Figure 2
Cubical levelset filtration of a greyscale image. (a) Example greyscale image with pixel intensity values between 0 (black) and 1 (white), as in the colour bar. (b) Some ordered steps of the cubical levelset filtration of the image in (a). As the threshold parameter \(\delta\) increases from 0 to 1, pixels (of intensity lower than \(\delta\)) are included as points (0-cubes), with two adjacent pixels joined by edges (1-cubes), and 4 adjacent pixels joined by squares (2-cubes). Note the ring we see in the image is captured by the third step of the filtration shown. (c) Persistence diagram for \(H_0\) and \(H_1\) of the image (a) with respect to the levelset filtration described in (b). The blue dots (\(H_0\)) correspond to connected pieces, which ‘die’ as they join one another. The final connected piece never ‘dies’, so has infinite persistence (marked as ‘\(\infty\)’). The green diamond (\(H_1\)) corresponds to a loop (b, panel 3) capturing the ring structure in the image (a). This loop disappears (‘dies’) as we add more pixels (b, panel 4).
