Abstract
In this article, the almost inner (resp. lower, upper) regularity of metrics on a group is proposed. In terms of these regularities, a sufficient and necessary condition for distance transforms to be Lipschitz-1 continuous is given, and relations between balls with different centres and radii are discussed. It turns out that the three almost regularities of metrics introduced and the results obtained here might provide useful tools in discrete analysis, mathematical morphology and image analysis, etc.
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Acknowledgements
The authors would like to express sincere thanks to the referees for their careful reading of the original and the revised manuscripts, their valuable suggestions and recommendations on the content and the references, and also for their pointing out errors and typos, which improve the article. The study is supported by the National Natural Science Foundation of China (Nos. 11671293, 11271282).
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Project supported by the National NSF of China (Nos. 11671293, 11271282).
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Guo, Q., Su, X. Almost Regular Metrics on Groups and Lipschitz-Continuity of Distance Transforms. J Math Imaging Vis 61, 1235–1242 (2019). https://doi.org/10.1007/s10851-019-00898-9
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DOI: https://doi.org/10.1007/s10851-019-00898-9