  Topological data analysis (TDA) is a rapidly developing collection of methods
for studying the shape of point cloud and other data types. One popular
approach, designed to be robust to noise and outliers, is to first use a
smoothing function to convert the point cloud into a manifold and then apply
persistent homology to a Morse filtration. A significant challenge is that this
smoothing process involves the choice of a parameter and persistent homology is
highly sensitive to that choice; moreover, important scale information is lost.
We propose a novel topological summary plot, called a persistence terrace, that
incorporates a wide range of smoothing parameters and is robust, multi-scale,
and parameter-free. This plot allows one to isolate distinct topological
signals that may have merged for any fixed value of the smoothing parameter,
and it also allows one to infer the size and point density of the topological
features. We illustrate our method in some simple settings where noise is a
serious issue for existing frameworks and then we apply it to a real data set
by counting muscle fibers in a cross-sectional image.
