  This paper presents a new algorithm for the lossy compression of scalar data
defined on 2D or 3D regular grids, with topological control. Certain techniques
allow users to control the pointwise error induced by the compression. However,
in many scenarios it is desirable to control in a similar way the preservation
of higher-level notions, such as topological features , in order to provide
guarantees on the outcome of post-hoc data analyses. This paper presents the
first compression technique for scalar data which supports a strictly
controlled loss of topological features. It provides users with specific
guarantees both on the preservation of the important features and on the size
of the smaller features destroyed during compression. In particular, we present
a simple compression strategy based on a topologically adaptive quantization of
the range. Our algorithm provides strong guarantees on the bottleneck distance
between persistence diagrams of the input and decompressed data, specifically
those associated with extrema. A simple extension of our strategy additionally
enables a control on the pointwise error. We also show how to combine our
approach with state-of-the-art compressors, to further improve the geometrical
reconstruction. Extensive experiments, for comparable compression rates,
demonstrate the superiority of our algorithm in terms of the preservation of
topological features. We show the utility of our approach by illustrating the
compatibility between the output of post-hoc topological data analysis
pipelines, executed on the input and decompressed data, for simulated or
acquired data sets. We also provide a lightweight VTK-based C++ implementation
of our approach for reproduction purposes.
