Security invariants in discrete transition systems Abstract. The Shadow semantics is a qualitative model for noninterference security for sequential programs. In this paper, we first extend the Shadow semantics to Event-B, to reason about discrete transition systems with noninterference security properties. In particular, we investigate how these security properties can be specified and proved as machine invariants. Next we highlight the role of security invariants during refinement and identify some common patterns in specifying them. Finally, we propose a practical extension to the supporting Rodin platform of Event-B, with the possibility of having some properties to be invariants-by-construction. Event-B [Abr10] is a formal modelling method for developing systems via step-wise refinement, based on firstorder logic and some typed set theory. The strength of the method is enhanced by the Rodin Platform (Rodin) [ABH + 10] for reasoning about Event-B models rigorously. Each machine, the basic construct in Event-B, corresponds to a discrete transitions system, with its properties defined as machine invariants, which need to be proved to hold always during the execution of the machine.. In [Mor06] , Morgan introduced the "Shadow Knows" framework for sequential programs, including an assertion language for expressing "knowledge" together with a weakest-precondition modal semantics, which can be used as the basis for ignorance-preserving refinement. An attractive property of this work is the possibility to translate (1) programs into standard statements and (2) properties into first-order logic (the "shadow form"), and to reason about (1) and (2) within the standard context. We presented a 3-agent model for the Dining Cryptographers, and our invariants (in particular for the parallel version) are often symmetric. It is clear that a more generic model parameterised by the number agents is desirable. In particular, assume that there are n cryptographers, the choices of the cryptographers can be represented by a single variable s as follows.. s ∈ 1 .. n → BOOL It is hence required to have a generalised version of exclusive-or and reasoning about properties of this operator. Note that here s is a single variable, rather than n different variables representing the choice of each individual cryptographer. As a result, the shadow H will be a set of total functions, each function represent a possible value of s. The first theoretical question we need to solve is how the shadow H is represented in the case where only part of s is hidden. For example, in the view of cryptographer i , only s(i ) is visible, whereas s(j ) is hidden for j i . Even in the case where s is all hidden, e.g., in the view of the waiter, it is also required to adapt the interpretation of complete ignorance accordingly. For example, considering we want to express the fact that we do not know the exact value of s(i ) for some i . Currently, using the notion of complete ignorance, the best of what we can express is < < s | s(i ) ∈ BOOL > > and its translation into the shadow form is as follows. ⇔ ∀t ·t(i ) ∈ BOOL ⇒ t ∈ H One-point rule However, this is obviously too strong compared to the property that we want to express. Basically, < < s | s(i ) ∈ BOOL > > states that the complete ignorance not only about s(i ) but also of all other cryptographers, and their possible combinations. Intuitively, the precise property that we want to express using H is as follows.. ∀e ·e ∈ BOOL ⇒ (∃s ·s ∈ H ∧ s(i ) e). The Shadow semantics is designed only for possibilistic (qualitative) reasoning about noninterference security. In [HMM + 11], it is showed that in some important class of security protocols, this (qualitative) reasoning can be soundly lifted to probabilistic (quantitative) context. We want to study the conditions (similar to those in [HMM + 11, Sect. 4]) under which this lifting is also valid in the context of discrete transition systems. We plan to extend Rodin to implement the tool support according to the proposal mentioned in Sect. 5. In particular, the connection with a model checker for Event-B such as ProB [LB08] will be investigated. We believe that the use of model checkers will complement the theorem proving task, in particular in verifying generic parameterised models.. Last but not least, we intend to apply our approach to other examples, such as Rivest's Oblivious Transfer [Riv99] . Note that the specifications will be our building blocks for reuse later, i.e. we are going to build more complex protocols using sub-protocols. This is illustrated in the work of McIver and Morgan [MM09] . For reusing specification, we propose to make use of techniques such as generic instantiation and design patterns [HFA09] for Event-B. Our work presented in this paper is strongly motivated by the work of Morgan [Mor06, Mor09] and built on the experience from [HMM + 11]. However, while the original work of Morgan concentrated on the ignorance preserving refinement of programs using mainly program algebra, we focus here on how the Shadow Knows framework fits into a development method such as Event-B. In particular, we showed that the Shadow Knows framework can be extended from reasoning about sequential programs to more general discrete transition systems (including concurrent or distributed systems).. We presented an extension to the Event-B method for handling security invariants: properties of systems constraining the knowledge of observers about some hidden variables. The state variables are split into the set of visible variables and hidden variables. The underlying logic of Event-B is extended with the "knows" operator K, where Kφ holds in the state where φ hold in every state compatible with the visible part of the state, the formal model text and the information about the execution path including the previous visible values and the order of executed events. We identify two patterns of security invariants to constraints the knowledge of the observer about hidden variables. Moreover, we propose the notion of invariant-by-construction and determine certain properties which fit into this category to reduce the number of obligations to be discharged.. For tool support, we propose an extension to Rodin. In particular, we consider multiple agents' point of view and generated different developments accordingly. A novel idea here the separation between functional model and shadow model, allowing different developments to share the functional part.. Security invariants in discrete transition systems 87 Our motivation starts from the work of Morgan [Mor06, Mor09] on the Shadow semantics for sequential programs. We extend the work to discrete transition systems, allowing us to formalise and reasoning about noninterference security for different types of systems including distributed and parallel ones. We use Event-B as the language to illustrate the extension, with the basic modelling elements are guarded events. Similar to the work in [Mor06] , we distinguish between two types of non-determinism: atomic non-determinism represented by non-deterministic event actions (assignments of the form :∈, :|), and composite non-determinism represented by the implicit choice between enabled events. The semantics of guarded events is designed based on the semantics given to conditional statement if G then S else T end in [Mor06] . In fact, a conditional statement usually modelled using two guarded events as follows.. The semantics of conditional statements can be deduced from our semantics of guarded events by combining the effect of the two events evt 1 , evt 2 . The link between the Shadow semantics and Event-B/Rodin is first explored in [HMM + 11]. There Rodin is used as a back-end for verifying the Shadow refinement of sequential programs by encoding the shadow sequential programs into Event-B. We go one step further in this paper to give a Shadow semantics to Event-B models themselves and use it to reason about discrete transition systems. We also translate ignorance-sensitive Event-B models into standard Event-B model via an additional shadow variable H . As a result, there are similarities between our work and [HMM + 11], including how the shadow variable H is updated for assignments of the form : , :∈. Ultimately, we moved away from purely verifying sequential programs into modelling and reasoning about discrete transition systems. In particular, we do not split the events into shadow and functional parts as described earlier in Sect. 4.2. While separation of shadow and functional updates make perfect sense for sequential programs, it introduces some complications in reasoning about discrete transition systems.. The logic of knowledge that we used is essentially identical to what described by Morgan in [Mor06] , subsequently inspired by the standard model for knowledge-based reasoning of Fagin et al. [FHMV95] . In particular, compared to [FHMV95, HO08, HO04], we only consider one agent (one point of view) at a time and the (v , h, H )-model only allow the h component to be varied in the underlying Kripke model [FHMV95] .. The example of the Dining Cryptographer is used to illustrate the Shadow semantics in [Mor09] . There its reasoning is based on the accompanying weakest precondition semantics. Subsequently, it is used as one of the illustrating examples in [HMM + 11]. We focus here on developing the specification of the problem and the algorithm within a development method like Event-B. In particular, we show that shadow invariants can be discovered during development as conditions for maintaining the consistency of the model. Compared to [HMM + 11] where invariants are generated according to some heuristics based on strongest post-conditions (often containing some redundant information), our invariants are added manually on demand and often simpler. Moreover, we identify two patterns for invariants which should help the developers in guiding their intuition when reasoning about non-interference security. This is of particular important for reasoning about discrete transition systems where invariants often play an important role for deriving the correctness of the formal models.. The Dining Cryptographers problem has been studied in [HO04] . Moreover, it has also been extensively analysed mostly using model checking techniques [vdMS04, vdM11, ABvdM10, RL07, KLN + 06]. In most of these works, the models of the protocol are often generic in terms of the number of cryptographers. We presented a 3-agent version of the protocol in this example. Later, we discuss the possibility of having a generic model in Sect. 7. A clear distinction between these fore-mentioned work and our work is also the difference between model checking and theorem proving. We develop our model gradually via refinement, starting with an abstract specification. Most of the existing work using model checkers involves some "implementation" models and having properties of the protocol verify directly on these concrete models. An abstraction of the Dining Cryptographers protocol is discussed in [ABvdM10] . However, in our opinion, their purpose of abstracting the protocol is different from our work. We present an abstract system capturing the essential properties of the protocol, whereas their abstract system is a means for optimising the model checking problem. In this section, we first give some background information on the Event-B modelling method. Afterwards we review the Shadow Knows framework [Mor06] including the accompanying logic for expressing "knowledge" (and its compliment "ignorance").

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


Fast maximum likelihood estimation using continuous-time neural point process models Abstract A recent report estimates that the number of simultaneously recorded neurons is growing exponentially. A commonly employed statistical paradigm using discretetime point process models of neural activity involves the computation of a maximum-likelihood estimate. The time to computate this estimate, per neuron, is proportional to the number of bins in a finely spaced discretization of time. By using continuous-time models of neural activity and the optimally efficient Gaussian quadrature, memory requirements and computation times are dramatically decreased in the commonly encountered situation where the number of parameters p is much less than the number of time-bins n. In this regime, with q equal to the quadrature order, memory requirements are decreased from O(np) to O(qp), and the number of floating-point operations are decreased from O(np 2 ) to O(qp 2 ). Accuracy of the proposed estimates is assessed based upon physiological consideration, error bounds, and mathematical results describing the relation between numerical integration error and numerical error affecting both parameter estimates and the observed Fisher information. A check is provided which is used to adapt the order of numerical integration. The procedure is verified in simulation and for hippocampal recordings. It is found that in 95 % of hippocampal recordings a q of 60 yields numerical error negligible with respect to parameter estimate standard error. Statistical inference using the proposed methodology is a fast and convenient alternative to statistical inference performed using a discrete-time point process model of neural activity. It enables the employment of the statistical methodology available with discrete-time inference, but is faster, uses less memory, and avoids any error due to discretization. A recent report estimates the number of recorded neurons to be growing exponentially (Stevenson and Kording 2011) . Exponential growth of data presents an increasingly important challenge to the effective application of contemporary statistical methods in the neurosciences, where computation times are beginning to exceed acceptable limits. The growing challenge is being met by recent developments within the neuroscience community. In Paninski et al. (2010) statespace models, Markov-chain Monte Carlo methods of neural decoding, and spatially varying firing rates are estimated in O(n) time, enabled by the banded structure of relevant matrices. Here n is the number of time-steps in the observation duration. In Citi et al. (2014) , computational efficiency is gained be replacing a Riemann approximation with an improved, but discrete, approximation to an integral appropriate for restricted but often realistic models of neural spiking activity.. In Ramirez and Paninski (2013) maximum likelihood estimates (MLE) computed for generalized linear models are hastened by replacing estimation with the log-likelihood with estimation using the expectation with respect to stochastic covariates of a stochastic log-likelihood. Here focus is placed upon the computational cost arising from the log-likelihood term that is nonlinear in the model parameters.. Similar to this latter work, in the proposed methodology focus is placed upon a computationally efficient approximation to an analagous nonlinear term in the log-likelihood of continuous-time point processes possessing an exponential conditional intensity model. This procedure uses a non-stochastic log-likelihood, consequently requiring no assumptions regarding a probabilistic description of model covariates. The analagous nonlinear term in the proposed methodology requires the integration of the rate, or conditional intensity, over the observation interval. By employing the optimal Gaussian quadrature rule to approximate this integral the number of evaluations of the conditional intensity when computing this integral reduces from n in the standard discrete-time situation (Truccolo et al. 2005) , to q where q n, ultimately reducing an O(np 2 ) computational cost to O(qp 2 ). 1 This computational complexity compares favorably with that of the discrete-time MLE counterparts studied in, for e.g. (Truccolo et al. 2005; Paninski 2004 ) which have a computational complexity of O(np 2 ).. Gaussian quadrature is known to be much faster than competitive methods in low-dimensional settings. 2 It has been known to the statistics community for a very long time, but has recently been largely ignored; possibly for three main reasons: (i) there is a community focus upon Markov Chain Monte-Carlo techniques in favor of classical quadrature, (ii) there is a current focus upon numerical integration in higher dimensions, and (iii) the standard Gaussian quadrature scheme is not adaptive -it is difficult to assess accuracy despite an expression for the numerical integration error (Kuonen 2003) .. In Genz and Kass (1991) , Genz and Kass (1997) methods based upon comparing successive Gaussian quadrature approximations of increasing orders are discussed. These methods are found to be promising but conservative, in the sense that the quadrature order in this scheme is often specified to be unnecessarily large. In Genz and Kass (1997) a method of adaptively determining the error as a linear combination of a theoretical error term with the difference between successively increasing orders of approximation is presented. In this scheme, the order of integration is increased until the error falls below a user defined threshold.. 1 Gaussian quadrature is a method of numerical integration first developed by Gauss. It is optimal for integrating polynomials (Davis and Rabinowitz 1967) . 2 Except for Clenshaw-Curtis quadrature. This quadrature scheme is a factor of 2 less computationally efficient at low quadrature orders q, and becomes computationally as efficient as q increases (Trefethen 2008). In the preparation of this manuscript, other work using Gaussian quadrature for the estimation of refractory neural point-process models was made known to the authors (Mena and Paninski 2014) .. In this work, based upon a physiological consideration bounding quadrature error, an empirical approximate error bound, and a theoretical expression linking quadrature error to parameter estimate error, quadrature order is increased until the parameter estimate error is small relative to either a physiologically meaningful scale or to parameter standard error. Despite often over-estimating the order of integration, up to an order of magnitude improvement in computation time can be realized for the setting n p. The paper begins in Section 2.1 with a review of maximum-likelihood estimation (MLE) for a continuoustime point process with an exponential model of the conditional intensity. In Section 2.2 background material on Gaussian quadrature is presented and in Section 2.3 the proposed approximate MLE for continuous-time point process models with exponential conditional intensities is presented. The computational and memory requirments needed to compute discrete-time and continuous-time MLEs is provided in Sections 2.4, 2.5 and 3. The effect of quadrature inaccuracy is quantified in Section 4, while the accuracy of Gaussian quadrature for computing the integral of a physiologically plausible conditional intensity is discussed in Section 5. The full proposed adaptive procedure is presented in Section 6. The proposed MLE is simulated in Section 7 on a synthetic time-cell, and the proposed procedure is applied to recordings of actual hippocampal time-cells in Section 8. The paper concludes with a discussion in Section 9. In Section 5 the accuracy of the proposed approximate MLE is addressed. By studying a plausible model of the neural refractory effect (Section 5.3), the existence of a plausible conditional intensity model that can be accurately integrated using Gaussian quadrature of modest order is established.. In Appendix C, a cheaply-computed bound on parameter estimate inaccuracy due to quadrature error is provided. Thus effects due to quadrature error can be monitored, and the quadrature order can be appropriately adjusted. Because the quadrature is computationally cheap, large changes in quadrature order can be employed.. In simulation (Section 7), and when analyzing actual data (Section 8), the time to compute the maximum likelihood estimates using Gaussian quadrature and the continuoustime point process models of neural activity (Section 2), Fig. 12 Example QQ-plots constructed using the discrete-time (plus symbol) and continuous-time (solid) point process models. The curves are computed using the data recorded from cell 5 in Fig. 11 . The two curves are nearly identical on the scale plotted. The thick red lines indicate approximate 95 percent confidence bounds is much faster than their discrete-time counter-parts. These results are conservative, in the sense that for all calculations, the Gaussian quadrature nodes and weights are re-computed. This time can be reduced, by storing these values offline, or by employing the methods presented in (Hale and Townsend 2013) . These approaches are left for future study. They will be most important for the small p and large q case.. Three limitations to this work are (i) the number of computations required depends upon the total number of spikes due to the first term in Eq. (2) (see Fig. 11 ), (ii) for non-identifiable log-likelihood functions, the standard issues arise, and (iii) when considering oscillatory conditional intensities, such as those investigated in Lepage et al. (2013) , limited speed-up may occur due to the fact that the integral is zero for an integer number of observed oscillatory periods. In case (ii), there exist parameter values that yield the same log-likleihood value (non-identifiable case), or similar values (approximately non-identifiable case, and often a numerically unstable case). These issues have been well documented in the statistics literature. They are discussed in books on generalized linear modeling. See for e.g., (McCullagh and Nelder 1999) . In case (iii), a simple two-step procedure is available: estimate the parameters assuming the integral in Eq. (2) is zero. Then with this parameter estimate as a starting guess, compute a few iterations including the integral.. In this work focus is placed upon significantly reducing n to q in the quadrature approximation to the integral of the conditional intensity appearing in the log-likelihood, Eq. (2), while maintaining sufficient quadrature accuracy. Emphasis is placed upon determining q such that quadrature error is acceptable.. In Section 6, a quadrature order selection criterion based upon a previously unreported approximation to the higher order derivative of the integrand (Appendix C) is introduced. Its use is motivated mathematically (Appendix C), its performance is demonstrated on a difficult integral (Table 2) , and its utility is demonstrated when analyzing actual neural recordings ( Fig. 10 and Table 4 ). It is expected that when used to analyze future neural recordings, this algorithm will be usefully employed to reduce the risk of unacceptable error due to Gaussian quadrature.. The statistical analysis of neural spiking is central to neuroscience. As experiments increase in size and complexity the speed at which these analyses can be conducted becomes increasingly important. In this work, by replacing discrete-time conditional intensity models with their continuous-time counter-parts and employing the incredibly efficient Gaussian quadrature, the computational complexity is reduced from O(np 2 ) to O(qp 2 ), memory requirements are reduced from O(np) to O(qp), and any discretization-error is eliminated. This is accomplished without reducing the available statistical methodology. The statistical methodology used to perform data analysis with discrete-time point process models of neural activity established in, for e.g., (Truccolo et al. 2005; Lepage et al. 2012 ) is applicable to the exponential family of continuous-time point process models used in this work.. A typical statistical analysis is an iterative exercise. By reducing the time to compute by even a factor of two, two weeks of model fitting becomes one. This contribution is expected to be a welcome addition to those performing statistical inference with point process models of neural activity.. Appendix B: Non-orderly discrete-time "Gaussian quadrature" process Gaussian quadrature might be used to compute the MLE associated with a discrete-time process. Based upon the Gaussian quadrature nodes t j , j = 1, . . . , q, consider the increment process, Y j = N(t j )−N(t j −1 ). Unlike in the previous discussion of a discrete-time point process, here the duration, j , of the j th increment is not constant, but rather equals t j − t j −1 . This duration may be relatively large, and the orderliness property of the process is not guaranteed. For a given sample path of the process, let the number of observed counts in the j th increment be y j . The associated log-likelihood, y , can be shown to equal, is provided in the following derivation. Begin by expandingλ q (t|β q ) in terms of the order q = 2q + x Legendre polynomials, where x is a user defined quantity specified in Section 6. The choice of x is discussed in Footnote 5 (Section 6). That is, compute the coefficients c j , j = 0, . . . , q − 1, using order q Gaussian quadrature:. Here j indexes the Legendre polynomials, while j indexes the roots, t j , of the q th order Legendre polynomial, L q . Then, . Here L (u ) j is the u th derivative of the order j Legendre polynomial. Because it can be shown, beginning with Rodriguez's formula, that L (q) n is equal to, . Equations (44), (45) and (47) . Continuing, . Here p 0 is a vector with identical entries equal to the constant L 0 , and t j , j = 1, 2, . . . , q are chosen as the roots of L q :. L q (t j ) = 0 , j = 1, . . . , q .. Thus specified, Equation (8) is exact when λ is a polynomial of order less than or equal to 2q − 1. To see this consider the integral of the order q polynomial z 2q−1 . Following Stoer and Bulirsch (2002) , this polynomial can be expressed as,. whereq (

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


Systematic

identification

of

transcriptional

and

post-transcriptional

regulations

in

human

respiratory

epithelial

cells

during

influenza

A

virus

infection

Background:

Respiratory

epithelial

cells

are

the

primary

target

of

influenza

virus

infection

in

human.

However,

the

molecular

mechanisms

of

airway

epithelial

cell

responses

to

viral

infection

are

not

fully

understood.

Revealing

genome-wide

transcriptional

and

post-transcriptional

regulatory

relationships

can

further

advance

our

understanding

of

this

problem,

which

motivates

the

development

of

novel

and

more

efficient

computational

methods

to

simultaneously

infer

the

transcriptional

and

post-transcriptional

regulatory

networks.

Results:

Here

we

propose

a

novel

framework

named

SITPR

to

investigate

the

interactions

among

transcription

factors

(TFs),

microRNAs

(miRNAs)

and

target

genes.

Briefly,

a

background

regulatory

network

on

a

genome-wide

scale

nodes

potential

interactions)

is

constructed

from

curated

knowledge

and

algorithm

predictions,

to

which

the

identification

of

transcriptional

and

post-transcriptional

regulatory

relationships

is

anchored.

To

reduce

the

dimension

of

the

associated

computing

problem

down

to

an

affordable

size,

several

topological

and

data-based

approaches

are

used.

Furthermore,

we

propose

the

constrained

LASSO

formulation

and

combine

it

with

the

dynamic

Bayesian

network

(DBN)

model

to

identify

the

activated

regulatory

relationships

from

time-course

expression

data.

Our

simulation

studies

on

networks

of

different

sizes

suggest

that

the

proposed

framework

can

effectively

determine

the

genuine

regulations

among

TFs,

miRNAs

and

target

genes;

also,

we

compare

SITPR

with

several

selected

state-of-the-art

algorithms

to

further

evaluate

its

performance.

By

applying

the

SITPR

framework

to

mRNA

and

miRNA

expression

data

generated

from

human

lung

epithelial

cells

in

response

to

virus

infection,

we

are

able

to

detect

the

activated

transcriptional

and

post-transcriptional

regulatory

relationships

as

well

as

the

significant

regulatory

motifs.Conclusion:

Compared

with

other

representative

state-of-the-art

algorithms,

the

proposed

SITPR

framework

can

more

effectively

identify

the

activated

transcriptional

and

post-transcriptional

regulations

simultaneously

from

a

given

background

network.

The

idea

of

SITPR

is

generally

applicable

to

the

analysis

of

gene

regulatory

networks

in

human

cells.

The

results

obtained

for

human

respiratory

epithelial

cells

suggest

the

importance

of

the

transcriptional,

post-transcriptional

regulations

as

well

as

their

synergies

in

the

innate

immune

responses

against

IAV

infection.

In

this

study,

we

proposed

a

novel

framework

for

systematic

identification

of

transcriptional

and

post-transcriptional

regulations

(SITPR)

using

curated

knowledge,

sequencebased

predictions

and

time-course

expression

data.

Briefly,

a

comprehensive

background

regulatory

network,

which

is

not

cell

type-or

disease-specific,

was

constructed

at

the

genome-wide

scale

using

information

databases

and

predictions

from

representative

algorithms.

After

dividing

the

background

network

into

smaller

modules,

we

further

proposed

the

constrained

LASSO

method

and

applied

it

with

the

dynamic

Bayesian

network

model

to

each

of

the

modules

to

determine

the

activated

regulatory

relationships

based

on

the

time-course

expression

data

of

mRNAs

and

miRNAs.

The

simulation

studies

clearly

suggest

the

superiority

of

the

proposed

framework

over

other

representative

approaches

in

the

context

of

identifying

activated

regulatory

relationships

from

a

given

background

network.

The

SITPR

framework

was

then

applied

to

real

data

to

identify

the

activated

regulatory

relationships

among

TFs,

miRNAs

and

target

genes

in

human

respiratory

epithelial

cells

in

response

to

influenza

virus

infection.

Different

from

many

existing

methods

for

inferring

gene

regulatory

network

that

are

purely

data-driven

we

incorporated

curated

knowledge

and

utilized

sequence

information

besides

time-course

gene

expression

data.

Including

curated

knowledge

can

improve

the

accuracy

of

identifying

genuine

regulatory

relationships

and

the

incorporation

of

predicted

potential

regulatory

interactions

among

TFs,

miRNAs

and

genes

allows

to

discover

novel

regulatory

relationships

between

regulators

and

targets.

Furthermore,

the

way

we

constructed

the

background

network

(e.g.,

using

predictions

of

sequence

binding

instead

of

associations

between

expression

data

and

the

use

of

the

constrained

LASSO

help

to

control

the

false

positive

rate

in

identifying

activated

regulatory

relationships,

as

suggested

in

Also,

since

we

considered

the

transcriptional

and

post-transcriptional

regulations

simultaneously,

the

identified

activated

regulatory

relationships

could

be

less

biased

It

should

be

mentioned

that,

limited

by

data

availability

and

algorithm

prediction

accuracy,

our

background

network

built

in

this

study

is

not

thorough;

however,

the

SITPR

framework

is

flexible

and

allows

us

to

update

the

background

network

whenever

new

knowledge

is

discovered

or

better

prediction

algorithms

are

devised..

Using

the

proposed

framework,

the

transcription

regulatory

landscape

was

depicted

for

human

respiratory

epithelial

cells

infected

by

influenza

A

virus.

The

transcriptional

and

post-transcriptional

regulations

were

simultaneously

identified

and

analyzed

for

airway

epithelial

cells

in

response

to

IAV

infection.

Besides

pairwise

interactions,

we

paid

particular

attention

to

regulation

motifs

and

their

synergistic

effects

on

gene

expressions.

We

identified

ten

regulation

motifs

using

FANDOM

and

selectively

presented

and

discussed

some

important

motifs

activated

during

IAV

infection,

which

could

suggest

potential

targets

for

influenza

treatment

or

vaccination.

It

should

be

addressed

that

the

regulation

motifs

we

defined

in

this

study

are

not

only

the

functional

building

blocks

but

also

the

topological

elements

in

a

complex

regulatory

network.

Identification

of

such

motifs

can

provide

the

basis

for

advanced

analyses

such

as

differential

network

biology

The

SITPR

is

also

applicable

to

a

pre-defined

gene

set.

For

example,

Figure

illustrates

the

activated

regulatory

relationships

in

the

KEGG

IAV

pathway,

in

which

we

included

the

genes

contained

in

this

pathway

as

well

as

the

neighboring

miRNAs

in

our

background

network.

One

can

tell

from

Figure

that

(signal

transducer

and

activator

of

transcription

directly

regulates

the

expressions

of

different

miRNAs,

target

genes

and

two

other

TFs,

suggesting

a

highly

influential

role

of

In

addition,

we

found

that

(interferon

regulatory

factor

and

form

a

co-regulation

motif.

and

(Janus

kinase

also

forms

another

co-regulation

motif.

and

are

known

to

be

the

key

regulators

in

the

'Jak-STAT

signaling

pathway',

which

plays

a

critical

role

in

regulating

immune

responses

The

co-regulation

motif

was

identified,

which

may

indicate

an

important

role

of

miRNA

in

controlling

the

production

of

the

chemokine

In

short,

one

obvious

benefit

of

analyzing

a

pre-defined

gene

set

is

that

the

network

dimension

is

significantly

reduced

such

that

SITPR

can

achieve

a

better

accuracy,

as

suggested

by

our

simulation

studies.

Loveday

et

al.

in

which

the

time-course

expression

data

used

in

this

study

were

generated,

also

conducted

a

regulation

study

on

a

pre-defined

gene

set.

However,

they

selected

hundreds

of

differentially

expressed

genes

and

only

focused

on

the

post-transcriptional

regulations

between

miRNAs

and

target

genes.

Also,

the

regulatory

relationships

identified

in

Loveday

et

al.

were

primarily

based

on

the

baseline

and

the

data

on

hour

Therefore,

although

there

exist

a

few

regulatory

relationships

identified

in

both

our

study

and

Loveday's

(e.g.,

and

the

majority

of

the

regulatory

interactions

identified

in

the

two

studies

are

different

due

to

all

the

reasons

described

above..

In

summary,

the

background

regulatory

network

constructed

in

this

study

provides

a

start

point

for

the

identification

of

condition-specific

regulatory

interactions.

Together

with

the

selected

dimension

reduction

techniques,

the

SITPR

framework

is

formed

to

achieve

a

(See

figure

on

previous

page.)

Figure

The

activated

regulatory

relationships

in

two

example

modules.

TFs,

miRNAs

and

genes

are

in

cyan,

magenta

and

green,

respectively.

The

up-regulation

and

down-regulation

are

labeled

in

red

and

blue,

respectively.

The

background

regulatory

relationships

which

are

not

activated

are

in

gray.

The

'arrow',

'T'

and

'diamond'

shapes

of

edge

terminals

denote

to

up-,

down-,

and

undetermined-regulations,

respectively.

(A)

This

module

contains

the

co-regulation

motifs

and

It

also

contains

some

two-node

regulatory

motifs,

e.g.,

and

(B)

This

module

contains

the

regulatory

motifs

'BRCA1'-'has-miR-28-5p'-'TUBB'/'POLR2A',

and

better

accuracy

at

an

affordable

computing

cost.

Also,

for

human

respiratory

epithelial

cells

in

response

to

influenza

A

virus

infection,

the

results

generated

using

SITPR

reveal

a

number

of

interesting

activated

regulatory

interactions

and

provide

useful

clues

for

future

experimental

design

and

validation.

Seasonal

and

pandemic

influenza

A

virus

(IAV)

continues

to

be

a

public

health

threat

and

to

exert

a

large

economic

burden

worldwide

IAV

RNA

segments

that

encode

the

hemagglutinin

(HA)

and

neuraminidase

(NA)

proteins

can

undergo

mutation

(antigenic

drift)

or

reassortment

(antigenic

shift),

resulting

in

new

viral

strains

that

humans

may

lack

the

heterologous

immunity

against

(e.g.,

the

pandemic

In

such

circumstances,

the

cell-mediated

and

humoral

immune

responses

are

primary,

and

a

better

understanding

of

the

molecular

mechanisms

of

immune

responses

to

IAV

infection

thus

becomes

necessary

to

the

development

of

more

effective

prevention

and

treatment

strategies.

Particularly,

the

regulations

of

cell

gene

expression

are

of

significant

importance

due

to

their

critical

roles

in

shaping

the

magnitude

and

timing

of

immune

responses

via

the

promotion

or

suppression

of

the

production

of

various

proteins

and

RNAs

(e.g.,

cytokines

and

miRNAs

Since

different

cell

populations

with

distinct

phenotypes

and

functions

are

expected

to

have

different

gene

regulation

profiles,

it

is

desirable

to

characterize

the

gene

regulations

for

each

cell

population

as

in

In

this

study,

the

respiratory

epithelial

cells

are

of

primary

interest

because

they

are

the

first

barrier

and

the

main

target

of

IAV

infection

in

human

A

number

of

previous

studies

have

measured

and

analyzed

the

gene

expression

profiles

in

respiratory

epithelial

cells

during

IAV

infection

however,

only

a

few

studies

investigated

the

gene

expression

regulators

like

transcription

factors

or

miRNAs

Several

studies

in

other

biological

disciplines

show

that

TFs

and

miRNAs

cooperatively

interact

with

each

other

and

both

of

them

should

be

considered

in

the

regulatory

network

model

To

our

best

knowledge,

simultaneous

identification

of

both

the

transcriptional

(TF)

and

post-transcriptional

(miRNA)

regulations

on

a

genome-wide

scale

in

human

respiratory

epithelial

cells

post

IAV

infection

has

not

been

sufficiently

addressed..

A

number

of

systems

biology

approaches

have

been

developed

to

understand

the

transcriptional

or

posttranscriptional

regulations

including

several

studies

particularly

for

influenza

virus

infection

For

example,

Butte

and

Kohane

considered

the

pair-wise

mutual

information

to

screen

the

gene

data

associations.

Friedman

et

al.

employed

the

Bayesian

network

model

to

capture

the

conditional

independencies

between

genes.

Yeung

et

al.

used

the

singular

value

decomposition

(SVD)

and

the

robust

regression

for

network

reverse

engineering,

given

the

sparsity

of

real

large-scale

networks.

Zhang

and

Horvath

proposed

the

weighted

gene

co-expression

network

analysis,

which

assigns

a

continuous

connection

weight

between

and

to

each

gene

pair.

Basso

et

al.

combined

the

mutual

information

with

the

data

processing

inequality

theory

to

analyze

the

gene

expression

profiles

in

human

B

cells.

Ordinary

differential

equation

models

have

also

been

considered

in

the

previous

studies

in

which

the

computing

efficiency

issues

were

addressed

for

highdimensional

dynamic

systems.

However,

as

evaluated

in

the

accuracy

of

these

existing

methods

are

not

satisfying,

and

the

reliability

of

network

reverse

engineering

from

gene

expression

data

needs

to

be

further

improved.

More

importantly,

two

key

issues

are

not

sufficiently

addressed

in

the

previous

studies:

first,

while

both

TFs

and

miRNAs

are

the

major

regulators

in

controlling

gene

expression

of

the

methods

mentioned

above

simultaneously

considered

both

transcriptional

and

post-transcriptional

regulations

such

that

the

inferred

interactions

could

be

biased;

second,

most

of

the

existing

approaches

are

purely

data-driven

and

thus

the

results

could

have

a

high

false

positive

rate

In

this

study,

we

propose

a

new

framework

called

SITPR,

which

stands

for

Systematic

Identification

of

Transcriptional

and

Post-transcriptional

Regulations,

to

identify

the

regulatory

relationships

by

exploiting

both

curated

knowledge

and

time-course

expression

data

of

mRNAs

and

miRNAs.

For

this

purpose,

a

background

regulatory

network

is

constructed

on

a

genome-wide

scale

by

collecting

the

experimentallyobserved

interactions

in

literature

or

public

databases

as

well

as

the

potential

regulatory

interactions

predicted

by

representative

algorithms

for

quantifying

the

sequence-selective

binding

feasibility

between

TFs/

miRNAs

and

genes

Since

this

background

network

is

not

cell

type

or

disease

specific,

only

part

of

the

interactions

in

this

background

will

be

activated

in

respiratory

epithelial

cells

in

response

to

influenza

infection.

To

identify

these

activated

regulatory

interactions,

we

first

employ

several

dimension

reduction

approaches,

including

the

community

detection

algorithm

based

on

network

modularity

the

time

delay

detection

method

the

functional

principal

component

analysis

(fPCA)

for

screening

differentially

expressed

genes

and

the

maximum

information

coefficient

(MIC)

After

dividing

the

large

network

into

much

smaller

modules,

we

propose

the

constrained

LASSO

method

and

combine

it

with

the

dynamical

Bayesian

network

(DBN)

model

to

detect

the

activated

regulatory

relationships

from

timecourse

expression

data

within

each

module.

Our

simulation

studies

suggest

that

the

proposed

framework

can

achieve

satisfying

performance

in

terms

of

sensitivity

and

specificity

for

networks

of

different

sizes.

We

then

apply

the

framework

to

the

real

experiment

data

and

analyze

the

topological

and

functional

features

of

the

regulatory

network

in

respiratory

epithelial

cells

in

response

to

influenza

virus

infection.

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


Algebraic tools for the construction of colored flows with boundary constraints Abstract. We give a linear time algorithm which, given a simply connected figure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed flow lines which match inputs and outputs according to the colors, in such a way that each edge of any cell is crossed by at most one line. The main tool is the notion of height function, previously introduced for tilings. It appears as an extension of the notion of potential of a flow in a planar graph.Mathematics Subject Classification. 05C25, 05C85. The motivations of the authors of this paper partially come from tiling problems: fix a finite set of basic tiles (which are called prototiles). Given a figure of the plane, does there exist a tiling of this figure with copies of the prototiles? In the affirmative, how can such a tiling be exhibited? Conway and Lagarias [2] defined tiling groups which are a very efficient tool for studying these problems. Their method gives a lot of necessary conditions for a simply connected figure to be tileable (see [7, 9] ).. This work has been extended by Thurston [11] who introduced (in some particular cases) the notion of height function associated with a tiling. Using this new notion, algorithms to tile a simply connected figure with "dominoes" (i.e. 2 × 1 rectangles), or with "calissons" (i.e. lozenges of unit side formed from two equilateral triangles sharing an edge), have been produced.. Thurston's ideas have been taken again and generalized, providing other results about tilings on regular lattices [5, 10] , but these ideas did not seem easy to apply to an irregular lattice: before this paper, the only result on a partially irregular lattice was the result of Chaboud [1] , which produced an algorithm for tiling with (generalized) dominoes formed from two cells of the lattice. This author only assumes that the cells of the lattice are 2-colorable and that all the cells of the lattice have the same number of edges.. In this paper, we prove that Thurston's method can be applied to solve a flow problem introduced here: given a figure F divided into cells, with colored arrows crossing the boundary of F , coming into F or going out of F , can we join coming in arrows and going out arrows by non-intersecting directed polygonal lines, with respect to colors, in such a way that any edge of any cell is crossed by at most one line? In the affirmative, how can such directed lines be exhibited? Notice that we do not assume that the structure of F has any regularity. This paper is divided as follows: in Section 1, we present our problem and give an algebraic interpretation of it. This algebraic model permits us to apply Thurston's method: introduction of a group function and a height function (Sect. 2), creation of an order on the solutions and study of minimal solutions according to this order (Sect. 3), construction of a minimal solution (Sect. 4). In Sections 5 and 6, we present two similar problems and the elements that permit us to produce the algorithms which solve them. We finish this paper by pointing out three facts which seem to be the most important contributions of this paper.. 1) In the special case when only one color is used, the problems seen above are flow problems (in the classical way), and the group function and height function here are the "potential function" introduced by Hassin ( [3, 4] ) to produce an algorithm of maximum flow on a planar network. Thus the algebraic notions developed in this paper are generalizations of the potential function studied before.. 2) The algorithms of this paper prove that the main reason for which Thurston's method [11] holds is not the regularity of the lattice used, since in our problems one can use an irregular figure. 3) This paper makes a bridge between flows problems and some tilings problems. They both appear as problems of labeling the edges of a cellular figure by elements of a group, with local constraints. We can now give an algorithm which either gives a solution of the CIO problem (if such a solution exists) or indicates that there is no solution. When a solution exists, this algorithm constructs a minimal solution S 0 .

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


Image Segmentation by Probabilistic Bottom-Up Aggregation and Cue Integration We present a bottom-up aggregation approach to image segmentation. Beginning with an image, we execute a sequence of steps in which pixels are gradually merged to produce larger and larger regions. In each step, we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. Our probabilistic formulation takes into account intensity and texture distributions in a local area around each region. It further incorporates priors based on the geometry of the regions. Finally, posteriors based on intensity and texture cues are combined using "a mixture of experts" formulation. This probabilistic approach is integrated into a graph coarsening scheme, providing a complete hierarchical segmentation of the image. The algorithm complexity is linear in the number of the image pixels and it requires almost no user-tuned parameters. In addition, we provide a novel evaluation scheme for image segmentation algorithms, attempting to avoid human semantic considerations that are out of scope for segmentation algorithms. Using this novel evaluation scheme, we test our method and provide a comparison to several existing segmentation algorithms. I MAGE segmentation algorithms aim at partitioning an image into regions of coherent properties as a means for separating objects from their backgrounds. As objects may be separable by any of a variety of cues, be it intensity, color, texture, or boundary continuity, many segmentation algorithms were developed and a variety of techniques were utilized including: clustering [1] , [2] , Markov random fields [3] , [4] , variational methods [5] , [6] , and level set methods [7] .. A different approach that has recently gained popularity is to apply graph algorithms to segmentation. Typically, given an image, a graph is constructed in which a node represents a pixel and weighted edge is used to encode an "affinity" measure between nearby pixels. The image is then segmented by minimizing a cost associated with partitioning the graph into subgraphs. In the simpler version, the cost is the sum of the affinities across the cut [8] , [9] . Other methods normalize the cut cost either by dividing it by the overall area or the boundary length of the segments [10] , [11] or by normalizing the cut cost with measures derived from the affinities within the segments [8] , [12] , [13] , [14] . A prominent example for the latter approach is the normalized-cut algorithm [15] , but attaining a globally optimal solution for the normalized-cut measure and similar measures is known to be NP-hard even for planar graphs. While polynomial time approximations to the normalized cuts are available, the computational requirement remains high. In an attempt to overcome this, [16] uses connections at different scales to decrease the complexity of the graph. Another multiscale approach, presented in [17] , utilizes a fast multilevel solver based on Algebraic Multigrid (AMG) to construct a hierarchy of segmentations where average image cues are estimated with minimal mixing of segments statistics. The average cues then influence the construction of further scales in the hierarchy to produce a more reliable segmentation.. Methods like [15] , [17] have been designed to utilize and combine multiple cues. Typically in such algorithms, each cue is handled by a separate module whose job is to assess the coherence of nearby pixels or regions according to that cue, and a segmentation decision is obtained by incorporating these similarities into a combined measure. Careful design of these modules, along with the use of appropriate optimization methods, has led to notable successes, but the challenge of reliably segmenting objects in a variety of natural images still lies ahead.. The utilization of multiple cues aggravates an old problem. In many multicue segmentation algorithms, each module comes with its own set of parameters, and those join an additional set of parameters intended to control the relative influence of each module. These parameters may depend nontrivially on the particular statistics of the input image, or even the statistics of different regions in the same image. While existing methods may be robust to changes in some of those parameters, segmentation results in many cases may depend critically on the proper assignment of parameter values. The common practice is to leave those parameters to be set by the user, but in effect most users leave the parameters in their default values. Allowing these parameters to automatically adapt to an image (or even locally to image portions) can greatly simplify the use of segmentation algorithms and potentially allow them to consistently provide better results. Indeed, recent algorithms attempt to achieve parameter-free segmentation, either by adapting to a specific class of images (e.g., [18] ) or, in the case of natural images, by relying on a training set that includes a variety of manually segmented images (e.g., [19] ). A different stream of work uses cluster analysis to estimate a global set of parameters (e.g., stability criteria in [20] ).. In this paper, we explore a different approach which relies primarily on local information available within the image to be segmented. We present a probabilistic approach to segmentation that is almost parameter free. Beginning with an image, we execute a sequence of steps in which pixels are gradually merged to produce larger and larger regions. In each step, we consider pairs of adjacent regions and provide a probability measure to assess whether or not they should be included in the same segment. We illustrate this method by constructing modules to handle intensity contrast and texture differences, and use an adaptively controlled "mixture of experts"-like approach to integrate the different cues and reach unified segmentation decisions. To illustrate the importance of adaptive, local cue integration consider the example in Fig. 1 , which shows two pairs of regions. The left pair can be distinguished by intensity cues, whereas the right pair of patches, which have similar texture, should be merged despite their different intensities.. Our approach is designed to work with bottom-up merge strategies for segmentation. A large number of methods approach segmentation using bottom-up merge strategies, beginning with the classic agglomerative clustering algorithm [21] to watershed [22] , [23] and region growing (including methods that use probabilistic approaches [24] , [25] ) to more recent algebraic multigrid inspired aggregation [17] . Merge algorithms generate a hierarchy of segments, allowing subsequent algorithms to choose between possible segmentation hypotheses. For implementation, we adapt the coarsening strategy introduced in [17] , as it enables incorporating, at every level of the hierarchy, measurements appropriate to the scale at that level.. Another contribution of our paper is a novel segmentation evaluation scheme that is suited for the evaluation of data-driven segmentation algorithms. Evaluating the results produced by segmentation algorithms is challenging, as it is difficult to come up with canonical test sets providing ground truth segmentations. This is partly because manual delineation of segments in everyday complex images can be laborious and often tends to incorporate semantic considerations which are beyond the scope of data-driven segmentation algorithms. In this paper, we propose an evaluation scheme that is based on an image data set which was specifically chosen such that the human annotations would avoid semantic considerations. We test our parameter-free approach on this database and compare our results to several existing algorithms. This paper is based on [26] , where we have expanded the evaluation test set. This results in a comprehensive segmentation evaluations scheme that accounts for changes in both scale and lowlevel cues. We have presented an approach to image segmentation that is almost parameter free. Our approach uses a bottom-up aggregation procedure in which regions are merged based on probabilistic considerations. The framework utilizes adaptive parametric distributions whose parameters are estimated locally using image information. Segmentation relies on an integration of intensity and texture cues, with priors determined by the geometry of the regions. The method is modular, and can readily be extended to handle additional cues. We further applied the method to a large database with manually segmented images and compared its performance to several recent algorithms. Our probabilistic framework is designed to work with any merge algorithm for segmentation. Here, we use the merge strategy suggested for the Segmentation by Weighted Aggregation (SWA) algorithm [17] , [32] , which employs a hierarchy construction procedure inspired by Algebraic Multigrid solutions for differential equations [33] . The SWA algorithm begins with a weighted graph representing image pixels, and in a sequence of steps creates a hierarchy of smaller ("coarse") graphs with soft relations between nodes at subsequent levels. The edge weights in the new graphs are determined by inheritance from previous levels and are modified based on regional properties. These properties are computed recursively as the merge process proceeds. Below we use the coarsening strategy of the SWA algorithm and modify it to incorporate our probabilistic framework. In particular, we use as edge weights the posterior probabilities defined in Section 2. We produce the coarser graphs using the coarsening strategy of SWA, but replace inheritance of weights by computing new posteriors. Overall, we achieve a method that is as efficient as the SWA algorithm, but relies on different, probabilistic measures to determine segmentation and requires almost no user-tuned parameters.

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


Formal Security Treatments for IBE-to-Signature Transformation: Relations among Security Notions In a seminal paper of identity based encryption (IBE), Boneh and Franklin [BF01] mentioned an interesting transform from an IBE scheme to a signature scheme, which was observed by Moni Naor. In this paper, we give formal security treatments for this transform and discover several implications and separations among security notions of IBE and transformed signature. For example, we show for such a successful transform, one-wayness of IBE is an essential condition. Additionally, we give a sufficient and necessary condition for converting a semantically secure IBE scheme into an existentially unforgeable signature scheme. Our results help establish strategies on design and automatic security proof of signature schemes from (possibly weak) IBE schemes. We also show some separation results which strongly support that one-wayness, rather than semantic security, of IBE captures an essential condition to achieve secure signature. Identity-based encryption (IBE) [Sh84, BF01] is a public key encryption scheme where a user's public key can be any bit string, such as an email address. Although IBE was originally advocated by Shamir [Sh84] to simplify public key and certificate management, it has now been shown a powerful tool in constructing various cryptographic applications: key-insulated encryption [DKXY02, BP02] , forward secure encryption [An97, CHK03] and public key encryption with keyword search [BDOP04] , etc. In this paper, we investigate an interesting application of IBE, whose observation was attributed to Naor, saying that "an IBE scheme can immediately be converted into a public key signature scheme" [BF01] .. Let us briefly recall Naor's observation. In IBE, a private key generator (PKG) uses his master key msk to issue a decryption key d which corresponds to an arbitrary bit string "ID". Here, msk can also be seen as a signing key of the PKG, and by letting ID = M , d becomes the PKG's signature for M where M is a message to be signed. The signature verification can be done by checking if d functions properly as a correct IBE decryption key for identity "M " by encrypting a random plaintext and checking if the ciphertext is decrypted to the original plaintext. We hereafter call such a transformation, the Naor Transform (NT), and N T (Π) denotes a signature scheme derived from an IBE scheme Π via NT (Refer Sec. 3 for a full description).. We investigate implications and separations among the notions of IBE and signature in detail. Our results provide generic security proofs for a wide range of Naor-transformed signatures. For example, we show that secure signatures can be generally derived from considerably weak IBE schemes.

O O O O O O O O B I 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 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


Generic multivariate model for color texture classification in RGB color space Abstract This paper presents a new method for modeling magnitudes of dual-tree complex wavelet coefficients, in the context of color texture classification. Based on the characterization of dependency between RGB color components, Gaussian copula associated with Generalized Gamma marginal function is proposed to design the multivariate generalized Gamma density (MG D) modeling. MG D has the advantages of genericity in terms of fitting over a variety of existing joint models. On the one hand, the generalized Gamma density function offers free-shape parameters to characterize a wide range of heavy-tailed densities, i.e., the genericity. On the other hand, the inter-component, inter-band dependency is captured by the Gaussian Copula which offers adapted flexibility. Moreover, this model leads to a closed form for the probabilistic similarity measure in terms of parameters, i.e., Kullback-Leibler divergence. By exploiting the separability between the copula and the mar- ginal spaces, the closed form enables us to minimize the computational time needed to measure the discrepancy between two Multivariate Generalized Gamma densities in comparison to other models which imply using a Monte Carlo method characterized by an expensive time computing. For evaluating the performance of our proposal, a K-nearest neighbor (KNN) classifier is then used to test the classification accuracy. Experiments on different benchmarks using color texture databases are conducted to highlight the effectiveness of the proposed model associated to the Kullback-Leibler divergence. For various practical applications in computer vision, texture analysis is a useful and key component for solving problems such as pattern recognition, classification or segmentation. Thus, in practical applications, an efficient modeling of the variation of intensity that characterizes the texture in the image is central. The relevance of the model determines to a large extent the effectiveness of the feature extraction and then the performance of the image processing method. However, as relevant as the model is, it is also important to propose an appropriate similarity measure and easy to use.. Many recent works devoted to texture stressed that accurate feature extraction can be achieved by statistical modeling of subband coefficients in the transformed domain of wavelets. A variety of wavelet decompositions can be applied, ranging from discrete wavelet transform (DWT), Steerable Pyramids, to the recent complex wavelet trans-forms [1] . Each of these transformations has some particularities, but all share the property sparsity of their subband distributions. Histograms are heavy tailed and pickily pronounced, and then need a sub-Gaussian modeling. To fulfill this need, in a pioneering work, Do and Vetterli [2] proposed the Generalized Gaussian Density (GGD) as an alternative to the Gaussian model. Recently, the magnitudes of complex subbands coefficients have been modeled by Weibull and Gamma distributions [3, 4] , leading to a considerable enhancement of the retrieval performances. In the same way, we have proposed the Generalized Gamma distribution as a model for the coefficient magnitudes issued from the DualTree Complex Wavelet Transform (DTCWT), for grey-level texture classification issue [5] . A more general work was done inspired by the Generalized Gamma density, but in the context of texture retrieval [6] .. In these previous works, only marginal distributions of subband histograms are taken into account. This is suitable for grey-level images when independence between subbands of the same scale is supposed. However, in the case of color textures, the dependence between color components is undeniable and must be taken into account by conceiving multivariate models, otherwise the system will suffer from the lack of this crucial information. The joint statistical modeling is then a welcome advantage for color texture retrieval or classification. Such framework has been considered previously by many authors. Verdoolaege et al. [7] proposed a multivariate Generalized Gaussian distribution (MGGD) for multiscale color texture retrieval. The model was devoted to describe dependence across color components while assuming independence among subbands of a single color component. In [8, 9] , Kwitt et al. treated the problem of joint modeling of complex coefficient magnitudes across subbands of different color components. Based on Gaussian copula and student t copula for modeling the dependence structure, in conjunction with Weibull and Gamma densities as parametric margin models, they achieved significant enhancements in the context of texture retrieval. Another Copula-based multivariate modeling was proposed by Sakji-Nsibi et al. [10] , for multicomponent image indexing using Gaussian Copula in conjunction with the GGD and Gamma densities. Again substantial improvement of classification results was obtained in comparison with the marginal modeling approach. However, texture databases are extremely increasing in terms of diversity and heterogeneity of textures, which limits the use of one model over others. Moreover, the lack of analytical expression for the KL divergence in the case of multivariate models based on copulas presents a shortcoming, since the alternative Monte Carlo-based approach [8, 10] is computationally expensive.. Based on these observations and to remedy the problem of complex nature and huge variety of textures in the databases, we propose a generic multivariate model that takes into account the strong dependency between RGB color components. To manage the diversity, the Generalized Gamma density is proposed associated to the Gaussian copula for capturing the dependence across color component subbands of a fixed and/or different pair of orientation at a fixed scale of decomposition, i.e., the MG D model. The choice of the Multivariate Generalized Gamma based on the Gaussian copula is justified by the existence of KL divergence for Gaussian copula and Generalized Gamma margin model, since we present a closed form of the KL divergence-based similarity measure.. Then, contribution in this work is threefold:. • First, concerning the dataset we consider an inter-band, inter-component dependency which seems to provide a rich information for the characterization process.. • Second, concerning the model we present MG D as a generic multivariate model to capture the rich information of dependency between R, G and B components subbands.. • Three, concerning the similarity measurement which is a bit hard to calculate in case of copulas-based models.. We present a closed-form expression of the KL divergence between two MG D densities which gives a big advantage over the Monte Carlo-based approach.. Experiments using the K-nearest neighbor (KNN) classifier show the superiority of the Multivariate Generalized Gamma model over the existing joint models, and highlight also the effectiveness of the proposed metric when compared with the Monte Carlo-based approach. This paper is organized as follows. In the next section, we give an overview of the multivariate statistical modeling in the RGB color space, we present the multivariate proposed model and we derive the Kullback-Leibler divergence. In Sect. 3, we provide texture classification results. Finally, we conclude in the Sect. 4. We have proposed a joint generic model for characterizing DTCWT coefficient magnitudes of color textures. MG D presents a pertinent color texture description in comparison with the marginal approach that assumes independence among color component subbands. MG D also presents a flexible modeling when compared with variety of joint models. We, further, proposed a simple, faster and closedform expression similarity measure using the independence between the marginals space and the dependence structure. The genericity of the proposed model and the pertinence of the similarity measure allowed us to achieve good improvement in terms of classification rate and computational time.. In future works, we would like to use copulas for characterizing color textures in luminance-chrominance color spaces such as L*a*b* or HSV. In such color spaces, the separation between luminance and chrominance information needs a multi-model characterization to improve classification rates.

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