Variational Regularisation for Inverse Problems

In this project, we introduce a new higher-order total directional variation (TDV) regulariser for inverse imaging problems by taking into account the image gradient weighted by the structural content.

We generalise previous work on the directional total variation, e.g. the definitions (DTV, wTV, dTV) given in Ehrhardt and Betcke (2016) and in Kongsgov and Dong (2017), by considering spatially dependent directions within the image domain and higher-order derivatives. Also, we extend previous work on total generalised variation (TGV) regulariser, see Bredies, Kunish and Pock (2010), Bredies (2013) and Bredies and Holler (2014), by showing that TGV is a special case of our regulariser.

We describe both the theoretical and the numerical details for its use within a variational formulation for solving inverse problems and give examples for the reconstruction of noisy images and videos, image zooming and the interpolation of scattered surface heigh value data from Digital Elevation Maps (DEM) and Atomic Force Microscopy (AFM).

Image Denoising

Image Zooming (4x)

Scattered Data Interpolation (DEM)

Scattered Data Interpolation (AFM)

Journal Articles

Parisotto, Simone; Masnou, Simon; Schönlieb, Carola-Bibiane: Higher-order Total Directional Variation: Analysis. In: SIAM Journal on Imaging Sciences, 13 (1), pp. 474–496, 2020. (Type: Journal Article | Abstract | Links)
Parisotto, Simone; Calatroni, Luca; Caliari, Marco; Schönlieb, Carola-Bibiane; Weickert, Joachim: Anisotropic osmosis filtering for shadow removal in images. In: Inverse Problems, 35 (5), 2019. (Type: Journal Article | Abstract | Links)
Parisotto, Simone; Lellmann, Jan; Masnou, Simon; Schönlieb, Carola-Bibiane: Higher-order total directional variation: Imaging Applications. In: (submitted, ArXiv: 1812.05023), 2019. (Type: Journal Article | Abstract | Links)
Burger, Martin; Korolev, Yury; Parisotto, Simone; Schönlieb, Carola-Bibiane: Total Variation Regularisation with Spatially Variable Lipschitz Constraints. In: (submitted, ArXiv: 1912.02768), 2019. (Type: Journal Article | Abstract | Links)

Inproceedings

Parisotto, Simone; Schönlieb, Carola-Bibiane: Total Directional Variation for Video Denoising. In: Lellmann J. Burger M., Modersitzki J (Ed.): Scale Space and Variational Methods in Computer Vision (SSVM 2019), Lecture Notes in Computer Science, Springer, 2019. (Type: Inproceedings | Abstract | Links)

PhD Theses

Parisotto, Simone: Anisotropic variational models and PDEs for inverse imaging problems. University of Cambridge, 2019. (Type: PhD Thesis | Abstract | Links)