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Two-Dimensional Compact Variational Mode Decomposition

Spatially Compact and Spectrally Sparse Image Decomposition and Segmentation

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Abstract

Decomposing multidimensional signals, such as images, into spatially compact, potentially overlapping modes of essentially wavelike nature makes these components accessible for further downstream analysis. This decomposition enables space–frequency analysis, demodulation, estimation of local orientation, edge and corner detection, texture analysis, denoising, inpainting, or curvature estimation. Our model decomposes the input signal into modes with narrow Fourier bandwidth; to cope with sharp region boundaries, incompatible with narrow bandwidth, we introduce binary support functions that act as masks on the narrow-band mode for image recomposition. \(L^1\) and TV terms promote sparsity and spatial compactness. Constraining the support functions to partitions of the signal domain, we effectively get an image segmentation model based on spectral homogeneity. By coupling several submodes together with a single support function, we are able to decompose an image into several crystal grains. Our efficient algorithm is based on variable splitting and alternate direction optimization; we employ Merriman–Bence–Osher-like threshold dynamics to handle efficiently the motion by mean curvature of the support function boundaries under the sparsity promoting terms. The versatility and effectiveness of our proposed model is demonstrated on a broad variety of example images from different modalities. These demonstrations include the decomposition of images into overlapping modes with smooth or sharp boundaries, segmentation of images of crystal grains, and inpainting of damaged image regions through artifact detection.

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Notes

  1. Throughout this paper, we will be using notation pertaining to images defined over continuous domains, albeit it is implicitly understood that numerical implementations will always make use of appropriate commonplace discretization and quantization.

  2. Similarly, in higher dimensions, a half-space of the frequency domain needs to be suppressed.

  3. To be more precise, the objective is convex in \({\mathbf {\upomega }}_k\) if we consider the analytic signal construction for \(u_{AS,k}\) fixed while optimizing for \({\mathbf {\upomega }}_k\).

  4. Note that the spectrum of \(u_k\) is complex-valued, so the process of “taking the first variation” is not self-evident. However, the functional is analytic in \({\hat{u}}_k\) and complex-valued equivalents to the standard derivatives do indeed apply.

  5. Again we omit iteration superscripts for \(A_i\), but it is understood that we always use the most recent estimate of a variable, \(A_i^{t+1}\) for \(i<k\) and \(A_i^t\) for \(i>k\).

  6. Here, t is understood as an artificial time introduced for the sole purpose of differential equation notation, but quantized into the discrete iterates of the scheme.

  7. Of course, the simpler 2D-VMD model only uses a subset of these parameters, for the support functions are fixed at \(A_k=1\) uniformly.

  8. MATLAB code available at http://bigwww.epfl.ch/demo/steerable-wavelets/.

  9. Obfuscated MATLAB p-code available at http://perso.ens-lyon.fr/nelly.pustelnik/Software/Toolbox_PHT_2D_v1.0.zip.

  10. Lower artifact threshold \(\delta \) and higher TV weight \({\gamma }_k\) might increase the mode cleanliness even further.

  11. Image used with permission, courtesy by Richard Wheeler, Sir William Dunn School of Pathology, University of Oxford, UK.

  12. Ibid.

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Acknowledgements

The authors are grateful to Jérôme Gilles for providing the synthetic texture image for Fig. 1; to Diana Yugay for providing peptide \(\beta \)-sheet images used in Figs. 6 and 7; and to Richard Wheeler for permitting the use of the colloidal crystal image of Fig. 12.

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  1. Dominique Zosso

    Present address: Department of Mathematical Sciences, Montana State University, 2-214 Wilson Hall, Box 172400, Bozeman, MT, 59717-2400, USA

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  1. Department of Mathematics, University of California, Los Angeles (UCLA), 520 Portola Plaza, Box 951555, Los Angeles, CA, 90095-1555, USA

    Dominique Zosso, Konstantin Dragomiretskiy & Andrea L. Bertozzi

  2. California NanoSystems Institute (CNSI), Department of Chemistry and Biochemistry, Department of Materials Science and Engineering, University of California, Los Angeles (UCLA), 570 Westwood Plaza, Building 114, Los Angeles, CA, 90095, USA

    Paul S. Weiss

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Correspondence to Dominique Zosso.

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This work was supported by the Swiss National Science Foundation (SNF) under Grants PBELP2-137727 and P300P2-147778, the UC Lab Fees Research Grant 12-LR-236660, and the W. M. Keck Foundation.

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Zosso, D., Dragomiretskiy, K., Bertozzi, A.L. et al. Two-Dimensional Compact Variational Mode Decomposition. J Math Imaging Vis 58, 294–320 (2017). https://doi.org/10.1007/s10851-017-0710-z

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