  Non-linear spectral decompositions of images based on one-homogeneous
functionals such as total variation have gained considerable attention in the
last few years. Due to their ability to extract spectral components
corresponding to objects of different size and contrast, such decompositions
enable filtering, feature transfer, image fusion and other applications.
However, obtaining this decomposition involves solving multiple non-smooth
optimisation problems and is therefore computationally highly intensive. In
this paper, we present a neural network approximation of a non-linear spectral
decomposition. We report up to four orders of magnitude ($\times 10,000$)
speedup in processing of mega-pixel size images, compared to classical GPU
implementations. Our proposed network, TVSpecNET, is able to implicitly learn
the underlying PDE and, despite being entirely data driven, inherits
invariances of the model based transform. To the best of our knowledge, this is
the first approach towards learning a non-linear spectral decomposition of
images. Not only do we gain a staggering computational advantage, but this
approach can also be seen as a step towards studying neural networks that can
decompose an image into spectral components defined by a user rather than a
handcrafted functional.
