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. 2023 Sep 14;13:15253. doi: 10.1038/s41598-023-41221-2

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© The Author(s) 2023

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Conceptual overview and application examples. (a) Main use case of the heatmap representation as an operational description of the target function in a learning system. (b) application of the heatmap representation to anatomical feature detection, boundary detection^46,47, skeletonization^33,48,49, line detection⁵⁰, segmentation⁵¹, and surgical tool detection in 3D (Sect. “Application to 2D and 3D representation problems”, Supplementary Material Sects. S3–S6).