Bound states of the Klein-Gordon equation for vector and scalar general Hulthen-type potentials in D-dimension We solve the Klein-Gordon equation in any D-dimension for the scalar and vector general Hulthén-type potentials with any l by using an approximation scheme for the centrifugal potential.Nikiforov-Uvarov method is used in the calculations. We obtain the bound state energy eigenvalues and the corresponding eigenfunctions of spin-zero particles in terms of Jacobi polynomials. The eigenfunctions are physical and the energy eigenvalues are in good agreement with those results obtained by other methods for D = 1 and 3 dimensions. Our results are valid for q = 1 value when l = 0 and for any q value when l = 0 and D = 1 or 3. The s-wave (l = 0) binding energies for a particle of rest mass m 0 = 1 are calculated for the three lower-lying states (n = 0, 1, 2) using pure vector and pure scalar potentials. In nuclear and high energy physics [1, 2] , it is necessary to obtain the exact bound state energy spectrum of non-relativistic or relativistic wave equations for various potentials. In relativistic quantum mechanics, the Klein-Gordon and Dirac wave equations are most frequently used to understand the dynamics of a particle in large class of potentials. Therefore, much works have been done to solve these relativistic wave equations for various radial and angular potentials. Unfortunately, some quantum mechanical equations with these potentials can be solved analytically only for s-wave equation (i.e., l = 0 states). Some other works have been done to study the l = 0 bound state solutions of the Klein-Gordon equation with pure scalar, pure vector and mixed vector and scalar potentials in various dimensional space using different methods. The scalar coupling constant is almost taken to be equal to the vector coupling constant (i.e., S 0 = V 0 ) [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] . For this case, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] . On the other hand, it has been shown that when S 0 ≥ V 0 , there exist real bound state solutions (i.e., the radial wave function must satisfy the boundary condition that it becomes zero when r → ∞ and also finite at r = 0). However, there are very few exact solvable cases [15] [16] [17] [18] [19] . The bound state solutions for the last case is obtained for the s-wave Klein-Gordon equation with the exponential [15] and some other classes of potentials [16] . Chen et al. [19] used the exponential function transformation approach along with an approximation for the centrifugal potential to find bound-state solutions of the Klein-Gordon equation with the vector and scalar Hulthén potentials for different l.. The D-dimensional Schrödinger and Klein-Gordon wave equations have been solved for various types of angular and radial dependent potentials using the Nikiforov-Uvarov (N-U) method [20] [21] [22] [23] [24] [25] . The exact bound state solutions for the Schrödinger, Klein-Gordon and Salpeter wave equations, in one-dimension (1D), with the generalized Woods-Saxon potential are also obtained [26] [27] [28] [29] . These solutions of the D-dimensional Klein-Gordon equation describing a spin-zero particle are obtained for the case V 0 = S 0 ring-shaped Kratzer-type and pseudoharmonic potentials [21, 22] . Furthermore, analytic solution of radial Schrödinger equation has been given for the l = 0 general Hulthén potential within an approximation to the centrifugal barrier term [25, 29] . Simsek and Egrifes [30] have investigated the reality of exact bound states of the 1D Klein-Gordon equation with complex and/or PT-symmetric non-Hermitian exponential-type like general Hulthén potential. Berkdemir [31] has also applied the method to solve the Klein-Gordon equation of a spin-zero particle for V (r) = S(r). Kratzer-type potentials. Qiang et al. [32] have given an approximate analytic solution of the l = 0 Klein-Gordon equation for the general Hulthén-type potential within the approximation given in [29] . Saad [33] has presented an approximate solution of the l = 0 bound states of the Klein-Gordon equation in D-dimensions with the general Hulthén-type potentials following the model used in [29, 32] . Chen et al. [34] have employed two semiclassical methods to determine the bound state energy spectrum of Klein-Gordon equation when S 0 ≥ V 0 . Dong et al [35] have also obtained the bound state solutions of the Schrödinger equation for an exponential-type potential following Refs. [29, 32] . Very recently, we have obtained the bound-state solutions for the D-dimensional Schrödinger equation with the ManningRosen potential [36] which can be reduced to the Hulthén potential and the exponential-type hyperbolic potential [37] using a novel approximation to the centrifugal term [29, 36, 37] .. The purpose of the present paper is to study the relativistic characteristics of the scalar and vector general Hulthén-type potentials written, respectively, as follows [32, 33] :. where V 0 and S 0 represent the coupling constants of the vector and scalar general Hulthén-type potentials, respectively, α is the screening range parameter, r 0 represents the spatial range and q is the deformation parameter [26] [27] [28] . This potential has been widely used in a number of areas of physics. In atomic physics, V 0 = Ze 2 /r 0 where Z is the atomic number. Equation (1) behaves like a Coulomb potential V C (r) = −Ze 2 /r when r ≪ r 0 , but decreases exponentially in the asymptotic region when (r ≫ r 0 ), so its capacity for bound state is smaller than the Coulomb potential. The Klein-Gordon equation has been solved analytically in [30] only for the s-wave. This equation cannot be solved analytically for l = 0 because of the centrifugal term, 1/r 2 . Therefore, for l = 0, we must use an approximation for the centrifugal term similar to other authors [29, [38] [39] [40] [41] [42] [43] This paper is organized as follows: In Section II, we will derive l = 0 state solutions within the approximation given in [29] for the D-dimensional Klein-Gordon equation describing spin-zero particle with the vector and scalar general Hulthén-type potentials. We further separate the wave equation into radial and angular parts. Section III is devoted to a brief description of the N-U method. In Section IV, we present general solutions to the radial and angular equations in D-dimensions. We also give discussions of the solution in various dimensions and the constraints for obtaining the real bound state solutions. Further, numerical values for the three lower-lying states when pure vector and pure scalar potentials are used with coupling constants V 0 = 0.25 and S 0 = 0.25 and unity mass particle. Finally, the relevant conclusions are given in Section V.

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


A Note on the Validity of Statistical Bootstrapping for Estimating the Uncertainty of Tensor Parameters in Diffusion Tensor Images Abstract-Diffusion tensors are estimated from magnetic resonance images (MRIs) that are diffusion-weighted, and those images inherently contain noise. Therefore, noise in the diffusion-weighted images produces uncertainty in estimation of the tensors and their derived parameters, which include eigenvalues, eigenvectors, and the trajectories of fiber pathways that are reconstructed from those eigenvalues and eigenvectors. Although repetition and wild bootstrap methods have been widely used to quantify the uncertainty of diffusion tensors and their derived parameters, we currently lack theoretical derivations that would validate the use of these two bootstrap methods for the estimation of statistical parameters of tensors in the presence of noise. The aim of this paper is to examine theoretically and numerically the repetition and wild bootstrap methods for approximating uncertainty in estimation of diffusion tensor parameters under two different schemes for acquiring diffusion weighted images. Whether these bootstrap methods can be used to quantify uncertainty in some diffusion tensor parameters, such as fractional anisotropy (FA), depends critically on the morphology of the diffusion tensor that is being estimated. The wild and repetition bootstrap methods in particular cannot quantify uncertainty in the principal direction (PD) of isotropic (or oblate) tensor. We also examine the use of bootstrap methods in estimating tensors in a voxel containing multiple tensors, demonstrating their limitations when quantifying the uncertainty of tensor parameters in those locations. Simulation studies are also used to understand more thoroughly our theoretical results. Our findings raise serious concerns about the use of bootstrap methods to quantify the uncertainty of fiber pathways when those pathways pass through voxels that contain either isotropic tensors, oblate tensors, or multiple tensors. We have presented a set of answers for two interrelated questions that are of central importance to the use of bootstrap methods in quantifying the uncertainty of tensor parameters. We have shown that the appropriate use of the wild and repetition bootstrap methods critically depends on the overall morphology of the tensor that is being estimated. We have shown in particular that quantifying the uncertainty of some diffusion tensor parameters, such as principal directions of isotropic and oblate tensors, is invalid when using these methods. We have demonstrated that the wild bootstrap is sensitive to the correct specification of the fitted tensor model. We have shown that use of bootstrap methods in voxel with multiple tensors cannot recover the principal direction of each tensor component within the voxel. Our findings suggest that use of the bootstrap methods to quantify the uncertainty of fiber pathways, when they pass through the voxels containing either oblate or isotropic tensors or multiple tensors, can be highly misleading. We have supported our findings using both simulation studies and statistical theory.. Many aspects of this work warrant further research. Perhaps the most pressing need is the development of new bootstrap methods that avoid the need for prior information of a tensor's true morphology. The development of new statistical methods for determining the number of tensors in a voxel and for accurately estimating their individual tensor parameters is also of crucial importance to improve the validity of fiber tracking algorithms. Finally, the probabilistic construction of fiber tracts using statistical methods will also be important for determining the confidence with which tractography algorithms identify specific fiber tracts within the brain.

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


Subspace Pursuit for Compressive Sensing Signal Reconstruction We propose a new method for reconstruction of sparse signals with and without noisy perturbations, termed the subspace pursuit algorithm. The algorithm has two important characteristics: low computational complexity, comparable to that of orthogonal matching pursuit techniques when applied to very sparse signals, and reconstruction accuracy of the same order as that of linear programming (LP) optimization methods. The presented analysis shows that in the noiseless setting, the proposed algorithm can exactly reconstruct arbitrary sparse signals provided that the sensing matrix satisfies the restricted isometry property with a constant parameter. In the noisy setting and in the case that the signal is not exactly sparse, it can be shown that the mean-squared error of the reconstruction is upper-bounded by constant multiples of the measurement and signal perturbation energies.Index Terms-Compressive sensing, orthogonal matching pursuit, reconstruction algorithms, restricted isometry property, sparse signal reconstruction. C OMPRESSIVE sensing (CS) is a sampling method closely connected to transform coding which has been widely used in modern communication systems involving large-scale data samples. A transform code converts input signals, embedded in a high-dimensional space, into signals that lie in a space of significantly smaller dimensions. Examples of transform coders include the well-known wavelet transforms and the ubiquitous Fourier transform.. Compressive sensing techniques perform transform coding successfully whenever applied to so-called compressible and/or -sparse signals, i.e., signals that can be represented by significant coefficients over an -dimensional basis. Encoding of a -sparse, discrete-time signal of dimension is accomplished by computing a measurement vector that consists of linear projections of the vector . This can be compactly described via Here, represents an matrix, usually over the field of real numbers. Within this framework, the projection basis is Manuscript assumed to be incoherent with the basis in which the signal has a sparse representation [1] . Although the reconstruction of the signal from the possibly noisy random projections is an ill-posed problem, the strong prior knowledge of signal sparsity allows for recovering using projections only. One of the outstanding results in CS theory is that the signal can be reconstructed using optimization strategies aimed at finding the sparsest signal that matches with the projections. In other words, the reconstruction problem can be cast as an minimization problem [2] . It can be shown that to reconstruct a -sparse signal , minimization requires only random projections when the signal and the measurements are noise-free. Unfortunately, the optimization problem is NP-hard. This issue has led to a large body of work in CS theory and practice centered around the design of measurement and reconstruction algorithms with tractable reconstruction complexity.. The work by Donoho and Candès et al. [1] , [3] - [5] demonstrated that CS reconstruction is, indeed, a polynomial time problem-albeit under the constraint that more than measurements are used. The key observation behind these findings is that it is not necessary to resort to optimization to recover from the underdetermined inverse problem; a much easier optimization, based on linear programming (LP) techniques, yields an equivalent solution, as long as the sampling matrix satisfies the so-called restricted isometry property (RIP) with a constant parameter.. While LP techniques play an important role in designing computationally tractable CS decoders, their complexity is still highly impractical for many applications. In such cases, the need for faster decoding algorithms-preferably operating in linear time-is of critical importance, even if one has to increase the number of measurements. Several classes of lowcomplexity reconstruction techniques were recently put forward as alternatives to LP-based recovery, which include group testing methods [6] , and algorithms based on belief propagation [7] .. Recently, a family of iterative greedy algorithms received significant attention due to their low complexity and simple geometric interpretation. They include the Orthogonal Matching Pursuit (OMP), the Regularized OMP (ROMP), and the Stagewise OMP (StOMP) algorithms. The basic idea behind these methods is to find the support of the unknown signal sequentially. At each iteration of the algorithms, one or several coordinates of the vector are selected for testing based on the correlation values between the columns of and the regularized measurement vector. If deemed sufficiently reliable, the candidate column indices are subsequently added to the current estimate of the support set of . The pursuit algorithms it-erate this procedure until all the coordinates in the correct support set are included in the estimated support set. The computational complexity of OMP strategies depends on the number of iterations needed for exact reconstruction: standard OMP always runs through iterations, and therefore its reconstruction complexity is roughly (see Section IV-C for details). This complexity is significantly smaller than that of LP methods, especially when the signal sparsity level is small. However, the pursuit algorithms do not have provable reconstruction quality at the level of LP methods. For OMP techniques to operate successfully, one requires that the correlation between all pairs of columns of is at most [8] , which by the Gershgorin Circle Theorem [9] represents a more restrictive constraint than the RIP. The ROMP algorithm [10] can reconstruct all -sparse signals provided that the RIP holds with parameter , which strengthens the RIP requirements for -linear programming by a factor of . The main contribution of this paper is a new algorithm, termed the subspace pursuit (SP) algorithm. It has provable reconstruction capability comparable to that of LP methods, and exhibits the low reconstruction complexity of matching pursuit techniques for very sparse signals. The algorithm can operate both in the noiseless and noisy regime, allowing for exact and approximate signal recovery, respectively. For any sampling matrix satisfying the RIP with a constant parameter independent of , the SP algorithm can recover arbitrary -sparse signals exactly from its noiseless measurements. When the measurements are inaccurate and/or the signal is not exactly sparse, the reconstruction distortion is upper-bounded by a constant multiple of the measurement and/or signal perturbation energy. For very sparse signals with , which, for example, arise in certain communication scenarios, the computational complexity of the SP algorithm is upper-bounded by , but can be further reduced to when the nonzero entries of the sparse signal decay slowly.. The basic idea behind the SP algorithm is borrowed from coding theory, more precisely, the order-statistic algorithm [11] for additive white Gaussian noise channels. In this decoding framework, one starts by selecting the set of most reliable information symbols. This highest reliability information set is subsequently hard-decision-decoded, and the metric of the parity checks corresponding to the given information set is evaluated. Based on the value of this metric, some of the low-reliability symbols in the most reliable information set are changed in a sequential manner. The algorithm can therefore be seen as operating on an adaptively modified coding tree. If the notion of "most reliable symbol" is replaced by "column of sensing matrix exhibiting highest correlation with the vector ," the notion of "parity-check metric" by "residual metric," then the above method can be easily changed for use in CS reconstruction. Consequently, one can perform CS reconstruction by selecting a set of columns of the sensing matrix with highest correlation that span a candidate subspace for the sensed vector. If the distance of the received vector to this space is deemed large, the algorithm incrementally removes and adds new basis vectors according to their reliability values, until a sufficiently close candidate word is identified. SP employs a search strategy in which a constant number of vectors is expurgated from the candidate list. This feature is mainly introduced for simplicity of analysis: one can easily extend the algorithm to include adaptive expurgation strategies that do not necessarily operate on fixed-sized lists.. In compressive sensing, the major challenge associated with sparse signal reconstruction is to identify in which subspace, generated by not more than columns of the matrix , the measured signal lies. Once the correct subspace is determined, the nonzero signal coefficients are calculated by applying the pseudoinversion process. The defining character of the SP algorithm is the method used for finding the columns that span the correct subspace: SP tests subsets of columns in a group, for the purpose of refining at each stage an initially chosen estimate for the subspace. More specifically, the algorithm maintains a list of columns of , performs a simple test in the spanned space, and then refines the list. If does not lie in the current estimate for the correct spanning space, one refines the estimate by retaining reliable candidates, discarding the unreliable ones while adding the same number of new candidates. The "reliability property" is captured in terms of the order statistics of the inner products of the received signal with the columns of , and the subspace projection coefficients.. As a consequence, the main difference between ROMP and the SP reconstruction strategy is that the former algorithm generates a list of candidates sequentially, without backtracing: it starts with an empty list, identifies one or several reliable candidates during each iteration, and adds them to the already existing list. Once a coordinate is deemed to be reliable and is added to the list, it is not removed from it until the algorithm terminates. This search strategy is overly restrictive, since candidates have to be selected with extreme caution. In contrast, the SP algorithm incorporates a simple method for re-evaluating the reliability of all candidates at each iteration of the process.. At the time of writing of this paper, the authors became aware of the related work by J. Tropp, D. Needell, and R. Vershynin [12] , describing a similar reconstruction algorithm. The main difference between the SP algorithm and the CoSAMP algorithm of [12] is in the manner in which new candidates are added to the list. In each iteration, in the SP algorithm, only new candidates are added, while the CoSAMP algorithm adds vectors. This makes the SP algorithm computationally more efficient, but the underlying analysis more complex. In addition, the restricted isometry constant for which the SP algorithm is guaranteed to converge is larger than the one presented in [12] . Finally, this paper also contains an analysis of the number of iterations needed for reconstruction of a sparse signal (see Theorem 6 for details), for which there is no counterpart in the CoSAMP study.. The remainder of the paper is organized as follows. Section II introduces relevant concepts and terminology for describing the proposed CS reconstruction technique. Section III contains the algorithmic description of the SP algorithm, along with a simulation-based study of its performance when compared with OMP, ROMP, and LP methods. Section IV contains the main result of the paper pertaining to the noiseless setting: a formal proof for the guaranteed reconstruction performance and the reconstruction complexity of the SP algorithm. Section V contains the main result of the paper pertaining to the noisy setting. Con-cluding remarks are given in Section VI, while proofs of most of the theorems are presented in the Appendix of the paper. We introduced a new algorithm, named subspace pursuit (SP), for low-complexity recovery of sparse signals sampled by matrices satisfying the RIP with a constant parameter . Also presented were simulation results demonstrating that the recovery performance of the algorithm matches, and sometimes even exceeds, that of the LP programming technique; and, simulations showing that the number of iterations executed by the algorithm for zero-one sparse signals and compressible signals is of the order .

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


Cell decomposition and definable functions for weak p-adic structures We develop a notion of cell decomposition suitable for studying weak padic structures (reducts of p-adic fields where addition and multiplication are not (everywhere) definable).

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


An Initial Seed Selection Algorithm for K-means Clustering of Georeferenced Data to Improve Replicability of Cluster Assignments for Mapping Application Abstract K-means is one of the most widely used clustering algorithms in various disciplines, especially for large datasets. However the method is known to be highly sensitive to initial seed selection of cluster centers. K-means++ has been proposed to overcome this problem and has been shown to have better accuracy and computational efficiency than k-means. In many clustering problems though -such as when classifying georeferenced data for mapping applications-standardization of clustering methodology, specifically, the ability to arrive at the same cluster assignment for every run of the method i.e.replicability of the methodology, may be of greater significance than any perceived measure of accuracy, especially when the solution is known to be non-unique, as in the case of k-means clustering. Here we propose a simple initial seed selection algorithm for k-means clustering along one attribute that draws initial cluster boundaries along the "deepest valleys" or greatest gaps in dataset. Thus, it incorporates a measure to maximize distance between consecutive cluster centers which augments the conventional k-means optimization for minimum distance between cluster center and cluster members. Unlike existing initialization methods, no additional parameters or degrees of freedom are introduced to the clustering algorithm. This improves the replicability of cluster assignments by as much as 100% over k-means and k-means++, virtually reducing the variance over different runs to zero, without introducing any additional parameters to the clustering process. Further, the proposed method is more computationally efficient than k-means++ and in some cases, more accurate.3 Clustering or classification of data into groups that represent some measure of homogeneity across a given variable range or values of multiple variables, is a much analyzed and studied problem in pattern recognition. K-means clustering is one of the most widely used methods for implementing a solution to this problem and for assigning data into clusters. The method in its initial formulation was first proposed by Mac Queen in 1967 (Mac Queen 1967) though the approximation developed by Lloyd in 1982 (Lloyd 1982 ) has proven to be most popular in application. The method assumes apriori knowledge of the number of clusters k and requires seeding with initial values of centers of these clusters in order to be implemented. These initial seed values have been shown to be an important determinant of the eventual assignment of data to clusters. In other words, k-means clustering is highly sensitive to the initial seed selection for the value of cluster centers (Peña, Lozano et al. 1999 ).. K-means++ has been proposed to overcome this problem and has been shown to produce a scale improvement in algorithm accuracy and computational efficiency or speed (Ostrovsky, Rabani et al. 2006; Arthur and Vassilvitskii 2007) . The algorithm assesses the performance of the initial seed selection based on the sum of square difference between members of a cluster and the cluster center, normalized to data size. While this is a worthwhile means of assessing method performance, it may be noted that in many clustering applications, the replicability of the resultant cluster assignment can be much more desirable than the homogeneity of the cluster perceived through an objective measure.. We encountered one such application of the clustering problem while trying to cluster georeferenced data into classes for mapping and visualization through a Geographical Information Systems (GIS) software suite. Commercially available GIS software ArcGIS for instance utilizes a proprietary modification of Jenks natural breaks algorithm (Jenks 1967) Where n is the data size of number of data points, k is the number of clusters and dist (di, cj) As can be seen in Equation 3, Jenks' algorithm not only searches for minimum distance between data points and centers of clusters they belong to but for maximum difference between cluster centers themselves (Jenks 1967 ).. If we are trying to develop a methodology for geo-processing; say a utility that studies the scaling characteristic of a city and models the distribution of sizes of housing within different size clusters, it can be essential to have a clustering mechanism that produces almost exactly similar results each time. Drawing inspiration from Jenks' algorithm, we propose an initial seed selection algorithm for k-means clustering that produces similar The methodology is novel in that unlike other initial seed selection algorithms, it does not introduce any new parameters (such as number of trials for k-means++) in the clustering algorithm thereby avoiding additional degrees of freedom. By clustering along the deepest valleys or highest gaps in the data series, the method introduces a measure of distance between cluster centers augmenting the k-means optimization for minimum distance between cluster center and cluster members. Additionally, unlike initialization algorithms like k-means++ there is no randomness involved in the algorithm and the initial clusters obtained are always the same. While the objective of development of this method is to produce more replicable results, the sums of squared differences between cluster members and cluster centers between the proposed method and k-means++ were compared and are juxtaposed in Table 1 . As can be seen in Table 1 , k-means++ in general continues to produce more accurate clustering using this methodology, though for two of the five datasets, our method produced better results. As shown in Table 2 , our proposed method is also significantly faster than k-means++, clustering as much as 89% faster than k-means++ in some cases. The advantage in clustering speed is obtained over the initial seed selection, where k-means++ takes significantly longer comparative to both, our proposed method and k-means [2]. Table 3 . As can be seen in all three cases, the variance was virtually reduced to zero using our method, which was at least a 90% improvement on k-means++ and k-means.

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


Novel coarse-to-fine dual scale technique for tuberculosis cavity detection in chest radiographs Although many lung disease diagnostic procedures can benefit from computer-aided detection (CAD), current CAD systems are mainly designed for lung nodule detection. In this article, we focus on tuberculosis (TB) cavity detection because of its highly infectious nature. Infectious TB, such as adult-type pulmonary TB (APTB) and HIV-related TB, continues to be a public health problem of global proportion, especially in the developing countries. Cavities in the upper lung zone provide a useful cue to radiologists for potential infectious TB. However, the superimposed anatomical structures in the lung field hinder effective identification of these cavities. In order to address the deficiency of existing computer-aided TB cavity detection methods, we propose an efficient coarse-to-fine dual scale technique for cavity detection in chest radiographs. Gaussian-based matching, local binary pattern, and gradient orientation features are applied at the coarse scale, while circularity, gradient inverse coefficient of variation and Kullback-Leibler divergence measures are applied at the fine scale. Experimental results demonstrate that the proposed technique outperforms other existing techniques with respect to true cavity detection rate and segmentation accuracy. Chest radiographs or chest X-ray (CXR) images are widely used to diagnose lung diseases such as lung cancer, tuberculosis (TB), and pneumonia. Due to the superimposed anatomical structures in the human chest, the CXR images are generally noisy and the diagnosis requires careful examination by experienced radiologists. Computer-aided detection (CAD) systems in chest radiography have therefore been developed to reduce the workload of radiologists. Ginneken et al. reviewed the CAD technological development in 2001 [1] and 2009 [2] . Developing a single system that looks into all abnormalities on a chest radiograph is practically impossible due to the widely different characteristics of abnormalities, and specific focus of the image processing algorithms. Therefore, the current CAD systems often aim at a single aspect, e.g., detection of lung cancer nodules. This strategy has been proved to be successful, and many effective algorithms have been developed for routine diagnostic procedures [2] .. A general CAD system framework is shown in Figure 1 . There are four modules in the system. First, a CXR image undergoes the preprocessing step, which generally includes image enhancement, noise removal, and lung field segmentation. In the next step, candidates that may contain abnormalities are coarsely detected using pattern recognition techniques. In the third step, features that can be used to identify abnormalities are identified from the candidates. Depending on the radiographic manifestation of the abnormalities, these features could be geometric, photometric, or textural. Finally, a classifier is applied to perform a high-level screening to reduce the false positive rate. An efficient CAD system relies on robust image processing, pattern recognition, and artificial intelligence techniques. For instance, a recent CAD system [3] designed for identifying lung nodules uses an active shape model for lung field segmentation, followed by a weighted multi-scale convergence-index filter for nodule candidates detection. To identify the nodules successfully, an adaptive distance-based threshold technique is applied to segment the contour of each candidate. The geometric, intensity, and gradient features are then extracted from the segmentation results. After the first level screening, a Fisher linear discriminant classifier is used on a subset of these features to perform the final detection.. Nodule detection has been the main focus in current CXR CAD systems. However, as Ginneken et al. pointed out [2] , there are other diseases, e.g. TB, that rely heavily on chest radiograph examination can benefit from the CAD systems. Infectious TB is still a public health problem in many countries [4] . Therefore, our research focus is on developing a CAD system for the diagnosis of infectious TB. The TB can be identified based on different radiographic patterns, such as cavity, airspace consolidation, and interstitial opacities [5] . A few existing CAD systems use texture analysis to detect interstitial changes [2] . However, the interstitial pattern is not a reliable radiographic cue for infectious TB. According to a recent research article on TB [6] , cavitation in the upper lung zone (ULZ) is a typical radiographic feature of APTB. So far, insufficient research has been done for efficient detection of TB cavities. Shen et al. [7] recently proposed a hybrid knowledge-guided (HKG) framework for TB cavity detection, which contains three major steps. In Step 1, the cavity candidates are detected using adaptive thresholding on the mean-shift clustered CXRs. In Step 2, a segmentation technique is applied to the candidates to generate contours of important objects present in the CXR image. In Step 3, the contour-based circularity and gradient inverse coefficient of variation (GICOV) features are extracted for the final cavity classification using a Bayesian classifier. Although, this technique provides a good performance, it has several limitations. First, due to cavity size variation and the occlusion from neighboring superimposed anatomical structures, the mean shift cluster result is sensitive to the parameter values used. Second, the adaptive threshold, which is a quadratic polynomial of GICOV score, does not perform well when the cavity boundary is weak. These two limitations lead to a high missing rate (MR) of true cavities. To overcome these problems, we propose a dual scale feature classification strategy for TB cavity detection in chest radiographs. First, a coarse feature classification step is performed to detect the cavity candidates by capturing the geometric, textural, and gradient features in the lung field. Second, a Hessian matrix-based technique is applied to enhance the cavity candidates, which leads to a more accurate contour segmentation. Finally, fine features based on the shape, edge, and region are extracted from the segmented contours for the final cavity classification. Experimental results show that the performance of the proposed candidates detection, segmentation, and cavity classification modules is superior compared to the results obtained using other related CAD systems.. The rest of this article is organized as follows. Section 2 explains the cavity pattern in CXRs. Section 3 describes our proposed method in detail. Section 4 reports and analyzes the performance of the proposed technique. Conclusion and future work are presented in Section 5. In this article, we proposed an efficient coarse-to-fine dual scale feature classification technique for TB cavity detection in chest radiographs. Experimental results demonstrate that the proposed technique outperforms existing methods in three aspects. First, a lower MR is achieved because in the proposed method local cavity region-related coarse features, such as geometric, textural, and gradient features, are taken into consideration. Second, edge-based segmentation becomes more accurate by incorporating HIE to enhance the contours. Third, the final cavity detection accuracy is greatly increased by introducing the fine scale feature classification using three types of contour-related features, which includes shape, edge, and region. This study contributes in the development of CAD systems for infectious TB diagnosis, because of the higher detection rate and lower MR compared to other techniques. Future work will focus on exploring novel algorithms to model other characteristics of infectious TB. Table 7 Cavity detection evaluation of D-Group Sensitivity (%) Specificity (%) Accuracy (%) Circularity + GICOV [7] 57.6 88 72.8. Circularity + GICOV + KLD 69.4 8 1 .6 755

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


On isomorphisms and density of NP and other complete sets If all NP complete sets are isomorphic under deterministic polynomial time mappings (p-isomorphic) then P ~ NP and if all PTAPE complete sets are p-isomorphic then P ~ PTAPE.We show that all NP complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets.Thus showing that, inspite of the radically different origins and attempted simplification of these sets, all the known NP complete sets are identical but for polynomially time bounded permutations.Furthermore, if all NP complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be NP complete, nor can any sparse language over an arbitrary alphabet be NP complete.We show that complete sets in EXPTI~ and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet.Similarly, we show that the hardest context-sensitive languages cannot be sparse.We also relate the existance of sparse complete sets to the existance of simple combinatorial circuits for the corresponding truncated recognition problem of these languages. During the past years the importance of the P = ~P? problem has been fully realized and today it is one of the most important problems in theoretical computer science [C,AHU,K,HS,SI] .. The importance of the P = NP? problem derives from the fact that NP, the family of languages accepted by non-deterministic Turing machines in polynomial time, contains complete problems to which all other problems in NP can be easily reduced and from the fact that very many problems of practical interest in computing are in NP and many of them are NP complete * This research has been supported in part by National Science Foundation Research Grants GJ-33171X and DR 75-09433. [AIIU,C,K,GJS,SA,U] .. Thus the search for fast algorithms for a bewildering variety of problems can be reduced to the search for a fast algorithm for a single problem. As a matter of fact, during the last years considerable effort has been expended in discovering new HP complete problems and it is quite impressive how many diverse problems from many different problem areas have turned out to be NP complete [AIIU,C,K, SA,U] .. Furthermore, among the known ~P complete problems some have been simplified and found still to be NP complete [GJS] .. In this paper we show that regardless of their origins and attempted simplifications, all the "known" NP complete sets are essentially the same set. ~re specifically, we prove that all the known HP complete sets are isomorphic under a deterministic polynomial time mappings.. Thus these NP complete sets, except for a deterministic polynomial time recoding, are identical.. The proof of this result follows from two technical lemmas which give necessary and sufficient conditions that a set is isomorphic under polynomial time mappings to a aiven NP complete set, say the set of all satisfiable Boolean functions in conjunctive normal form.. To establish Dolynomial time isomorphisms (p-isomorphism) between NP complete sets we just have to check that these sets satisfy the sufficient conditions of our lemmas, which turn out to be easy to verify for all the NP comnlete sets found in the literature.. We exhibit the proof of the existence of p-isomorphism for the best known NP complete problems and they can be easily supDlied for the other ~P complete problems which have been described up to date in the literature. Since so far no ~P complete problems have been found which are not p-isomorphic and since all attempts to construct such sets have failed, we are forced to conjecture that all NP complete sets are isomorphic under deterministic polynomial time mappings.. It should be observed that a proof of this conjecture implies that P ~ NP.. To see this, we just have to note that P = NP iff every non-empty finite set is NP complete.. Since finite sets cannot be isomorphic to infinite sets, the isomorphism of all NP complete sets implies that P # NP. As a matter of fact, P ~ NP iff all ~P complete sets are isomorphic under recursive mappings.. It still could happen that P # NP but that there exist NP complete sets which are not p-isomorphic.. We conjecture that this is not the case.. By the same methods we also show that all the known PTAPE complete sets are isomorphic under deterministic polynomial time mappings.. Furthermore, if all PTAPE complete sets are p-isomorphic then P ~ PTAPE, since P = PTAPE iff every non-empty finite set is PTAPE complete.. Next we look at the density of NP and PTAPE complete sets.. We say that a set A, A ~ Z*, is p-sparse iff the number of elements in A u~ tO length n is bounded by a polynomial in n.. It is easily seen that the known NP and PTAPE complete sets are not p-sparse and that they cannot be p-isomorphic to p-sparse sets.. We suspect that neither NP nor PTAPE complete sets can be p-sparse. Note that a proof of p-sparse sets cannot be NP nor PTAPE complete would prove that P ~ NP and P # PTAPE.. On the other hand, we show that p-sparse sets cannot be complete in EXPTIME and EXPTAPE, as first observed by A. Meyer [M] .. Our proof actually shows that in EXPTI~ there exist sets which are not p-sparse and whose reduction to another set must be one-one almost everywhere, thus, no p-sparse set can be EXPTI~ complete.. The corresponding result also holds for EXPTAPE and more complex time and tape bounded families of languages.. It is still an open problem whether the EXPTI~ and EXPTAPE complete sets are all p-isomorphic, respectively.. It should be observed that the existence of a p-sparse complete set for NP or PTAPE would imply the existence of combinatorial circuits of polynomial complexity for the solution of the corresponding truncated recognition problems.. It actually has been shown by Meyer [M] that there exist circuits of polynomial complexity for the truncated problem of an NP complete set iff there is a sparse oracle set with which the corresponding NP complete problem can be solved in deterministic polynomial time.. Equivalently, the existence of p-sparse complete sets for NP (or PTAPE) implies that we could prepare a tape ~table) growing only polynomially in n such that all NP (or PTAPE) problems could be solved in deterministic polynomial time using a fixed tape (for table-look up).. Thus the existence of sparse NP complete sets would permit, for all practical purposes, the recognition of NP sets in deterministic polynomial time (using a precomputed, polynomially long tape segment).. This seems to be quite unlikely, and the above mentioned results show that this is not the case for EXPTIME and EXPTAPE: there does not exist any sparse set to which complete problems in EXPTI~ and EXPTAPE can be reduced in polynomial time.. Finally, we turn to context-sensitive languages.. We say (following R. Book) that a context-sensitive language L is hardest if every other context-sensitive language can be reduced to L by a lineartime mapping.. It is well known that hardest context-sensitive languages exist [HH] and that hardest context-free languages also exist [GR] .. Clearly, the context-sensitive languages are contained in P or NP iff a hardest csl is in P or NP, respectively.. Similarly, the deterministic context-sensitive languages are equal to the non-deterministic contextsensitive lanquages iff a hardest csl is a deterministic csl.. We prove that no p-sparse language can be a hardest csl and show that all known hardest cs!'s are p-isomorphic.. These results easily generalize to hardest languages of other families of tape bounded languages. A number of interesting and apparently difficult problems suggest themselves immediately from this work.. As we have noted, if all NP complete problems are p-isomorphic then P ~ NP and if all PTAPE complete problems are p-isomorphic then P ~ PTAPE.. Thus, the question whether all NP and PTAPE complete sets, respectively, are p-isomorphic could be a very important and hard question.. Similarly, the problem about the existence of sparse complete sets for NP and PTAPE seems very difficult and could help solve the P=NP=PTAPE? problem.. What about EXPTIME and EXPSPACE complete problems?. We know that they cannot be sparse, are they all p-isomorphic?. Similarly, are hardest context-sensitive languages all p-isomorphic?

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


The competitive advantage of nations: An application to academia Within the field of bibliometrics, there is sustained interest in how nations "compete" in terms of academic disciplines, and what determinants explain why countries may have a specific advantage in one discipline over another. However, this literature has not, to date, presented a comprehensive structured model that could be used in the interpretation of a country's research profile and academic output. In this paper, we use frameworks from international business and economics to present such a model. Our study makes four major contributions. First, we include a very wide range of countries and disciplines, explicitly including the Social Sciences, which unfortunately are excluded in most bibliometrics studies. Second, we apply theories of revealed comparative advantage and the competitive advantage of nations to academic disciplines. Third, we cluster our 34 countries into five different groups that have distinct combinations of revealed comparative advantage in five major disciplines. Finally, based on our empirical work and prior literature, we present an academic diamond that details factors likely to explain a country's research profile and competitiveness in certain disciplines. Within the international business discipline, scholars have long reflected upon what industries and nations specialize in, and what the explanation behind their international competitive advantages might be (see, e.g. Porter, 1990; Grant, 1991; Rugman and D'Cruz, 1993) . Within the field of bibliometrics, there is sustained interest in how nations "compete" in terms of academic disciplines, and what determinants explain why countries may have a specific advantage in one discipline over another (see, e.g., Frame, 1977; Braun et al., 1995a/b; BIE, 1996; Kozlowski et al., 1999; Garg, 2003; Horta and Veloso, 2007) . Clearly academic publishing is not a traditional competitive zero-sum game, i.e. if academics from one country publish in a particular journal, it doesn't mean academics from other countries cannot publish there. However, there is certainly an element of competition involved as most journals have page limits and not all papers can be published 1 . Therefore, one can assume that when a country publishes more in certain disciplines in comparison to others, it has a competitive advantage in these disciplines.. Understanding academic competitiveness matters, since academic output has risen worldwide over the past two decades, and higher-education institutions put increasing emphasis on the research performance of their academic staff, which in turn is a way to assess institutions' competitiveness in various academic fields. In this paper, we consider country-level competitiveness in various disciplines, with the understanding that it is higher education institutions that compete and engage in specific actions to maintain their competitive position (Bertsch, 2000) .. There is a large number of quantitative analyses of research productivity. Yet very few engage in a comparative analysis of how academically competitive individual countries are in specific disciplines in relation to others, and, importantly, what might explain this competitive advantage. Studies that have compared countries include reports aimed at aiding policy-makers (e.g. May, 1997; King, 2004; FWF, 2007) , or broad-ranging cross-country and cross-discipline studies that provide descriptive comparisons (e.g. Braun et al., 1995a/b; Yang et al., 2012) , rather than a more systematic analysis of elements that might constitute a country's competitiveness in individual fields of study.. Existing studies have identified some determinants explaining variation in research output and quality. The main determinants put forward include country size, level of economic development (GDP or GDP per capita) (May, 1997; King, 2004; Rousseau and Rousseau, 1998; Inonu, 2003) , financial investments (public expenditure for R&D) (May, 1997; King, 2004; Zhou and Leydesdorff, 2006; FWF, 2007) , competitive promotion of basic research such as knowledge infrastructure (for instance, the example of CERN would explain Switzerland's success for publications in physics), and incentive structures within research institutions (Almeida et al., 2009 ). These determinants centre on selected economic and institutional considerations, and the literature has not, to date, presented a comprehensive structured model that could be used in the interpretation of a country's academic competitiveness.. In this paper, we use frameworks from the international business and economics disciplines to present such a model. Our study makes four major contributions. First, we include a very wide range of countries and disciplines, explicitly including the Social Sciences, which unfortunately are excluded in most bibliometrics studies (see also Harzing, 2013) . Second, we apply theories of revealed comparative advantage and the competitive advantage of nations to academic publishing. Third, we cluster our 34 countries into five different groups that have distinct combinations of revealed comparative advantage in five major disciplines. Finally, based on our empirical work and prior literature, we present an academic diamond that details factors likely to explain a country's research profile and competitiveness in certain disciplines. Different countries have very different research profiles, and international business theories can be used very effectively to analyse these differences. Our results were based upon 21 disciplines and 34 countries, with a comparison between two key time periods (1994-2004 and 2002-2012) . Countries with the largest publication output included the USA, Canada, Japan, China and four European countries (Germany, Italy, France, UK), but this changes significantly when comparing publication output in relation to the size of the country. In this case, small economies move up the ranking. All countries increased their output over time, with some demonstrating a much faster increase -notably developing or newly-developed economies (China, Turkey, South Korea, Singapore and Brazil) showed the fastest increase. Research profiles for individual countries, however, remained very stable when comparing publication output over the two time periods.. We identified groups of countries with similar RCAs in Social Sciences, Physical Sciences, Engineering Sciences, Environmental Sciences or Biomedical Sciences. Using Porter's diamond, we argued that key determinants explain country groupings and specialization. Group 1 has RCA in the Social Sciences. A highly developed knowledge infrastructure (factor condition), first-mover advantage with longestablished institutions (strategy and rivalry), sophisticated demand (demand condition) and related industries (such as publishing) were useful in explaining why countries fell in this group. Group 2 comprises countries with an advantage in the Physical Sciences, but a rather balanced research profile. In this case, socio-political factors, historical patterns and strong institutional conditions are key explanations. Group 3 presented a balanced research profile with no strong RCA, and is illustrative of high income economies with advanced specialized factors. Group 4 comprised Asian countries with RCA in the Engineering Sciences. This research profile is explained by high capital resources (factor conditions), universities' goals, rules and incentives (strategy and rivalry), as well as economic development objectives and government support. Group 5 was composed of the two neighboring coun-tries, Russia and Ukraine, with high RCA in Physical Sciences and moderate RCA in Engineering Sciences, where socio-political factors, historical patterns and strong institutional conditions are key explanations. Group 6a and 6b demonstrate high RCA in the Environmental Sciences. Countries within this group benefit from strong natural basic resources linked to high related demand from local industry.. Our analysis highlighted a number of key points. Firstly, we find that to explain scientific advantage in nations, a more complex framework is needed to extend arguments provided in previous studies (e.g. May, 1997; King, 2004; Rousseau and Rousseau, 1998; Inonu, 2003) , which focused primarily on country size, economic and financial considerations. Instead, in this paper, we find that a number of key factors should be jointly considered and that a concept borrowed from the international business discipline -Porter's diamond -helps in this approach. Importantly, we have shown that various factors should be distinguished, separating the presence of natural or human resources from capital resources (in the form of government funding), coupled with a distinction between basic and advanced factors. Basic factors refer to availability of natural resources that would explain a specialization and advantages in, say, environmental sciences, whilst advanced factors refer to factors such as knowledge infrastructure available in individual countries. Secondly, our analysis also pointed to the key role played by institutions. Institutional heritage in individual countries, regional institutions (such as the European Union) and the role of national government support and guidance can, in selected cases, be instrumental in explaining the scientific advantage of nations.. It is noticeable in our sample that high-income economies tend to present a balanced research profile, explained by the presence of advanced specialized factors, as well as strong institutional environment. Newly-developed or emerging economies show strong RCA in Engineering Sciences (which as discussed above was historically based upon economic development strategies) or Environmental Sciences (linked to existing natural resources), whilst transition economies can be found to have strong RCA in Physical Sciences. In addition, although all countries increased their scientific output overall, the highest increases occurred in developing and newly-developed countries, mostly in Asia.. To conclude, we show that whilst all four quadrants of the diamond provide explanations behind strength in scientific outputs overall, specific factors vary in the explanation of the scientific advantage of nations, and the role of government is important in a number of cases, although through different means and actions. Combining existing knowledge with the results of this present study, we propose an academic diamond (see Figure 2 ).

O 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 B I I O O O O O O O B I O O O O O O O O O O O O O O O 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 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 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 B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 Singular Value Filter: A General Filter Design Strategy for PCA-Based Signal Separation in Medical Ultrasound Imaging Abstract-A general filtering method, called the singular value filter (SVF), is presented as a framework for principal component analysis (PCA) based filter design in medical ultrasound imaging. The SVF approach operates by projecting the original data onto a new set of bases determined from PCA using singular value decomposition (SVD). The shape of the SVF weighting function, which relates the singular value spectrum of the input data to the filtering coefficients assigned to each basis function, is designed in accordance with a signal model and statistical assumptions regarding the underlying source signals. In this paper, we applied SVF for the specific application of clutter artifact rejection in diagnostic ultrasound imaging. SVF was compared to a conventional PCA-based filtering technique, which we refer to as the blind source separation (BSS) method, as well as a simple frequency-based finite impulse response (FIR) filter used as a baseline for comparison. The performance of each filter was quantified in simulated lesion images as well as experimental cardiac ultrasound data. SVF was demonstrated in both simulation and experimental results, over a wide range of imaging conditions, to outperform the BSS and FIR filtering methods in terms of contrast-to-noise ratio (CNR) and motion tracking performance. In experimental mouse heart data, SVF provided excellent artifact suppression with an average CNR improvement of 1.8 dB (P 0 05) with over 40% reduction (P 0 05) in displacement tracking error. It was further demonstrated from simulation and experimental results that SVF provided superior clutter rejection, as reflected in larger CNR values, when filtering was achieved using complex pulse-echo received data and non-binary filter coefficients.

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


Topology optimization of an acoustic metamaterial with negative bulk modulus using local resonance During the past decade, materials that display novel properties in the acoustic realm, so-called acoustic metamaterials, have attracted much attention, since these properties can provide promising opportunities to design new acoustic devices that cannot be made with natural materials. Although acoustic metamaterials that exhibit negative mass density or negative bulk modulus, and double-negative acoustic metamaterials, have been obtained experimentally by trial and error, our aim is to develop a topology optimization method for the direct design of acoustic metamaterials, based on the concept of local resonant mechanisms, which ensures that the lattice constant is orders of magnitude functionally smaller than the corresponding sonic wavelength, and avoids unwanted effects of Bragg scattering mechanisms. This paper proposes a level set-based topology optimization method for the structural design of acoustic metamaterials that achieve an extremely negative bulk modulus at certain prescribed frequencies. Level set-based topology optimization methods can directly provide clear boundaries in optimal configurations that avoid the presence of grayscales. The optimization problem is formulated for a two-dimensional wave propagation problem, with the objective being to minimize the effective bulk modulus at chosen target frequencies. An effective medium description based on S-parameters is introduced to describe the acoustic metamaterial. Finite element method (FEM) is used to solve the Helmholtz equation for acoustic waves, sensitivities are obtained with the adjoint variable method (AVM), and a reaction-diffusion equation is used to update the level set function. Several numerical examples with prescribed target frequencies and different initial shapes are provided to demonstrate that the proposed method can provide clear, optimized structures for the design of negative bulk modulus acoustic metamaterials. This paper presented a level set-based topology optimization method applied to the design of negative bulk modulus acoustic metamaterials based on localized resonant mechanisms, using a functional scale. We obtained the following results.. (1) The optimization problem was formulated to minimize the effective bulk modulus.. A homogenization method based on S-parameters was applied to compute the effective bulk modulus of the metamaterials. A level set-based topology optimization method, using on a reaction-diffusion equation, was successfully applied to obtain clear boundary expressions of the optimal configurations. (2) Based on the formulation of the optimization problem, an optimization algorithm was constructed. FEM was used to solve the governing equation of acoustic wave propagations, and to update the level set function. The Adjoint Variable Method was used when computing sensitivity analyses. (3) Several examples were presented to confirm that the proposed method successfully finds optimized material distributions at different target frequencies, and with different initial configurations. It was also confirmed that a two-step optimization method enabled the handling of effective bulk modulus curves with unusual characteristics. The proposed method should facilitate the design of acoustic metamaterials without the need for physical experiments or trial and error approaches. We hope to extend the current two-dimensional cases to three-dimensional or three material phase problems in the future, which may broaden and improve the prospects for developing effective applications of acoustic metamaterials.

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


Wyner-Ziv video coding for wireless lightweight multimedia applications Wireless video communications promote promising opportunities involving commercial applications on a grand scale as well as highly specialized niche markets. In this regard, the design of efficient video coding systems, meeting such key requirements as low power, mobility and low complexity, is a challenging problem. The solution can be found in fundamental information theoretic results, which gave rise to the distributed video coding (DVC) paradigm, under which lightweight video encoding schemes can be engineered. This article presents a new hashbased DVC architecture incorporating a novel motion-compensated multi-hypothesis prediction technique. The presented method is able to adapt to the regional variations in temporal correlation in a frame. The proposed codec enables scalable Wyner-Ziv video coding and provides state-of-the-art distributed video compression performance. The key novelty of this article is the expansion of the application domain of DVC from conventional video material to medical imaging. Wireless capsule endoscopy in particular, which is essentially wireless video recording in a pill, is proven to be an important application field. The low complexity encoding characteristics, the ability of the novel motion-compensated multi-hypothesis prediction technique to adapt to regional degrees of temporal correlation (which is of crucial importance in the context of endoscopic video content), and the high compression performance make the proposed distributed video codec a strong candidate for future lightweight (medical) imaging applications. Traditional video coding architectures, like the H.26x [1] recommendations, mainly target broadcast applications, where video content is distributed to multiple users, and focus on optimizing the compression performance. The source redundancy is exploited at the encoder by means of predictive coding. In this way, traditional video coding implies joint encoding and decoding of video. Namely, the encoder produces a prediction of the source and then codes the difference between the source and its prediction. Motion-compensated prediction in particular, a key algorithm to achieve high compression performance by removing the temporal correlation between successive frames in a sequence, is very effective but computationally demanding.. The need for highly efficient video compression architectures maintaining lightweight encoding remains challenging in the context of wireless video capturing devices that have only modest computational capacity or operate on limited battery life. The solution to reduce the encoding complexity can be found in the fundamentals of information theory, which constitute an original coding perspective, known as distributed source coding (DSC). The latter stems from the theory of Slepian and Wolf [2] on lossless separate encoding and joint decoding of correlated sources. Subsequently, Wyner and Ziv [3] extended the DSC problem to the lossy case, deriving the rate distortion function with side information at the decoder. Driven by these principles, the distributed, alias Wyner-Ziv, video coding paradigm has arisen [4, 5] .. Unlike traditional video coding, in distributed video coding (DVC), the source redundancies are exploited at the decoder side, implying separate encoding and joint decoding. Specifically, a prediction of the source, named side information, is generated at the decoder by using the already decoded information. By expressing the statistical dependency between the source and the side information in the form of a virtual correlation channel, e.g. [4] [5] [6] [7] [8] , compression can be achieved by transmitting parity or syndrome bits of a channel code, which are used to decode the source with the aid of the side information. Hence, computationally expensive tasks, like motion estimation, could be relocated to the decoder, allowing for a flexible sharing of the computational complexity between the encoder and the decoder and enabling the design of lightweight encoding architectures.. DVC has been recognized as a potential strategic component for a wide range of lightweight video encoding applications, including visual sensor networks and wireless low-power surveillance [9, 10] . A unique application of particular interest in this article is wireless capsule endoscopy a . Conventional endoscopy, like colonoscopy or gastroscopy, has proven to be an indispensable tool in the diagnosis and remedy of various diseases of the digestive track. Significant advances in miniaturization have led to the emergence of wireless capsule endoscopy [11] . At the size of a large pill, a wireless capsule endoscope comprises a light source, an integrated chip video camera, a radio telemetry transmitter and a limited lifespan battery. The small-scale nature of the recording device forces severe constraints on the required video coding technology, in terms of computational complexity, operating time, and power consumption. Moreover, since the recorded video is used for medical diagnosis, high-quality decoded video at an efficient compression ratio is of paramount importance.. Generating high-quality side information plays a vital role in the compression performance of a DVC system. Conversely to traditional predictive coding, in DVC the original frame is not available during motion estimation, since this is performed at the decoder. Producing accurate motion-compensated predictions at the decoder for a wide range of video content, while at the same time constraining the encoding complexity and guaranteeing high compression performance, poses a major challenge. This problem becomes even more intricate in the largely unexplored application of DVC in wireless capsule endoscopy, in which the recorded video material contains extremely irregular motion, due to low frame acquisition rates and the erratic movement of the capsule along the gastrointestinal track. Towards tackling this challenge, this study presents a novel hash-based DVC architecture.. First and foremost, this study paves the road for the application of DVC systems in lightweight medical imaging where the proposed codec achieves high compression efficiency with the additional benefit of low computational encoding complexity. Second, the proposed Wyner-Ziv video codec incorporates a novel motion-compensated multi-hypothesis prediction scheme, that supports online tuning to the spatial variations in temporal correlation in a frame by obtaining information from the coded hash in case temporal prediction is unreliable. Third, this article includes a thorough experimental evaluation of the proposed hashbased DVC scheme on (i) conventional test sequences, numerous (ii) traditional endoscopic as well as (iii) wireless capsule endoscopic video content. The experimental results show that the proposed DVC outperforms alternative DVC schemes, including DISCOVER, the hashbased DVC from [12] and our previous study [13] , as well as conventional codecs, namely, Motion JPEG and H.264/AVC Intra [1] . Four, this article incorporates a detailed analysis of the encoding complexity and buffer size requirements of the proposed system.. The rest of the article is structured as follows. Section 2 covers an overview of Slepian-Wolf and Wyner-Ziv coding and their instantiation in DVC. Section 3 describes two application scenarios, both relevant to DVC in general and the proposed video codec in particular. Our novel DVC codec is explained in Section 4 and experimentally evaluated in Section 5, using conventional test sequences as well as endoscopic test video. Section 6 draws the conclusions of this study. Motivated by the strict prerequisites of wireless lightweight multimedia applications, such as wireless capsule endoscopy, this article has introduced a novel video codec based on the principles of Wyner-Ziv coding. The proposed codec maintains low encoding complexity and facilitates quality and temporal scalability. Intrinsically, the proposed codec achieves high compression performance by embracing a novel hash-driven motion estimation technique, which generates accurate side information at the decoder. The presented technique performs motion-compensated multi-hypothesis prediction, enabling adaptation to the regional variations in temporal correlation in a frame. Concrete experimentation using various conventional and endoscopic test video sequences has confirmed the superior compression performance of the proposed codec against several state-of-the-art traditional and Wyner-Ziv video codecs. In effect, in conventional and endoscopic test video material significant Bjøntegaard rate savings of up to 32.13 and 43.37% over the state-of-the-art have been obtained.. Endnotes a This paper has been presented in part at the IEEE International Conference on Image Processing, Brussels, Belgium, September 2011 [51] . b Motion JPEG is based on the JPEG coding standard [52] and includes a file format that can handle multiple JPEG images. Unlike the Motion JPEG 2000 standard [53] , no standard specification has been defined for Motion JPEG, and hence only proprietary solutions are available (e.g., support in Microsoft AVI files, Apple Quicktime format or the RFC 2435 spec that describes how Motion JPEG can be supported by an RTP stream). c To simplify the presentation, the LDPC code is assumed systematic. d The experimental results of DISCOVER [26] have been obtained using the executable of the DISCOVER codec which is available on the projects website [26] . e Given an RD point and a number of iterations, the process starts from a specific hash QP value (QP_hash = QP_key+1), and calculates the total and the hash rate, and the resulting RSD of the decoded frames (both key and WZ frames). If the RSD is lower that a strict threshold, the QP and the rate values are stored; otherwise they are discarded. Next, the hash QP is increased and the algorithm continuous till it reaches a given number of iterations. Out of the retained QPs, the one which minimizes the total rate is chosen as the best for the specific rate point. In case of equal total rates the highest QP value is selected. f These volunteers presented no evidence of gastrointestinal pathologies. g These sequences were transformed to the YUV 4:2:0 format supported by the proposed codec. h Nowadays more powerful processors exist to be deployed in devices at the size of the decoder of a capsule endoscope. For instance, the Apple A5 processor of iPhone 4S has an ARM Cortex-A9 MPCore 32-bit multicore processor at 800 MHz, 2.5DIMPS/MHz/core. i All these figures correspond to Soft-Input Soft Output (SISO) decoders.. Abbreviations AVC: advanced video coding; CPU: central processing unit; CRC: cyclic redundancy check; DCT: discrete cosine transform; DISCOVER: distributed coding for video services; DJSCC: distributed joint source-channel coding; DVC: distributed video coding; DSC: distributed source coding; GOP: group of pictures; HPS: hash-predictor-selection; JPEG: joint photographic experts group; LDPCA: low-density parity-check accumulate; LLR: log likelihood ratio; MCI: motion-compensated interpolation; OBME: overlapped block motion estimation; PRISM: power-efficient robust high-compression syndrome-based multimedia coding; PSNR: peak signal-to-noise ratio; RSD: relative standard deviation; QCIF: quarter common intermediate format; QM: quantization matrix; QP: quality parameter; RD: rate-distortion; SAD: sum of the absolute differences; TDWZ: transform-domain Wyner-Ziv.

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