Adaptive variable selection for extended Nijboer–Zernike aberration retrieval via lasso

Abstract In this paper, we propose extended Nijboer–Zernike (ENZ) method for aberration retrieval by incorporating lasso variable selection method which can improve the accuracy of aberration retrieval. The proposed model is computed by the state-of-art algorithm of the Bregman iterative algorithm (Bregman, 1967 [1] ; Cai et al., 2008 [2] ; Yin et al., 2008 [3] ) for L 1 minimization problem with adaptive regularized parameter choice based on the strategy (Ito et al., 2011 [4] ). Numerical simulations for real world and simulated phase data validate the effectiveness of the proposed ENZ AR via lasso.

[1]  Dimitri Mawet,et al.  Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO , 2012 .

[2]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[3]  Wotao Yin,et al.  Iteratively reweighted algorithms for compressive sensing , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[4]  Thomas Bonesky Morozov's discrepancy principle and Tikhonov-type functionals , 2008 .

[5]  L. Bregman The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[6]  Anat Levin,et al.  User Assisted Separation of Reflections from a Single Image Using a Sparsity Prior , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  Joseph J. M. Braat,et al.  Estimating resist parameters in optical lithography using the extended Nijboer-Zernike theory , 2006 .

[8]  A. Janssen,et al.  Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[9]  Bangti Jin,et al.  Heuristic Parameter-Choice Rules for Convex Variational Regularization Based on Error Estimates , 2010, SIAM J. Numer. Anal..

[10]  A. Janssen Extended Nijboer-Zernike approach for the computation of optical point-spread functions. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[11]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

[12]  Michel Verhaegen,et al.  Modal-based phase retrieval for adaptive optics. , 2015, Journal of the Optical Society of America. A, Optics, image science, and vision.

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

[14]  Stephen J. Wright,et al.  Sparse reconstruction by separable approximation , 2009, IEEE Trans. Signal Process..

[15]  Bangti Jin,et al.  Iterative parameter choice by discrepancy principle , 2012 .

[16]  A. Janssen,et al.  Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory , 2013 .

[17]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[18]  Joseph J. M. Braat,et al.  High-NA aberration retrieval with the Extended Nijboer-Zernike vector diffraction theory , 2006 .

[19]  M. Hestenes Multiplier and gradient methods , 1969 .

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

[21]  Jian-Feng Cai,et al.  Linearized Bregman iterations for compressed sensing , 2009, Math. Comput..

[22]  Nicolas A. Roddier Atmospheric wavefront simulation using Zernike polynomials , 1990 .

[23]  Joseph J. M. Braat,et al.  Aberration retrieval using the extended Nijboer-Zernike approach , 2003 .

[24]  J. J. M. Braat,et al.  Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach , 2005 .

[25]  Sven van Haver,et al.  Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[26]  Yin Zhang,et al.  Fixed-Point Continuation for l1-Minimization: Methodology and Convergence , 2008, SIAM J. Optim..

[27]  Xinyue Liu,et al.  A three-dimensional point spread function for phase retrieval and deconvolution. , 2012, Optics express.

[28]  Peter Dirksen,et al.  Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[29]  Kazufumi Ito,et al.  A Regularization Parameter for Nonsmooth Tikhonov Regularization , 2011, SIAM J. Sci. Comput..