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Digital Hyperplane Fitting

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Combinatorial Image Analysis (IWCIA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 12148))

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Abstract

This paper addresses the hyperplane fitting problem of discrete points in any dimension (i.e. in \(\mathbb {Z}^d\)). For that purpose, we consider a digital model of hyperplane, namely digital hyperplane, and present a combinatorial approach to find the optimal solution of the fitting problem. This method consists in computing all possible digital hyperplanes from a set \(\mathbf {S}\) of n points, then an exhaustive search enables us to find the optimal hyperplane that best fits \(\mathbf {S}\). The method has, however, a high complexity of \(O(n^d)\), and thus can not be applied for big datasets. To overcome this limitation, we propose another method relying on the Delaunay triangulation of \(\mathbf {S}\). By not generating and verifying all possible digital hyperplanes but only those from the elements of the triangulation, this leads to a lower complexity of \(O(n^{\lceil \frac{d}{2} \rceil + 1})\). Experiments in 2D, 3D and 4D are shown to illustrate the efficiency of the proposed method.

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Notes

  1. 1.

    Supposing it needs 1 \(\upmu \)s for generating and testing one hyperplane, then it takes about \(3*10^{7}\) years to find the optimal solution.

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Author information

Authors and Affiliations

  1. LORIA, Université de Lorraine, Nancy, France

    Phuc Ngo

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  1. Phuc Ngo
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Correspondence to Phuc Ngo .

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Editors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Tibor Lukić

  2. School of Business, SUNY Fredonia, Fredonia, NY, USA

    Reneta P. Barneva

  3. SUNY Buffalo State, Buffalo, NY, USA

    Valentin E. Brimkov

  4. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Lidija Čomić

  5. Department of Information Technology, Uppsala University, Uppsala, Sweden

    Nataša Sladoje

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Ngo, P. (2020). Digital Hyperplane Fitting. In: Lukić, T., Barneva, R., Brimkov, V., Čomić, L., Sladoje, N. (eds) Combinatorial Image Analysis. IWCIA 2020. Lecture Notes in Computer Science(), vol 12148. Springer, Cham. https://doi.org/10.1007/978-3-030-51002-2_12

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  • DOI: https://doi.org/10.1007/978-3-030-51002-2_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-51001-5

  • Online ISBN: 978-3-030-51002-2

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