Persistence itself is a relatively simple but powerful concept. To study the topology of a function, one can measure how the topology of its excursion sets (i.e. the set of points with value higher than a given threshold) evolves when the threshold is continuously and monotonically changing. Whenever the threshold crosses the value of a critical point, the topology of the excursion change. Supposing that the threshold is sweeping the values of a 1D function from high to low, whenever it crosses the value of a maximum, a new component appears in the excursion, while two components merge (i.e. one is destroyed) whenever the threshold crosses the value of a minimum. This concept can be extended to higher dimensions (i.e. creation/destruction of hole, spherical shells, ....) and in general, whenever a topological component is created at a critical point, the critical point is labeled positive, while it is labeled negative if it destroys a topological component. Using this definition, topological components of a function can be represented by pairs of positive and negative critical points called persistence pairs. The absolute difference of the value of the critical points in a pair is called its persistence : it represents the lifetime of the corresponding topological component within the excursion set.

Figure 1: Persistence pairs

Upper part: Two functions with identical topology (i.e. two peaks). Lower part: Changes in topology of excursion sets (sets of point with value lower than a decreasing threshold, see main text above). In 1D, new components are created at maxima and destroyed at minima. Two persistence pairs represented by green lines account for the two peaks in each function. The length of the green lines corresponds to the persistence of the pairs, which correctly accounts for the the fact that A has a small bump on the top of a peak (one high and one low persistence pair), while B has two peaks (two high persistence pairs).

The concept of persistence is powerful because it yields a simple way to measure how robust topological components are to local modifications of a function values. Indeed, noise can only affect a function's topology by creating or destroying topological components of persistence lower that its local amplitude. Therefore, it suffice to know the amplitude of noise to decide which components certainly belong to an underlying function and which may have been affected (i.e. created or destroyed) by noise. In DisPerSE, a persistence threshold can be specified (see options -nsig and -cut of mse) to remove topological components with persistence lower than the threshold and therefore filter noise from the Morse-Smale complex (see figure 2 below).

Figure 1: Morse-smale complexsimplification

Simplification of a low persistence pair (formed by maximum 19 and saddle point 18). Disks represent critical points (minima, saddle-points and maxima in blue, green and red respectively) and black lines show the arcs of the Morse-Smale complex.

A very useful way to set the persistence threshold is to plot a persistence diagram, in which all the persistence pairs are represented by points with coordinates the value at the critical points in the pair (see option -interactive of mse or see pdview and read the tutorial section to learn how to compute and use persistence diagrams).