A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography

In this paper, we consider 3D Bioluminescence tomography (BLT) source reconstruction from Poisson data in three dimensional space. With a priori information of sources sparsity and MAP estimation of Poisson distribution, we study the minimization of Kullback-Leihbler divergence with ℓ1 and ℓ0 regularization. We show numerically that although several ℓ1 minimization algorithms are efficient for compressive sensing, they fail for BLT reconstruction due to the high coherence of the measurement matrix columns and high nonlinearity of Poisson fitting term. Instead, we propose a novel greedy algorithm for ℓ0 regularization to reconstruct sparse solutions for BLT problem. Numerical experiments on synthetic data obtained by the finite element methods and Monte-Carlo methods show the accuracy and efficiency of the proposed method.

[1]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[2]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[3]  Robert D. Nowak,et al.  A statistical multiscale framework for Poisson inverse problems , 2000, IEEE Trans. Inf. Theory.

[4]  H. Engl,et al.  CORRIGENDUM: The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis , 2007 .

[5]  Geoffrey McLennan,et al.  Practical reconstruction method for bioluminescence tomography. , 2005, Optics express.

[6]  Gabriele Steidl,et al.  Deblurring Poissonian images by split Bregman techniques , 2010, J. Vis. Commun. Image Represent..

[7]  Mário A. T. Figueiredo,et al.  Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization , 2009, 2009 IEEE/SP 15th Workshop on Statistical Signal Processing.

[8]  Raymond H. Chan,et al.  Multilevel algorithm for a Poisson noise removal model with total-variation regularization , 2007 .

[9]  T. Mihaljevic,et al.  Near-infrared fluorescent type II quantum dots for sentinel lymph node mapping , 2004, Nature Biotechnology.

[10]  Robert D. Nowak,et al.  A Bayesian multiscale framework for Poisson inverse problems , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[11]  J. Tropp,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, Commun. ACM.

[12]  M. Schweiger,et al.  The finite element method for the propagation of light in scattering media: boundary and source conditions. , 1995, Medical physics.

[13]  Martin Burger,et al.  Bregman-EM-TV Methods with Application to Optical Nanoscopy , 2009, SSVM.

[14]  M. Jiang,et al.  Uniqueness theorems in bioluminescence tomography. , 2004, Medical physics.

[15]  K. Lange,et al.  EM reconstruction algorithms for emission and transmission tomography. , 1984, Journal of computer assisted tomography.

[16]  Mohamed-Jalal Fadili,et al.  A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations , 2008, IEEE Transactions on Image Processing.

[17]  Singiresu S. Rao.  The finite element method in engineering , 1982 .

[18]  Rebecca Willett,et al.  Performance bounds on compressed sensing with Poisson noise , 2009, 2009 IEEE International Symposium on Information Theory.

[19]  Ge Wang,et al.  The First Bioluminescence Tomography System for Simultaneous Acquisition of Multiview and Multispectral Data , 2006, Int. J. Biomed. Imaging.

[20]  Gregory B. Passty.  Ergodic convergence to a zero of the sum of monotone operators in Hilbert space , 1979 .

[21]  Wenxiang Cong,et al.  Mathematical theory and numerical analysis of bioluminescence tomography , 2006 .

[22]  Roummel F. Marcia,et al.  Compressed Sensing Performance Bounds Under Poisson Noise , 2009, IEEE Transactions on Signal Processing.

[23]  L. Shepp,et al.  Maximum Likelihood Reconstruction for Emission Tomography , 1983, IEEE Transactions on Medical Imaging.

[24]  T. Chan,et al.  Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information. , 2009, Optics express.

[25]  Ge Wang,et al.  Mathematical Study and Numerical Simulation of Multispectral Bioluminescence Tomography , 2006, Int. J. Biomed. Imaging.

[26]  Jeffrey A. Fessler,et al.  Sparsity regularization for image reconstruction with Poisson data , 2009, Electronic Imaging.

[27]  S R Arridge,et al.  Recent advances in diffuse optical imaging , 2005, Physics in medicine and biology.

[28]  Jie Tian,et al.  A multilevel adaptive finite element algorithm for bioluminescence tomography. , 2006, Optics express.

[29]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[30]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[31]  L. Shepp,et al.  A Statistical Model for Positron Emission Tomography , 1985 .

[32]  Weimin Han,et al.  Bioluminescence Tomography: Biomedical Background, Mathematical Theory, and Numerical Approximation. , 2008, Journal of computational mathematics : an international journal on numerical methods, analysis and applications.

[33]  B. Rice,et al.  Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging. , 2007, Journal of biomedical optics.

[34]  Arion F. Chatziioannou,et al.  In Vivo Mouse Bioluminescence Tomography with Radionuclide-Based Imaging Validation , 2010, Molecular Imaging and Biology.

[35]  Hongkai Zhao,et al.  Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity. , 2010, Optics express.

[36]  Hongkai Zhao,et al.  Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization. , 2010, Optics express.

[37]  Rebecca Willett,et al.  This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms—Theory and Practice , 2010, IEEE Transactions on Image Processing.

[38]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[39]  José M. Bioucas-Dias,et al.  Restoration of Poissonian Images Using Alternating Direction Optimization , 2010, IEEE Transactions on Image Processing.

[40]  S. Mallat,et al.  Adaptive greedy approximations , 1997 .

[41]  I. Csiszár.  Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[42]  Hua-bei Jiang,et al.  Three-dimensional bioluminescence tomography with model-based reconstruction. , 2004, Optics express.

[43]  A. Chatziioannou,et al.  Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study , 2005, Physics in medicine and biology.

[44]  Bernie Mulgrew,et al.  IEEE Workshop on Statistical Signal Processing , 2005 .

[45]  Abu-Bakr Al-Mehdi,et al.  Increased Depth of Cellular Imaging in the Intact Lung Using Far-Red and Near-Infrared Fluorescent Probes , 2006, Int. J. Biomed. Imaging.