Some Constacyclic Codes Over Z2k and Binary Quasi-cyclic Codes The concept of negacyclic code was recently introduced in Wolfmann (IEEE Trans. Inform. Theory 45 (1999) 2527-2532, in which some relations between the negacyclic codes and their Gray map images are proved. In this note, for k ¿ 1 an isometry ' k between codes over Z 2 k+1 and codes over Z4 is introduced and used to give a generalization of the Gray map equivalent to the one given in Carlet (IEEE Trans. Inform. Theory 44 (1998) 1543-1547). Furthermore, by means of this isometry, the concept of negacyclic codes is extended to codes over the ring Z 2 k+1 , obtaining a class of constacyclic codes referred to as hpo-cyclic codes (half plus one-cyclic codes). A characterization of these codes in terms of their images under ' k is given. It is also proved that the generalized Gray map image of an hpo-cyclic code is a binary distance invariant (not necessarily linear) quasi-cyclic code. Finally, some linear hpo-cyclic codes are discussed and a few examples are given. ? Let R be a commutative ring with identity and let n be a positive integer. For some ÿxed unit of R, let be the automorphism on R n given by (a 0 ; a 1 ; : : : ; a n−1 ) = ( a n−1 ; a 0 ; a 1 ; : : : ; a n−2 ):. Recall that a subset C of R n is a constacyclic code of length n if there exists a unit of R such that it is invariant under the automorphism , that is, (C) = C. If = 1 the code is said to be cyclic.. Constacyclic linear codes of length n over R can be identiÿed as ideals in the quotient ring R[x]=(x n − ) via the isomorphism from R n to R[x]=(x n − ) deÿned by (a 0 ; a 1 ; : : : ; a n−1 ) → a 0 + a 1 x + · · · + a n−1 x n−1 :. In the following we will restrict the ring R to Z 2 k+1 , the ring of integers modulo 2 k+1 , where k is a positive integer. For the special case = −1 and k = 1, the negashift map, , over the ring Z 4 of integers modulo 4 is introduced in [12] as the permutation on the module Z n 4 , given as (a 0 ; a 1 ; : : : ; a n−1 ) = (−a n−1 ; a 0 ; a 1 ; : : : ; a n−2 ) and a negacyclic code of length n over Z 4 is deÿned as a subset C of Z n 4 such that (C) = C.. For any positive integer k, if the unit in Z 2 k+1 is equal to 2 k + 1 (half of 2 k+1 plus one), then a subset C ⊆ Z n 2 k+1 such that (C) = C will be called an hpo-cyclic code. For k ¿ 1, an isometry ' k between codes over Z 2 k+1 and codes over Z 4 is deÿned and used, together with the usual Gray map (i.e., on the ring Z 4 ), to give an equivalent generalization of the Gray map, as introduced in [3] . A characterization of the hpo-cyclic codes in terms of their images under the isometry ' k is given. It is also shown that the generalized Gray map image of an hpo-cyclic code of length n is a binary distance invariant (not necessarily linear) quasi-cyclic code, of length 2 k n. Finally, some linear hpo-cyclic codes are discussed and a few examples are given.

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The State of the Art in Flow Visualization: Dense and Texture-Based Techniques Abstract Flow visualization (FlowVis) is one of the classic subfields of visualization, covering a rich variety of applications, from the automotive industry, aerodynamics, turbomachinery design, to weather simulation, meteorology, climate modeling, ground water flow, and medical visualization. Consequently, the spectrum of FlowVis solutions is very rich, spanning multiple technical challenges: 2D vs. 3D solutions and techniques for steady or time-dependent data.. Bringing many of those solutions in linear order (as necessary for a text like this) is neither easy nor intuitive. Several options of subdividing this broad field of literature are possible. Hesselink et al., for example, addressed the problem of how to categorize techniques in their 1994 overview of research issues 24 and consider dimensionality as a means to classify the literature. In the following, several aspects are discussed on an abstract level before literature is addressed directly.

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Any-time probabilistic switching model using bayesian networks Modeling and estimation of switching activities remain to be important problems in low-power design and fault analysis. A probabilistic Bayesian Network based switching model can explicitly model all spatio-temporal dependency relationships in a combinational circuit, resulting in zero-error estimates. However, the space-time requirements of exact estimation schemes, based on this model, increase with circuit complexity [1, 2] . This paper explores a non-simulative, Importance Sampling based, probabilistic estimation strategy that scales well with circuit complexity. It has the any-time aspect of simulation and the input pattern independence of probabilistic models. Due to drastic growth in semiconductor technology, the number of gates per chip have increased enormously. This increase in chip density, together with decrease in feature sizes have made power dissipation a major issue in VLSI circuits. Average power consumption in circuits depends on three factors: static leakage power, short circuit power, and dynamic power. As device sizes and transistor threshold scales down, power lost due to the leakage cu rrent will increase and will become sizable component of total power dissipation. However, the impact of dynamic power on total power consumption will remain to be substantial. The dynamic power consumed at the gate is given by 0 5V 2 dd f clk C load Sw´xµ, where V dd is supply voltage, f clk is clock frequency, C load is the load capacitance, and Sw´xµ is the switching activity at the node. Thus, modeling and estimation of switching activities remain to be important problems in low-power design and fault analysis.. This work is concentrated at the logic or gate level switching activity estimation. Estimation of switching activity requires the Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISLPED'04, August 9-11, 2004 knowledge of input statistics, connectivity of the circuit, the correlation between nodes, the gate type, and the gate delays, ultimately making the estimation process a complex procedure. The correlations among the nodes affect switching activity. It has been established that for zero-delay model of a combinational circuit, only first order temporal correlation is exhibited [11] , because signals possess first order Markov property. Thus, it is sufficient to consider just first order temporal correlation, but all high order spatial correlations to model all spatio-temporal dependencies in the combinational circuit. The non-simulative probabilistic techniques [4, 5, 7, 8] use knowledge about input statistics to probabilistically estimate the switching activity of internal nodes making the technique pattern-insensitive. In a later effort to capture higher order correlation approximately, Marculescu et al. in [6] handled higher order correlation as a composition of pair-wise correlations.. Recently, we proposed a novel model [1, 2] , for switching activity estimation in combinational circuits using Probabilistic Bayesian Networks [3] , that captures both the temporal and spatial dependencies in a comprehensive manner, resulting in zero-error estimates. Bayesian Networks are a Directed Acyclic Graph (DAG) representations, whose nodes represent random variables and the links denote direct dependencies, quantified by conditional probabilities of a node given the state of its parents. The DAG structure models the joint probability over a set of random variables in a compact manner. Bayesian Networks are exciting probabilistic inferencing models for VLSI circuits, particularly due to the following reasons.. 1. Like any causal models, such as the traditional Binary Decision Diagrams (BDDs), Bayesian Networks models conditional independencies. However, Bayesian network has the least complex network structure in that is a minimal, compact independence map (I-map) of all the independencies among the underlying random variables. This minimal DAG structure is exploited to construct efficient probabilistic updating. 2. It can accommodate input correlation, temporal, and spatial correlations efficiently. 3. Bayesian Networks are unique probabilistic causal model in capturing the induced dependence between independent parents of a node given an observed state for the node.. Bayesian network models of switching activity are inherently zero-error models. However, the space-time complexity of exact estimation schemes increase with circuit complexity. For instance, the inference scheme that we used in [1, 2] , which was a cluster based scheme, resulted exact estimates, however, it was memory intensive. So, for complex circuit, we had to resort to partitioning schemes, resulting in an approximate model of the switching activities in terms of a set of loosely coupled cascaded Bayesian Net-86

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ICP Registration using Invariant Features This paper investigates the use of Euclidean invariant features in a generalization of iterative closest point registration of range images. Pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances. It is shown that under ideal noise-free conditions, correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone. In addition, monotonic convergence to at least a local minimum is shown to hold for this method. When noise is present, a method that automatically sets the optimal relative contribution of features and positions is described. This method trades o error in feature values due to noise against error in positions due to misalignment. Experimental results show that using invariant features decreases the probability of being trapped in a local minimum, and is most e ective for di cult registration problems where the scene is very small compared to the model. Building 3D models of real world objects for reverse engineering, facility mapping, and computer graphics applications typically requires three stages: a data capture stage which samples the 3D world using a range camera, a data registration stage which aligns the various 3D views, and a data merge stage which simpli es the aligned views into parametric models. The goal of the registration stage is to nd the relative position and orientation of each view with respect to each other view. This paper addresses the use of features for improving the probability of convergence of a popular solution to the registration problem, the iterative closest point registration algorithm. We present a theoretical basis for the use of invariant features, an automatic method for selecting the tradeo between features and positions, and an experimental evaluation demonstrating improved convergence.. Range image registration is typically accomplished using a variant of the iterative closest point algorithm ICP 4 . ICP is an asymmetric iterative descent procedure which seeks to minimize the sum of the squared distances between all points in one of the views the scene and their closest points in the other view the model. When a scene and a model can be represented as two point sets with known correspondences, the rigid motion that best aligns the scene in a least square sense can be solved in closed form according to the method of Faugeras and Hebert 12 or the method of Horn 15 . Traditional registration methods construct the correspondence sets by extracting salient features from the scene and model, and perform a search procedure to match the features. In ICP registration, however, Besl and McKay solve the correspondence problem by assuming that the scene is approximately aligned with the model, and therefore that each scene point corresponds with its closest model point 4 . Zhang extended ICP to include robust statistics and adaptive thresholding to handle outliers and occlusions 29 . Masuda and Yokoya use ICP with random sampling and a least median square error measurement that is robust to a partially overlapping scene 19 . Chen and Medioni independently developed an approach similar to ICP, which minimizes the sum of squared distance between scene points and a local planar approximation of the model 7 . Correspondences are formed by projecting the scene points onto the model in the direction of their normal vectors rather than selecting the closest point. Dorai et al. extend the method of Chen and Medioni to an optimal weighted least squares framework 9 . These methods have been extended to make simultaneous registration over multiple views possible 2, 25, 11 .. Since ICP is an iterative descent algorithm, it requires a good initial estimate in order to converge to the global minimum. A fully automated registration algorithm can choose to use multiple initial conditions sampled randomly or uniformly throughout the search space in order to ensure that the goal is found 6 . The search space is large, however, requiring many initial conditions. Therefore several researchers have used features, either alone or together with positions, in order to improve the registration. Chua and Jarvis use principal curvatures to constrain a heuristic search for correspondences 8 . Higuchi, et al. build a spherical map of curvature values called an SAI for each view of an object 14 . The SAI are registered by rotating the spheres until the curvature values are aligned. Feldmar and Ayache perform a ne registration by minimizing the combined distance between positions, surface normals and curvatures 13 . Thirion uses crest lines to extract extremal points and their associated Darboux frames, which are matched in an ICP-like fashion 26 . Soucy and Ferrie locally register surface patches by minimizing the distance between Darboux frames over an entire neighborhood 24 . Yang and Allen minimize a scaled product of positional and curvature distances 28 . VandenWyngaerd, et al. match bitangent curve pairs, which are pairs of curves that share the same tangent plane, between two views for rigid and a ne registration 27 . Johnson uses invariants derived from the spin-image, a histogram of distances and angles to nearby surface points, to perform recognition and registration of 3D range maps 18, 17 .. Our work investigates a particular instance of the feature-based ICP approach, which we call Iterative Closest Points using Invariant F eatures ICPIF 23 . This method chooses nearest neighbor correspondences according to a distance metric which is a scaled sum of the positional and feature distances. We show that under ideal, noise free conditions, correct correspondences are chosen at least as often using ICPIF as they would be using traditional ICP. In addition, we show that ICPIF converges monotonically to a local minimum in the same manner as traditional ICP. An automatic method for computing the tradeo between positions and features is demonstrated. Experimental evidence demonstrates that ICPIF converges to the goal state in fewer iterations than traditional ICP, and that it converges to the goal state for more initial transformations. Section 2 of this paper introduces the ICP algorithm and three di erent i n variant v alues that may be computed directly from range data. Section 3 introduces ICPIF, an extension of the ICP algorithm that uses features to improve the correspondence search. In section 4 we analyze the noise-free properties of ICPIF, and in section 5 we describe the tradeo s between position and feature values under a Gaussian noise model. Section 6 presents experimental results on simulated and real range data, and section 7 presents concluding remarks. This paper presents ICPIF, a fully automatic range image registration method that uses shape features in conjunction with point positions to register range images without the need for a user supplied initial estimate. Theoretical results show that under noise-free conditions, ICPIF chooses the correct pointwise correspondences at least as well as ICP, and that monotonic convergence to a local minimum is preserved. Convergence to the ground truth registration occurs more often and in fewer iterations than traditional ICP. The relative weights of the feature and positional components can be controlled by trading o error in feature values caused by noise against error in positions caused by misalignment. This is accomplished using a calibration-time estimation of feature noise and fully automatic runtime estimation of misalignment. Experimental results on real and synthetic images suggest that for some alignment problems, matching can be performed using features alone, while for larger alignment problems, a blend of position and features may be better. To test the e cacy of the ICPIF algorithm, we h a ve run experiments on the synthetic and real data sets shown in gure 7. The Cruiser data set is a synthetic data set constructed using the Z bu er output of the Radiance rendering software, and the Odetics data set contains range images taken with an Odetics laser range nder atop a mobile platform. For the Odetics images, distortion and scaling have been corrected using the default sensor calibration parameters 16 , but no ltering or noise reduction was performed. From each o f these data sets we h a ve selected both an easy" test case and a hard" test case. The easy test case seeks to register the scene within a model that is only slightly larger than the scene, while the hard test case uses a model that is considerably larger than the scene. By larger we mean that the sensor captures a wider eld of view; there is no change of scale between the scene and model data. Three moment i n variants, three spherical harmonics invariants and two curvature invariants were computed at each point in both images using the method described in section 2. A 9 9 9 uniform Cartesian sampling of the space centered at the image point w as used to perform the integration. The size of the enclosing sphere was set to approximately 2 of the size of the model, and was held xed over all images within a given set.

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


Reducing the number of ancilla qubits and the gate count required for creating large controlled operations In this paper we show that it is possible to adapt a qudit scheme for creating a controlled-Toffoli created by Ralph et al. [Phys. Rev. A 75 011213] to be applicable to qubits. While this scheme requires more gates than standard schemes for creating large controlled gates, we show that with simple adaptations it is directly equivalent to the standard scheme in the literature. This scheme is the most gate-efficient way of creating large controlled unitaries currently known, however it is expensive in terms of the number of ancilla qubits used. We go on to show that using a combination of these standard techniques presented by Barenco et al. [Phys. Rev. A 52 3457 (1995)] we can create an n-qubit version of the Toffoli using less gates and the same number of ancilla qubits as recent work using computer optimization. This would be useful in any architecture of quantum computing where gates are cheap but qubit initialization is expensive.Making a unitary controlled on other qubits is an essential task for many algorithms in quantum computing [1] [2] [3] . In this paper we focus on a particular problem, making a unitary which is already controlled on one qubit, controlled on n − 1 further qubits. These highly controlled unitaries (i.e. unitaries controlled on more than one other qubit) are useful in numerous quantum algorithms including the oracle in the binary welded tree algorithm [4] and quantum simulation [2, 3] . Barenco et al.[5] outlined several techniques to make controlled unitaries in 1995, this work was expanded on in Nielsen and Chuang [6] to provide a technique for making a n-qubit version of the CNOT gate using 14n − 13 operations and n − 2 ancilla. Other work has explored this problem in the context of using computational algorithms to optimize circuit layout using the decomposition procedures proposed by Barenco et al. [5] and known commutation relations [7] [8] [9] [10] [11] [12] . However, while these techniques are useful if we have native controlled-square-root-not gates, in the case where the only native two qubit operation is a CNOT or C-Phase they perform worse than the technique in Nielsen and Chuang [6] for the same number of ancilla.An interesting alternative technique was proposed by Ralph et al.[13] and implemented experimentally by Lanyon et al. [14] who used additional levels in one of the subsystems of the controlled gate to reduce the overall number of operations required. In this work we will show that when converted to qubits this technique is directly equivalent to the one from Nielsen and Chuang [6] using the same number of operations, and requiring the same number of ancilla. We go on to show that by using the techniques from Nielsen and Chuang [6] combined with other decomposition procedures from Barenco et al. [5] it is possible to reduce the number of ancilla qubits required from n − 2 to 2 √ n − 1 at the expense of double the number of operations. Our new techniques requires less operations for the same number of ancilla qubits when compared to existing decomposition schemes if we assume both schemes have the ability of perform a controlled square root of NOT gate [7] [8] [9] [10] [11] [12] .For the rest of this work we will represent a CNOT gate controlled on n-qubits as C n X, and a generally local unitary controlled on n qubits as C n U. Given the ability to perform general local unitaries, and a CNOT gate we can make a Toffoli gate using nine local unitaries and six CNOT gates [15] . However, if are prepared to accept an approximate Toffoli gate, we can use the Margolus gate, which is equivalent to a Toffoli gate and a controlled controlled phase. This procedure requires three CNOT gates and eight local unitaries [16, 17] . An alternative set of decomposition procedures are used if we assume the ability to perform the CV gate in a single operation. Here a standard Toffoli uses two CNOT gates and three CV gates [5, 12] . A more efficient decomposition is the Peres gate [11, 18] , which is the equivalent of a Toffoli gate with an additional CNOT. This decomposition uses only one CNOT gate, and three CV gates. In this work we can use the more efficient decomposition procedures because our Toffoli gates are arranged symmetrically with no other operations in between them.Ralph et al. [13] and Lanyon et al. [14] demonstrate that by using a qutrit they can generate a Toffoli gate using only three CNOT gates, two standard Pauli X operations, and two implementations of a three-level version of the Pauli X operation. One requirement of Lanyon et al. is that the CNOT gates act trivially on the |2 level of the qutrit. Replacing the qutrit with two qubits would require each CNOT gate to be replaced with a Toffoli gate as shown in fig 1. A qubit version of the Lanyon Toffoli gate is therefore impossible since we would need to consume three Toffoli gates to build a single Toffoli gate.

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Deriving robust counterparts of nonlinear uncertain inequalities In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters. Robust Optimization (RO) has become an important field in the last decade. For a comprehensive treatment of RO we refer to [5] or the recent survey [10] . The number of applications of RO has increased rapidly in recent years. Moreover, the RO methodology has recently been implemented into (commercial) mathematical modelling and optimization systems (e.g. AIMMS [1] , ROME [14] , and YALMIP [18] ). The goal of RO is to immunize an optimization problem against uncertain parameters in the problem. Such uncertain parameters may arise as a result of estimation or rounding errors in the parameter values, or due to implementation errors. Therefore, a so-called uncertainty region for the uncertain parameters is defined, and then it is required that the constraints should hold for all parameter values that reside in the uncertain region. For several optimization problems, and for several choices of this uncertainty region, the so-called Robust Counterpart (RC) can be formulated as a tractable optimization problem. For example the robust counterpart for a linear programming problem with polyhedral or ellipsoidal uncertainty regions can be reformulated as a linear programming or conic quadratic programming problem, respectively. We refer to [5] for an extensive treatment of these cases.. In this paper we show how Fenchel duality can be used to construct the RC for a nonlinear constraint that is concave in the uncertain parameters. This approach has several advantages.. First of all, it shows that for new classes of optimization problems a tractable robust counterpart can be derived. Secondly, for linear programming it provides an easy way to construct the RC, e.g. in case the uncertainty region is the intersection of several regions. In this paper many examples are given, some being of direct applications to known problems, some being candidates for future applications.. We also analyze the tractability of the resulting robust counterparts and show that many RCs can be written as linear, quadratic, or conic quadratic constraints, or else they can be shown to admit a self-concordant barrier function. The latter implies that the optimization problem is solvable in polynomial-time (see [20] ).. We now discuss other literature on deriving robust counterparts for nonlinear inequalities following the description in [5] . Exact tractable reformulations for Conic Quadratic Problems are derived when the uncertainty is a simple interval or unstructured norm-bounded. Approximation results are derived for structured norm-bounded uncertainty or intersection of ellipsoids. For Semidefinite Problems exact results are known for unstructured norm-bounded uncertainty, and approximation results for structured norm-bounded uncertainty. In [3] an explicit tractable reformulation is derived for a robust inequality that is linear in the uncertain parameters and the uncertainty set is given by convex inequalities. This is a special case of the type of robust optimization problems that we study in this paper. This paper is organized as follows. In Section 2 we derive the RC of a general nonlinear robust constraint by using Fenchel duality. The RC involves calculating the support function of the uncertainty region, which is treated in Section 3, and calculating the conjugate function of the constraint function, which is treated in Section 4, using among other methods the theory of Conic Quadratic representable (CQr) functions ( [6] ). Many concrete examples are treated in Sections 3 and 4. In Section 5 we treat the complexity of the resulting RCs. In the appendices, we recall known theorems on conjugate functions, support functions, Fenchel duality, Conic Quadratic duality and Conic Quadratic representability. Those results will be used in the paper and for the sake of clarity the lemmas and theorems in the appendix will be unambiguously numbered as such.. Notation. Throughout this paper we use the following notation.. Let f : R n → R be a closed convex function with dom(f ) = {x | f (x) < ∞}. Let g : R n → R be a closed concave function with dom(g) = {x | g(x) > −∞}. The convex conjugate of f is defined as f * (y) = sup x∈dom(f ). The concave conjugate of g is defined as g * (y) = inf x∈dom(g). For a function g(., .) of two vector variables, g * (., .) will denote the partial concave conjugate with respect to the first variable.. A special conjugate function is the support function, which is the conjugate of the indicator function. The indicator function on the set S is defined as:. Then the conjugate function of δ(x|S),. is the so-called support function of the set S.. The adjoint function of f : R + −→ R is defined as:. Note that f ♦ (x) is convex if f (x) is convex.. Let A be a linear transformation from R n to R m , then the function gA is defined by (gA)(x) = g(Ax).. The relative interior of a set S is denoted by ri(S).. Throughout the paper a subindex i denotes a scalar (the ith element of the vector) and a superindex denotes a vector.

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Alternative DRM formulations In this paper, we present two DRM formulations. For a pseudo-Poisson equation, if the right hand side is a linear operation on the dependent variable, we can derive a new DRM formulation. In comparison with the traditional DRM formulation for the same equation, the new one is much easier and more efficient. For the axisymmetric Poisson equation, we construct a DRM formalation by using the linear axisymmetric radial basis function. The particular solution involved is written in a closed form, and thus speeds up the evaluation of the particular solution. A few numerical examples demonstrate the accuracy and efficiency of these formulations. The boundary element method (BEM) is now a powerful alternative numerical technique for solving partial differential equations (PDEs). For homogeneous PDEs, only the boundary discretisation is necessary. For inhomogeneous PDEs, however, the integral equation involves domain integrals. To avoid domain integrals, the dual reciprocity method (DRM) was proposed [5] . This method actually divides the solution into two parts: a particular solution of the inhomogeneous PDE plus a solution of its homogeneous counterpart. Since particular solutions to complex inhomogeneities are very difficult or even impossible to obtain, the inhomogeneity is normally represented by a series expansion in terms of simpler functions for which particular solutions can be (easily) determined [11] . Furthermore, the DRM has been extended to deal with nonlinear problems, heterogeneous problems, variable coefficient problems and time-dependent problems.. In this paper, we first consider the following equation. where 6 is a given function. Conceptually this is nothing special, but it is extremely difficult to find a particular solution to the axisymmetric Laplacian for the traditional radial basis functions (RBFs) being the right hand side. We turned to the axisymmetric RBFs.. The paper is organised as follows: In Section 2, we recall the essentials of the DRM and show how equation (1) is solved by the DRM traditionally. A new formulation for equation (1) is derived in Section 3, which is better than the traditional one. In Section 4, an axisymmetric DRM formulation is established for equation (2) . A few numerical examples are shown in Section 5. Comments and conclusions are given in Section 6. Two DRM formulations are derived and their efficiency and accuracy are demonstrated by a number of numerical examples. For a pseudo-Poisson equation with the right hand side being a linear operation on the unknown, the new DRM formulation is easier and more efficient.. In the axisymmetric DRM formulation, the particular solution is written in a closed form by using the linear axisymmetric radial basis function. Obviously this formulation can be extended to axisymmetric nonlinear problems and axisymmetric time-dependent problems.

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Correlation-based Algorithm for Multi-Dimensional Single-Tone Frequency Estimation In this paper, parameter estimation for a R-dimensional (R-D) single cisoid with R ≥ 2 in additive white Gaussian noise is addressed. By exploiting the correlation of the data samples, we construct R single-tone sequences which contain the R-D frequency parameters. Based on linear prediction and weighted linear squares techniques, two proposals are developed for fast and accurate frequency estimation from each constructed sequence. The two devised estimators are proved to be asymptotically unbiased while their variances achieve Cramér-Rao lower bound when the signal-to-noise ratio and/or data length tend to infinity. Computer simulations are also included to compare the proposed approach with conventional R-D harmonic retrieval schemes in terms of mean square error performance and computational complexity. A fast and accurate multi-dimensional single frequency estimation approach based on correlation has been devised. Two estimators, C-1 and C-2, which are ideal for high SNR and/or large sample conditions, are proposed.. The main idea in the algorithm development is to construct single-tone sequences which contain the frequency parameters by exploiting correlation of the observed data. Frequency estimation for each dimension is performed separately by using LP and WLS techniques. It is proved that the variances of the C-1 and C-2 estimators attain CRLB in the presence of white Gaussian noise, when the SNR and/or data length tend to infinity. Furthermore, the proposed correlation approach is more computationally attractive than the AIQML, UE and IMDF schemes, and provides superior estimation accuracy at higher SNRs. As a future work, we will extend the correlation-based methodology to estimation of multiple tones.

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An empirical study on the implementation and evaluation of a goal-driven software development risk management model The significance of software is growing everyday as it becomes a major part of enterprise business. Building a quality software product in the shortest possible time to satisfy the global market demand gives an enterprise a competitive advantage. However, uncertainties and risks exist at every stage of a software development project. These can have an extremely high influence on the success of the final software product. Early risk management practice is effective to identify and control such risks and contributes effectively towards the project success. Despite of several risk management approaches being presented in the literature, a detailed guideline that explains where to integrate risk management activities into the software development project is still missing. Moreover, although empirical investigations focus on identifying potential risk factors on software projects, very little effort has been directed towards the evaluation of the overall impact of a risk management method to a software project. This paper contributes towards these two limitations. In particular, we present a Goal-driven Software Development Risk Management Model (GSRM) and its explicit integration into requirements engineering phase and an empirical investigation result of applying GSRM into an on-going software development project. The result clearly shows that when risk factors are beyond the control of the project manager and project environment then it is difficult to control these risks. Project scope affects all dimensions of risk. We conclude that a formal early stage risk management practice is not only provides early warning related to the problems exist in project but also effectively contributes to the overall project success. Software projects, by inherent nature, contain a significant numbers of uncertainties relating to time-to-market, budget and schedule estimation, technology evolution, and stakehold-ers expectations. Failure to control these uncertainties imposes potential risks to a project. Therefore software engineering domain is more prone to risks compared to many other engineering domains. Software risk management can be used as a tool to manage these risks and to reason under high degree of uncertainties involved. For an effective risk management practice, it needs to be included at early phase of the development process [1] . There exist several risk management methods for managing risks in software projects [2, 3, 4, 5, 6] . Practitioners and reseachers agreed that risk management practice should be included as early as possible during the development. But, there is still lack of comprehensive and detailed guidelines that explain how to integrate risk management activities at the early development stages [7, 8, 1] . Several study results exist in the literature on identifying risk factors in software project but only a few reports are available on evaluating the impact of an overall risk management method into a software project [5] . Research studies results indicate that risk management is not well applied in practice [9] and practitioners are more concerned on the tangible development cost which provides direct benefits in terms of project deliverables [7] . For a successful project, it is difficult to prove that any part of the resulting product is influenced by software risk management [10] . Therefore, it is necessary to integrate risk management approaches into the early development and to create awareness among the practitioners about the impact of risk management practice on software project.. Within the above context, the novel contributions of this paper are i) a risk management method using goal-driven approach; ii) explicit integration of the risk management method into requirements engineering phase; and iii) empirically evaluate the impact of the risk management method into software development project. The presented risk management method is based on KAOS goal modelling language [11] . In particular, GSRM adopts goal and obstacle concepts from the KAOS goal modelling language and extends with risk assessment and treatment [1] . Therefore, GSRM is based from the existing work of goal modelling language. The reason for choosing goal modelling language for risk management is that goals and risks are complementary entities of a software project. A risk is usually defined as negation to single or multiple goals or loss of attainment of corresponding objectives [11] . The goal-driven approach anchors the risk management activities and allows to trace and rationalize the risk factors, events and control actions with respect to the goals. Furthermore, Goal-orient Requirements Engineering(GORE) has long been recognised in requirements engineering community to elicit, analyse, document the requirements. It allows GSRM to explicitly integrates into early development phase. An advantage of considering risk management during the requirements engineering phase is to contribute for the development of robust and complete requirements specification document. Because requirement problems are one of the main causes of project failure [12] . These problems are expensive and persist throughout the life cycle of the software product [11] . But requirements engineering lays the foundation for a successful project development regarding cost and quality [13] .. The methodology is explained with the aid of a carefully designed case study for the development of an automation system in a public sector organisation (Ministry of Planning Commission) in Bangladesh under the e-governance project. The main goal of empirical investigation is to evaluate the effectiveness of GSRM and in particular impact of an early risk management practice on software development project. Our work, combines a case study method with action research so that identified treatment actions can be used to control the potential project risks. The results from the case study outline the impact of GSRM on the project and compare the identified results with the other results to generalize the findings. We believe these contributions enhance the state of the art considering the explicit 2 integration of risk management into early development and demonstrate the impact of risk management into software development project. The remainder of the paper is structured as follows. Section 3 provides an overview of the goal-driven risk management model. Section 2 outlines the state of the art about risk management methods and risk factors. Section 4 outlines the integration of GSRM into requirements engineering. Section 5 demonstrates the evaluation results on the implementation of the GSRM into a software project. This section discusses the evaluation approach, study context, GSRM process and lessons learned from the study context by implementing GSRM. Section 7, we provide a critical discussion of the various parts of GSRM and outline some of our experiences from the studied project. Section 8 provides validity of the study results. Finally Sect. 9 concludes the paper and presents directions for the future work. Every software project is unique and undertakes multidimensional tasks during the course of development, deployment, and maintenance. Early risk management is always effective to meet the project goals and to produce a more reliable and dependable product. We believe that GSRM potentially contributes in this direction.. In this paper, we presented GSRM method and reported experience from its usefulness through an empirical evaluation by combining case study with action research. Our studied results show that goal-driven approach is suitable for risk management and risk management is well integrated into requirements engineering. The results indicate that GSRM is a practical and reasonable risk management method in industrial context. The goal oriented view made ease to understand and communicate the concepts of GSRM and a tutorial session is adequate for this purpose. We have noted our experience and insight gained and lessons learned from the case study. To generalize our result we compared the study results with the similar study results. We believe this study is useful to improve GSRM and to inform and motivate practitioners about the effectiveness of integrating risk management activities from the early phase of the development. However replicate case study is necessary so that the validity of the results can be more generalized and the risk management method can be further improved based on the case study observation. We are currently working on defining a comprehensive application guidelines for the GSRM, so that it can provide better support for risk analysis into a software project. We are also planning to develop a goal-risk taxonomy. A goal-risk taxonomy identifies and classifies goals and risks that are associated with certain project characteristics and links with possible potential control actions in controlling specific types of risks. Project managers can reuse the taxonomy for any upcoming project. Some factors within the course of the product life cycle such as deployment, operation, and maintenance do not get adequate attention at an early stage. But later these factors can cause a real danger for the project. These factors need further investigation. We would like to integrate a continuous risk management activities for analysing these factors. In this section, we report the findings based on the observations from the studied planning ministry's automation project. We compare our case study findings with other study results to generalize our finding. We also include a summary of the overall impact of GSRM into software project.. 7.1. Lessons Learned 7.1.1. Usefulness of the GSRM Goal-driven approach for risk management. Goal-driven approach for risk management is reasonably beneficial during the early development. A project contains goals. Goals make it easy to communicate with the users, project sponsors, and development team members. Goal identification and modelling: The main users of the studied project were government officials who do not had adequate knowledge of the purpose of automation due to lack of IT competence but they demonstrated what were their expectations from the project. From the user perspective, several goals are identified such as user satisfaction, successful training, and system usage. From the project team perspective, goals were emphasized more on issues related to project completion, training, user participation and reputation. However, some participants observed that goal modelling was difficult to perform and required more experience. This section focuses on existing work that relates to our approach. We first discuss the risk management framework then study results on identifying risk factors and their impact on software project and barriers in risk management practice.

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


Longitudinal Image Registration With Temporally-Dependent Image Similarity Measure Abstract-Longitudinal imaging studies are frequently used to investigate temporal changes in brain morphology and often require spatial correspondence between images achieved through image registration. Beside morphological changes, image intensity may also change over time, for example when studying brain maturation. However, such intensity changes are not accounted for in image similarity measures for standard image registration methods. Hence, 1) local similarity measures, 2) methods estimating intensity transformations between images, and 3) metamorphosis approaches have been developed to either achieve robustness with respect to intensity changes or to simultaneously capture spatial and intensity changes. For these methods, longitudinal intensity changes are not explicitly modeled and images are treated as independent static samples. Here, we propose a model-based image similarity measure for longitudinal image registration that estimates a temporal model of intensity change using all available images simultaneously. We proposed a new model-based similarity measure which allows the deformable registration of longitudinal images with appearance change. This method can account for the intensity change over time and enables the registration method to recover the deformation due only to changes in morphology. We compared the model-based approach to mutual information and Fig. 17 . Landmark distance mismatch for four subjects out of 9. Each row shows a single subject. The last column shows the target images and landmarks (cyan). Columns 1-3 are the source images: each target-source landmark pair is marked as a circle (red: mutual information; yellow: SSR) with size proportional to the distance (mismatch) after registration between the landmark pair (the smaller circle is always on top). That is, the size of the circles are proportional to registration accuracy (smaller is more accurate) in that particular location. demonstrated that it can achieve higher accuracy than mutual information in cases when there is a large appearance change between source and target images. We used a logistic model of intensity change and an elastic deformation model, however, the formulation is general and can be used with any other appearance or deformation model. In the future we will investigate the use of prior models to inform the estimation step in regions with high uncertainty (e.g., due to poor initial alignment) and combine SSR with a registration method incorporating spatio-temporal regularization of displacements.

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


Nonintersecting subspaces based on finite alphabets It was shown in [1] and [2] that in a multiple antenna noncoherent channel the coding problem is equivalent to one of packing subspaces (codewords) according to a certain notion of distance. In particular, [1] showed that the diversity order (the asymptotic slope of the error probability with respect to SNR) depends on the dimension of the intersection of the subspaces. Therefore in order to construct maximal diversity codes we need to find "nonintersecting" subspaces, that is, subspaces which intersect only at the origin. Maximal diversity differential space-time codes have been constructed in [3] (and references therein). These constructions do not impose explicit constraints on the number of different symbols used to define the codewords, so the transmitted symbols could in principle use a large alphabet. The main question addressed in the present paper is the construction of codewords (i.e. nonintersecting subspaces) subject to the constraint that the elements of the codewords use symbols from a fixed, small constellation.Let the number of transmitting antennas be Mt and the number of receiving antennas be Mr. We can write the received signal y(k) ∈ C Mr over m time instants asThe channel matrix H(k) ∈ C Mr ×Mt is assumed to be time-invariant over the block of m symbols. The column vector x(k) ∈ C Mt is the channel input, Es is the signal power per transmitting antenna, and z(k) ∈ C Mr is zero-mean i.i.d. Gaussian noise with E [z(k)z(k) H ] = N0I. Furthermore, we assume that the receiver will not attempt to estimate the channel matrix H, i.e. that we have a noncoherent receiver. We assume that the elements of H are i.i.d with distribution C N (0, 1) (zero-mean complex Gaussian with unit variance). Then the code design criterion derived in [1] shows that the diversity order of the space-time code is given bySince the focus of this paper is on constructing maximal diversity (MtMr) space-time codewords X, the criterion implies that we need distinct codewords to be nonintersecting. Furthermore, we impose the constraint that the elements of X are taken from a fixed constellation.Definition 1 Let F be a field. A codeword or subspace will mean an Mt-dimensional subspace of F m . Two subspaces Π1 and Π2 are said to be nonintersecting over F if Π1 ∩ Π2 = {0}.Suppose that we restrict the entries in the matrices Π1 and Π2 to belong to a finite subset A ⊆ F, called the alphabet. The question is the following: given Mt, m and a finite alphabet A ⊆ F, how many subspaces can we find which are generated by vectors from A m and which are pairwise nonintersecting over F?We investigate this question for two kinds of alphabets: (a) A is a finite field F, and (b) Mt = 2 and A ⊆ C m is a set of complex roots of unity. The question of nonintersecting subspaces over a finite field is a well-studied question in finite geometry. When the alphabet is a finite field, the main idea is to lift those constructions 1 This work was carried out during F. E. Oggier's visit to AT&T Shannon Labs during the summer of 2003. She thanks the Fonds National Suisse, Bourses et Programmes d'Échange for support.to the complex field to obtain codes for the noncoherent multiple antenna problem. for PSK constellations we use a different approach.We assume that the alphabet A = F = GF (q), the finite field of q elements, where q is a power of a prime p. Let V denote the vector space GF (q) m .Theorem 1 The number of pairwise nonintersecting Mtdimensional subspaces of V is at most (q m − 1)/(q Mt − 1).Using projective geometry, we show that:Theorem 2 The bound of Theorem 1 can be attained whenever Mt divides m, and only in these cases.Furthermore, it is straightforward to construct nonintersecting subspaces meeting the bound given in Theorem 1: suppose Mt divides m, and consider the fields F0 = GF (q), F1 = GF (q Mt ), F2 = GF (q m ). Then F0 ⊆ F1 ⊆ F2. By regarding GF (q m ) as a vector space of dimension m over GF (q) we can identify F2 with V . Similarly we can regard F1 as an Mt-dimensional subspace of V . The desired result is now obtained by partitioning F * 2 into (multiplicative) cosets of F * 1 . This construction shows that the codewords are nonintersecting over the finite field. We also show that the subspaces constructed in Theorem 2 remain nonintersecting when lifted to the complex field. Therefore this construction gives full diversity order when the elements of the codewords are restricted to come from a finite field.We assume that the alphabet A consists of the set of complex 2 r -th roots of unity, that is, A = {exp(2πij/2 r ), 0 ≤ j < 2 r }, for some r ≥ 1. Let µ = exp(2πi/2 r ) be a primitive 2 r -th root of unity.There is a trivial upper bound:Theorem 3 Let A be the set of 2 r roots of unity, r ≥ 1. Then the number of pairwise nonintersecting planes is at most 1 2 |A| m−1 = 2 (m−1)r−1 .In the other direction we prove:Theorem 4 Assume r ≥ 1 and that m ≥ 2 is even. There exist N = |A| m−2 = 2 (m−2)r pairwise nonintersecting planes in C m defined using the complex 2 r -th roots of unity.Theorems 2 and 4 are our main results.[1] B. M. Hochwald and T. L. Marzetta, Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading, IEEE Transactions on Information Theory, 46 (No. 2, 2000), 543-564.[2] L. Zheng and D. N. C. Tse, Communication on the Grassmann manifold: a geometric approach to the noncoherent multiple-antenna channel, IEEE

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 I 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 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 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 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 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


The marketplace of high-performance computing In this paper we analyze the major trends and changes in the High Performance Computing HPC market place since the beginning of the journal`Parallel Computing'. The initial success of vector computers in the seventies was driven by r a w performance. The introduction of this type of computer systems started the area of Supercomputing'. In the eighties the availability of standard development e n vironments and of application software packages became more important. These criteria determined next to performance the success of MP vector systems especially at industrial customers. MPPs became successful in the early nineties due to their better price performance ratios which w as enabled by the attack of the`killer-micros'. In the lower and medium market segments the MPPs were replaced by microprocessor based SMP systems in the middle of the nineties. This success was the basis for the emerging cluster concepts for the very high end systems. In the last few years only the companies which h a ve e n tered the emerging markets for massive parallel database servers and nancial applications attract enough business volume to be able to support the hardware development for the numerical high end computing market as well. Success in the traditional oating point i n tensive engineering applications seems to be no longer su cient for survival in the market. The Only Thing Constant Is Change" | Looking back on the last decades this seem certainly to be true for the market of High-Performance Computing systems HPC. This market was always characterized by a rapid change of vendors, architectures, technologies and the usage of systems. Despite all these changes the evolution of performance on a large scale however seems to be a very steady and continuous process. Moore's Law is often cited in this context. If we plot the peak performance of various computers of the last 5 decades in gure 1 which could have been called the`supercomputers' of there time 4,2 we indeed see how w ell this law holds for almost the complete lifespan of modern computing. On average we see an increase in performance of two magnitudes of order every decade. In this paper we analyze the major trends and changes in the HPC market for the last three decades. For this we focus on systems which had at least some commercial relevance. Historical overviews with di erent focus can be found in 8,9 .. In the second half of the seventies the introduction of vector computer systems marked the beginning of modern Supercomputing. These systems o ered a performance advantage of at least one order of magnitude over conventional systems of that time. Raw performance was the main if not the only selling argument. In the rst half of the eighties the integration of vector system in conventional computing environments became more important. Only the manufacturers which provided standard programming environments, operating systems and key applications were successful in getting industrial customers and survived. Performance was mainly increased by improved chip technologies and by producing shared memory multi processor systems.. Fostered by several Government programs massive parallel computing with scalable systems using distributed memory got in the focus of interest end of the eighties. Overcoming the hardware scalability limitations of shared memory systems was the main goal. The increase of performance of standard micro processors after the RISC revolution together with the cost advantage of large scale productions formed the basis for the Attack of the Killer Micro". The transition from ECL to CMOS chip technology and the usage of o the shelf" micro processor instead of custom designed processors for MPPs was the consequence.. Traditional design focus for MPP system was the very high end of performance.. In the early nineties the SMP systems of various workstation manufacturers as well as the IBM SP series which targeted the lower and medium market segments gained great popularity. Their price performance ratios were better due to the missing overhead in the design for support of the very large con gurations and due to cost advantages of the larger production numbers. Due to the vertical integration of performance it was no longer economically feasible to produce and focus on the highest end of computing power alone. The design focus for new systems shifted to the market of medium performance systems.. The acceptance of MPP system not only for engineering applications but also for new commercial applications especially for database applications emphasized di erent criteria for market success such as stability of system, continuity of the manufacturer and price performance. Success in commercial environments is now a new important requirement for a successful Supercomputer business. Due to these factors and the consolidation in the number of vendors in the market hierarchical systems build with components designed for the broader commercial market are currently replacing homogeneous systems at the very high end of performance. Clusters build with components of the shelf also gain more and more attention.

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


Data Fusion Trees for Detection: Does Architecture Matter? We consider the problem of decentralized detection in a network consisting of a large number of nodes arranged as a tree of bounded height, under the assumption of conditionally independent and identically distributed (i.i.d.) observations. We characterize the optimal error exponent under a Neyman-Pearson formulation. We show that the Type II error probability decays exponentially fast with the number of nodes, and the optimal error exponent is often the same as that corresponding to a parallel configuration. We provide sufficient, as well as necessary, conditions for this to happen. For those networks satisfying the sufficient conditions, we propose a simple strategy that nearly achieves the optimal error exponent, and in which all non-leaf nodes need only send 1-bit messages.Index Terms-Decentralized detection, error exponent, sensor networks.M OST of the decentralized detection literature has been concerned with characterizing optimal detection strategies for particular sensor configurations; the comparison of the detection performance of different configurations is a rather unexplored area. We bridge this gap by considering the asymptotic performance of bounded height tree networks. We analyze the dependence of the optimal error exponent on the network architecture, and characterize the optimal error exponent for a large class of tree networks. The problem of optimal decentralized detection has attracted a lot of interest over the last 25 years. Tenney and Sandell [1] are the first to consider a decentralized detection system in which each of several sensors makes an observation and sends a summary (e.g., using a quantizer or other "transmission function") to a fusion center. Such a system is to be contrasted to a centralized one, where the raw observations are transmitted directly to the fusion center. The framework introduced in [1] involves a "star topology" or "parallel configuration": the fusion center is regarded as the root of a tree, while the sensors are the leaves, directly connected to the root. Several pieces of work follow, e.g., [2]-[12], all of which study the Manuscript We have studied the asymptotic detection performance of tree networks with bounded height, under a Neyman-Pearson criterion. Similar to the parallel configuration, we have shown that the optimal Type II error probability decays exponentially fast with the number of nodes. In addition, we have shown that if leaves dominate (i.e., ), the network can achieve the same performance as if all nodes were transmitting directly to the fusion center. We also provided a simple strategy, in which all leaves use the same transmission function, and all other nodes act as 1-bit relays, which achieves the optimal error exponent to any desired accuracy. The sufficient conditions are easy to achieve in cases of practical interest, hence, a system designer can obtain the optimal performance while ensuring that the network is energy efficient. Once the sufficient conditions are satisfied, the architecture of the network no longer affects its detection error exponent. On the other hand, we also showed that for the practically interesting case, where , the sufficient conditions are also necessary. Thus, in a network where the leaves do not dominate, the error decay rate will be worse than that of a parallel configuration, and will actually depend on the particular network architecture.. Needless to say, our conclusions only hold for the particular setting and criterion we have employed. One issue that has not been touched upon is that, with a relay network, a significantly larger value of may be required before the asymptotic error exponent yields a good approximation. Moreover, in practice, it would be wasteful to have only the leaves make observations, if is not large enough. Furthermore, under a Bayesian criterion, the same performance as the parallel configuration can no longer be achieved, although exponential decay is still possible [28] . Finally, the more realistic case where the i.i.d. assumption is violated, remains unexplored, with work mainly limited to the parallel configuration [29] [30] [31] [32] [33] [34] .. Future work includes characterizing the asymptotically optimal performance of tree networks without the bounded height constraint. We would like to understand the rate at which the error probability decays, and its dependence on the rate at which the height of the tree increases. Another intriguing question, which has been left unanswered, is whether the inequality is always true under the bounded height assumption, when every node is constrained to sending the same number of bits.

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