Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections

We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator obtained by the least squares fit has the smallest operator norm, and therefore is most stable with respect to noisy measurements. We then construct the operator with the smallest possible quasi-optimality constant, which yields the most stable reconstruction with respect to a systematic error appearing before the sampling process (model uncertainty). We describe how to vary continuously between the two reconstruction methods, so that we can trade stability for quasi-optimality. As an application we study the reconstruction of a compactly supported function from nonuniform samples of its Fourier transform.

[1]  Kannan Ramchandran,et al.  Distributed Sampling for Dense Sensor Networks: A "Bit-Conservation Principle" , 2003, IPSN.

[2]  Anthony T. Patera,et al.  A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics , 2015 .

[3]  L. Carleson,et al.  The Collected Works of Arne Beurling , 1989 .

[4]  Tom Fleischer,et al.  Applied Functional Analysis , 2016 .

[5]  Ben Adcock,et al.  On Stable Reconstructions from Nonuniform Fourier Measurements , 2013, SIAM J. Imaging Sci..

[6]  H. Ogawa,et al.  Pseudoframes for Subspaces with Applications , 2004 .

[7]  Yonina C. Eldar.  Sampling with Arbitrary Sampling and Reconstruction Spaces and Oblique Dual Frame Vectors , 2003 .

[8]  A. Hansen,et al.  Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements , 2014, 1411.0300.

[9]  Jae-Hun Jung,et al.  Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon , 2004 .

[10]  Jared Tanner,et al.  Optimal filter and mollifier for piecewise smooth spectral data , 2006, Math. Comput..

[11]  K. Gröchenig,et al.  Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces , 2000 .

[12]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[13]  Wolfgang Dahmen,et al.  Data Assimilation in Reduced Modeling , 2015, SIAM/ASA J. Uncertain. Quantification.

[14]  Yonina C. Eldar.  Sampling Without Input Constraints: Consistent Reconstruction in Arbitrary Spaces , 2004 .

[15]  Ben Adcock,et al.  A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases , 2010, 1007.1852.

[16]  Yonina C. Eldar,et al.  Oblique dual frames and shift-invariant spaces , 2004 .

[17]  Ben Adcock,et al.  Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients , 2014, Math. Comput..

[18]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[19]  Akram Aldroubi,et al.  Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces , 2001, SIAM Rev..

[20]  Ben Adcock,et al.  On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate , 2012, ArXiv.

[21]  Wai-Shing Tang,et al.  Oblique projections, biorthogonal Riesz bases and multiwavelets in Hilbert spaces , 1999 .

[22]  Ben Adcock,et al.  Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon , 2010, 1011.6625.

[23]  J. Steinberg.  Oblique projections in Hilbert spaces , 2000 .

[24]  H. Landau.  Necessary density conditions for sampling and interpolation of certain entire functions , 1967 .

[25]  R. Lewitt.  Reconstruction algorithms: Transform methods , 1983, Proceedings of the IEEE.

[26]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[27]  Anne Gelb,et al.  Robust reprojection methods for the resolution of the Gibbs phenomenon , 2006 .

[28]  Emanuele Viterbo,et al.  Performance of Linear Field Reconstruction Techniques With Noise and Uncertain Sensor Locations , 2007, IEEE Transactions on Signal Processing.

[29]  Karlheinz Gröchenig.  Non-Uniform Sampling in Higher Dimensions: From Trigonometric Polynomials to Bandlimited Functions , 2001 .

[30]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[31]  Jae-Hun Jung,et al.  Inverse Polynomial Reconstruction of Two Dimensional Fourier Images , 2005, J. Sci. Comput..

[32]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

[33]  Jean-Pierre Antoine,et al.  Weighted and Controlled Frames: Mutual Relationship and First Numerical Properties , 2010, Int. J. Wavelets Multiresolution Inf. Process..

[34]  Rosemary A. Renaut,et al.  On Reconstruction from Non-uniform Spectral Data , 2010, J. Sci. Comput..

[35]  Karlheinz Gröchenig,et al.  Fast Local Reconstruction Methods for Nonuniform Sampling in Shift-Invariant Spaces , 2002, SIAM J. Matrix Anal. Appl..

[36]  Karlheinz Gröchenig,et al.  Pseudospectral Fourier reconstruction with the modified Inverse Polynomial Reconstruction Method , 2010, J. Comput. Phys..

[37]  Hidemitsu Ogawa,et al.  Pseudoframes for subspaces with applications , 1998, Optics & Photonics.

[38]  Kristian Kirsch,et al.  Methods Of Modern Mathematical Physics , 2016 .

[39]  Yonina C. Eldar,et al.  Generalized Shift-Invariant Systems and Frames for Subspaces , 2005 .

[40]  Don Buckholtz.  Hilbert space idempotents and involutions , 1999 .

[41]  Jae-Hun Jung,et al.  On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon , 2007, J. Comput. Phys..

[42]  R. DeVore,et al.  Data assimilation and sampling in Banach spaces , 2016, 1602.06342.

[43]  Ben Adcock,et al.  A Stability Barrier for Reconstructions from Fourier Samples , 2012, SIAM J. Numer. Anal..

[44]  Yonina C. Eldar,et al.  General Framework for Consistent Sampling in Hilbert Spaces , 2005, Int. J. Wavelets Multiresolution Inf. Process..

[45]  Gerald Matz,et al.  Distributed Field Reconstruction in Wireless Sensor Networks Based on Hybrid Shift-Invariant Spaces , 2012, IEEE Transactions on Signal Processing.

[46]  Yoo Young Koo,et al.  Duals of Frame Sequences , 2009 .

[47]  R. DeVore,et al.  Data Assimilation in Banach spaces , 2016 .

[48]  Karlheinz Gröchenig,et al.  On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions , 1996 .

[49]  J. Romero,et al.  Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates , 2016, Applied and Computational Harmonic Analysis.

[50]  Eitan Tadmor,et al.  Filters, mollifiers and the computation of the Gibbs phenomenon , 2007, Acta Numerica.

[51]  Y. Maday,et al.  A generalized empirical interpolation method : application of reduced basis techniques to data assimilation , 2013, 1512.00683.

[52]  Yonina C. Eldar,et al.  Characterization of Oblique Dual Frame Pairs , 2006, EURASIP J. Adv. Signal Process..

[53]  Jae-Hun Jung,et al.  Towards the resolution of the Gibbs phenomena , 2003 .

[54]  G. Corach,et al.  Sampling theory, oblique projections and a question by Smale and Zhou , 2006 .

[55]  M. L. Eaton.  Multivariate statistics : a vector space approach , 1985 .

[56]  Thomas Strohmer,et al.  Numerical analysis of the non-uniform sampling problem , 2000 .

[57]  Kannan Ramchandran,et al.  High-Resolution Distributed Sampling of Bandlimited Fields With Low-Precision Sensors , 2007, IEEE Transactions on Information Theory.

[59]  Ben Adcock,et al.  Generalized sampling: extension to frames and inverse and ill-posed problems , 2013 .

[60]  Jackie Ma.  Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets , 2014 .

[61]  Kristian Seip,et al.  On the Connection between Exponential Bases and Certain Related Sequences in L2(− π,π) , 1995 .

[62]  Karlheinz Gröchenig,et al.  Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type , 1999, Math. Comput..

[63]  Daniel B. Szyld,et al.  The many proofs of an identity on the norm of oblique projections , 2006, Numerical Algorithms.

[64]  Anthony T. Patera,et al.  The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation , 2015 .

[65]  Gitta Kutyniok,et al.  Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements , 2015, SIAM J. Math. Anal..

[66]  P. Balázs.  Basic definition and properties of Bessel multipliers , 2005, math/0510091.

[67]  J. M. Whittaker.  Interpolatory function theory , 1935 .

[68]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[69]  Yonina C. Eldar,et al.  General Framework for Consistent Sampling in Hilbert Spaces , 2005, Int. J. Wavelets Multiresolution Inf. Process..

[70]  Ole Christensen,et al.  Frames and Bases , 2008 .

[71]  Ben Adcock,et al.  Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem , 2013, SIAM J. Math. Anal..

[72]  A. Ron.  Review of An introduction to Frames and Riesz bases, applied and numerical Harmonic analysis by Ole Christensen Birkhäuser, Basel, 2003 , 2005 .

[73]  Ben Adcock,et al.  Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples , 2014, 1405.3111.

[74]  Laurent Jacques,et al.  Stereographic wavelet frames on the sphere , 2005 .