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Fitting cylinders computation with an application to measuring 3D shapes

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Abstract

This paper observes a fitting cylinders problem for 3D shapes. The method presented defines two cylinders that fit well with the shape considered. These cylinders are easy and fast to compute. Would the 3D shape considered be digitized, i.e. represented by the set of voxels, the computation is asymptotically optimal. Precisely, the time required for the computation is \({{\mathcal {O}}}(N)\), where N is the number of voxels inside the shape. Next, we show how these fitting cylinders can be used to measure 3D shapes. More precisely, we define a new 3D shape measure that numerically evaluates how mach a shape given looks like a cylinder. Interestingly, both fitting cylinders have to be used to define such a measure—just one of them is not sufficient. The new measure is invariant with respect to translation, rotation, and scaling transformations, and ranges over the interval [0; 1],  and takes the value 1 if and only if the shape considered is a perfect cylinder. It is robust and simple to compute.

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Now new data is provided by the authors, in this manuscript.

Notes

  1. Here in, by a cylinder we mean a 3D body, not a cylindrical surface.

  2. This is a cubic equation, whose solutions can be given in an explicit (but slightly complicated) form, by using Cardano’s result.

  3. Notice that the fitting cylinders, in Fig. 2 are presented by their oval surfaces only — not as 3D closed bodies, as they are. This has been done for a better visualization purpose.

  4. Notice that a comparison with just one of these fitting cylinders does not lead to the measure satisfying the desirable properties like those listed in the Theorem 1. This is, for example, because \({{\mathcal {C}}}_a(S) = {{\mathcal {C}}}_b(S)= 1\) cannot be true for an arbitrary shape (cylinder) S. More precisely, \({{\mathcal {C}}}_a(S) = {{\mathcal {C}}}_b(S)= 1\) would imply that S coincides with both fitting cylinders corresponding to the parameters a and b. This is not possible.

  5. We still use the maximum operator in (35) and (36), but there are no more than 8 choices for \(\beta \). Just to mention that the number of \(\alpha \) values in (29) and (30) can be selected to be arbitrary large.

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Acknowledgements

The authors wish to hank to Dr. Carlos Martinez-Ortiz for his help during the preparation of this paper.

Funding

This work is partially supported by the Serbian Ministry of Science and Technology.

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Authors and Affiliations

  1. Serbian Academy of Sciences, Mathematical Institute, Belgrade, Serbia

    Joviša Žunić

  2. School of Computer Science and Informatics, Cardiff University, Cardiff, UK

    Padraig Corcoran

Authors
  1. Joviša Žunić
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  2. Padraig Corcoran
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Correspondence to Joviša Žunić.

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Communicated by Wei GONG.

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Žunić, J., Corcoran, P. Fitting cylinders computation with an application to measuring 3D shapes. Comp. Appl. Math. 42, 207 (2023). https://doi.org/10.1007/s40314-023-02348-0

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