Abstract
We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image.
Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores.
Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.
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Notes
- 1.
The astrophysics community refers to the Euler characteristic curve as the genus.
- 2.
We review related work at the end of the paper.
- 3.
We thank Reinhold Erben and Stephan Handschuh from Vetmeduni Vienna for providing micro-CT scans of rat vertebrae.
- 4.
The second condition is implied by the first condition since we allow only consecutive integers as interval endpoints in the definition of cells.
- 5.
Throughout this paper we use “voxel” as multidimensional generalization of “pixel”.
- 6.
Another interpretation of voxel data is via the dual complex (voxels become vertices) using the lower star filtration. The way we use appears more natural in image processing context. The two approaches yield similar but not necessarily identical Euler characteristic curves.
- 7.
Defining cells as products of closed intervals implies \((3^d-1)\)-connectivity for the voxels of the thresholded images. This corresponds to 8-connectivity for 2D images.
- 8.
where \(\chi \left( \tilde{f}^{-1}\left( \left( -\infty ,-1\right] \right) \right) =\chi (\emptyset )=0\).
- 9.
In lexicographical order a voxel at position \((i_1,\dots ,i_d)\) succeeds a voxel at position \((j_1,\dots ,j_d)\) if \(i_k>j_k\) for the first k where \(i_k\) and \(j_k\) differ.
- 10.
The input size is \(\log _2(m)n\).
- 11.
Most of the images are available at www.byclb.com/TR/Muhendislik/Dataset.aspx.
References
Bauer, U., Kerber, M., Reininghaus, J.: Distributed computation of persistent homology. In: 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 31–38. SIAM (2014)
Colley, W.N., Richard Gott III, J.: Genus topology of the cosmic microwave background from WMAP. Mon. Not. R. Astron. Soc. 344(3), 686–695 (2003)
Dean, J., Ghemawat, S.: Mapreduce: simplified data processing on large clusters. Commun. ACM 51(1), 107–113 (2008)
Delgado-Friedrichs, O., Robins, V., Sheppard, A.: Skeletonization and partitioning of digital images using discrete morse theory. IEEE Trans. Pattern Anal. Mach. Intell. 37(3), 654–666 (2015)
Dyer, C.R.: Computing the euler number of an image from its quadtree. Comput. Graph. Image Process. 13(3), 270–276 (1980)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society (2010)
Gonzalez, R., Wintz, P.: Digital Image Processing. Addison-Wesley Publishing Co. Inc., Reading (1977)
Gray, S.B.: Local properties of binary images in two dimensions. IEEE Trans. Comput. 100(5), 551–561 (1971)
Günther, D., Reininghaus, J., Wagner, H., Hotz, I.: Efficient computation of 3D Morse–Smale complexes and persistent homology using discrete Morse theory. Vis. Comput. 28(10), 959–969 (2012). Springer
Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004)
Odgaard, A., Gundersen, H.: Quantification of connectivity in cancellous bone, with special emphasis on 3-d reconstructions. Bone 14(2), 173–182 (1993)
Pikaz, A., Averbuch, A.: An efficient topological characterization of gray-levels textures, using a multiresolution representation. Graph. Models Image Process. 59(1), 1–17 (1997)
Rhoads, J.E., Gott, J.R., Postman, M.: The genus curve of the abell clusters. Astrophys. J. 421, 1–8 (1994)
Richardson, E., Werman, M.: Efficient classification using the euler characteristic. Pattern Recognit. Lett. 49, 99–106 (2014)
Robins, V., Wood, P.J., Sheppard, A.P.: Theory and algorithms for constructing discrete morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)
Saha, P.K., Chaudhuri, B.B.: A new approach to computing the euler characteristic. Pattern Recognit. 28(12), 1955–1963 (1995)
Ségonne, F., Pacheco, J., Fischl, B.: Geometrically accurate topology-correction of cortical surfaces using nonseparating loops. IEEE Trans. Med. Imaging 26(4), 518–529 (2007)
Snidaro, L., Foresti, G.: Real-time thresholding with euler numbers. Pattern Recognit. Lett. 24(9–10), 1533–1544 (2003)
Sossa-Azuela, J.H., et al.: On the computation of the euler number of a binary object. Pattern Recognit. 29(3), 471–476 (1996)
Verri, A., Uras, C., Frosini, P., Ferri, M.: On the use of size functions for shape analysis. Biol. Cybern. 70(2), 99–107 (1993)
Wagner, H., Chen, C., Vuçini, E.: Efficient computation of persistent homology for cubical data. In: Workshop on Topology-based Methods in Data Analysis and Visualization (2011)
Ziou, D., Allili, M.: Generating cubical complexes from image data and computation of the euler number. Pattern Recognit. 35(12), 2833–2839 (2002)
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Heiss, T., Wagner, H. (2017). Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images. In: Felsberg, M., Heyden, A., Krüger, N. (eds) Computer Analysis of Images and Patterns. CAIP 2017. Lecture Notes in Computer Science(), vol 10424. Springer, Cham. https://doi.org/10.1007/978-3-319-64689-3_32
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