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Euler characteristic surfaces

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  • In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily in the context of random fields. The goal of this paper, is to present the extension of using the Euler characteristic in higher dimensional parameter spaces. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.

    Mathematics Subject Classification: Primary: 62R40, 68T09; Secondary: 55N31.

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  • Figure 1.  From left to right: $ 3\times 3 $ gray-scale image, full cubical complex $ Q $ of the image in a), finite point set in $ \mathbb{R}^2 $, and Delaunay complex $ D $ of the points in (c)

    Figure 2.  Filtrations of the example data in Figure 1, for two given set of values $ \{ a_s \}_{s = 1}^m $. In both (a) and (b) the filtration parameters $ a_s $ are displayed above each subcomplex. The last subcomplexes in the sequence are the full cubical complex $ Q $ and the Delaunay complex $ D $

    Figure 3.  The plot in (a) is the Euler characteristic curve of the image in Figure 1a, where sublevel sets are taken at all values between $ 0 $ and $ 255 $. The plot in (b) is the Euler characteristic curve of the points in Figure 1c, where the sublevel sets are taken at values $ a_s $ such that there is a one simplex difference between $ Q_{a_{i-1}} $ and $ Q_{a_s} $ for each $ i \in [1, m] $

    Figure 4.  Contour plot of $ S_{(h_{M_1}, h_{M_2})} $ of a pair of random images $ M_1 $, $ M_2 $ with correlation coefficient equal to $ 0.1 $ in (a) and equal to $ 0.8 $ in (b). The difference between these two Euler surfaces is the Euler terrain in (c)

    Figure 5.  The realtionship between the ECT and the ECS for functions on $ \mathbb{R}^2 $. For every direction $ \alpha $, the ECT computes the Euler characteristic curve based on the filtration arising from the height function in that direction. In the standard setup, it would be a direction on the sphere $ \mathbb{S}^2 $, but it could equivalently be all "directions'' in $ \mathbb{S}^1\times \mathbb{R} $. The 2-dimensional Euler surface we consider is a direction and the function sublevel sets, so the Euler surface can be thought of as representing the slice shown

    Figure 6.  Euler characteristic changes produced by adding an elementary cube of maximal dimension in a two-dimensional cubical complex $ Q_{s-1, t} $. In (a) the change is equal to $ -3 $, while in (b) it is $ +1 $

    Figure 7.  The images in (a) are eight of the twenty-four patterns in the $ \texttt{OUTEX_TC_00000} $ test suite. The images in (b) are a random selection of the $ 70,000 $ handwritten digits from the MNIST database

    Figure 8.  Points sampled from two Clayton copula distributions. In (a) $ \theta = 1 $, while in (b) $ \theta = 5 $

    Figure 9.  The average Euler characteristic surfaces of random images generated with random points as in Figure 8. Each point represents the function values of a voxel (in a 3D image) generated from Clayton copula distributions to have uniform marginal distributions but different joint distributions. In (a), the surface was obtained with $ \theta = 1 $, and in (b) with $ \theta = 5 $. The absolute value of the difference of the surfaces in (a) and (b) is in (c)

    Figure 10.  Point processes, corresponding to a Poisson point process in (a) and Hawkes cluster process in (b)

    Figure 11.  Average Euler characteristic surfaces obtained from the point processes in Figure 10. In (a) and (b) the average surfaces of points obtained from a Poisson and Hawkes cluster process respectively. In (c) the Euler terrain representing their difference. In (d) the black area represent regions of the parameter space where the two average surfaces in (a) and (b) are significantly different

    Figure 12.  Examples of the complex built at the radius and density corresponding to Region A in Figure 11(c) for a cluster process (a, b) and the Poisson process (c, d)

    Figure 13.  OCTA scans from Control, DR and NoDR patients. Changes to the microvasculature are apparent with disease progression. For example, the vessel density reduces and the foveal avascular zone (FAZ), which is the black regions in the middles of the image, is enlarged and distorted with less circular shape

    Figure 14.  Example of the ECS with $ M_2 $ being the complement image

    Figure 15.  Example of the ECS with $ M_2 $ being the radial gradient image

    Figure 16.  Examples of regions of interest in the terrains of OCTAGON

    Figure 17.  Examples of level-sets of images of the control (a, b) and DR (c, d) corresponding to Region B

    Figure 18.  Examples of sublevel sets with the radial mask shown in yellow with the control shown in (a, b) and DR (c, d) corresponding to Region A

    Table 1.  Classification results obtained with Euler characteristic based feature vectors and logistic regression

    $ \texttt{Outex_TC_00000} $
    Features Avg test accuracy
    Euler curve - pixel intensity $ 91.08\pm 1.57 $ %
    Euler surface - pixel intensity, Laplacian $ \mathbf{96.29}\pm 1.24 $ %
    MNIST
    Features Test accuracy
    Euler curve - pixel intensity $ 33.63 $ %
    Euler surface - pixel intensity, top/bottom gradient $ \mathbf{71.79} $ %
     | Show Table
    DownLoad: CSV

    Table 2.  Tables of classification performances in the Control vs. DR study

    NHS Lothian, Control vs. DR
    Baseline Our approach VGG16
    Overall Acc $0.72 \pm 0.03$ $0.81 \pm 0.04$ 0.84 $\pm$ 0.07
    Sen (Control) 0.94 $\pm$ 0.05 0.94 $\pm$ 0.05 $0.88 \pm 0.07$
    Spe (Control) $0.30 \pm 0.06$ $0.60 \pm 0.06$ 0.77 $\pm$ 0.09
    AUC $0.75 \pm 0.06$ 0.88 $\pm$ 0.03 0.88 $\pm$ 0.12
     
    OCTAGON, Control vs. DR
    Baseline Our approach VGG16
    Overall Acc $0.82 \pm 0.04$ 0.87 $\pm$ 0.04 $0.84 \pm 0.07$
    Sen (Control) $0.87 \pm 0.04$ 0.96 $\pm$ 0.04 1.00 $\pm$ 0.00
    Spe (Control) 0.71 $\pm$ 0.08 0.71 $\pm$ 0.08 $0.53 \pm 0.20$
    AUC $0.87 \pm 0.04$ 0.91 $\pm$ 0.03 0.94 $\pm$ 0.06
     | Show Table
    DownLoad: CSV

    Table 3.  Table of classification performances with transfer learning

    Controls NoDR DR
    Our approach VGG16 Our approach VGG16 Our approach VGG16
    ACC 0.68 $ \pm $ 0.05 0.78 $ \pm $ 0.05 0.81 $ \pm $ 0.04 $ 0.72 \pm 0.04 $ 0.76 $ \pm $ 0.02 0.77 $ \pm $ 0.04
    SEN $ 0.79 \pm 0.05 $ 0.90 $ \pm $ 0.05 0.56 $ \pm $ 0.06 $ 0.20 \pm 0.13 $ 0.26 $ \pm $ 0.14 0.55 $ \pm $ 0.11
    SPE 0.57 $ \pm $ 0.09 0.67 $ \pm $ 0.11 0.90 $ \pm $ 0.04 0.88$ \pm $ 0.05 0.90 $ \pm $ 0.02 $ 0.86 \pm 0.05 $
    AUC 0.80 $ \pm $ 0.04 0.90 $ \pm $ 0.15 0.70 $ \pm $ 0.04 0.67$ \pm $ 0.28 0.86 $ \pm $ 0.06 $ 0.75 \pm 0.22 $
     | Show Table
    DownLoad: CSV

    Table 4.  Correlation results for NHS Lothian and OCTAGON

    EC ($ p $-value), NHS Lothian EC ($ p $-value), OCTAGON
    (VD, EC(VD)) $ 0.86 (1.65 \times 10^{-12}) $ $ 0.20 (0.20) $
    (FAZ, EC(FAZ)) $ 0.55 (2.52 \times 10^{-4}) $ $ 0.57 (7.13 \times 10^{-5}) $
     | Show Table
    DownLoad: CSV
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