Nonsmooth Convex Optimization for Structured Illumination Microscopy Image Reconstruction

In this paper, we propose a new approach for structured illumination microscopy image reconstruction. We first introduce the principles of this imaging modality and describe the forward model. We then propose the minimization of nonsmooth convex objective functions for the recovery of the unknown image. In this context, we investigate two data-fitting terms for Poisson-Gaussian noise and introduce a new patch-based regularization method. This approach is tested against other regularization approaches on a realistic benchmark. Finally, we perform some test experiments on images acquired on two different microscopes.

[1]  Gabriele Steidl,et al.  First order algorithms in variational image processing , 2014, ArXiv.

[2]  Karen O. Egiazarian,et al.  BM3D Frames and Variational Image Deblurring , 2011, IEEE Transactions on Image Processing.

[3]  Ayan Chakrabarti,et al.  Image Restoration with Signal-dependent Camera Noise , 2012, ArXiv.

[4]  Nelly Pustelnik,et al.  Epigraphical projection and proximal tools for solving constrained convex optimization problems , 2012, Signal Image Video Process..

[5]  OsherStanley,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[6]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[7]  Nelly Pustelnik,et al.  A Nonlocal Structure Tensor-Based Approach for Multicomponent Image Recovery Problems , 2014, IEEE Transactions on Image Processing.

[8]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[9]  Nelly Pustelnik,et al.  Non-smooth convex optimization for an efficient reconstruction in structured illumination microscopy , 2014, 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI).

[10]  L. Cohen,et al.  Non-local Regularization of Inverse Problems , 2008, ECCV.

[11]  Faming Liang,et al.  Statistical and Computational Inverse Problems , 2006, Technometrics.

[12]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[13]  Anne Sentenac,et al.  Structured illumination microscopy using unknown speckle patterns , 2012, Nature Photonics.

[14]  M. Gustafsson,et al.  Extended resolution fluorescence microscopy. , 1999, Current opinion in structural biology.

[15]  Souleymen Sahnoun,et al.  A 2-D spectral analysis method to estimate the modulation parameters in structured illumination microscopy , 2014, 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI).

[16]  M. Gustafsson.  Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy , 2000, Journal of microscopy.

[17]  Fionn Murtagh,et al.  Image Processing and Data Analysis - The Multiscale Approach , 1998 .

[18]  Jong Chul Ye,et al.  Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery , 2013, Scientific Reports.

[19]  Laurent Condat.  A Simple Trick to Speed Up and Improve the Non-Local Means , 2010 .

[20]  P. L. Combettes,et al.  Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators , 2011, Set-Valued and Variational Analysis.

[21]  Michael Unser,et al.  Hessian Schatten-Norm Regularization for Linear Inverse Problems , 2012, IEEE Transactions on Image Processing.

[22]  Jie Yin,et al.  Image reconstruction for structured-illumination microscopy with low signal level. , 2014, Optics express.

[23]  M. Elad,et al.  $rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation , 2006, IEEE Transactions on Signal Processing.

[24]  A. Bruckstein,et al.  K-SVD : An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation , 2005 .

[25]  Wotao Yin,et al.  Splitting Methods in Communication, Imaging, Science, and Engineering , 2017 .

[26]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[27]  Laurent Condat,et al.  A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms , 2013, J. Optim. Theory Appl..

[28]  Rainer Heintzmann,et al.  Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating , 1999, European Conference on Biomedical Optics.

[29]  I. Csiszár.  A class of measures of informativity of observation channels , 1972 .

[30]  H. Leonhardt,et al.  A guide to super-resolution fluorescence microscopy , 2010, The Journal of cell biology.

[31]  Jian Yu,et al.  A Dictionary Learning Approach for Poisson Image Deblurring , 2013, IEEE Transactions on Medical Imaging.

[32]  Nikos Komodakis,et al.  Playing with Duality: An overview of recent primal?dual approaches for solving large-scale optimization problems , 2014, IEEE Signal Process. Mag..

[33]  Hari Shroff,et al.  Resolution Doubling in Live, Multicellular Organisms via Multifocal Structured Illumination Microscopy , 2012, Nature Methods.

[34]  F. J. Anscombe,et al.  THE TRANSFORMATION OF POISSON, BINOMIAL AND NEGATIVE-BINOMIAL DATA , 1948 .

[35]  L H Schaefer,et al.  Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach , 2004, Journal of microscopy.

[36]  Patrick Bouthemy,et al.  Patch-Based Nonlocal Functional for Denoising Fluorescence Microscopy Image Sequences , 2010, IEEE Transactions on Medical Imaging.

[37]  Bang Công Vu,et al.  A splitting algorithm for dual monotone inclusions involving cocoercive operators , 2011, Advances in Computational Mathematics.

[38]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[39]  J. Moreau.  Proximité et dualité dans un espace hilbertien , 1965 .

[40]  Vincent Loriette,et al.  Bayesian Estimation for Optimized Structured Illumination Microscopy , 2012, IEEE Transactions on Image Processing.

[41]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[42]  Antonin Chambolle,et al.  An introduction to continuous optimization for imaging , 2016, Acta Numerica.

[43]  Laurent Condat,et al.  Discrete Total Variation: New Definition and Minimization , 2017, SIAM J. Imaging Sci..

[44]  Guy Gilboa,et al.  Nonlocal Operators with Applications to Image Processing , 2008, Multiscale Model. Simul..

[45]  Daniela Calvetti,et al.  Hypermodels in the Bayesian imaging framework , 2008 .

[46]  M. Gustafsson,et al.  Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination. , 2008, Biophysical journal.

[47]  Clemens F. Kaminski,et al.  A joint Richardson—Lucy deconvolution algorithm for the reconstruction of multifocal structured illumination microscopy data , 2015, 2015 Conference on Lasers and Electro-Optics (CLEO).

[48]  Jérôme Idier,et al.  Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations. , 2016, Journal of the Optical Society of America. A, Optics, image science, and vision.

[49]  Hugues Talbot,et al.  A Convex Approach for Image Restoration with Exact Poisson-Gaussian Likelihood , 2015, SIAM J. Imaging Sci..

[50]  K. O’Holleran,et al.  Optimized approaches for optical sectioning and resolution enhancement in 2D structured illumination microscopy , 2014, Biomedical optics express.

[51]  J. Goodman.  Introduction to Fourier optics , 1969 .

[52]  Karel Fliegel,et al.  Three-dimensional super-resolution structured illumination microscopy with maximum a posteriori probability image estimation. , 2014, Optics express.

[53]  Nelly Pustelnik,et al.  Epigraphical splitting for solving constrained convex formulations of inverse problems with proximal tools , 2012, 1210.5844.

[54]  Laurent D. Cohen,et al.  Non-local Regularization of Inverse Problems , 2008, ECCV.