Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

Iterative TV minimization on the graph

Pages: 941 – 968



Japhet Niyobuhungiro (Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda)

Eric Setterqvist (Faculty of Mathematics, University of Vienna, Austria)

Freddie Åström (Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany)

George Baravdish (Department of Science and Technology, Linköping University, Norrköping, Sweden)


We define the space of functions of bounded variation (BV) on the graph. Using the notion of divergence of flows on graphs, we show that the unit ball of the dual space to BV in the graph setting can be described as the image of the unit ball of the space $\ell^\infty$ by the divergence operator. Based on this result, we propose a new iterative algorithm to find the exact minimizer for the total variation (TV) denoising problem on the graph. The proposed algorithm is provable convergent and its performance on image denoising examples is compared with the Split Bregman and Primal-Dual algorithms as benchmarks for iterative methods and with BM3D as a benchmark for other state-of-the-art denoising methods. The experimental results show highly competitive empirical convergence rate and visual quality for the proposed algorithm.


total variation, ROF model on the graph, Split Bregman, Primal-dual, BM3D

2010 Mathematics Subject Classification

65K10, 68U10, 93B40

The research of Japhet Niyobuhungiro was partially supported by The World Academy of Sciences (TWAS), RGA No. 17−356 RG/MATHS/AF/AC I−FR3240297744. Eric Setterqvist acknowledges support by the Austrian Science Fund (FWF) within the national research network S117 “Geometry + Simulation”, subproject 4.

Received 8 January 2018

Accepted 28 February 2019

Published 25 October 2019