Stability of a Shape Transform for Curves

The Euler characteristic transform (ECT) is an operation which takes shapes as input and returns integer valued functions. The ECT is used in practice to study shapes for two reasons. First, no two well-behaved shapes have the same ECT, meaning that a shape is described entirely by its ECT. Second, it is straightforward to compare the ECTs of shapes quantitatively. It is important to verify that the ECT is stable, i.e. that changing an input shape slightly will not drastically affect the resulting ECT. We present a novel stability result for the ECTs of curves and show how our results can be leveraged to estimate the ECT of an underlying curve in the presence of noisy data.