Convex Optimization approach to signals with fast varying instantaneous frequency

Applied and Computational Harmonic Analysis

Volumn 44, Issue 1, 2018, Pages 89-122

  Kowalski, Matthieu ^a,b   Meynard, Adrien ^a   Wu, Hau tieng ^c  

^a UNIV PARIS SUD   (France)

^b CEA SACLAY   (France)

^c UNIVERSITY OF TORONTO   (Canada)

Author keywords

Chirp factor; Convex optimization; FISTA; Instantaneous frequency; Time frequency analysis

Indexed keywords

APPROXIMATION ALGORITHMS; CONVEX OPTIMIZATION; ITERATIVE METHODS; TIME VARYING NETWORKS;

CHIRP FACTORS; FISTA; INSTANTANEOUS FREQUENCY; ITERATIVE SHRINKAGE-THRESHOLDING ALGORITHMS; NUMERICAL APPROXIMATIONS; OPTIMIZATION APPROACH; TIME FREQUENCY ANALYSIS; TIME-FREQUENCY REPRESENTATIONS;

OPTIMIZATION;

EID: 84975166544     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2016.03.008     Document Type: Article

References (54)

1. Auger, F., Chassande-Mottin, E., Flandrin, P., Making reassignment adjustable: the Levenberg-Marquardt approach. International Conference on Acoustics, Speech and Signal Processing, ICASSP, March 2012, 2012, IEEE, 3889–3892.
2. Auger, F., Flandrin, P., Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Trans. Signal Process. 43:5 (May 1995), 1068–1089.
3. Balazs, P., Dörfler, M., Jaillet, F., Holighaus, N., Velasco, G., Theory, implementation and applications of nonstationary Gabor frames. J. Comput. Appl. Math. 236:6 (2011), 1481–1496.
4. Beck, A., Teboulle, M., Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18:11 (2009), 2419–2434.
5. Beck, A., Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2:1 (2009), 183–202.
6. Bolte, J., Sabach, S., Teboulle, M., Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146:1–2 (2014), 459–494.
7. A. Chambolle, C. Dossal, On the convergence of the iterates of FISTA, Preprint hal-01060130, September 2014.
8. Chassande-Mottin, E., Auger, F., Flandrin, P., Time-frequency/time-scale reassignment. Wavelets and Signal Processing Appl. Numer. Harmon. Anal., vol. 10, 2003, Birkhäuser Boston, Boston, MA, 233–267.
9. Chen, Y.-C., Cheng, M.-Y., Wu, H.-T., Nonparametric and adaptive modeling of dynamic seasonality and trend with heteroscedastic and dependent errors. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 (2014), 651–682.
10. Chui, C.K., Lin, Y.-T., Wu, H.-T., Real-time dynamics acquisition from irregular samples – with application to anesthesia evaluation. Anal. Appl., 2015, 10.1142/S0219530515500165 accepted for publication.
11. Chui, C.K., Mhaskar, H.N., Signal decomposition and analysis via extraction of frequencies. Appl. Comput. Harmon. Anal. 40:1 (2016), 97–1360.
12. Cicone, A., Liu, J., Zhou, H., Adaptive local iterative filtering for signal decomposition and instantaneous frequency analysis. preprint arXiv:1411.6051, 2014.
13. Cicone, A., Zhou, H., Multidimensional iterative filtering method for the decomposition of high-dimensional non-stationary signals. preprint arXiv:1507.07173, 2015.
14. Coifman, R.R., Steinerberger, S., Nonlinear phase unwinding of functions. J. Fourier Anal. Appl., 2016 in revision.
15. Combettes, P.L., Wajs, V.R., Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4:4 (2005), 1168–1200.
16. Daubechies, I., Lu, J., Wu, H.-T., Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Appl. Comput. Harmon. Anal. 30 (2011), 243–261.
17. Daubechies, I., Maes, S., A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models. Wavelets in Medicine and Biology, 1996, 527–546.
18. Daubechies, I., Wang, Y., Wu, H.-T., ConceFT: Concentration of frequency and time via a multitapered synchrosqueezing transform. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 374(2065), 2016, 10.1098/rsta.2015.0193 accepted for publication, 2015.
19. De Livera, A.M., Hyndman, R.J., Snyder, R.D., Forecasting time series with complex seasonal patterns using exponential smoothing. J. Amer. Statist. Assoc. 106:496 (2011), 1513–1527.
20. Deledalle, C., Vaiter, S., Peyré, G., Fadili, J., Dossal, C., Proximal splitting derivatives for risk estimation. J. Phys.: Conf. Ser., 386(1), 2012, 012003.
21. Dragomiretskiy, K., Zosso, D., Variational mode decomposition. IEEE Trans. Signal Process. 62:2 (2014), 531–544.
22. Flandrin, P., Time-Frequency/Time-Scale Analysis. Wavelet Analysis and its Applications, vol. 10, 1999, Academic Press Inc.
23. Flandrin, P., Time frequency and chirps. Proc. SPIE, vol. 4391, 2001, 161–175.
24. Galiano, G., Velasco, J., On a non-local spectrogram for denoising one-dimensional signals. Appl. Math. Comput. 244 (2014), 1–13.
25. Gilles, J., Empirical wavelet transform. IEEE Trans. Signal Process. 61:16 (2013), 3999–4010.
26. 1-minimization: methodology and convergence. SIAM J. Optim. 19:3 (2008), 1107–1130.
27. Hou, T., Shi, Z., Data-driven time-frequency analysis. Appl. Comput. Harmon. Anal. 35:2 (2013), 284–308.
28. Hou, T., Shi, Z., Sparse time-frequency representation of nonlinear and nonstationary data. Sci. China Math. 56:12 (2013), 2489–2506.
29. Hou, T.Y., Shi, Z., Adaptive data analysis via sparse time-frequency representation. Adv. Adapt. Data Anal. 3:1/2 (2011), 1–28.
30. Huang, C., Wang, Y., Yang, L., Convergence of a convolution-filtering-based algorithm for empirical mode decomposition. Adv. Adapt. Data Anal. 1:4 (2009), 561–571.
31. Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.-C., Tung, C.C., Liu, H.H., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 454:1971 (1998), 903–995.
32. Huang, Z., Zhang, J., Zhao, T., Sun, Y., Synchrosqueezing s-transform and its application in seismic spectral decomposition. IEEE Trans. Geosci. Remote Sens. PP:99 (2015), 1–9.
33. Jahangir, A., Lee, V., Friedman, P.A., Trusty, J.M., Hodge, D.O., et al. Long-term progression and outcomes with aging in patients with lone atrial fibrillation: a 30-year follow-up study. Circulation 115 (2007), 3050–3056.
34. Kodera, K., Gendrin, R., Villedary, C., Analysis of time-varying signals with small BT values. IEEE Trans. Acoust. Speech Signal Process. 26:1 (Feb 1978), 64–76.
35. Li, C., Liang, M., A generalized synchrosqueezing transform for enhancing signal time-frequency representation. Signal Process. 92:9 (2012), 2264–2274.
36. Li, P.-C., Sheu, Y.-L., Laughlin, C., Chu, S.-I., Dynamical origin of near- and below-threshold harmonic generation of Cs in an intense mid-infrared laser field. Nat. Commun., 6, 2015.
37. Lin, L., Wang, Y., Zhou, H., Iterative filtering as an alternative for empirical mode decomposition. Adv. Adapt. Data Anal. 1:4 (2009), 543–560.
38. Liu, C., Hou, T.Y., Shi, Z., On the uniqueness of sparse time-frequency representation of multiscale data. Multiscale Model. Simul. 13:3 (2015), 790–811.
39. Loris, Ignace, On the performance of algorithms for the minimization of 1-penalized functionals. Inverse Probl., 25(3), 2009, 035008.
40. Mann, S., Haykin, S., The chirplet transform: physical considerations. IEEE Trans. Signal Process. 43:11 (1995), 2745–2761.
41. Morozov, V.A., On the solution of functional equations by the method of regularization. Sov. Math., Dokl. 7:1 (1966), 414–417.
42. Oberlin, T., Meignen, S., Perrier, V., An alternative formulation for the empirical mode decomposition. IEEE Trans. Signal Process. 60:5 (2012), 2236–2246.
43. Oberlin, T., Meignen, S., Perrier, V., Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations. IEEE Trans. Signal Process. 63:5 (March 2015), 1335–1344.
44. Pustelnik, N., Borgnat, P., Flandrin, P., Empirical mode decomposition revisited by multicomponent non-smooth convex optimization. Signal Process. 102:0 (2014), 313–331.
45. Ricaud, B., Stempfel, G., Torrésani, B., An optimally concentrated Gabor transform for localized time-frequency components. Adv. Comput. Math. 40 (2014), 683–702.
46. Sheu, Y.-L., Wu, H.-T., Hsu, L.-Y., Exploring laser-driven quantum phenomena from a time-frequency analysis perspective: a comprehensive study. Opt. Express 23 (2015), 30459–30482.
47. Stockwell, R.G., Mansinha, L., Lowe, R.P., Localization of the complex spectrum: the S transform. IEEE Trans. Signal Process. 44:4 (1996), 998–1001.
48. Tavallali, P., Hou, T., Shi, Z., Extraction of intrawave signals using the sparse time-frequency representation method. Multiscale Model. Simul. 12:4 (2014), 1458–1493.
49. Tseng, P., Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Program. 125:2 (2010), 263–295.
50. Villanic, C., Topics in Optimal Transportation. Graduate Studies in Mathematics, 2003, American Mathematical Society.
51. Wu, H.-T., Instantaneous frequency and wave shape functions (I). Appl. Comput. Harmon. Anal. 35 (2013), 181–199.
52. Wu, Z., Huang, N.E., Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 1 (2009), 1–41.
53. Yang, H., Synchrosqueezed wave packet transforms and diffeomorphism based spectral analysis for 1D general mode decompositions. Appl. Comput. Harmon. Anal. 39 (2014), 33–66.
54. Zangwill, W.I., Nonlinear Programming: A Unified Approach, vol. 196. 1969, Prentice-Hall, Englewood Cliffs, NJ.