Compressed Sensing and Electron Microscopy

Compressed sensing (CS) is a relatively new approach to signal acquisition which has as its goal to minimize the number of measurements needed of the signal in order to guarantee that it is captured to a prescribed accuracy. It is natural to inquire whether this new subject has a role to play in electron microscopy (EM). In this chapter, we shall describe the foundations of CS and then examine which parts of this new theory may be useful in EM.

[1]  Wang-Q Lim,et al.  Compactly Supported Shearlets , 2010, 1009.4359.

[2]  Christian Bender,et al.  Error Criteria for Numerical Solutions of Backward SDEs , 2010 .

[3]  Armin Iske,et al.  Curvature analysis of frequency modulated manifolds in dimensionality reduction , 2011 .

[4]  Wolfgang Dahmen,et al.  Fast high-dimensional approximation with sparse occupancy trees , 2011, J. Comput. Appl. Math..

[5]  E. Candes,et al.  11-magic : Recovery of sparse signals via convex programming , 2005 .

[6]  Andrej Yu. Garnaev,et al.  On widths of the Euclidean Ball , 1984 .

[7]  Steffen Dereich,et al.  Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction , 2011, 1101.1369.

[8]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[9]  L. S. Johnson,et al.  Optimal Entropy Encoders For Mining MultiplyResolved Data , 2000 .

[10]  Daniela Rosca,et al.  A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform , 2011, IEEE Transactions on Image Processing.

[11]  G. Kutyniok,et al.  Construction of Compactly Supported Shearlet Frames , 2010, 1003.5481.

[12]  Erich Novak,et al.  Optimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2 and other norms , 2004, J. Complex..

[13]  Erich Novak,et al.  The Curse of Dimensionality for Monotone and Convex Functions of Many Variables , 2010, 1011.3680.

[14]  I. Daubechies,et al.  Tree Approximation and Optimal Encoding , 2001 .

[15]  Ronald A. DeVore,et al.  Image compression through wavelet transform coding , 1992, IEEE Trans. Inf. Theory.

[16]  O. L. Krivanek,et al.  Sub-ångstrom resolution using aberration corrected electron optics , 2002, Nature.

[17]  Wolfgang Hackbusch,et al.  Tensorisation of vectors and their efficient convolution , 2011, Numerische Mathematik.

[18]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[19]  Deanna Needell,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, ArXiv.

[20]  O. Ernst,et al.  ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS , 2011 .

[21]  S. Muthukrishnan,et al.  Approximation of functions over redundant dictionaries using coherence , 2003, SODA '03.

[22]  Wolfgang Dahmen,et al.  Adaptive Petrov-Galerkin Methods for First Order Transport Equations , 2011, SIAM J. Numer. Anal..

[23]  Christian Bender,et al.  Dual pricing of multi-exercise options under volume constraints , 2011, Finance Stochastics.

[24]  K. Bredies,et al.  Regularization with non-convex separable constraints , 2009 .

[25]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[26]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[27]  G. Teschke,et al.  Compressive sensing principles and iterative sparse recovery for inverse and ill-posed problems , 2010 .

[28]  Bin Dong,et al.  Fast Linearized Bregman Iteration for Compressive Sensing and Sparse Denoising , 2011, ArXiv.

[29]  R. DeVore,et al.  Instance-optimality in probability with an ℓ1-minimization decoder , 2009 .

[30]  Winfried Sickel,et al.  Best m-term aproximation and tensor product of Sobolev and Besov spaces-the case of non-compact embeddings , 2010 .

[31]  Earl J. Kirkland,et al.  Advanced Computing in Electron Microscopy , 1998 .

[32]  Karsten Urban,et al.  Adaptive Wavelet Methods on Unbounded Domains , 2012, Journal of Scientific Computing.

[33]  Lars Grasedyck,et al.  Polynomial Approximation in Hierarchical Tucker Format by Vector – Tensorization , 2010 .

[34]  Reinhold Schneider,et al.  On manifolds of tensors of fixed TT-rank , 2012, Numerische Mathematik.

[35]  Wang-Q Lim,et al.  Shearlets on Bounded Domains , 2010, 1007.3039.

[36]  Reinhold Schneider,et al.  An analysis for the DIIS acceleration method used in quantum chemistry calculations , 2011 .

[37]  R. DeVore,et al.  Compressed sensing and best k-term approximation , 2008 .

[38]  Markus Hansen,et al.  On tensor products of quasi-Banach spaces , 2010 .

[39]  P. Maass,et al.  An analysis of electrical impedance tomography with applications to Tikhonov regularization , 2012 .

[40]  H. Harbrecht,et al.  On the low-rank approximation by the pivoted Cholesky decomposition , 2012 .

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

[42]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[43]  Nigel D. Browning,et al.  Practical aspects of atomic resolution imaging and analysis in STEM , 1999 .

[44]  E. Novak,et al.  Optimal Order of Convergence and (In)Tractability of Multivariate Approximation of Smooth Functions , 2009 .

[45]  Wolfgang Dahmen,et al.  Adaptivity and variational stabilization for convection-diffusion equations∗ , 2012 .

[46]  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 .

[47]  Armin Iske,et al.  Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction , 2010, SIAM J. Sci. Comput..

[48]  G. Teschke,et al.  Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints , 2010 .

[49]  Lutz Kämmerer,et al.  Interpolation lattices for hyperbolic cross trigonometric polynomials , 2012, J. Complex..

[50]  Stephan Dahlke,et al.  An adaptive wavelet method for parameter identification problems in parabolic partial differential equations , 2022 .

[51]  Stanley Osher,et al.  Development and Optimization of Regularized Tomographic Reconstruction Algorithms Utilizing Equally-Sloped Tomography , 2010, IEEE Transactions on Image Processing.

[52]  Gabriele Steidl,et al.  Shearlet Coorbit Spaces: Compactly Supported Analyzing Shearlets, Traces and Embeddings , 2011 .

[53]  P. Midgley,et al.  Electron tomography of nanoparticle catalysts on porous supports: A new technique based on Rutherford scattering , 2001 .

[54]  Christian Bender,et al.  Primal and Dual Pricing of Multiple Exercise Options in Continuous Time , 2011, SIAM J. Financial Math..

[55]  Klaus Ritter,et al.  Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains , 2010, 1011.1814.

[56]  Sadegh Jokar,et al.  Sparse recovery and Kronecker products , 2010, 2010 44th Annual Conference on Information Sciences and Systems (CISS).

[57]  Konstantin Grella,et al.  Sparse tensor spherical harmonics approximation in radiative transfer , 2011, J. Comput. Phys..

[58]  H. Yserentant,et al.  On the Complexity of the Electronic Schrödinger Equation , 2022 .

[59]  Klaus Ritter,et al.  Derandomization of the Euler scheme for scalar stochastic differential equations , 2012, J. Complex..

[60]  Tobias Jahnke,et al.  On Reduced Models for the Chemical Master Equation , 2011, Multiscale Model. Simul..

[61]  Emmanuel J. Candès,et al.  NESTA: A Fast and Accurate First-Order Method for Sparse Recovery , 2009, SIAM J. Imaging Sci..

[62]  K. Ritter,et al.  Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations , 2022 .

[63]  Winfried Sickel,et al.  Best m-term approximation and Lizorkin-Triebel spaces , 2011, J. Approx. Theory.

[64]  H. Sawada,et al.  STEM imaging of 47-pm-separated atomic columns by a spherical aberration-corrected electron microscope with a 300-kV cold field emission gun. , 2009, Journal of electron microscopy.

[65]  Vladimir Temlyakov,et al.  CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS , 2022 .

[66]  Wolfgang Dahmen,et al.  Super-Resolution Image Reconstruction by Nonlocal Means Applied to High-Angle Annular Darkfield Scanning Transmission Electron Microscopy (HAADF-STEM) , 2009 .

[67]  Stephan Dahlke,et al.  Adaptive wavelet methods and sparsity reconstruction for inverse heat conduction problems , 2010, Adv. Comput. Math..

[68]  G. Teschke,et al.  A compressive Landweber iteration for solving ill-posed inverse problems , 2008 .

[69]  Winfried Sickel,et al.  Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings , 2012 .

[70]  R. Schneider,et al.  The Alternating Linear Scheme for Tensor Optimisation in the TT Format , 2022 .

[71]  N. Baba,et al.  An auto-tuning method for focusing and astigmatism correction in HAADF-STEM, based on the image contrast transfer function. , 2001, Journal of electron microscopy.

[72]  G. Plonka The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data , 2009 .

[73]  Bernd Kabius,et al.  Electron microscopy image enhanced , 1998, Nature.

[74]  Erwan Faou,et al.  Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets , 2009, SIAM J. Sci. Comput..

[75]  Andreas Zeiser,et al.  Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schrödinger Equation , 2012 .

[76]  I. Daubechies,et al.  Iteratively reweighted least squares minimization for sparse recovery , 2008, 0807.0575.

[77]  Wang-Q Lim,et al.  Image Separation Using Shearlets , 2011 .

[78]  W. Hackbusch,et al.  Black Box Low Tensor-Rank Approximation Using Fiber-Crosses , 2009 .

[79]  Dirk A. Lorenz,et al.  Beyond convergence rates: exact recovery with the Tikhonov regularization with sparsity constraints , 2010, 1001.3276.

[80]  N. F. F. Ebecken,et al.  Data mining II , 2000 .

[81]  E. Novak,et al.  On the power of function values for the approximation problem in various settings , 2010, 1011.3682.

[82]  R. DeVore,et al.  Instance optimal decoding by thresholding in compressed sensing , 2008 .

[83]  Daniel Rudolf,et al.  Error bounds for computing the expectation by Markov chain Monte Carlo , 2009, Monte Carlo Methods Appl..

[84]  Ting Sun,et al.  Single-pixel imaging via compressive sampling , 2008, IEEE Signal Process. Mag..

[85]  S. Dereich,et al.  Constructive quantization: Approximation by empirical measures , 2011, 1108.5346.

[86]  D. Lorenz,et al.  Greedy solution of ill-posed problems: error bounds and exact inversion , 2009, 0904.0154.

[87]  Y. Nesterov A method for unconstrained convex minimization problem with the rate of convergence o(1/k^2) , 1983 .

[88]  Fred J. Hickernell,et al.  Deterministic multi-level algorithms for infinite-dimensional integration on RN , 2011, J. Complex..

[89]  Christian Bender,et al.  Least-Squares Monte Carlo for Backward SDEs , 2012 .

[90]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[91]  I. Daubechies,et al.  Tree Approximation and Encoding , 1999 .

[92]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

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

[94]  Harry Yserentant,et al.  The mixed regularity of electronic wave functions multiplied by explicit correlation factors , 2011 .

[95]  Michael Elad,et al.  L1-L2 Optimization in Signal and Image Processing , 2010, IEEE Signal Processing Magazine.