Sparse Reconstruction Algorithm for Nonhomogeneous Counting Rate Estimation

One of the main objectives of nuclear spectroscopy is the estimation of the counting rate of unknown radioactive sources. Recently, we proposed an algorithm based on a sparse reconstruction of the time signal in order to estimate precisely this counting rate, under the assumption that it remained constant over time. Computable bounds were obtained to quantify the performances. This approach, based on a postprocessed approach of a non-negative sparse regression of the time signal, performed well even when the activity of the source was high. The purpose of this paper is to present an extension of the previous method for an activity varying over time. It relies on the same preliminary sparse reconstruction. However, the postprocessed and plug-in steps are made differently to fit the nonhomogeneous framework. The adapted bounds are presented, and results on simulations illustrate the advantages and limitations of this method.

[1]  Todd P. Coleman,et al.  A computationally efficient method for modeling neural spiking activity with point processes nonparametrically , 2007, 2007 46th IEEE Conference on Decision and Control.

[2]  José Carlos Príncipe,et al.  Sequential Monte Carlo Point-Process Estimation of Kinematics from Neural Spiking Activity for Brain-Machine Interfaces , 2009, Neural Computation.

[3]  Todd P. Coleman,et al.  Using Convex Optimization for Nonparametric Statistical Analysis of Point Processes , 2007, 2007 IEEE International Symposium on Information Theory.

[4]  Emery N Brown,et al.  Computing Confidence Intervals for Point Process Models , 2011, Neural Computation.

[5]  Petr Lánský,et al.  Parameters of Spike Trains Observed in a Short Time Window , 2008, Neural Computation.

[6]  Alfred O. Hero,et al.  Time-delay estimation for filtered Poisson processes using an EM-type algorithm , 1994, IEEE Trans. Signal Process..

[7]  Jeffrey A. Fessler,et al.  Asymptotic Source Detection Performance of Gamma-Ray Imaging Systems Under Model Mismatch , 2011, IEEE Transactions on Signal Processing.

[8]  Raúl E. Sequeira,et al.  Intensity estimation from shot-noise data , 1995, IEEE Trans. Signal Process..

[9]  Tingting Zhang,et al.  Nonparametric Inference of Doubly Stochastic Poisson Process Data via the Kernel Method. , 2011, The annals of applied statistics.

[10]  Matthew A. Wilson,et al.  Construction of Point Process Adaptive Filter Algorithms for Neural Systems Using Sequential Monte Carlo Methods , 2007, IEEE Transactions on Biomedical Engineering.

[11]  Jean-Yves Tourneret,et al.  Joint Segmentation of Multivariate Astronomical Time Series: Bayesian Sampling With a Hierarchical Model , 2007, IEEE Transactions on Signal Processing.

[12]  Raúl E. Sequeira,et al.  Blind intensity estimation from shot-noise data , 1997, IEEE Trans. Signal Process..

[13]  Peter A. W. Lewis,et al.  Statistical Analysis of Non-Stationary Series of Events in a Data Base System , 1976, IBM J. Res. Dev..

[14]  Kazuo Yana,et al.  Estimation of the density of the filtered Poisson impulse process: a parametric approach , 1993, IEEE Trans. Signal Process..

[15]  Y. Salimpour,et al.  Extended Kalman filtering of point process observation , 2010, 2010 Annual International Conference of the IEEE Engineering in Medicine and Biology.

[16]  Todd P. Coleman,et al.  A Computationally Efficient Method for Nonparametric Modeling of Neural Spiking Activity with Point Processes , 2010, Neural Computation.

[17]  Arnaud Rivoira,et al.  A consistent nonparametric spectral estimator for randomly sampled signals , 2004, IEEE Transactions on Signal Processing.

[18]  H.W. Kraner,et al.  Radiation detection and measurement , 1981, Proceedings of the IEEE.

[19]  Hugues Talbot,et al.  An EM Approach for Time-Variant Poisson-Gaussian Model Parameter Estimation , 2014, IEEE Transactions on Signal Processing.

[20]  Michael Unser,et al.  MMSE Estimation of Sparse Lévy Processes , 2013, IEEE Transactions on Signal Processing.

[21]  Ulrike Goldschmidt,et al.  An Introduction To The Theory Of Point Processes , 2016 .

[22]  T. Trigano,et al.  On nonhomogeneous activity estimation in Gamma spectrometry using sparse signal representation , 2011, 2011 IEEE Statistical Signal Processing Workshop (SSP).

[23]  R. Tibshirani,et al.  Least angle regression , 2004, math/0406456.

[24]  Josep F. Oliver,et al.  Singles-prompts-randoms: Estimation of spurious data rates in PET , 2012, 2012 IEEE Nuclear Science Symposium and Medical Imaging Conference Record (NSS/MIC).

[25]  G. Knoll.  Radiation Detection And Measurement, 3rd Ed , 2009 .

[26]  Sylvie Huet,et al.  High-dimensional regression with unknown variance , 2011, 1109.5587.

[27]  Hiroyuki Mino,et al.  Parameter estimation of the intensity process of self-exciting point processes using the EM algorithm , 2001, IEEE Trans. Instrum. Meas..

[28]  P. Diggle.  A Kernel Method for Smoothing Point Process Data , 1985 .

[29]  S. Mendelson,et al.  Regularization and the small-ball method I: sparse recovery , 2016, 1601.05584.

[30]  Nonparametric Bayesian Estimation of Censored Counter Intensity from the Indicator Data , 2006 .

[31]  Stéphane Chrétien,et al.  Sparse Recovery With Unknown Variance: A LASSO-Type Approach , 2011, IEEE Transactions on Information Theory.

[32]  Emmanuel J. Candès,et al.  A Probabilistic and RIPless Theory of Compressed Sensing , 2010, IEEE Transactions on Information Theory.

[33]  Yaacov Ritov,et al.  Sparse Regression Algorithm for Activity Estimation in $\gamma$ Spectrometry , 2010, IEEE Transactions on Signal Processing.

[34]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[35]  M. C. Jones,et al.  A Brief Survey of Bandwidth Selection for Density Estimation , 1996 .

[36]  Michael Unser,et al.  Stochastic Models for Sparse and Piecewise-Smooth Signals , 2011, IEEE Transactions on Signal Processing.

[37]  Ronald W. Schafer,et al.  A three-state biological point process model and its parameter estimation , 1998, IEEE Trans. Signal Process..