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doi: 10.3934/fods.2021027
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Euler characteristic surfaces

Gabriele Beltramo 1, , Primoz Skraba 1,, , Rayna Andreeva 2, , Rik Sarkar 2, , Ylenia Giarratano 3, and  Miguel O. Bernabeu 3,

1. 

School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK
2. 

School of Informatics, University of Edinburgh, Edinburgh, EH8 9AB, UK
3. 

Centre for Medical Informatics, Usher Institute, University of Edinburgh, Edinburgh, EH16 4UX, UK

* Corresponding author

Received  February 2021 Revised  September 2021 Early access November 2021

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.

Keywords: Topological data analysis, multiparameter filtrations, Euler characteristic, Euler characteristic curves, image classification.
Mathematics Subject Classification: Primary: 62R40, 68T09; Secondary: 55N31.
Citation: Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, doi: 10.3934/fods.2021027
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show all references

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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)
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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 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 $
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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 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] $
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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)
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$ \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 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
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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 $
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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
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Figure 8.  Points sampled from two Clayton copula distributions. In (a) $ \theta = 1 $, while in (b) $ \theta = 5 $
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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 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)
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Figure 10.  Point processes, corresponding to a Poisson point process in (a) and Hawkes cluster process in (b)
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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 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
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Figure 11(c) for a cluster process (a, b) and the Poisson process (c, d)">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)
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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
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Figure 14.  Example of the ECS with $ M_2 $ being the complement image
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Figure 15.  Example of the ECS with $ M_2 $ being the radial gradient image
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Figure 16.  Examples of regions of interest in the terrains of OCTAGON
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Figure 17.  Examples of level-sets of images of the control (a, b) and DR (c, d) corresponding to Region B
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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
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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} $ %
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$ \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} $ %
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
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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
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 $
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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 $
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}) $
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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}) $
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