  The total variation (TV) flow generates a scale-space representation of an
image based on the TV functional. This gradient flow observes desirable
features for images, such as sharp edges and enables spectral, scale, and
texture analysis. Solving the TV flow is challenging; one reason is the the
non-uniqueness of the subgradients. The standard numerical approach for TV flow
requires solving multiple non-smooth optimisation problems. Even with
state-of-the-art convex optimisation techniques, this is often prohibitively
expensive and strongly motivates the use of alternative, faster approaches.
Inspired by and extending the framework of physics-informed neural networks
(PINNs), we propose the TVflowNET, an unsupervised neural network approach, to
approximate the solution of the TV flow given an initial image and a time
instance. The TVflowNET requires no ground truth data but rather makes use of
the PDE for optimisation of the network parameters. We circumvent the
challenges related to the non-uniqueness of the subgradients by additionally
learning the related diffusivity term. Our approach significantly speeds up the
computation time and we show that the TVflowNET approximates the TV flow
solution with high fidelity for different image sizes and image types.
Additionally, we give a full comparison of different network architecture
designs as well as training regimes to underscore the effectiveness of our
approach.
