Abstract
In this paper we propose a variational method to adaptively decompose an image into few different modes of separate spectral bands, which are unknown before. A popular method for recursive one dimensional signal decomposition is the Empirical Mode Decomposition algorithm, introduced by Huang in the nineties. This algorithm, as well as its 2D extension, though extensively used, suffers from a lack of exact mathematical model, interpolation choice, and sensitivity to both noise and sampling. Other state-of-the-art models include synchrosqueezing, the empirical wavelet transform, and recursive variational decomposition into smooth signals and residuals. Here, we have created an entirely non-recursive 2D variational mode decomposition (2D-VMD) model, where the modes are extracted concurrently. The model looks for a number of 2D modes and their respective center frequencies, such that the bandlimited modes reproduce the input image (exactly or in least-squares sense). Preliminary results show excellent performance on both synthetic and real images. Running this algorithm on a peptide microscopy image yields accurate, timely, and autonomous segmentation - pertinent in the fields of biochemistry and nanoscience.
Keywords
- Empirical Mode Decomposition
- Synthetic Image
- Intrinsic Mode Function
- Intrinsic Mode Function
- Quadratic Penalty
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work is supported by the National Science Foundation (NSF) under grant DMS-1118971, UC Lab Fees Research grant 12-LR-236660, the Swiss National Science Foundation (SNF) grant P300P2_147778, and the W. M. Keck Foundation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bertsekas, D.P.: Multiplier methods: A survey. Automatica 12(2), 133–145 (1976)
Bertsekas, D.P.: Constrained optimization and Lagrange Multiplier methods, vol. 1. Academic Press, Boston (1982)
Bülow, T., Sommer, G.: A Novel Approach to the 2D Analytic Signal. In: Solina, F., Leonardis, A. (eds.) CAIP 1999. LNCS, vol. 1689, pp. 25–32. Springer, Heidelberg (1999)
Candes, E.J., Donoho, D.L.: Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges. In: Curve and Surface Fitting, pp. 105–120 (1999)
Claridge, S.A., Thomas, J.C., Silverman, M.A., Schwartz, J.J., Yang, Y., Wang, C., Weiss, P.S.: Differentiating Amino Acid Residues and Side Chain Orientations in Peptides Using Scanning Tunneling Microscopy. Journal of the American Chemical Society (November 2013)
Clausel, M., Oberlin, T., Perrier, V.: The Monogenic Synchrosqueezed Wavelet Transform: A tool for the Decomposition/Demodulation of AM-FM images (November 2012), http://arxiv.org/abs/1211.5082
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics 41(7), 909–996 (1988)
Daubechies, I., Lu, J., Wu, H.T.: Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Applied and Computational Harmonic Analysis 30(2), 243–261 (2011)
Do, M., Vetterli, M.: Pyramidal directional filter banks and curvelets. In: Proceedings of the 2001 International Conference on Image Processing, vol. 2, pp. 158–161. IEEE (2001)
Dragomiretskiy, K., Zosso, D.: Variational Mode Decomposition. IEEE Transactions on Signal Processing 62(3), 531–544 (2014)
Gilles, J.: Multiscale Texture Separation. Multiscale Modeling & Simulation 10(4), 1409–1427 (2012)
Gilles, J.: Empirical Wavelet Transform. IEEE Transactions on Signal Processing 61(16), 3999–4010 (2013)
Gilles, J., Tran, G., Osher, S.: 2D Empirical Transforms. Wavelets, Ridgelets, and Curvelets Revisited. SIAM Journal on Imaging Sciences 7(1), 157–186 (2014)
Guo, K., Labate, D.: Optimally Sparse Multidimensional Representation Using Shearlets. SIAM Journal on Mathematical Analysis 39(1), 298–318 (2007)
Hestenes, M.R.: Multiplier and Gradient Methods. Journal of Optimization Theory and Applications 4(5), 303–320 (1969)
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. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 454, 903–995 (1971)
Labate, D., Lim, W.Q., Kutyniok, G., Weiss, G.: Sparse Multidimensional Representation using Shearlets. In: Papadakis, M., Laine, A.F., Unser, M.A. (eds.) Optics & Photonics, pp. 1–9. International Society for Optics and Photonics (August 2005)
Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7), 674–693 (1989)
Nocedal, J., Wright, S.J.: Numerical optimization, 2nd edn. Springer, Berlin (2006)
Nunes, J., Bouaoune, Y., Delechelle, E., Niang, O., Bunel, P.: Image analysis by bidimensional empirical mode decomposition. Image and Vision Computing 21(12), 1019–1026 (2003)
Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained optimization. Mathematical Programming 5(1), 354–373 (1973)
Schmitt, J., Pustelnik, N., Borgnat, P., Flandrin, P.: 2D Hilbert-Huang Transform. In: Proc. Int. Conf. Acoust., Speech Signal Process. (2014)
Schmitt, J., Pustelnik, N., Borgnat, P., Flandrin, P., Condat, L.: 2-D Prony-Huang Transform: A New Tool for 2-D Spectral Analysis, p. 24 (April 2014), http://arxiv.org/abs/1404.7680
Lee, T.S.: Image representation using 2D Gabor wavelets. IEEE Transactions on Pattern Analysis and Machine Intelligence 18(10), 959–971 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Dragomiretskiy, K., Zosso, D. (2015). Two-Dimensional Variational Mode Decomposition. In: Tai, XC., Bae, E., Chan, T.F., Lysaker, M. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2015. Lecture Notes in Computer Science, vol 8932. Springer, Cham. https://doi.org/10.1007/978-3-319-14612-6_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-14612-6_15
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14611-9
Online ISBN: 978-3-319-14612-6
eBook Packages: Computer ScienceComputer Science (R0)