem, to which the efficient Hungarian algorithm as such cannot be applied anymore.

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


Domain transfer via cross-domain analogy Analogical learning has long been seen as a powerful way of extending the reach of one's knowledge. We present domain transfer via analogy (DTA) as a method for learning new domain theories via cross-domain analogy. Our model uses analogies between pairs of textbook example problems, or worked solutions, to create a domain mapping between a familiar and a new domain. This mapping allows us to initialize a new domain theory. After this initialization, another analogy is made between the domain theories themselves, providing additional conjectures about the new domain. We present two experiments in which our model learns rotational kinematics by an analogy with translational kinematics, and vice versa. We compare these learning rates against a version of the system that is incrementally given the correct domain theory. Cognitive scientists have long argued that cross-domain analogy is an important element in people"s adaptability to new situations. It has been studied in the contexts of learning new domains quickly (Gentner & Gentner 1983; Gentner 2003) , producing paradigm shifts in scientific thought (Gentner et al. 1997; Holyoak & Thagard 1989; Falkenhainer 1988) , and future reasoning in new domains (Rand et al. 1989) . While analogies are powerful, Gick and Holyoak (1980) found that people have difficulties spontaneously producing cross-domain analogies, but they succeed when given hints. Collins & Gentner (1987) report that successful cross-domain analogies require a known base domain and a domain mapping. A domain mapping consists of correspondences between the objects and relationships of the two domains. Additional evidence of the utility of cross-domain analogies comes from textbook authors, who routinely use them to explain concepts. The linear kinematics section of the textbook used for this work (Giancoli 1991) contains eight worked through examples, or worked solutions. These include instantiations of all four linear kinematics equations. In the rotational kinematics section, however, there are only two worked solutions, neither of which utilizes two of the equations necessary for completing problems from this chapter. The summary section of the rotational motion chapter invites the learner to use analogy to fill in the details: "The dynamics of rotation is analogous to the dynamics of linear motion" (p. 197, Giancoli 1991) . Our model seeks to understand this kind of learning.. One method students use to learn physics is by completing problem sets. Afterwards, students compare their reasoning against worked solutions to refine their understanding. Our model, domain transfer via analogy (DTA), uses worked solutions as a starting point for learning a new domain theory through multiple cross-domain analogies. Given a known base domain with worked solutions and a worked solution from a new domain, DTA begins by learning the domain mapping. Our model uses this domain mapping to initialize the new target domain theory and constrain an analogy between the base and target domain theories. This analogy results in inferences about the new domain that are added to the agent"s knowledge after verification. This paper describes DTA, which uses existing computational models of analogy and similarity basedretrieval to learn new domain theories.. We begin by discussing the structure-mapping theory of analogy and the computational models used in this work. Next, we describe the representations for the problems, worked solutions and domain theories our model uses. We then outline the DTA method of domain transfer and illustrate this process with an example. We evaluate our model via two experiments in which our model learns rotational kinematics by an analogy with translational kinematics, and vice versa. We compare these learning rates against a version of the system that is incrementally provided the correct domain theory. We close with a discussion of related work and implications for future work. We have shown that a domain theory for solving physics problems can be learned via cross-domain analogies, using our DTA method for cross-domain transfer. DTA is very general; it should work with any domain theory where reasoning involves schema-like domain theories and where analogous worked solutions are available. Furthermore, our experiments demonstrate that such analogical learning can be very efficient. In fact, when the two domains are sufficiently similar, DTA can outperform a baseline system that is incrementally given the correct domain theory. The process of constructing domain mappings by exploiting structural similarities in worked solutions, and using that mapping to import theories from one domain to another, is, we believe, a general and powerful process.. There are several directions we intend to pursue next. First, we have only tested our model with learning encapsulated histories, so we want to extend it to handle other types of domain knowledge, such as model fragments and modeling knowledge. Currently, we are developing a method for learning modeling decisions via generalization and incorporating it into our domain transfer system. Second, while DTA has an explicit verification step, this does not guarantee that all the learned knowledge is correct. Therefore, we plan to implement model-based diagnostic strategies to debug our analogicallyderived domain theories, similar to the strategies used by de Koning et al. (2000) to diagnose misconceptions in student models. Finally, we plan to explore a broader range of domain pairs, including domains which are quite distant such as those found in system dynamics (Olsen 1966) . While linear and rotational mechanics are quite similar, the analogy between mechanics and electricity, two superficially different domains, is quite systematic. This will explore the extent that similarities in worked solution structure can learn an adequate domain mapping. Finally, an analogy between mechanics and heat flow will stress that only certain aspects of the domain should transfer, as there are important non-analogous aspects between these domains. 2 In both experiments, domain transfer via analogy outperformed the spoon-fed baseline system. DTA learned faster and achieved a ceiling that was the same or higher than the baseline. An analysis of the few linear kinematics transfer failures indicates that certain sub-event structures and time intervals increase the difficulty in generating an appropriate domain mapping. Linear kinematics problems "b" and "d" both contained such structures. One requirement for successful transfer in these scenarios is that an objectRotation statement in the rotational kinematics worked solution must correspond with a primaryObjectMoving statement in the linear kinematics worked solution. In order for this to occur, the entities which make up these expressions must already be in alignment. Given the structure and number of the events and time intervals in these two problems, the events listed in these statements may differ from the events and time intervals referenced in the quantities and equations. Therefore, one aspect of our future work is to incorporate rerepresentation strategies (Yan et al. 2003) to bring these worked solutions into better alignment with the analogous rotational kinematics worked solutions. The added complexity of the linear kinematics problems also slowed the baseline learning system.. Another interesting result of these experiments is the difference in the utility of retrieving an additional worked solution if the first one fails. In experiment 1, this part of the algorithm was critical to the analogy system"s perfect transfer. On the other hand, in experiment 2, the additional retrievals never produced an adequate domain mapping. This supports our intuition that the decision to retrieve additional analogs should be controlled by the system based upon its goals. There are three main branches of related work: AI systems of analogy and case based reasoning, cognitive science models of cross-domain analogy, and integrated reasoning systems working towards human-level capabilities. The majority of analogical and case based systems focus on using previous experiences to guide future problem-solving (Koloder 1991) . This work has occurred in a variety of domains including transportation (Veloso & Carbonell 1993) , physics problem solving (van Lehn 1998), inductive theorem proving (Melis & Whittles 1999) and thermodynamics problem-solving (Ouyang & Forbus 2006 ). These systems use examples to provide search control knowledge in order to solve future problems faster. DTA differs by focusing on learning schema-like domain knowledge, in this case encapsulated histories.

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


A new similarity measure for nonlocal filtering in the presence of multiplicative noise This paper presents a new similarity measure and nonlocal filters for images corrupted by multiplicative noise. The considered filters are generalizations of the nonlocal means filter of Buades et al., which is known to be well suited for removing additive Gaussian noise. To adapt this filter to different noise models, the involved patch comparison has first of all to be performed by a suitable noise dependent similarity measure. To this purpose, a recently proposed probabilistic measure for general noise models by Deledalle et al. is studied. This measure is analyzed in the context of conditional density functions and its properties are examined for images corrupted by additive and multiplicative noise. Since it turns out to have unfavorable properties for multiplicative noise, a new similarity measure is deduced consisting of a probability density function specially chosen for this type of noise. The properties of this new measure are studied theoretically as well as by numerical experiments. To finally obtain nonlocal filters, a weighted maximum likelihood estimation framework is applied, which also incorporates the noise statistics. Moreover, the weights occurring in these filters are defined using the new similarity measure and different adaptations are proposed to further improve the results. Finally, restoration results for images corrupted by multiplicative Gamma and Rayleigh noise are presented to demonstrate the very good performance of these nonlocal filters. In 2005, Buades et al. proposed the nonlocal (NL) means filter [4] , which is based on the following idea: Each pixel f i of a given noisy image f is compared together with its neighboring pixels to other image patches. For each comparison a weight is assigned depending on the similarity of the image patches. The restored pixel u i is then the weighted average of the central pixels of these patches using the obtained weights. For a discrete image f ∈ R m,n , N = mn, we have in detail. with. . If the image patches with centers f i , f j are given by f i+I , resp. f j+I for I denoting an appropriate index set, then the weights are obtained by. Here, the parameter h > 0 is used to control the amount of filtering. The vector g a = (g a,k ) k∈I represents usually a sampled two dimensional Gaussian kernel with mean zero and standard deviation a steering the influence of neighboring pixels on the weight.. In the past five years this filter has been extensively studied and further improved in various directions. An analysis and comparison with other state-of-the-art image denoising methods as well as an overview of recent developments in this area can for example be found in [5] . Among other improvements, several authors proposed different approaches to adapt the nonlocal means filter to noise statistics. In [24, 25] Mäkitalo et al. studied variance-stabilizing transformations and their inverses which can be applied to data corrupted by Poisson and multiplicative Rayleigh noise before and after the standard NL means approach is applied. Kervrann et al. proposed the so-called Bayesian NL means filter [21] which gave a first possibility to incorporate the statistics of the noise directly into the design of the filter. In [6] this filter has been applied to remove speckle noise in ultrasound images. An approach for Rician noise was presented in [37] . Another generalization of the original NL means filter in a probabilistic framework was given by Deledalle et al. in [8] . Here, a central step was to incorporate the noise statistics in a suitable way into the weight definition of their filters. To illustrate the basic idea, the weights of the NL means filter can be written in the form. with s N L (x, y) := exp(−|x − y| 2 ).. Consequently, they can be constructed by taking the product over s N L (f i+k , f j+k ) g a,k h for all pairs of pixels f i+k and f j+k of the two image patches. The function s N L : R × R → (0, 1] can be viewed as a similarity measure, where s N L (f i+k , f j+k ) is supposed to be close to one if the original noise free pixels belonging to f i+k and f j+k have been the same and it should be close to zero if not. For images corrupted by additive Gaussian noise, s N L is known to perform well. Unfortunately, it can be far from optimal for other types of noise. Hence, the challenge is to find a suitable noise adapted similarity measure, which can cope with different types of noise. The similarity measure proposed for general noise models in [8] was demonstrated to perform well for images corrupted by additive Gaussian noise, noise following a Nakagami-Rayleigh distribution and Poisson noise studied in [9] .. The aim of this paper is to present a new similarity measure specially designed for comparing data in the presence of multiplicative noise. This type of noise occurs for example often as speckle [2, 17, 36] produced by various imaging systems. Real life applications include but are not limited to laser, ultrasonic [34] and synthetic aperture radar (SAR) imaging [23] . In case of fully developed speckle, the magnitude of the complex observations of these imaging devices can usually be modeled as corrupted by multiplicative Rayleigh noise, cf. [17, 23] . As a consequence, the noise present in the square of the magnitude, the so-called intensity, is exponentially distributed. To improve the quality of such data, a common approach in SAR imaging is to average independent intensity observations of the same scene to obtain socalled multilook data, which is then contaminated by multiplicative Gamma noise. Hence, we will mainly concentrate on Rayleigh and Gamma noise in our examples, although also other noise distributions could be considered, see, e.g. [22] . In contrast to recent image restoration methods, see, e.g., [1, 10, 11, 22, 33, 38] , we use our new measure to define suitable nonlocal filters for removing multiplicative noise in images. This paper extends results presented in the conference proceedings of SSVM 2011 [35] . To start with, we revisit the similarity measure proposed in [8] in Section 2 and analyze it in the framework of conditional density functions. Moreover, we study its properties for images corrupted by additive and multiplicative noise. Since it turns out to be well suited for additive noise but to have unfavorable properties for multiplicative noise, we deduce our new measure given by a noise dependent density function in Section 3. The advantages of this measure are shown theoretically as well as by different examples and experiments. Next, we deduce our nonlocal filters by maximum likelihood estimation in Section 4 and define the involved weights using our new similarity measure. Moreover, we present different modifications to further improve the results. The very good performance of our new filters is demonstrated for images corrupted by multiplicative Gamma and Rayleigh noise in Section 5 and finally, we end with conclusions in Section 6.. 2 The similarity measure of Deledalle et al.. To start with, we revisit the similarity measure applied by Deledalle, Denis and Tupin in [8] .. To fix the notation, all random variables are named with capital letters and are supposed to be real-valued, continuous, and defined on a fixed probability space (Ω, F, P ). Moreover, p X : R → R ≥0 stands for the density of the random variable X. For any x ∈ R with p X (x) > 0, the conditional density of a random variable Y given X = x, is defined by. see, e.g., [18, p. 104] . It holds that. which shows the connections between a conditional density and the corresponding conditional probability. Note that Appendix A contains a collection of results from probability theory, which will be used in this and the subsequent sections.. In the following, we suppose that the noisy pixels f i are realizations of independent continuous random variables F i and the corresponding original noise free pixels u i are realizations of independent identically distributed random variables U i , i = 1, . . . , N . Moreover, let all f i be contaminated by the same noise model with equal parameters. Since for the following considerations we need only two pixels, we set N = 2 for simplicity. Now, to measure whether u 1 = u 2 by the noisy observations f 1 , f 2 , Deledalle et al. suggest to use a so-called 'similarity probability' denoted by p(θ 1 = θ 2 |f 1 , f 2 ). In their paper, θ i is a parameter depending deterministically on u i and we consider θ i = u i , i = 1, 2. Since, in general, it is not clear what the probability or even conditional density function of. is, see, e.g., [18, p. 111], we start by interpreting the 'similarity probability' as a conditional density: In [8] it is set to be. where we need to have p F i (f i ) > 0, i = 1, 2 and define S := supp(p U i ). By the definition of the conditional density it holds that. Furthermore, we obtain by Theorem A.1 and the independence of (U 1 , F 1 ) and (U 2 , F 2 ) that. and thus,. Inserting this in (4) shows with (3) and the independence of. For this reason, we will in the following refer to the 'similarity probability' by the conditional density on the right hand side. By (4) it can also be expressed in the form. Since in general no knowledge about the distribution of the random variables U i is given, Deledalle et al. propose to neglect the densities p U i and p F i , i = 1, 2 and to consider only. This measure is very close to the one investigated for block matching in [26] . One may ask if s DDT can also be interpreted in terms of a conditional density function similarly to (5) . For the case of additive noise with S = R the answer is yes as we will see in the next subsection. In this paper we have presented a new noise adapted similarity measure for comparing data corrupted by multiplicative noise. This measure is deduced by an appropriate probability density function and its good properties have been studied theoretically as well as by numerical examples. Moreover, we have shown how this measure can be incorporated into the weight Original image (1500 × 1500), copyright [28] , with values in [1, 2047] (left) and noisy version corrupted by multiplicative Rayleigh noise with θ = 1 (right). Bottom: Results by our nonlocal filter (25) using (27) with l = 7, ω = 21, a = 2, h = 0.4, q = 0.6 (left) and after an additional updating step with l = 5, a = 1.5, d = 0.05 (right). Note that the displayed images have been subsampled to the size 500 × 500 to better meet the standard screen and printer resolutions. The PSNR and MAE values have again been averaged over the results for 500 noisy realizations of the original image. definition of nonlocal filters, which leads to very good denoising results as demonstrated for images corrupted by multiplicative Gamma and Rayleigh noise. Nevertheless, the use of our measure is not restricted to nonlocal filtering. It could also be incorporated into appropriate variational methods or used for different applications such as inpainting and segmentation, see, e.g., [14, 15, 16, 33] . Alternatively, it could also be considered for block matching in registration problems, cf. [26] . These are only a few other fields of application, which might be topics of future research.

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