Abstract
The aim of this paper is to present a new algorithm to calculate the confinement tree of an image - also known as component tree or dendrone - for which we can prove that its worst-case complexity is O(n log n) where n is the number of pixels. More precisely, in a first part, we present an algorithm which separates the different kinds of operations - which we call scanning, fusion, propagation, and attribute operations - such that we can separately apply complexity analysis on them and such that all operations except propagation stay in O(n). The implementation of the propagation operations is presented in a second part, first in O(n2n), where nn is the number of nodes in the tree (n n = n). This is suficient if the number of pixels is much larger than the number of nodes (n n << n). Else, we show how to obtain O(n n log n n ) complexity for propagation. We construct two example images to investigate the behavior of two known algorithms for which we can show worst-case complexity of O(n2 log n) and O(n2), respectively, and we compare it to our algorithm. Finally, a practical evaluation will be opposed to the theoretical results. Several variations of the implementation will show which operations are time consuming in practice.
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Mattes, J., Demongeot, J. (2000). Efficient Algorithms to Implement the Confinement Tree. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds) Discrete Geometry for Computer Imagery. DGCI 2000. Lecture Notes in Computer Science, vol 1953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44438-6_32
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DOI: https://doi.org/10.1007/3-540-44438-6_32
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