Further results on geometric properties of a family of relative entropies

This paper extends some geometric properties of a one-parameter family of relative entropies. These arise as redundancies when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the Kullback-Leibler divergence. They satisfy the Pythagorean property and behave like squared distances. This property, which was known for finite alphabet spaces, is now extended for general measure spaces. Existence of projections onto convex and certain closed sets is also established. Our results may have applications in the Rényi entropy maximization rule of statistical physics.

Reconstruction from incomplete data in cone-beam tomography

A method for reconstruction of an object f(x) x=(x,y,z) from a limited set of cone-beam projection data has been developed. This method uses a modified form of convolution back-projection and projection onto convex sets (POCS) for handling the limited (or incomplete) data problem. In cone-beam tomography, one needs to have a complete geometry to completely reconstruct the original three-dimensional object. While complete geometries do exist, they are of little use in practical implementations. The most common trajectory used in practical scanners is circular, which is incomplete. It is, however, possible to recover some of the information of the original signal f(x) based on a priori knowledge of the nature of f(x). If this knowledge can be posed in a convex set framework, then POCS can be utilized. In this report, we utilize this a priori knowledge as convex set constraints to reconstruct f(x) using POCS. While we demonstrate the effectiveness of our algorithm for circular trajectories, it is essentially geometry independent and will be useful in any limited-view cone-beam reconstruction.

Fienup Algorithm With Sparsity Constraints: Application to Frequency-Domain Optical-Coherence Tomography

We address the problem of reconstructing a sparse signal from its DFT magnitude. We refer to this problem as the sparse phase retrieval (SPR) problem, which finds applications in tomography, digital holography, electron microscopy, etc. We develop a Fienup-type iterative algorithm, referred to as the Max-K algorithm, to enforce sparsity and successively refine the estimate of phase. We show that the Max-K algorithm possesses Cauchy convergence properties under certain conditions, that is, the MSE of reconstruction does not increase with iterations. We also formulate the problem of SPR as a feasibility problem, where the goal is to find a signal that is sparse in a known basis and whose Fourier transform magnitude is consistent with the measurement. Subsequently, we interpret the Max-K algorithm as alternating projections onto the object-domain and measurement-domain constraint sets and generalize it to a parameterized relaxation, known as the relaxed averaged alternating reflections (RAAR) algorithm. On the application front, we work with measurements acquired using a frequency-domain optical-coherence tomography (FDOCT) experimental setup. Experimental results on measured data show that the proposed algorithms exhibit good reconstruction performance compared with the direct inversion technique, homomorphic technique, and the classical Fienup algorithm without sparsity constraint; specifically, the autocorrelation artifacts and background noise are suppressed to a significant extent. We also demonstrate that the RAAR algorithm offers a broader framework for FDOCT reconstruction, of which the direct inversion technique and the proposed Max-K algorithm become special instances corresponding to specific values of the relaxation parameter.

Minimization Problems Based on Relative alpha-Entropy I: Forward Projection

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative alpha-entropies (denoted I-alpha), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative alpha-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative alpha-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Renyi or Tsallis entropy principle. The minimizing probability distribution (termed forward I-alpha-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse I-alpha-projection is studied.

A Convex Optimization Framework for Almost Budget Balanced Allocation of a Divisible Good

We address the problem of allocating a single divisible good to a number of agents. The agents have concave valuation functions parameterized by a scalar type. The agents report only the type. The goal is to find allocatively efficient, strategy proof, nearly budget balanced mechanisms within the Groves class. Near budget balance is attained by returning as much of the received payments as rebates to agents. Two performance criteria are of interest: the maximum ratio of budget surplus to efficient surplus, and the expected budget surplus, within the class of linear rebate functions. The goal is to minimize them. Assuming that the valuation functions are known, we show that both problems reduce to convex optimization problems, where the convex constraint sets are characterized by a continuum of half-plane constraints parameterized by the vector of reported types. We then propose a randomized relaxation of these problems by sampling constraints. The relaxed problem is a linear programming problem (LP). We then identify the number of samples needed for ``near-feasibility'' of the relaxed constraint set. Under some conditions on the valuation function, we show that value of the approximate LP is close to the optimal value. Simulation results show significant improvements of our proposed method over the Vickrey-Clarke-Groves (VCG) mechanism without rebates. In the special case of indivisible goods, the mechanisms in this paper fall back to those proposed by Moulin, by Guo and Conitzer, and by Gujar and Narahari, without any need for randomization. Extension of the proposed mechanisms to situations when the valuation functions are not known to the central planner are also discussed. Note to Practitioners-Our results will be useful in all resource allocation problems that involve gathering of information privately held by strategic users, where the utilities are any concave function of the allocations, and where the resource planner is not interested in maximizing revenue, but in efficient sharing of the resource. Such situations arise quite often in fair sharing of internet resources, fair sharing of funds across departments within the same parent organization, auctioning of public goods, etc. We study methods to achieve near budget balance by first collecting payments according to the celebrated VCG mechanism, and then returning as much of the collected money as rebates. Our focus on linear rebate functions allows for easy implementation. The resulting convex optimization problem is solved via relaxation to a randomized linear programming problem, for which several efficient solvers exist. This relaxation is enabled by constraint sampling. Keeping practitioners in mind, we identify the number of samples that assures a desired level of ``near-feasibility'' with the desired confidence level. Our methodology will occasionally require subsidy from outside the system. We however demonstrate via simulation that, if the mechanism is repeated several times over independent instances, then past surplus can support the subsidy requirements. We also extend our results to situations where the strategic users' utility functions are not known to the allocating entity, a common situation in the context of internet users and other problems.

Daily relative humidity projections in an Indian river basin for IPCC SRES scenarios

A two-stage methodology is developed to obtain future projections of daily relative humidity in a river basin for climate change scenarios. In the first stage, Support Vector Machine (SVM) models are developed to downscale nine sets of predictor variables (large-scale atmospheric variables) for Intergovernmental Panel on Climate Change Special Report on Emissions Scenarios (SRES) (A1B, A2, B1, and COMMIT) to R (H) in a river basin at monthly scale. Uncertainty in the future projections of R (H) is studied for combinations of SRES scenarios, and predictors selected. Subsequently, in the second stage, the monthly sequences of R (H) are disaggregated to daily scale using k-nearest neighbor method. The effectiveness of the developed methodology is demonstrated through application to the catchment of Malaprabha reservoir in India. For downscaling, the probable predictor variables are extracted from the (1) National Centers for Environmental Prediction reanalysis data set for the period 1978-2000 and (2) simulations of the third-generation Canadian Coupled Global Climate Model for the period 1978-2100. The performance of the downscaling and disaggregation models is evaluated by split sample validation. Results show that among the SVM models, the model developed using predictors pertaining to only land location performed better. The R (H) is projected to increase in the future for A1B and A2 scenarios, while no trend is discerned for B1 and COMMIT.

Image reconstruction with laterally truncated projections in helical cone-beam CT: Linear prediction based projection completion techniques

Lateral or transaxial truncation of cone-beam data can occur either due to the field of view limitation of the scanning apparatus or iregion-of-interest tomography. In this paper, we Suggest two new methods to handle lateral truncation in helical scan CT. It is seen that reconstruction with laterally truncated projection data, assuming it to be complete, gives severe artifacts which even penetrates into the field of view. A row-by-row data completion approach using linear prediction is introduced for helical scan truncated data. An extension of this technique known as windowed linear prediction approach is introduced. Efficacy of the two techniques are shown using simulation with standard phantoms. A quantitative image quality measure of the resulting reconstructed images are used to evaluate the performance of the proposed methods against an extension of a standard existing technique.

Reliable Means of Diagnosis and Serovar Determination of Blood-Borne Salmonella Strains: Quick PCR Amplification of Unique Genomic Loci by Novel Primer Sets

Typhoid fever is becoming an ever increasing threat in the developing countries. We have improved considerably upon the existing PCR-based diagnosis method by designing primers against a region that is unique to Salmonella enterica subsp. enterica serovar Typhi and Salmonella enterica subsp. enterica serovar Paratyphi A, corresponding to the STY0312 gene in S. Typhi and its homolog SPA2476 in S. Paratyphi A. An additional set of primers amplify another region in S. Typhi CT18 and S. Typhi Ty2 corresponding to the region between genes STY0313 to STY0316 but which is absent in S. Paratyphi A. The possibility of a false-negative result arising due to mutation in hypervariable genes has been reduced by targeting a gene unique to typhoidal Salmonella serovars as a diagnostic marker. The amplified region has been tested for genomic stability by amplifying the region from clinical isolates of patients from various geographical locations in India, thereby showing that this region is potentially stable. These set of primers can also differentiate between S. Typhi CT18, S. Typhi Ty2, and S. Paratyphi A, which have stable deletions in this specific locus. The PCR assay designed in this study has a sensitivity of 95% compared to the Widal test which has a sensitivity of only 63%. As observed, in certain cases, the PCR assay was more sensitive than the blood culture test was, as the PCR-based detection could also detect dead bacteria.

A Branch and Bound Algorithm for a Transportation Type Problem with Piecewise Linear Convex Objective Function

A branch and bound type algorithm is presented in this paper to the problem of finding a transportation schedule which minimises the total transportation cost, where the transportation cost over each route is assumed to be a piecewice linear continuous convex function with increasing slopes. The algorithm is an extension of the work done by Balachandran and Perry, in which the transportation cost over each route is assumed to beapiecewise linear discontinuous function with decreasing slopes. A numerical example is solved illustrating the algorithm.

Non-linear Transceiver Designs with Imperfect CSIT Using Convex Optimization

In this paper, we consider non-linear transceiver designs for multiuser multi-input multi-output (MIMO) down-link in the presence of imperfections in the channel state information at the transmitter (CSIT). The base station (BS) is equipped with multiple transmit antennas and each user terminal is equipped with multiple receive antennas. The BS employs Tomlinson-Harashima precoding (THP) for inter-user interference pre-cancellation at the transmitter. We investigate robust THP transceiver designs based on the minimization of BS transmit power with mean square error (MSE) constraints, and balancing of MSE among users with a constraint on the total BS transmit power. We show that these design problems can be solved by iterative algorithms, wherein each iteration involves a pair of convex optimization problems. The robustness of the proposed algorithms to imperfections in CSIT is illustrated through simulations.

Conservation law with the flux function discontinuous in the space variable- II - Convex-concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex-concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine-Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine-Hugoniot condition by a generalized Rankine-Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L-1. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented. (C) 2006 Elsevier B.V. All rights reserved.

Proximinality in Banach spaces

In this paper, we study approximatively τ-compact and τ-strongly Chebyshev sets, where τ is the norm or the weak topology. We show that the metric projection onto τ-strongly Chebyshev sets are norm-τ continuous. We characterize approximatively τ-compact and τ-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in τ-almost locally uniformly rotund spaces. We also prove some stability results on approximatively τ-compact and τ-strongly Chebyshev subspaces.

Electrochemical insertion of Sr2+ ions onto nano delta-MnO2 particles

Manganese dioxide is known to be an important electroactive material for supercapacitors. Generally, delta-MnO2 is subjected to electrochemical characterization studies in aqueous electrolytes of Na2SO4. It exhibits capacitance behaviour in the potential range between 0 and 1.0 V vs. SCE (saturated calomel electrode). In the present study, it is shown that delta-MnO2 exhibits capacitance behaviour in Sr(NO3)(2) electrolytes also. The suitable potential range in this electrolyte is also found to be 0-1.0 V. Specific capacitancemeasured in Sr(NO3)(2) electrolyte is 192 F g(-1). X-ray photoelectron spectroscopy data confirm that Sr2+ ions get inserted onto delta-MnO2 anoparticles. (C) 2010 Elsevier B.V. All rights reserved.

Coding for Two-User Gaussian MAC with PSK and PAM Signal Sets

Constellation Constrained (CC) capacity regions of a two-user Gaussian Multiple Access Channel(GMAC) have been recently reported. For such a channel, code pairs based on trellis coded modulation are proposed in this paper with MPSK and M-PAM alphabet pairs, for arbitrary values of M,toachieve sum rates close to the CC sum capacity of the GMAC. In particular, the structure of the sum alphabets of M-PSK and M-PAMmalphabet pairs are exploited to prove that, for certain angles of rotation between the alphabets, Ungerboeck labelling on the trellis of each user maximizes the guaranteed squared Euclidean distance of the sum trellis. Hence, such a labelling scheme can be used systematically,to construct trellis code pairs to achieve sum rates close to the CC sum capacity. More importantly, it is shown for the first time that ML decoding complexity at the destination is significantly reduced when M-PAM alphabet pairs are employed with almost no loss in the sum capacity.

Peroxomonosulphate initiated graft copolymerization of aniline onto poly(propylene) fibre - A kinetic approach

Graft copolymerization of poly(aniline) (PANI) onto poly(propylene) (PP) fibre was carried out in aqueous acidic medium under nitrogen atmosphere by using peroxomonosulphate (PMS) as a lone initiator. The non-conducting fibre was now made into a conducting one through the chemical grafting of PANI units onto the PP fibre backbone. The content of PANI in the backbone was found to vary while varying the [ANI], [PMS] and amount of PP fibre. Various graft parameters were evaluated. The chemical grafting of PANI onto PP fibre was confirmed by conductivity measurements.