TV Transform & Nonlinear Spectral Theory

A new framework is proposed for variational analysis and processing. It defines a functional-based nonlinear transform and inverse-transform. The framework is developed in the context of total-variation (TV), but it can be generalized to other functionals.

An eigenfunction, with respect to the subdifferential of the functional, such as a disk in the TV case, yields an impulse in the transform domain. This can be viewed as a generalization of known spectral approaches, based on linear algebra, which are extensively used in image-processing, e.g. for segmentation.

Following the Fourier intuition, a spectrum can be computed to analyze dominant scales in the image. Moreover, new nonlinear low-pass, high-pass and band-pass filters can be designed with full contrast and edge preservation.


Movie explained: Top (from left): image layers, phi bands (phi(t) = u_{tt}*t). Bottom: Ideal TV High-pass-filter, ideal TV Low-pass-filter at different scales. Ideal is in the sense of eigenfunctoin preservation.

Related papers

  1. R. Nossek, G. Gilboa, “Flows generating nonlinear eigenfunctions”, accepted to J. of Scientific Computing, 2017.
  2. G. Gilboa, “Semi-inner-products for convex functionals and their use in image decomposition”, Journal of Mathematical Imaging and Vision (JMIV), Vol. 57, No. 1, pp. 26-42, 2017.
  3. M Benning, M. Moeller, R. Nossek, M. Burger, D. Cremers, G. Gilboa, C. Schoenlieb, “Nonlinear Spectral Image Fusion”, In International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), 2017.
  4. M. Benning, G. Gilboa, J.S. Grah, C. Bibiane Schönlieb, “Learning Filter Functions in Regularisers by Minimising Quotients.” In International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), pp. 511-523. Springer, 2017.
  5. M. Burger, G. Gilboa, M. Moeller, L. Eckardt, D. Cremers, “Spectral decompositions using one-homogeneous functionals”, SIAM J. Imaging Sciences, Vol. 9, No. 3, pp. 1374-1408, 2016.
  6. M. Benning, G. Gilboa, C.B. Schönlieb, “ Learning parametrised regularisation functions via quotient minimisation”. PAMM-16, 16(1), 933-936, 2016.
  7. G. Gilboa, M. Moeller, M. Burger, “Nonlinear spectral analysis via one-homogeneous functionals – overview and future prospects”, accepted to the Journal of Mathematical Imaging and Vision (JMIV), Vol. 56, No. 2, pp 300–319, 2016.
  8. D. Horesh, G. Gilboa, “Separating surfaces for structure-texture decomposition using the TV transform”, IEEE Trans. Image Processing, Vol. 25, No. 9, pp. 4260 – 4270, 2016.
  9. G. Gilboa, “A total variation spectral framework for scale and texture analysis”. SIAM Journal on Imaging Sciences, Vol. 7, No. 4, pp. 1937–1961,  2014.
  10. G. Gilboa, “Nonlinear Band-Pass Filtering Using the TV Transform”, Proc. European Signal Processing Conference (EUSIPCO-2014), Lisbon, p. 1696 – 1700, 2014.
  11. G. Gilboa, “A spectral approach to total variation”, Scale-Space and Variational Methods”, SSVM 2013, LNCS 7893, p. 36-47, 2013.
  12. G. Gilboa, CCIT Report 833, Technion, 2013.