Art’s hidden topology: A window into human perception
Fig 3
Visualisations of the filtration process.
(a) ‘barcodes’ – each structure corresponds to a line segment parallel to the axis of the filtration parameter r, which begins when the structure appears (i.e., at point rb, birth) and ends when it disappears (at point rd, death), marked by two vertical lines for clarity; (b) persistence diagram – persistence is also representable via a two-dimensional scatter plot diagram with coordinates (rb, rd) representing births and deaths of the shape’s parts and holes on two orthogonal filtration axes marked by the horizontal blue lines orthogonal to respective axes. Naturally, points on this diagram occupy only the area above the main diagonal, (c) and (d) persistence landscape – connecting each of the birth/death points to the diagonal with vertical and horizontal lines, as shown for the uppermost point, green point on b, we get a system of ‘pyramids’ or isosceles right triangles. After rotating the persistence diagram by , it becomes a persistence landscape drawn separately for dim 0 – separate connected components (Fig 2c) and dim 1 – holes (Fig 2d). Persistence landscapes, in general, consist of several layers, where the n-th layer is the n-th highest value at a given point r of filtration. In (c), the top (0-th) layer is constructed from the ‘pyramid’ built on the largest barcode that lasts throughout the filtration (this pyramid is greater than all others at any value of r), the 1st and 2nd layers are constructed from the pyramids built on 2 shorter barcodes – whose corresponding ‘pyramids’ intersect, resulting in 3 layers (a full definition of layers is provided in S3 Appendix).