In this work, we develop a new method to detect slow-moving landslides from a
time series of displacement maps, coming either from optical image correlation or In-
SAR. The developped method is based on the sparsity charcateristics of the landslide
signal in space. We suggest to use the most recent techniques in optimization, such
as the Monotone+Lipschitz Forward-Backward-Forward (M+L FBF) algorithm [1],
using specific constraints from slow landslide, like their monotonic aspect through
time. This approach leads to a mathematical formulation of the problem. The effi-
ciency of this approach is demonstrated through comparison with a field inventory
of slow landslides collected on the Colca valley in Peru. A database of about 30 land-
slides is thus realized over an area of 300 km2, using a time-series of displacement
over 27 years (1986-2013) generated by time-series correlation of SPOT1-5/Pléiades
images. The results shows that all important landslides, already known from field
investigations or previous research, are detected. The detection also include more
than 250 % other small and unknown landslides. We finally analyze the displace-
ment time-series from the different landslides in terms of natural forcings.