Robust crossings detection in noisy signals using topological signal processing

Figure 1.  An example point cloud in $ \mathbb{R} $ is given at the top of the figure. The persistence diagram points $ (a_i-a_{i-1}) $ are given at the bottom left, drawn at the location of the $ a_{i-1} $ value. Note that high-value points in this diagram occur at splits in the point cloud. Then at right, we show the histogram of sorted persistence points based on their death-time

Figure 2.  A pulse signal showing the $ p $ and $ q $ notation. The $ P $ points are those on the top row; the $ Q $ on the bottom. For a threshold $ \mu = 3\Delta t $, we see entries $ (p_{i-1},p_i) $ from $ \operatorname{dgm} P $ and $ (q_{j-1},q_j) $ from $ \operatorname{dgm} Q $

Figure 3.  An example using $ x(t) = \sin (t)+\sin (10 / 3 t) $. The intervals returned by the algorithm are $ [r_1,r_2] $ and $ [r_5,r_6] $ (without a uncertainty flag); and $ [r_3,r_4] $ and $ [r_7,r_8] $ (with an uncertainty flag). On the right, the persistence histogram is shown. The threshold was chosen to be $ \mu = 0.4 $, but different choices of $ \mu $ will result in different zero-crossing intervals

Figure 4.  Sorted persistence points for $ x_{9} $ (using SNR $ = 15 $ dB) with statistical measures for outlier detection plotted. Here, $ f $ is the sampling frequency

Figure 5.  Application of Isolation Forest on sorted persistence points of $ x_{4} $. Here, $ f $ is the sampling frequency

Figure 6.  True roots versus brackets returned at a sampling frequency of 25 Hz for (a) $ x_{2} $, (b) $ x_{5} $, (c) $ x_{11} $ and (d) $ x_{12} $. The legend in (d) applies to all subfigures

Figure 7.  True roots versus estimated roots returned at a sampling frequency of 1000 Hz for $ x_{8} $ (left) and $ x_{14} $ (right) at low, medium and high SNR. The legend in (f) applies to all subfigures

Figure 8.  True roots versus estimated roots for $ x_{2} $ at a sampling frequency of 1000 Hz. Part (a) shows the algorithm missing a root due to high noise. The legend in (b) applies to both subfigures

Figure 9.  Double-pendulum and zero-crossings for experimental data in Fig. 13 by Myers et al. [21]

Figure 10.  Relative errors at low, medium and high SNR from both algorithms at $ f = 500 $ Hz for the first roots in the interval

Figure 11.  Relative error and time taken for convergence by Molinaro's algorithm (left in each) and 0-D Persistence (right in each) for $ x_{12} $. Relative errors are for the first roots in the interval

Figure 12.  (a) 2D signal from function $ z = -x^2 + 1 $ with zero-crossings expected at $ x = 1 $ and $ x = -1 $. (b) Binarization of the input signal into $ P $ and $ Q $ (corresponds to Step 1 in Algorithm 1). (c) Persistence diagrams for $ P $ and $ Q $ point clouds (replaces the calculation of $ \Delta p_i $ and $ \Delta q_i $ in Step 3 of Algorithm 1). (d) Extraction of high 0-D persistence locations into sets $ I_P $ and $ I_Q $ (equivalent to Step 3 in Algorithm 1). (e) Interleaving of the thresholded sets to give set $ R $ (equivalent to Step 4 and 5 in Algorithm 1). (f) Returned brackets bounding $ x \approx 1 $ and $ x \approx -1 $