Abstract
We present the results of numerical simulation for image segmentation based on the chain distance clustering algorithm. The key issue is the use of the p-adic metric, where p > 1 is a prime number, at the scale of levels of brightness (pixel wise). In previous studies the p-adic metric was used mainly in combination with spectral methods. In this paper this metric is explored directly, without preparatory transformations of images. The main distinguishing feature of the p-adic metric is that it reflects the hierarchic structure of information presented in an image. Different classes of images match with in general different prime p (although the choice p = 2 works on average). Therefore the presented image segmentation procedure has to be combined with a kind of learning to select the prime p corresponding to the class of images under consideration.
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Notes
- 1.
This is the nearest neighbor chain algorithm, that, with the reciprocal or mutual nearest neighbor; this is the standard efficient implementation for all standard agglomerative hierarchical clustering algorithms.
- 2.
This (thresholding chains or relation representations, from, e.g., single link clustering, or the minimal spanning tree) has been used for image segmentation in some previous work. A review that includes some of this work is available, e.g., in [28].
- 3.
We noted that future work would consider other color encoding beyond RGB (YUV, etc.); also dynamic range, here 256, to be extended.
- 4.
However, the following comment can be useful to clarify the situation with computational speed up. Clustering that uses a p-adic distance can be considered by analogy with (1) creating a hierarchical clustering, and then (2) cutting the hierarchy to produce a partition. This analogy is more the case if the hierarchy is a contiguity-constrained one. The computational requirements that are referred to quite negatively as regards this work may not be all that bad, especially if one makes comparison relative to straightforward, Euclidean distance or related, segmentation. A further comparison in future work is likely to be beneficial too: the use of wavelet transforms—hierarchical and hence ideally represented p-adically for image and signal segmentation.
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Khrennikov, A., Kotovich, N. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: Toni, B. (eds) New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-55612-3_6
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