关于高斯低通滤波器的超分辨分析
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摘要
对于经过高斯低通滤波的信号,通过求解一类凸优化模型稳定地恢复该信号的高频信息.当信号满足一定的分离条件时,给出了误差估计的界,从理论上证明了求解凸优化方法的稳定性.理论的证明依赖于压缩感知中的对偶理论.一个显著的差异在于高斯低通滤波器并不满足压缩感知中对于测量矩阵的要求,例如相关性,约束等距性质等.
In this paper, for the signal which is convoluted by Gaussian low-pass filters, the high frequency information of the signal is recovered stably by solving a convex optimization model. When the signal satisfies a certain separation condition, the stability of the convex optimization problem is proved theoretically. The proof depends on the duality certification in compressed sensing. A significant difference is that Gaussian low-pass filters do not satisfy the conditions of measurement matrix in compressive sensing, such as mutual coherence, restricted isometry property, etc.
In this paper, for the signal which is convoluted by Gaussian low-pass filters, the high frequency information of the signal is recovered stably by solving a convex optimization model. When the signal satisfies a certain separation condition, the stability of the convex optimization problem is proved theoretically. The proof depends on the duality certification in compressed sensing. A significant difference is that Gaussian low-pass filters do not satisfy the conditions of measurement matrix in compressive sensing, such as mutual coherence, restricted isometry property, etc.
引文
[1] Donoho D L. Compressed sensing[J]. IEEE Transactions on Information Theory, 2006, 52:1289-1306.
[2] Candes E J, Tao T. Decoding by linear programming[J]. IEEE Transactions on Information Theory, 2005, 51:4203-4215.
[3] Donoho D L. Superresolution via sparsity constraints[J]. SIAM Journal on Mathematical Analysis, 1992, 23(5):1309-1331.
[4] Candes E J, Fernandezgranda C. Super-resolution from noisy data[J]. Journal of Fourier Analysis and Applications, 2012, 19(6):1229-1254.
[5] Candes E J, Fernandezgranda C. Towards a mathematical theory of super-resolution[J]. Communications on Pure and Applied Mathematics, 2012, 67(6):906-956.
[6] Morgenshtern V I, Candes E J. Super-resolution of positive sources:the discrete setup[J]. SIAM Journal on Imaging Sciences, 2015, 9(1):412-444.
[7] Bendory T, Dekel S, Feuer A. Exact recovery of dirac ensembles from the projection onto spaces of spherical harmonics[J]. Constructive Approximation, 2014, 42(2):183-207.
[8] Bendory T, Dekel S, Feuer A. Exact recovery of non-uniform splines from the projection onto spaces of algebraic polynomials[J]. Journal of Approximation Theory, 2014, 182(1):7-17.
[9] Duval V, Peyre G. Exact support recovery for sparse spikes deconvolution[J]. Foundations of Computational Mathematics, 2015, 15(5):1315-1355.
[10]许志强.压缩感知[J].中国科学A辑,2012, 42(9):865-877.
[11] Bernstein B, Fernandez-Granda C. Deconvolution of point sources:a sampling theorem and robustness guarantees[J]. arXiv preprint arXiv:1707.00808, 2017.
[12] Azais J, De Castro Y, Gamboa F. Spike detection from inaccurate samplings[J]. Applied and Computational Harmonic Analysis, 2013, 38(2):177-195.
[13] Oppenheim A W, Willsky A S, Hamid S. Signals and Systems[M]. 2nd ed., Englewood Cliffs:Prentice Hall, 1996.
[14] Hansen P C, Nagy J G, Leary D P. Deblurring images:matrices, spectra and filtering[J]. Journal of Electronic Imaging, 2006, 17(1):019901.
[15] Rauhut H. Stability results for random sampling of sparse trigonometric polynomials[J]. IEEE Transactions on Information Theory, 2008, 54(12):5661-5670.
[16] Aldroubi A, Grochenig K. Nonuniform sampling and reconstruction in shift-invariant spaces[J].SIAM Review, 2001, 43(4):585-620.
[17] Duval V,Peyre G. Exact support recovery for sparse spikes deconvolution[J]. Foundations of Computational Mathematics, 2015, 15(5):1315-1355.
[18] Tang G, Bhaskar B N, Recht B. Near minimax line spectral estimation[J]. IEEE Transactions on Information Theory, 2013, 61(1):499-512.
[2] Candes E J, Tao T. Decoding by linear programming[J]. IEEE Transactions on Information Theory, 2005, 51:4203-4215.
[3] Donoho D L. Superresolution via sparsity constraints[J]. SIAM Journal on Mathematical Analysis, 1992, 23(5):1309-1331.
[4] Candes E J, Fernandezgranda C. Super-resolution from noisy data[J]. Journal of Fourier Analysis and Applications, 2012, 19(6):1229-1254.
[5] Candes E J, Fernandezgranda C. Towards a mathematical theory of super-resolution[J]. Communications on Pure and Applied Mathematics, 2012, 67(6):906-956.
[6] Morgenshtern V I, Candes E J. Super-resolution of positive sources:the discrete setup[J]. SIAM Journal on Imaging Sciences, 2015, 9(1):412-444.
[7] Bendory T, Dekel S, Feuer A. Exact recovery of dirac ensembles from the projection onto spaces of spherical harmonics[J]. Constructive Approximation, 2014, 42(2):183-207.
[8] Bendory T, Dekel S, Feuer A. Exact recovery of non-uniform splines from the projection onto spaces of algebraic polynomials[J]. Journal of Approximation Theory, 2014, 182(1):7-17.
[9] Duval V, Peyre G. Exact support recovery for sparse spikes deconvolution[J]. Foundations of Computational Mathematics, 2015, 15(5):1315-1355.
[10]许志强.压缩感知[J].中国科学A辑,2012, 42(9):865-877.
[11] Bernstein B, Fernandez-Granda C. Deconvolution of point sources:a sampling theorem and robustness guarantees[J]. arXiv preprint arXiv:1707.00808, 2017.
[12] Azais J, De Castro Y, Gamboa F. Spike detection from inaccurate samplings[J]. Applied and Computational Harmonic Analysis, 2013, 38(2):177-195.
[13] Oppenheim A W, Willsky A S, Hamid S. Signals and Systems[M]. 2nd ed., Englewood Cliffs:Prentice Hall, 1996.
[14] Hansen P C, Nagy J G, Leary D P. Deblurring images:matrices, spectra and filtering[J]. Journal of Electronic Imaging, 2006, 17(1):019901.
[15] Rauhut H. Stability results for random sampling of sparse trigonometric polynomials[J]. IEEE Transactions on Information Theory, 2008, 54(12):5661-5670.
[16] Aldroubi A, Grochenig K. Nonuniform sampling and reconstruction in shift-invariant spaces[J].SIAM Review, 2001, 43(4):585-620.
[17] Duval V,Peyre G. Exact support recovery for sparse spikes deconvolution[J]. Foundations of Computational Mathematics, 2015, 15(5):1315-1355.
[18] Tang G, Bhaskar B N, Recht B. Near minimax line spectral estimation[J]. IEEE Transactions on Information Theory, 2013, 61(1):499-512.