Euclidean distance degree conjectures for point multiview varieties

For generic arrangements of cameras, let MnN,h\mathcal{M}_n^{N,h} denote the point multiview variety with nn cameras, ambient dimension parameters (N,h)(N,h), and let EDD\mathrm{EDD} denote its Euclidean distance degree. Point multiview EDD conjectures.

For n1,EDD(Mn3,3)=92n3212n2+11n4,for n4,EDD(Mn3,1)=92n3392n2+22n4,for n1,EDD(Mn2,2)=92n2132n+3.\begin{aligned} \text{For }n\ge1,\quad &\mathrm{EDD}(\mathcal{M}_n^{3,3})=\frac{9}{2}n^3-\frac{21}{2}n^2+11n-4,\\ \text{for }n\ge4,\quad &\mathrm{EDD}(\mathcal{M}_n^{3,1})=\frac{9}{2}n^3-\frac{39}{2}n^2+22n-4,\\ \text{for }n\ge1,\quad &\mathrm{EDD}(\mathcal{M}_n^{2,2})=\frac{9}{2}n^2-\frac{13}{2}n+3. \end{aligned}

These formulas are suggested by numerical homotopy-continuation computations, while the preceding formulas for other point multiview varieties are stated as known theorems.

Sources & referencesView supporting material

Primary source

Timothy Duff and Felix Rydell, “Metric Multiview Geometry – a Catalogue in Low Dimensions”, arXiv:2402.00648 (2024).

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