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Point Source Super-resolution Via Non-convex \(L_1\) Based Methods

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Abstract

We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1, 2) of the Rayleigh length (physical resolution limit), \(L_1\) minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the \(L_1\) certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two \(L_1\) based nonconvex penalties, the difference of \(L_1\) and \(L_2\) norms (\(L_{1-2}\)) and capped \(L_1\) (C\(L_1\)), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global \(L_1\) solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in finding a sparse solution satisfying the constraints more accurately.

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Notes

  1. Denote \(.*\) be entry-wise multiplication, and \(|\cdot |\) be entry-wise absolute value (Note \(\Vert \cdot \Vert _1\) is the standard \(L_1\) norm).

References

  1. Aubel, C., Stotz, D., Bölcskei, H.: A theory of super-resolution from short-time fourier transform measurements. Tech. rep., arXiv preprint arXiv:1509.01047 (2015)

  2. Azais, J.M., De-Castro, Y., Gamboa, F.: Spike detection from inaccurate samplings. Appl. Comput. Harmon. Anal. 38(2), 177–195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borman, S., Stevenson, R.L.: Super-resolution from image sequences—a review. In: Midwest Symposium on Circuits and Systems, pp. 374–378 (1998)

  4. Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Candès, E.J., Fernandez-Granda, C.: Super-resolution from noisy data. J. Fourier Anal. Appl. 19(6), 1229–1254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Candès, E.J., Fernandez-Granda, C.: Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math. 67(6), 906–956 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 10(14), 707–710 (2007)

    Article  Google Scholar 

  8. Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 3869–3872 (2008)

  9. Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of \(L_{2}{-}L_{p}\) minimization. SIAM J. Sci. Comput. 32(5), 2832–2852 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. De-Castro, Y., Gamboa, F.: Exact reconstruction using Beurling minimal extrapolation. J. Math. Anal. Appl. 395(1), 336–354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demanet, L., Nguyen, N.: The recoverability limit for superresolution via sparsity. Tech. rep., arXiv preprint arXiv:1502.01385 (2015)

  12. Donoho, D.L.: Superresolution via sparsity constraints. SIAM J. Math. Anal. 23(5), 1309–1331 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Donoho, D.L., Johnstone, I.M., Hoch, J.C., Stern, A.S.: Maximum entropy and the nearly black object. J. R. Sat. Soc. Series B (Methodological), 41–81 (1992)

  15. Duval, V., Peyré, G.: Exact support recovery for sparse spikes deconvolution. Found. Comput. Math. 15(5), 1–41 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems. Springer, Berlin (2007)

    MATH  Google Scholar 

  17. Fannjiang, A., Liao, W.: Coherence pattern-guided compressive sensing with unresolved grids. SIAM J. Imaging Sci. 5(1), 179–202 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fernandez-Granda, C.: Super-resolution of point sources via convex programming. Tech. rep., arXiv preprint arXiv:1507.07034 (2015)

  19. Goodman, J.W.: Introduction to Fourier Optics. Roberts and Company Publishers, Englewood (2005)

    Google Scholar 

  20. Lai, M.J., Xu, Y., Yin, W.: Improved iteratively reweighted least squares for unconstrained smoothed lq minimization. SIAM J. Numer. Anal. 5(2), 927–957 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lou, Y., Osher, S., Xin, J.: Computational aspects of constrained L1–L2 minimization for compressive sensing. In: Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 169–180. Springer (2015)

  22. Lou, Y., Yin, P., He, Q., Xin, J.: Computing sparse representation in a highly coherent dictionary based on difference of l1 and l2. J. Sci. Comput. (2014). doi:10.1007/s10915-014-9930-1

    MathSciNet  Google Scholar 

  23. Mallat, S., Yu, G.: Super-resolution with sparse mixing estimators. IEEE Trans. Image Process. 19(11), 2889–2900 (2010)

    Article  MathSciNet  Google Scholar 

  24. Marquina, A., Osher, S.: Image super-resolution by TV-regularization and Bregman iteration. J. Sci. Comput. 37(3), 367–382 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Morgenshtern, V.I., Candès, E.J.: Super-resolution of positive sources: the discrete setup. Tech. rep., arXiv preprint arXiv:1504.00717 (2015)

  26. Park, S.C., Park, M.K., Kang, M.G.: Super-resolution image reconstruction: a technical overview. IEEE Signal Process. Mag. 20(3), 21–36 (2003)

    Article  Google Scholar 

  27. Peleg, D., Meir, R.: A bilinear formulation for vector sparsity optimization. Signal Process. 88(2), 375–389 (2008)

    Article  MATH  Google Scholar 

  28. Pham-Dinh, T., Le-Thi, H.A.: Convex analysis approach to dc programming: Theory, algorithms and applications. Acta Math. Vietnam. 22(1), 289–355 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Pham-Dinh, T., Le-Thi, H.A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pham-Dinh, T., Le-Thi, H.A.: The dc (difference of convex functions) programming and dca revisited with dc models of real world nonconvex optimization problems. Ann. Oper. Res. 133(1–4), 23–46 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Protter, M., Elad, M., Takeda, H., Milanfar, P.: Generalizing the non-local-means to super-resolution reconstruction. IEEE Trans. Image Process. 18(1), 36–51 (2009)

    Article  MathSciNet  Google Scholar 

  32. Shahram, M., Milanfar, P.: Statistical and information-theoretic analysis of resolution in imaging. IEEE Trans. Inf. Theory 8(52), 3411–3437 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, X., Pan, W., Zhu, Y.: Likelihood-based selection and sharp parameter estimation. J. Am. Stat. Assoc. 107(497), 223–232 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tang, G., Bhaskar, B.N., Recht, B.: Near minimax line spectral estimation. IEEE Trans. Inf. Theory 61(1), 499–512 (2015)

    Article  MathSciNet  Google Scholar 

  35. Tropp, J.: Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50, 2231–2242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tropp, J., Gilbert, A.: Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Z., Liu, H., Zhang, T.: Optimal computational and statistical rates of convergence for sparse nonconvex learning problems. Ann. Stat. 42(6), 2164 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xu, Z., Chang, X., Xu, F., Zhang, H.: \(L_{1/2}\) regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. 23, 1013–1027 (2012)

    Article  Google Scholar 

  39. Yang, J., Wright, J., Huang, T., Ma, Y.: Image super-resolution via sparse representation. IEEE Trans. Image Process. 19(11), 2861–2873 (2010)

    Article  MathSciNet  Google Scholar 

  40. Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of \(l_1{-}l_2\) for compressed sensing. SIAM J. Sci. Comput. 37, A536–A563 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, T.: Multi-stage convex relaxation for learning with sparse regularization. In: Advances in Neural Information Processing Systems, pp. 1929–1936 (2009)

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Acknowledgments

The authors would like to thank C. Fernandez-Granda at New York University, V. Morgenshtern at Stanford University, J.-S. Pang at University of Southern California, and M. Yan at Michigan State University for helpful discussions.

Funding Yifei Lou was partially supported by NSF Grant DMS-1522786. Penghang Yin and Jack Xin were partially supported by NSF Grants DMS-1222507 and DMS-1522383.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, 75080, USA

    Yifei Lou

  2. Department of Mathematics, UC Irvine, Irvine, CA, 92697, USA

    Penghang Yin & Jack Xin

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  1. Yifei Lou
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Correspondence to Yifei Lou.

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Lou, Y., Yin, P. & Xin, J. Point Source Super-resolution Via Non-convex \(L_1\) Based Methods. J Sci Comput 68, 1082–1100 (2016). https://doi.org/10.1007/s10915-016-0169-x

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