Abstract
This paper aims to present a new method of translating labeled 3D scans of biological tissues into Generalized Maps (nGmaps). Creating such nGmaps from labeled images is a solved problem in 2D and 3D using incremental algorithms. We present a new approach that works in arbitrary dimensions. To achieve this in an effective manner, we perform the necessary operations implicitly using theory rather than explicitly in memory. First we define implicit nGmaps. We then present a scheme to construct said nGmap representing an nD pixel/voxel-grid implicitly. Thirdly we give a description of the process needed to reduce such implicit nGmap. We demonstrate that our implicit approach is able to reduce nGmaps in a fraction of otherwise necessary memory.
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Notes
- 1.
In this paper we use pixel as generic term for any dimension, i.e. including voxels in 3D and hypervoxels in 4D.
- 2.
Because the grid is infinite, the construction technically is not an nGmap. One can modify \(D := {\mathbb {Z}}_k^n \times {\mathbb {N}}_{<2^n\cdot n!}\) using the cyclic group \({\mathbb {Z}}_k\) for some sufficiently large number k. The nGmap then represents a grid on a large torus and D is finite. When implementing D in code using for example 32-bit ints, this automatically happens with \(k = 2^{32}\).
References
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Acknowledgements
This project was supported by the Vienna Science and Technology Fund (WWTF), project LS19-013. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).
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Bogner, F., Hladůvka, J., Kropatsch, W. (2022). Implicit Encoding and Simplification/Reduction of nGmaps. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_10
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DOI: https://doi.org/10.1007/978-3-031-19897-7_10
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