Confidence regions for a persistence diagram of a single image with one or more loops

Figure 1.  (A) Grey scale image of a protein ring formed around a wound where there was no toxin (control) injected into the cell. (B) Grey scale image of a protein ring formed around a wound where a toxin (C3) was injected into the cell

Figure 2.  (A) Partitions of an underlying simulated pattern into background $ \mu_0 $, loop $ \mu_1 $, and interior of the loop $ \mu_{1*} $. (B) Partitions of the data image (observed) into into background, the loop, and the interior of the loop. Each $ Z^k $ denotes pixel intensity value for partition $ k = \{0,1,1*\} $ and each $ F_k $ is a distribution from which the pixel was drawn. (C) The simplicial complex $ \mathcal{K}_{3778} $ built on $ (\mathcal{M}^{\sigma})^{[3778,\infty)} $ contains one connected component and five loops; 3778 is the birth time of the true loop (loop number 1). (D) The simplicial complex $ \mathcal{K}_{3000} $ built on $ (\mathcal{M}^{\sigma})^{[3000,\infty)} $ contains five connected components and six loops. (E) The simplicial complex $ \mathcal{K}_{2512} $ built on $ (\mathcal{M}^{\sigma})^{[2512,\infty)} $ contains two connected components and no loops, where 2512 is the death time of the large loop born at $ \mathcal{K}_{3778} $

Figure 3.  (A) The persistence diagram of the underlying pattern in Figure 2A which has only one loop (red triangle) and one connected component (black dots). (B) The persistence diagram of the data in Figure 2B with loops (red triangles) and connected components (black dots)

Figure 4.  Illustration of simplicial complexes (purple) built on the upper-level sets at the birth time and death time of an image with a loop, with $ \mu_1 = 4000 $ and $ \mu_{1*} = 3000 $. (A) The simplicial complex $ \mathcal{K}_{b_j} = \mathcal{K}_{3597} $ at the birth of the loop where the black dot is the pixel with intensity value equal to 3597, which is the upper-level set threshold associated with the birth of the loop. The white rectangles indicate the pixels of $ \mathcal{M}^{\sigma}_1 $. (B) The simplicial complex $ \mathcal{K}_{d_j} = \mathcal{K}_{2593} $ at the death of the loop where the black dot is the pixel with intensity value equal to 2593, which is the upper-level set threshold associated with the death of the loop. The white rectangles, which indicate the pixels of $ \mathcal{M}^{\sigma}_{1*} $, appear light purple due to the overlaying two-simplices in $ \mathcal{K}_{d_j} $

Figure 5.  Boxplots illustrating estimated birth (A) and death (B) times of loops using parTDA (red) and tTDA (blue), based on 100 iid images within each factor Level of loop thickness or dimensions. The true death and birth times are indicated by the horizontal solid black lines. The tTDA estimates have a strong negative bias with higher variability, while the proposed parTDA estimates appear to be unbiased with lower variability

Figure 6.  Confidence regions and coverage before (misclassified) and after (corrected) Algorithm 2 has been applied. The misclassified segmentation $ \hat e $ has six pixels incorrectly classified. (A) Confidence regions for 100 images at noise level $ \sigma = \{50,100,300 \} $ are shown using $ \hat e $ (misclassified) and $ \hat e_{\text{new}} $ (corrected). The green dots indicate the true death and birth time location. (B) The coverage of the 95% confidence regions for $ \sigma = \{10, 50,100,200,300 \} $ for misclassified (red) and corrected (blue) segmentations, using 100 iid images

Figure 7.  death and birth estimates (A) and confidence regions (B) of 100 images across four noise levels, $ \sigma = \{50,150,250,350 \} $. (A) Point estimates for $ (\mu_{1*}, \mu_1) $ using tTDA (triangle) and parTDA (asterisk), estimates of $ (\tilde \mu_{1*}, \tilde \mu_1) $ using sTDA (plus), and the true death and birth time of the manifold (black circle). (B) The $ 95 $% confidence regions for $ (\mu_{1*}, \mu_1) $ using parTDA and $ (\tilde \mu_{1*}, \tilde \mu_1) $ using sTDA

Figure 8.  (A) An example of simulated images with multiple loops. (B) The confidence regions for the first two most persistent loops (blue and red) where the green points are the death and birth times of the true loops

Figure 9.  (A) The top row displays the images for the control cell $ \mathcal{M}^{\text{control}}_t $ and the bottom row is the images for the C3 cell $ \mathcal{M}^{\text{C3}}_t $. The columns represent different time points, $ t_1 $, $ t_{10} $, $ t_{20} $, and $ t_{30} $. (B) Image of $ \mathcal{M}^{\text{C3}}_{t_{15}} $ segmented by $ e^{\text{C3}}_{15} $ where the black lines are the edges

Figure 10.  Estimated persistences of the C3 and Control cell images from $ t = \{t_8,\ldots,t_{28}\} $. (A) The parTDA death and birth times are shown on the persistence diagram along with confidence regions for both the C3 cell (right) and the Control cell (left). The black line is the diagonal line $ b = d $. (B) Persistence is plotted over time for the C3 cell (solid line with points) and the Control cell (dashed line with triangles) using parTDA (red), sTDA (purple), and tTDA (light blue)