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.
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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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