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The Chiral Domain of a Camera Arrangement

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Abstract

We introduce the chiral domain of an arrangement of cameras \(\mathcal {A} = \{A_1,..., A_m\}\) which is the subset of \(\mathbb {P}^3\) visible in \(\mathcal {A}\). It generalizes the classical definition of chirality to include all of \(\mathbb {P}^3\) and offers a unifying framework for studying multiview chirality. We give an algebraic description of the chiral domain which allows us to define and describe the chiral version of Triggs’ joint image. We then use the chiral domain to re-derive and extend prior results on chirality due to Hartley.

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Notes

  1. The broken arrow (\(\dashrightarrow \)) and the phrase “rational map” mean here that the domain of the map A is not actually \(\mathbb {P}^3\) but rather \(\mathbb {P}^3{\setminus }\{\mathbf {c}\} \).

  2. Again, the broken arrow (\(\dashrightarrow \)) and the words “rational map” refer to the fact that the domain of the map \(\varphi _\mathcal {A}\) is not \(\mathbb {P}^3\) but rather \(\mathbb {P}^3{\setminus }\{\mathbf {c}_1,\ldots ,\mathbf {c}_m\}\).

  3. Recall that the topology we use on \(\mathbb {P}^n\) is induced by the Euclidean topology on \(\mathbb {R}^{n+1} {\setminus }\{0\}\). This induces a topology on the product of real projective spaces \((\mathbb {P}^2)^m\). Explicitly, a set \(U_1 \times U_2 \times \ldots \times U_m \subseteq (\mathbb {P}^2)^m\) is open if and only if the sets \(U_i \subseteq \mathbb {P}^2\) are all open sets.

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Acknowledgements

We thank Tomas Pajdla for discussions at the start of this project and for pointers to the chirality literature.

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Authors and Affiliations

  1. Google Inc., Seattle, USA

    Sameer Agarwal

  2. Department of Mathematics, University of Washington, Seattle, USA

    Andrew Pryhuber & Rekha R. Thomas

  3. Fakultät für Mathematik und Informatik, Universität Leipzig, Leipzig, Germany

    Rainer Sinn

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  1. Sameer Agarwal
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  2. Andrew Pryhuber
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  3. Rainer Sinn
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  4. Rekha R. Thomas
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Correspondence to Andrew Pryhuber.

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Pryhuber and Thomas were partially supported by the NSF grant DMS-1719538.

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Agarwal, S., Pryhuber, A., Sinn, R. et al. The Chiral Domain of a Camera Arrangement. J Math Imaging Vis 64, 948–967 (2022). https://doi.org/10.1007/s10851-022-01101-2

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