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Object-image correspondence for algebraic curves under projections. (English) Zbl 1278.14044

A projection \(P\) from \(\mathbb R^3\) to \(\mathbb R^2\) can be described by a linear fractional transformation \[ x=\frac{p_{11}z_1+p_{12}z_2+p_{13}z_3+p_{14}}{p_{31} z_1 + p_{32} z_2 + p_{33} z_3 + p_{34}},\quad y=\frac{p_{21} z_1 + p_{22} z_2 + p_{23} z_3 + p_{24}}{p_{31} z_1 + p_{32} z_2 + p_{33} z_3 + p_{34}} \] where \((z_1 , z_2 , z_3)\) denote coordinates in \(\mathbb R^3\), \((x, y)\) denote coordinates in \(\mathbb R^2\) and \(p = (p_{ij})_{3\times4}\) are real parameters of the projections under certain conditions for central or parallel projections. In the paper, the authors consider the object-image problem for algebraic curves. Instead of comparing curves directly under projections, they study the group-equivalence classes of a given curve under affine or projective algebraic group action. Such group-equivalence problem is solved by the author by introducing the notions of classifying sets of rational differential invariants and signature maps. This method provides a computationally efficient way to establish a correspondence between a space curve and a planar curve by significantly reducing the number of parameters. Their algorithm is present for rational algebraic curves.
Precisely, let \(Z\) and \(X\) are given rational curves in \(\mathbb R^3\) and \(\mathbb R^2\) respectively, then the image-object problem is whether there is a projection \(P\) as above such that \(P (Z) = X\). This problem requires in general elimination of 14 parameters for central projections and 10 for parallel projections. After considering the group actions of \(X\) and studying its group-equivalence classes, the author successfully define rational signature maps \(S|_X :\mathbb R \dasharrow\mathbb R^2\) and \(S|_Z :\mathbb R^4\dasharrow\mathbb R^2\). They show that the object-image problem is equivalent to compare the images of \(S|_X\) and \(S|_Z\), which requires elimination of 5 parameters for central projections and 4 for parallel projections. The case of non-rational curves and finite points are also discussed in the paper.

MSC:

14H50 Plane and space curves
14Q05 Computational aspects of algebraic curves
14L24 Geometric invariant theory
53A55 Differential invariants (local theory), geometric objects
68T45 Machine vision and scene understanding
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