The following problem, arising from medical imaging, is addressed: Suppose that $T$ is a known tetrahedron in $\R^3$ with centroid at the origin. Also known is the orthogonal projection $U$ of the vertices of the image $\phi T$ of $T$ under an unknown rotation $\phi$ about the origin. Under what circumstances can $\phi$ be determined from $T$ and $U$?
The perspective-n-point problem, often abbreviated PnP, is the problem of determining the position of a camera from the perspective images of n given points. The problem has been widely investigated during the last few decades, using several traditional camera models, such as projective (see, for example, [9]), orthographic (see, for example, [11]), or weak perspective (i.e., scaled orthographic, see [2,10]), and focusing on various aspects (such as small values of n).
While the solution of a specific instance of PnP is often an application of elementary geometry, understanding the configuration space-for example, classifying which configurations admit a given number of solutions-involves challenging nonlinear aspects (cf. [6,13] and the references therein). Indeed, it was not until recently that Faugère et al [6] (partially) classified the configurations for the perspective-3-point problem via the discriminant variety, using extensive computations.
Our point of departure is a paper by Robinson, Hemler and Webber [14], who, motivated by an application in imaging, studied the perspective-4-point problem for the orthographic camera model. The problem is as follows. A given tetrahedron T in R 3 with vertices p (1) , . . . , p (4) has been transformed by an unknown (direct) rigid motion φ. Also given is the image U = {u (1) , . . . , u (4) } of the set of vertices of φT under a parallel projection onto the xy-plane in an unknown direction w ∈ S 2 . The problem is to find φ and w.
In [14] it is observed that one may as well take the parallel projection to be the orthogonal projection π z onto the xy-plane. It is also noted that then φ can only be determined up to a vertical translation, because such a translation does not change U. Since φ is the composition of a rotation about the origin and a translation, it suffices to determine the rotation and the horizontal component of the translation. The authors of [14] make the assumption that it is known which projection comes from which vertex of T , that is, they assume that u (i) = π z φp (i) , i = 1, . . . , 4. Under this labeling assumption, they show that the rotation and horizontal shift can be determined.
Our purpose here is to study this problem when the labeling assumption is removed, and to provide a systematic foundational study from the viewpoint of nonlinear computational geometry (see, for example, [1,5,12]).
Clearly, the centroid of the vertices of φT must lie on the vertical line through the known centroid of U. From this, we make two conclusions. Firstly, the horizontal shift can always be determined, so we may assume that φ is a rotation about the origin. Secondly, if such a rotation φ can be determined when the centroid of T is at the origin, then it can also be determined when the centroid of T is located elsewhere. Thus our problem can be stated in the following form.
Suppose that T is a known tetrahedron in R 3 with vertices p (1) , . . . , p (4) and centroid at the origin. Also known is the orthogonal projection U = {u (1) , . . . , u (4) } onto the xy-plane of the vertices of the image φT of T under an unknown rotation φ about the origin. Under what circumstances can we determine φ from T and U?
Obviously, if T has nontrivial automorphisms-for example, if T is regular-then φ cannot be uniquely determined. Now let T be an arbitrary tetrahedron in R 3 with vertices p (1) , . . . , p (4) . Suppose that the images φ (p (1) + p (2) )/2 and φ (p (3) + p (4) )/2 under φ of the midpoints of the opposite edges [p (1) , p (2) ] and [p (3) , p (4) ] are contained in the z-axis. Then a rotation ψ of φT by π about the z-axis results in a tetrahedron ψφT whose vertices also project onto U. In this case U forms the vertices of a parallelogram in the xy-plane, so U has a symmetry (rotation by π about its center).
These preliminary remarks show that in general φ cannot be determined if T or U has extra symmetries. A general goal is to understand if it can be uniquely determined otherwise, and if not, to find those T and U that do allow φ to be determined.
The relation between our problem and the one considered in [14] can be made clearer if we regard the labels of the vertices of T as having been permuted by an unknown permutation σ of {1, 2, 3, 4}, so that u (i) is the projection of φp (σ(i)) , i = 1, . . . , 4. Then the problem in [14] corresponds to the case when σ is the identity.
In this paper, we deal with both uniqueness and reconstruction. Our focus is on the geometry of the problem, in particular, the configuration space of all tetrahedra leading (for a given rotation) to the same set of projection points as the original tetrahedron. By decomposing this space into the union of the spaces corresponding to the various types of permutations involved, we can treat the configuration questions from a linear algebra point of view. Then, using some nonlinear symbolic methods, we precisely classify situations where the dimension of the configuration
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